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[quote="PenguinIntegral"]Let me tow the Libertarian party line by saying that we shoud not have a death penelty, ever. The government should not control whether you live or die, ever.[/quote]

But sth I have to say about it is that making someone loosing his life is something like making someone loosing his freedom. No one can decide whether you have to die or not, but a murderer decides. I mean, you are told that you don't have to kill, and you do it, and then we (citizens) have to feed you?

We have two views:

_The victim's familiy: How do you think they feel? They're life is broken, and they have to pay their bills to the government in order to the killer to eat without working... how about that?

_The killer and his familiy: Everybody make mistakes, but killing? It's like a mistake you CAN'T do... So, if I was to "cool" to kill someone, I must be that "cool" to understand that I don't deserve living, and I won't contribute nothing to the world... what do you feel by being the son of a murderer? It's not a father that is an example

I find it hard to accept that the government should ever have a right to decide whether or not anyone should live or die, but let's take an example:

A 25 year old man kills/rapes/tortures 25 or so people over 18 months. He is convicted. The government cannot reasonably, ever, let him free, as part of their duty to the population. So either the use the death penalty, or they keep him locked up for the remaining 50-60 years of his life. I have heard once that it can cost up to £30,000 (roughly $55,000) per year to keep a prisoner in jail. So, over his lifetime, this prisoner will have cost the taxpayer over $2 million, on top of all the money spent on his education, and in all that time he will have earnt very little and contributed very little in tax himself.

What should the government do? Kill him or let him drain their funds?

Hard labour camps are the way to go!

[quote="José"]I'd like to know what do you think about death penalty (if you think yes, say when)[/quote]
 :huh:
Um I don't think that's a yes or no question(its not even a question :ninja: ) and what do you mean by when?

as for my opinion:
I think we should just have death penalties. I mean if we stick a guy in prison for life, thats the same as killing him, only more torturous.

[quote="junggi"] I mean if we stick a guy in prison for life, thats the same as killing him, only more torturous.[/quote]
The hells no.

[quote]
Hard labour camps are the way to go![/quote]
Never ruled those out.

The state is generally ineffcient. Lots of people are convicted of capital crimes and killed, and some are innocent. On a tangent, isn't even one wrongful death too many?

[quote="tim1234133"]
What should the government do? Kill him or let him drain their funds?

[/quote]
Nice false dilemma.
[quote]
This should not be taken to mean that the right of individuals or agents of individuals, including the state, lack a right to defendthemselves against an imminent danger. People certainly have the right to kill other people who pose a direct and immediate threat to their own safety, but once a criminal, even a murderer, is in custody and placed on trial he or she rarely poses such a direct and immediate threat. Killing him or there, therefore, is more akin to cold blooded murder than an act of self-defense.

Similarly, this should not be taken as a call for the sort of coddling of serious criminals advocated by some misguided activists (usually of Leftist persuasions). A prison system that wasn't bogged down with the overwhelming number of drug and other consensual crimes could easily handle the permanent, lifetime incarceration of those who intentionally commit murder. Those falsely accused of such crimes could be released and compensated by the state when the error is brought to light -- an option unavailable to those killed by the state.

The historical record of the 20th century is quite clear that giving any group of people the power to kill those who don't pose an immediate and direct threat inevitably leads that group to apply its power against the innocent, whether intentionally or not. The power to kill those who don't pose a direct and immediate threat should be one firmly opposed by libertarians.
[/quote]

death penalty is completely wrong!  it doesn't bring any victims back to life, and with all the trials and things to make sure the accused individual is truly guilty, it takes as long as 20 years, and during that time, it costs more than keeping the person in jail for life.  maybe christians and other religious people think less of life because they believe in some afterlife, but as an atheist, all i think happens after death is [i]void[/i].  what gives a system the right to send an individual to that void?  what gives anyone the right to murder anyone else, even under legal laws?  it's all the worse that these particular murderers are entire groups of people who call their assasinations "executions" and pretend they're justified under a "legal" system.  to conclude, here's part of my xanga, written either about mussaoui or stanley tookie williams:

"consider the death penalty: the cold contemplation and planning of the death of an individual.  and so the american impassively watches the execution of another human, and thinks to himself proudly, "the world is now a safer place," and hence derives pleasure from legalized assasination!  and through the plotting and enjoyment of murder, he is as good as a terrorist.  good job america."

[quote="wobster109"]what gives a system the right to send an individual to that void?[/quote]
what gives a system the right to send an individual to jail forever and keep him bored for 50 years?

I really dont think a death penalty should be given.....prisoners need to be used for doing something constructive.

[quote="wobster109"]the system can detain people to keep them from hurting others.  because the accused had no right to commit the crime in the first place.

"What does the law say? You will not kill! How does it say it? By killing!" ---Victor Hugo[/quote]

May I go further by saying "What does the law say? You will not do bad! How does it say it? By doing bads!"

A familiar dialogue in dormitories:
"Silent please!"
"Aren't you talking now?"

[quote="PenguinIntegral"][quote="junggi"][quote="wobster109"]the system can detain people to keep them from hurting others.  because the accused had no right to commit the crime in the first place.[/quote]

You can detain from killing too  :wink:[/quote]
Sentence does not parse. Rephrase.[/quote]
errr
Killing can detain the bad people from harming other people

the problem is, jose, that you aren't reading what i'm posting.

[quote="Treething"]i doubt either of you have problems finding money for food so you really can't be completely objective[/quote]

Not me thanks God, but yes some in my neighborhood and some of my friends too

[quote="José"]I'd like to know what do you think about death penalty (if you think yes, say when)

I'm not sure myself, so I'd like to know your opinions.

Something I want to say is that I don't think that the answer "No, I'm aganist it because some deads by this law were free" would count, because they're the minimum, and, you like it or not, you have to trust your law, so, if someone thinks that death penalty for homicide is OK, then the % of not guilty killed people would be really short...

Please, post your opinion, and don't deflect the theme[/quote]

i think it is interesting to learn about.

it depends if it is good or bad

[quote="perfect_radio"]Let's put the problem your way: José, what happens if an innocent person gets the death penalty (I doubt it that you can always be $100 \%$ sure that someone really did it)? What then? How would his family feel? He won't be killed by some psycho, he'd be killed by a court's order. What if it'd happen to one of your friends?[/quote]

And what if one of my friends were killed?

Nevertheless, you have to trust your law system. If not, this topic is useless

[quote="Malcolmk133"][quote="wobster109"]about the subject of terrorists, although it's probably irrelevant: why is it that when bin laden kills americans, it's horrible "terrorism," but when u.s. soldiers go in and shoot down other soldiers, they're considered "heros?"  why is it that human society, at this advanced stage, must still resolve political issues by pouring money and resources into shooting down others, when children are hungry and people are poor?[/quote]

Bin Laden and terroristst kill innocent Western civilians on purpose.

American Soldiers kill other soldiers.[/quote]

LOL aren't you naive, you obviously have no idea about the atrocities American soldiers commit. You should see the death toll for civilians in Iraq. We don't care about the tens and thousands of civilians being killed in Iraq but when 1 possible-killer American Soldier gets killed, we get all pissed.

True, american soldiers did horrible things with some Irak's soldiers, that was terrible (there are lots of photos that can prove it). So, remember that they're not angels

[quote="wizard2378"]I post to raise several things.

First, a trivial point, Penguin the term is to toe the line.  (Point is when military folks would put their toes on a line drawn on the floor.)  If you only hear the expression and it is not explained, it is difficult.  You are articulate and I would hate for some to discount your ideas based on this simple misunderstanding.

You have all raise many good arguments and several that don't hold up.  The cost of trials is not a good reason to oppose the death penalty, it is just an artifact of the political fight that many opposing the death penalty are allowed to raise for everyone scheduled to be killed, regardless of their merits or their personal desire to continue to fight their conviction.

In the US the Constitution requires that any criminal be convicted "beyond a reasonable doubt."  This means that any doubt of someone's guilt that is based on reason, rather than a feeling or speculation would require a juror or judge to not find the defendant guilty.  So unless we have judges and juries not following their legal constraints, in the US we have only those we are certain are guilty.  Before I receive a broadside of examples of convictions being overturned, see a later paragraph about the reason that the US criminal justice system is biased strongly in favor of the criminal.  (France assumes that a defendant is guilty of the crime and leave the burden of proof on the criminal.  They have fewer acquittals.)  (The former Soviet Union concluded that every criminal defendant was guilty and the trial focused on whether the criminal accepted his guilt and was therefore not needing as severe punishment as those who continued to deny their guilt.  They had no acquittals.)

However, this is not the final story, each rule is biased with this ten to one ratio.  The eventual balance for most criminal trials ends up with a 100 or 1,000 to 1 balance of rules.  For example, if the police know someone in a train car killed a person and they still have the murder weapon on them, the police can detain and search the people in the car only if there are three or fewer people in the car.  If there are four, the infringement of rights to search three uninvolved people outweighs the public benefit of finding the murder weapon and capturing the real criminal at this stage.  The police can follow the people and continue to investigate and may find proof later, but this type of rule will completely eliminate many such people from further risk of criminal conviction.

While the US courts are defy attempts for folks to define the constitutional limits in rational terms, it appears from the cases that the balance struck over a long time that the balance of type I and type II errors are drawn where the allowing 10 defendants that are known to be guilty free if the rule will also prevent one not guilty conviction is viewed as appropriate.

The US founding fathers (law makers) were mostly criminals, perhaps wrongfully charged, but they had seen what the British criminal justice system could do and they were determined to avoid convicting the not guilty, even at the real risk of letting those actually guilty go to commit crimes again (the overall US reconviction rate is extremely high).  There essentially has been no system developed that reduces the re-conviction rate.  The only variables that have been shown to be statistically significant are getting the person married or getting the person over age 35.

If one rejects religion, how can one morally distinguish the act of murder a cow to get meat that one desires for a hamburger from murdering a person to get that same hamburger.  (I know the laws everywhere make this distinction, I'm asking morally what's the difference?)

In the Christian/ Jewish/ Moslem morality (same God, same morality on this point) why is murdering a person wrong?  Many people are surprised that the answer is because all people are made in God's image.  Cows are not.  Murdering a person is a form of an attack on God.

Note that the discussion wrongfully quoted the Bible as prohibiting killing.  It doesn't.  It prohibits murder.  Many times in the Bible, God tells people that they must kill people.  Look up Saul, first king if Israel.  His kingdom was torn from him and given to David over his disobedience in failing to kill everything and everyone that God (through the prophet Samuel) told him to kill.  See 1 Samuel 15:3 where the instructions are to kill all the men, women, children and infants in the Amalekite nation.

Look up the time when Israel disobeyed God and made a golden calf when Moses was getting the ten stone tablets on Mount Sinai, God first wanted to destroy all of Israel and start over with Moses holding the position of Abraham.  God relented but had the Levites (the ones that came over to the side of God) get their swords and go through the camp of Israel “each killing his brother and friend and neighbor.”  Exodus 32:27 Three thousand were killed that day.  (More than the number of US solders killed in the war on terror in five years.)  Jesus himself said, “Do not suppose that I have come to bring peace to the earth.  I did not come to bring peace, but a sword.”  God said many things in the Bible, but never, “Do not kill.”

Again this is an interesting topic.  One that has been visited by many folks over many years.  The arguments presented here are typical.  People have very strong views on this subject, sometimes for reasons that are more emotional than logical.  I find that most important decisions are made emotionally and merely justified rationally – this is no different in this area.

I hope this posting has helped.[/quote]
That's a really, nice, insightful post. However, I must go soon, and I shall respond to it later tonight.

Also, you could you point out the place where I misused that phrase? I can go back and edit it.

Hi, I also found your post very interesting, I've been away for a while so I might just be arriving a little late for the discussion. Anyway, I still have some doubts about what you said and I do disagree with some of the things you said.

[quote="wizard2378"]
You have all raise many good arguments and several that don't hold up.  The cost of trials is not a good reason to oppose the death penalty, it is just an artifact of the political fight that many opposing the death penalty are allowed to raise for everyone scheduled to be killed, regardless of their merits or their personal desire to continue to fight their conviction.
[/quote]

I wouldn't say as much since it seems to me that thorough all this discussion, financial arguments have been used both to support the death penalty and to condemn it. Nevertheless, I am with you if you meant that money shouldn't be the primary issue in this discussion. I think that the meaning of taking the life of a human being should. Also I think that before worrying about costs one should worry about the extent of power over human lives a legislature has. I also understand, coming from an underdeveloped country myself, that the economy and specially the goverment funds are not things to be taken lightly. But then again I wonder how much money are we trying to save here. In other words, how many people are we going to sentence to death instead of life imprisonment? Cause even the US at the moment, are executing way less than a hundred citizens a year, we're talking as far as one knows, of one of the countries with greatest incidence. I don't think that you're saving THAT amount of money by having to feed sixty people less in Argentina. And I don't think you're suggesting either to harden the law THAT much. It's clearly not a valid argument to state that with that money you could feed sixty poor people in your three hundred million inhabitants country. And what would you do? A lottery or something? Social programs and benefits is something the government must accomplish at a national or regional level, it's not personal stuff.

[quote="wizard2378"]
In the US the Constitution requires that any criminal be convicted "beyond a reasonable doubt."  This means that any doubt of someone's guilt that is based on reason, rather than a feeling or speculation would require a juror or judge to not find the defendant guilty.  So unless we have judges and juries not following their legal constraints, in the US we have only those we are certain are guilty.  Before I receive a broadside of examples of convictions being overturned, see a later paragraph about the reason that the US criminal justice system is biased strongly in favor of the criminal.  (France assumes that a defendant is guilty of the crime and leave the burden of proof on the criminal.  They have fewer acquittals.)  (The former Soviet Union concluded that every criminal defendant was guilty and the trial focused on whether the criminal accepted his guilt and was therefore not needing as severe punishment as those who continued to deny their guilt.  They had no acquittals.)
[/quote]

I have nothing ot say here since know nothing about Law. I was searching through the US Constitution and didn't find the 'beyond reasonable doubt' article. Could you please tell me which one is it? I would also love to check the differences between the french justice system and that of the US. I might have naively thought that France, above all France! with all their human rights  tradition and pride about themselves, wouldn't assume such a thing as 'the defendant is guilty until proven otherwise' but again, this is what I would have thought.

[quote="wizard2378"]
However, this is not the final story, each rule is biased with this ten to one ratio.  The eventual balance for most criminal trials ends up with a 100 or 1,000 to 1 balance of rules.  For example, if the police know someone in a train car killed a person and they still have the murder weapon on them, the police can detain and search the people in the car only if there are three or fewer people in the car.  If there are four, the infringement of rights to search three uninvolved people outweighs the public benefit of finding the murder weapon and capturing the real criminal at this stage.  The police can follow the people and continue to investigate and may find proof later, but this type of rule will completely eliminate many such people from further risk of criminal conviction.
[/quote]

Another question, I suppose then that the high rate of executions in the US (obviously compared to other countries with similar population and who also have the death penalty)is a consequence of the much higher rate of crimes, is it not?

[quote="wizard2378"]
If one rejects religion, how can one morally distinguish the act of murder a cow to get meat that one desires for a hamburger from murdering a person to get that same hamburger.  (I know the laws everywhere make this distinction, I'm asking morally what's the difference?)
[/quote]

Well, there we begin to disagree,... or maybe not. Let's see. In first place I should say that you seem to pose the question as if there were an evident answer, or as if it were clearly a support to the neccesity of religion in modern life. Well there are people who have actually answered your question and based their morality in their answers in a way you might find unexpected. Because to the question 'how can one morally distinguish the act of murder a cow to get meat that one desires for a hamburger from murdering a person to get that same hamburger?' they have, as you seem to predict, answered 'you can't'. And therefore, I DON'T start killing people around in order to have my hamburger, but STOP killing cows. Indeed if you don't think there's a god that made you special in some way, you have an issue answering the question you pose and acting accordingly. I have actually argued with people who defend this opinion and I've come up with the conviction that it is not at all a trivial discussion.
That said, I truly do not think one must renounce to the pleasure of a nice steak and still be an atheist. For instance I could argue that most animals do not feed on their own species, at least not as a common practice.  Maybe in extreme conditions, but on very extreme conditions we would feed on ourselves too, clearly we wouldn't kill each other(I hope), but if there happens to be a dead human around, believe me, it will soon enough turn into the next hamburger. And it occurs to me that the reason for which must carnivorous animals do not practice cannibalism is because as a species, it doesn't benefits them so evolution logically leads them not to do so.
But the strongest argument I think is none of the above but the fact that Homo sapiens IS  a special species in the world, whether religions exists or not. Humans are by far (far is not far enouhg, read interstellar) the most socially developed, amazingly complex, deeply evolved  species, all of which is kinda obvious, and all of which is not in my opinion, due to any god. And when you think of killing a human being to found your petit hot dog business, you're thinking of putting to an end to one example of this incredible, extraordinary kind of creature capable by itself to achieve things no other species as a whole could. Indeed, humans are the only species capable of, if there is for instance a fire, subordinate the most basic instincts of survival to the reasoned thought that one needs some help back there, and although at first your feet might want to escape, many humans stop and turn back to confront danger, humans is the only species who has given their lives plentifully in order to save another one of their kind. Humans are the only species that is not driven solely or to a very big extent by instinct but by reason. Humans are the only species in the whole history of the earth that has had the fate of every other species in their hands, a remind that we are not good enough yet is that we don't know what to do with them.  En fin, humans can rethink about founding the hot dog business if what it takes is to kill another human, not by religion or anything else, but because is not worth a human life. Nor it is any other such enterprise, and this is so fundamental that it should be stated in the law, right where it is. And so although our morality was shaped by ancient religions, today we want to keep it as it is in that regard. In other words, I don't think any advanced civilisation and similar to humans up to some extent, will have a morality different than ours with respect to killing their own, I clarify similar to humans because one could imagine civilisations with every characteristic one could think of.

[quote="wizard2378"]
In the Christian/ Jewish/ Moslem morality (same God, same morality on this point) why is murdering a person wrong?  Many people are surprised that the answer is because all people are made in God's image.  Cows are not.  Murdering a person is a form of an attack on God.
[/quote]

I am of the surprised ones. Never thought I could be attacking God in some way. Don't think it still.

[quote="wizard2378"]
Note that the discussion wrongfully quoted the Bible as prohibiting killing.  It doesn't.  It prohibits murder.  Many times in the Bible, God tells people that they must kill people.  Look up Saul, first king if Israel.  His kingdom was torn from him and given to David over his disobedience in failing to kill everything and everyone that God (through the prophet Samuel) told him to kill.  See 1 Samuel 15:3 where the instructions are to kill all the men, women, children and infants in the Amalekite nation.
[/quote]

I understand we could be speaking two different languages, and get nowhere. The Old Testament and evolution are not so easy to reconcile if you take both to the dot.
To finally state my opinion, I think DP should be abolished at least in the developed countries, and gradually in the world. I don't think the arguments proDP given are good enough, mainly for the same reasons that ZetaX, wobster, etc have said. You seem to imagine very easily one of your parents killed by a murderer but rather difficult to imagine your parents being murderers themselves. I know we all probably come from nice, balanced homes, but if there's a victim, there's a killer and it could also be one of those you love. The people in history with less respect for life, Hitler for instance, have also had bonds.  I see this is not a strong argument, just one that has not been made. It does support the stronger argument of the value of one human being as I have said above, one murderer that has redeemed itself is worth in my opinion ten or a hundred or the amount you need of others who have not. In that sense, I know prisons are hardly effective, by putting all wasps together they most likely will build themselves a hive won't they?.
Anyway, let's see if this thread goes back to life

Ursula

If someone is convinced to death and he is ill, will he recive any medical help or will be left to die?

[quote="wizard2378"]
In the US the Constitution requires that any criminal be convicted "beyond a reasonable doubt."  This means that any doubt of someone's guilt that is based on reason, rather than a feeling or speculation would require a juror or judge to not find the defendant guilty.  So unless we have judges and juries not following their legal constraints, in the US we have only those we are certain are guilty.  Before I receive a broadside of examples of convictions being overturned, see a later paragraph about the reason that the US criminal justice system is biased strongly in favor of the criminal.  (France assumes that a defendant is guilty of the crime and leave the burden of proof on the criminal.  They have fewer acquittals.)  (The former Soviet Union concluded that every criminal defendant was guilty and the trial focused on whether the criminal accepted his guilt and was therefore not needing as severe punishment as those who continued to deny their guilt.  They had no acquittals.)
[/quote]

Your information about foreign countries is not very accurate. First, there is presumption of innocence in France. Here is a quote from wikipedia:
[quote]
In France, article 9 of the Declaration of the Rights of Man and of the Citizen, of constitutional value, says "Every man is supposed innocent until having been declared guilty." and the preliminary article of the code of criminal procedure says "any suspected or prosecuted person is presumed to be innocent until his guilt has been established". The jurors' oath reiterates this assertion.
[/quote]
Second, formally the burden of proof of guilt  was on the prosecution in the Soviet Union. But in practice this  did not work, and acquittals were rare - but there were some.


Everyone knows the typical distance from a point to a ray formula -- just take the perpendicular to the ray that intersects the point and find its distance.

Now how would you do this in 3D?

Paraphrased, given a ray whose direction in 3D space is specified by <a, b, c> (a unit vector) and whose origin is specified by <x, y, z>, find the minimum distance from this ray to a point specified by <d, e, f>

Very easy u write a plane equation that passes through that(d,e,f) point whose coefficients a,b,c correspond to the direction of the ray (a,b,c), the plane will be  P aX+bY+cZ=K(k depends on the point in question in this case a*d +b*e+c*f), this plane is perpendicular(normal) to the ray, u compute by using the parametric 3D ray equation the piercing point of that ray in the plane P and then very simply use the point-to-point distance, and here's your distance.
If I'm not clear write me a PM,I'll give u good illustrated example .


In how many ways can 6 boys and 4 girls be seated in a row containing 15 seats such that no two girls sit together(i.e. a girl may only sit beside an empty seat or besides a boy)


Source: College of Charleston Math Meet 2004

Let $x$ be a real number between $.9$ and $1$. List $x,y,z$ in order of increasing magnitude where: $y=x^x$, and $z=x^y$.

Yeah, it's fairly easy, one just has to remember that when a number between $0$ and $1$ is raised to a power, it gets smaller, what I did was that I picked $x=.95$. Then $y=.95^{.95}$ which is smaller than $x$ and $y$ would equal something like $.90$. Then $z=.95^{.90}$ which is smaller than $y$.

Thus: $x>y>z$

[quote="arp"]i think this should be x, z, y[/quote]

Explanation?

"one just has to remember that when a number between 0 and 1 is raised to a power, it gets smaller"
This is only true if it is raised to a power greater than 1.  If x is raised to a power between 0 and 1 and x itself is also between zero and one, the result must be greater than x.  Maybe i don't understand the problem.

It's x<y<z.
Here's my explanation...
It's irrelevent which number you take, the results will be the same, and x, y, z<1 will always hold because n^n=1 if and only if n=1: for any other number n^0=1 and 0^0=...dunno, should =1 i guess but there's a contradiction with 0...
Anyways...take x=0.95.  Then $y=0.95^{0.95}$.  We know that if one takes any number^1, one gets that number, and if one takes a number^0, one gets 1.  The exponential function is either strictly increasing or strictly decreasing, depending on the base.  Thus, x^x must then be closer to 1 than x is because 0<x<1.  So x<y.  Same principle dictates that y<z.  Then, x<y<z.

are you sure that $.95^{.95}$ is closer to $1$ than $.95$??

EDIT: Oh my god!!! You're right! (just checked it in my calculator) :o  :o

However, With $x=.95$, then $y=.952439558...$ and $z=.952320....$ so $y>z>x$.

Thus, the smaller the power a number between $0$ and $1$ the bigger the number. I guess it is because raised to the $0$ power, it's equal to $1$. Hmmm...anybody wanna give a really good explanation for why this is?

think about $(\frac{1}{4})^{\frac{1}{2}} = \frac{1}{2}$ and $\frac{1}{4} < \frac{1}{2}$

So when you raise a value that is less than one to a power that is less than one, you get a value that is greater than your original value.

Yes, it's a bit strange but I guess with fractional numbers, things are completely the opposite, the smaller the power, the greater the number.

i just thought of x as 1/(1/x) so if we raise this to x (let x=m/r and m<r) we have (1/(1/x)^z)^1/m which will obviously produce a smaller denominator than 1/x; thus y is larger than x.  same idea can be used on z

K8, it is x<z<y i believe

I just checked it for x=0.95 and got y=0.95243956, z=0.95476294...
I mean, you know that x^x still less that 1, so using the logic, it just increases more.
arp, where did you get the ^z from?  and the ^1/m?  It would make sense that x^x isn't rational, so there is no denominator...you don't have to make it that complicated...
Just think for a moment.  Any number^(0<something<1)=a number which is closer to 1 than that number, because number^1=number, and number^0=1.  However y<1 as well, so z should be even greater.
Try it with a number.

EDIT: Oh, okay, now I see what the problem was.  I thought $z=y^x$  :D  My mistake...
In that case, it is $x<z<y$, arp.  $x^{number}$   such that $x<number<1$  makes a number which is greater than $x^1$ but less than $x^x$.
My bad...

ahh i just mistyped it
"i just thought of x as 1/(1/x) so if we raise this to x (let x=m/r and m<r) we have (1/(1/x)^z)^1/m which will obviously produce a smaller denominator than 1/x; thus y is larger than x. same idea can be used on z"

i was letting x=z/r at first then realized it was a bad idea because z is a variable in the problem, and i forgot to change the second part when i redefined x.  i meant to write ((1/(1/x)^m)^1/r.  We can see from this that the result will be greater than 1/(1/x) because we know r is greater than m.  I was just trying to explain my reasoning but i screwed it up..oh well

Yeah, but I now see what you mean.  y is smaller than 1, so (1/x)^y=1/x^y>1/x


Three schools have $n$ students each. We know that every student knows at least $n+1$ students from the other two schools. Show that there are three students who know each other (assume $A$ knows $B$ iff $B$ knows $A$).

It's taken from [url=http://www.math.columbia.edu/~zare/prob_all.html]here[/url].

Let $S_1,S_2,S_3$ be the three schools. Among all the students let $A$ be the one who have the maximal number, say $k$, of acquintances from the same school. Wlog, we may assume that $A $ belongs to $S_1$ and that he has $k$ acquintances in $S_2$.
Since $S_2$ only contains $n$ students and that $A$ has $n+1$ acquintances, it follows that $A$ knows someone from $S_3$, say $C$.

Now, if $C$ does not know any of the $k$ people from $S_2$ that $A$ knows, it follows that $C$ only knows at most $n-k$ people from $S_2$ so that he has to know at least $k+1$ people from $S_1$, which contradicts the maximality of $k$.

Therefore, $C$ knows at least one person, say $B$, from $S_2$ that $A$ knows, and we are done.

Pierre.


function
integration
real analysis
real analysis unsolved

Suppose that $ f$ is integrable with respect to an increasing function $ g$ on $ J\equal{}[a,b]$ and let $ F$ be defined for $ x \in J$ by

$ F(x) \equal{} \int_{a}^{x} fdg$

Prove that
a) if $ g$ is continuous at $ c$, then $ F$ is continuous at $ c$.
b) if $ f$ is positive, then $ F$ is increasing.

I think I have an idea for a, but not for b) any help?

thanks!

for $ g$-continuous at $ c$, how does$ \parallel{}g(x)-g(c)\parallel{}<\epsilon$ show $ \parallel{}F(x)-F(c)\parallel{}<\epsilon$?

One guess I had was that, since I m given the definition of continuity, would I have to expand the $ %Error. "intf" is a bad command.
$ and show something with the partitions of $ g$?(i.e. that they are $ \epsilon$ small?)

I think I got b)

I assumed $ y> x$ on $ J$

then $ F(y)\minus{}F(x) \equal{} \int_{a}^{y}fdg \minus{} \int_{a}^{x}fdg$ which gives

$ \int_{a}^{y}fdg \plus{} \int_{x}^{a}fdg \equal{} \int_{x}^{y} fdg$ ;if we look at the Riemann-Stieltjes sum:

then $ \sum_{k\equal{}1}^{n} f(\xi_{k})[g(x_{k})\minus{}g(x_{k\minus{}1})] \ge 0$ since $ f(x)$ is positive and $ g(x)$ is increasing.

therefore, $ F(y)\minus{}F(x) \ge 0 \rightarrow F(y) \ge F(x)$.

p.s. how do you "hide" possible solutions when posting?


geometry
circumcircle
Euler
symmetry
projective geometry
geometry proposed

Let ABC be a triangle, and P, Q and R three points on a line g.

Let $A_PB_PC_P$ be the circumcevian triangle of the point P with respect to triangle ABC.
Let $A_QB_QC_Q$ be the circumcevian triangle of the point Q with respect to triangle $A_PB_PC_P$.
Let $A_RB_RC_R$ be the circumcevian triangle of the point R with respect to triangle $A_QB_QC_Q$.

Prove that the lines $AA_R$, $BB_R$ and $CC_R$ concur at a point on the line g.

[i]Note.[/i] The [i]circumcevian triangle[/i] of a point P with respect to a triangle ABC is defined as the triangle XYZ, where X, Y, Z are the points of intersection of the lines AP, BP, CP, respectively, with the circumcircle of triangle ABC (different from the points A, B, C).

This may be easy, but it is definitely useful. Generalizing Treegoner's lemma in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=22549#p22549]http://www.mathlinks.ro/Forum/viewtopic.php?t=6417 post #11[/url] (Lemma 4 in my [url=http://de.geocities.com/darij_grinberg/Schroeder/Schroeder.html#7][Schroeder7][/url]), it should shed more light on why so many perspectors inhabit the Euler line.

  Darij

This problem is defintely nice!  :)

And I'm glad to prove it: (of course, by "point chasing")
- Pascal's theorem applied to the cyclic hexagon $AA_{P}A_{Q}BB_{P}B_{Q}$ gives us $AA_{P}\cap BB_{P}$, $A_{P}A_{Q}\cap B_{P}B_{Q}$, $A_{Q}B \cap B_{Q}A$ collinear, or $P,Q,A_{Q}B \cap B_{Q}A$ collinear, or $A_{Q}B \cap B_{Q}A \in g$
- Pascal's theorem applied to the cyclic hexagon $AA_{R}A_{Q}BB_{R}B_{Q}$ gives us $AA_{R}\cap BB_{R}$, $A_{R}A_{Q}\cap B_{R}B_{Q}$, $A_{Q}B \cap B_{Q}A$ collinear, or $AA_{R}\cap BB_{R}, R, A_{Q}B \cap B_{Q}A$ collinear. But we know that $R, A_{Q}B \cap B_{Q}A \in g$. Then $AA_{R}\cap BB_{R}\in g$.

In the same way we prove that $BB_{R}\cap CC_{R}\in g$, $CC_{R}\cap AA_{R}\in g$.
Finally: $AA_{R}\cap BB_{R}\cap CC_{R}= S \in g$.

I like the symmetry of the proof of this problem... I think one could write it without drawing any figure.

Let me just refraze this lemma in projective involutions. First note, that for every involution $f$ on a conic holds, that all lines $f(X)X$ coincide in one point. I call this point the center of such involution.

[b]Ping Pong lemma:[/b] Let $\omega$ be a conic and $p$ a line. Then have $A$, $B$, $C$ points on $p$ and not on $\omega$. Then we consider involutions $\varphi_A$, $\varphi_B$, $\varphi_C$ on $\omega$ with centers $A$, $B$, $C$. Then the composition $\varphi_A\circ\varphi_B\circ\varphi_C$ is an involution on $\omega$ and has center on $p$.

For proof, first use Pascal to show, that this composition is an involution. To show that it has center on $p$, we can use Pascal as above, or WLOG $\omega$ is an circle. Then $\varphi_A$ is some an inversion with center $A$. Analogically $\varphi_B$ and $\varphi_C$. Hence their composition has to be symmetrical around $p$.

[quote=riadok] To show that it has center on $p$, we can use Pascal as above, or WLOG $\omega$ is an circle. Then $\varphi_A$ is some an inversion with center $A$. Analogically $\varphi_B$ and $\varphi_C$. Hence their composition has to be symmetrical around $p$.[/quote]

I can't get this part of the solution. Like why should the compositions of the three inversions be same as some other inversion with center on $p$ ?
[rule]
Also I found a different way to prove that the involution has center on $p$. We get that all lines $Xf(X)$ pass through some fixed point $T$. Now just consider the case when $X \in p$, to get $T \in p$.

When $p$ and $\omega$ don't intersect, maybe the argument can still be fixed by working in $\mathbb C \mathbb P^2$, but I am not sure exactly how.

It's because we first use pascal to prove that the composition is an involution, which in complex plane is the same as inversion. Now we just study which exact inversion $\psi$ this compostion is. For that we look at the line $p$. We see that image of this line under $\psi$ is again the same line, which is enough to conclude that the center of this inversion lies on this line.


Solve equation:
1)$ e^x\equal{}1\plus{}x$
 2)$ (x\plus{}3)ln(2\plus{}x)\equal{}2x\plus{}2$
3)$ sinx(4\minus{}cosx)\equal{}3x(0<x<3)$
4)$ 3^x\plus{}5^x\equal{}6x\plus{}2$

hello,to your first equation obviously is $ x=0$ a solution. Let us consider $ f(x)=e^x-1-x$. $ x=0$ is the only solution of
the equation $ f^'(x)=e^x-1=0$, $ x=0$ gives a local minimum.Further we have $ \lim_{x \to \pm \infty} f(x)=+\infty$, hence we got $ x=0$ is the only solution of your equation.
Sonnhard.

2) $ (x\plus{}3)ln(2\plus{}x)\equal{}2x\plus{}2$

Condition of the equation is $ x>\minus{}2$.

$ x\equal{}\minus{}1$ is a solution.

The equation is equivalent to $ ln(2\plus{}x)\equal{}\frac{2x\plus{}2}{x\plus{}3}$.
Set $ f(x)\equal{}ln(2\plus{}x)\minus{}\frac{2x\plus{}2}{x\plus{}3}\Rightarrow f(x)$ continues in $ (\minus{}2;\plus{}\infty)$.
$ f'(x)\equal{}\frac{1}{x\plus{}2}\minus{}\frac{4}{(x\plus{}3)^2}\equal{}\frac{(x\minus{}1)^2}{(x\plus{}2)(x\plus{}3)^2}\geq 0$, $ f'(x)\equal{}0 \Leftrightarrow x\equal{}1$  $ \Rightarrow f(x)$ increases in $ (\minus{}2;\plus{}\infty)$.

Otherwise, $ \lim_{x\rightarrow \minus{}2^\plus{}}{f(x)}\equal{}\minus{}\infty$  and $ \lim_{x\rightarrow \plus{}\infty}{f(x)}\equal{}\plus{}\infty$.

Thus, the equation has an unique solution $ x\equal{}\minus{}1$.


induction
arithmetic sequence
algebra proposed
algebra

The sequence $\{a_n\}_{n}$ satisfies the relations $a_1=a_2=1$ and for all positive integers $n$,
\[ a_{n+2} = \frac 1{a_{n+1}} + a_n . \]
Find $a_{2004}$.

It is very interesting!

Since we have $a_{n+2}a_{n+1}-a_{n+1}a_{n}=1, $ the sequence $a_{n+1}a_n $ is arithmetic progression with comon difference 1.
so we obtain
$a_{n+1}a_{n}=a_{2}a_{1}+1\cdot(n-1).$ i.e. $a_{n}a_{n+1}=n$

Substituting this recursive formula for $n=1,3,5,7,\cdot\cdot\cdot,2001,2003$  we obtain

$a_{1}a_{2}=1,a_{3}a_{4}=3,a_{5}a_{6}=5,a_{7}a_{8}=7,\cdot\cdot\cdot,a_{2001}a_{2002}=2001,a_{2003}a_{2004}=2003$,respectively.

Multiplying both sides of each equatios,we have

$a_{1}a_{2}a_{3}\cdot\cdot\cdot a_{2003}a_{2004}=1\cdot3\cdot5\cdot\cdot\cdot\cdot 2001\cdot2003$           　　　　　　　　　　　　　　　　①

Similaly,substituting for $n=2,4,6,\cdot\cdot\cdot,2000,2002$ we obtain

$a_{2}a_{3}=2,a_{4}a_{5}=4,a_{6}a_{7}=6,\cdot\cdot\cdot, a_{2000}a_{2001}=2000,a_{2002}a_{2003}=2002,$ respctively.

Therefore we obtain

$a_{2}a_{3}a_{4}\cdot\cdot\cdot ,a_{2002}a_{2003}=2\cdot4\cdot6\cdot\cdot\cdot2000\cdot2002$　　　　　　　　　　　　　　　　　②

Dividing ①　ｂｙ　②　of both sides,we yield ,from  $a_{1}=1$,

$a_{2004}=\frac{1\cdot3\cdot5\cdot\cdot\cdot 2001\cdot2003}{2\cdot4\cdot6\cdot\cdot\cdot 2000\cdot2002}$.

Thank you

kunny

Thank you for confirmation of my answer ;) :)

We just have this recursion obtained
$a_{n+1}a_{n}=a_{n}a_{n-1}$ which when combined with the  initial values
just gives $a_{n}=\frac{n-1}{a_{n-1}}$ which now just gives
$a_{n}=\frac{(n-1)!!}{(n-2)!!}$
and so we are done,I don't simplify this further.


AMC
USA(J)MO
USAMO
blogs
geometry
geometric transformation
reflection

[quote="bubka"]Funny how so called maths lovers sell their maths knowledge to make money.[/quote]
I have an intense urge to stab you.

Funny how so called animal lovers get money to take care of animals.

[quote="bubka"]Funny how so called maths lovers sell their maths knowledge to make money.[/quote]

How's about this forum?  :o  :P

[quote="bubka"]Funny how so called maths lovers sell their maths knowledge to make money.[/quote]

Amazingly stupid how you expect people to not make a living. :furious:

[quote="bubka"]Funny how so called maths lovers sell their maths knowledge to make money.[/quote]

Wow.  Get a life.

[quote="bubka"]Funny how so called maths lovers sell their maths knowledge to make money.[/quote]
It's called a job, genius.

Aren't the prices a litle bit ...ummm...too much?Because i'm sure not everyone can afford that,poor or even middle class geniuses who need a little bit of guiding must be really sad for being in such a condition.Anyways,just a thought i had in my mind for a long time.This must have already been discussed over and all maybe....

Oh and i'm not supporting bubka,just an independent thought.

Well, let's compare prices:

Taking a 4 credit hour class at a college near me: $\$$ 1204
And that's just for one semester.  You also have to consider gas and the time it takes to get there.

EPGY upper level courses : from $\$$ 525 to $\$$ 700

And then if you look at summer programs, you won't be able to find one for less than a 1000 usually.  (Of course, a lot of the money goes toward room & board.)

I'm not saying that the prices are cheap (they certainly aren't).  What I [i]am[/i] saying is that they are rather reasonable for what you get.  A lot of time is put into developing WOOT.  The instructors are high caliber. etc

bubka, do you have a specific purpose?  Or are you just flaming for the heck of it?  It wasn't merely a 'simple' remark.  The implication in your post was that the instructors are here just to take our money.  Questioning the motives of the people who made this forum, those ones who have revolutionized problem solving education, the ones who have provided tons of free resources, spent hours and hours of their time, wrote an increasing number of books and so forth...questioning them is not taken lightly especially when you do it out of place and in a totally disgraceful manner.

I posted some thoughts on pricing in [url=http://www.artofproblemsolving.com/Forum/weblog.php?w=1]my blog[/url] if anyone cares to read about the difficulties we face (and mistakes we may have made) in pricing.  (I don't really address the issue that launched discussion of people 'selling their math knowledge for money'.  Isn't that what nearly everyone in any intellectual pursuit does?  At least, in well-functioning economies...)

[quote="joml88"]Why would you question their motives?  [i]Everyone[/i] needs money in this world.  If the instructors only cared about money, they could certainly be making much, much more doing something else.

Additionally, they do a lot of things for the mathematics community that they don't charge for.  The best example that they sincerely care about math and not money is that they financed the USAMO expansion.

And really, if you think about it, AoPS has a monopoly on their business and [i]could[/i] be charging a LOT more than what they do.[/quote]

If Mr. Rusczyk really cared about money more than education, he would be off making a fortune at D.E. Shaw.  Instead, he chose to stay here and create AoPS.  Bubka, I bet if you had the same choice...

[quote="bubka"]Funny how so called maths lovers sell their maths knowledge to make money.[/quote]

Funny how doctors sell health services to make money.

Funny how engineers don't give cars, computers, and appliances away for free.

Funny how writers charge for books.

Funny how airlines charge for tickets.

Funny how houses cost money.

Funny how people who enjoy making clothes sell what they make.

Funny how musicians charge ticket prices.

Funny how engineers expect a salary.

Funny how restaurants charge for food and a clean eating environment.

Funny how teachers get paid.

We are of course sympathetic to students who have a harder time affording our services.  In fact, we could charge more and make more money doing what we do.

As a student, I was in the boat with those who couldn't afford services outside of school: college courses (which are much more expensive), summer camps, or personal tutors.  My family lived at times below the poverty line and there weren't nearly so many free resources available on the internet.  But I was still so vastly better off than most people in the world.  I mention that only because it leaves me with a desire to make those resources available to more people.  And I have ever confidence that we do just that and are getting better at it very quickly.

We have discussed many times in the office that a day will come when we can offer financial assistance to students.  That won't happen before we prove that this kind of alternate educational resources company is economically sustainable.  In the meantime however, we've produced an array of free resources that can help highly interested students in so many ways.

What we are doing is [b]bringing down the cost[/b] of additional educational resources.  We wouldn't have gotten this far were that not the case because nobody would come take classes with us or buy our books if there were no better alternatives.  But understanding that means taking an economic look at educational services.  They're never free, though tax-funded services can appear that way at times.  And you're much better off with providers of a service who believe in what they offer -- who love the subject.

[quote="joml88"]Well, let's compare prices:

Taking a 4 credit hour class at a college near me: $\$$ 1204
And that's just for one semester.  You also have to consider gas and the time it takes to get there.
[/quote]
I know! College classes are WAY more expensive.  Plus, in my case, there won't be a differential equations or linear algebra or proofs  class I can take with my schedule, so I'll probably do WOOT! this year because it's cheaper than other options I have and I need proof-writing practice my last year before I go off to college.

[quote="bubka"]Funny how a simpleton's simple remarks which just points out explicitly a tacitly accepted truth lead to such vehement and clamorous protests among both the elite and the multitude.[/quote]

whoa! you have a thesaurus too!

[quote="ragnarok23"][quote="mysmartmouth"][quote="bubka"]Hey no! Please keep this in Round Table. This is not meant for G&FF.[/quote]

If this goes to G&FF, bubka will lose like 20% of his posts.  MOVE IT!![/quote]
:rotfl:  :rotfl:  :rotfl:[/quote]

Nice avatar ragnarok!  Might wanna share with Shobhit before he loses his avatar privileges (like I did a long time ago).

[quote="Xantos C. Guin"][quote="bubka"]Ok ok. I got into India's maths summer camp. The fares were fully reimbursed, board and lodging was free, and whatever I lost towards other expenses I got back in the daily allowance of a dollar.

So much for your summer programme.[/quote]

A few questions before I make up my opinion about India's maths summer camp.

1) What language are the classes taught in?

2) What contests do the classes help prepare you for?

3) What credentials do the instructors have?

3) Is the room better or worse than the dorms at a typical lower cost American college. (like Penn State, which had AC, a few electric outlets, lights, windows, two closets, two small beds, two desks, two mirrors, and a microwave.)

4) Is the food edible?

5) Is there a non-vegetarian option for food?
---------------------------------------------------------------------------------------------------------------------
[quote="bubka"]Good. Now I will accept the second part. Communism [i]is[/i] better than any other system of government in the world.[/quote]

Communism works great on paper and computer simulations where the people are 100% obedient and loyal and don't need food to survive. In communism, there is no motivation to work hard. If you are a circus performer with a 170 rubels per show salary, you get 170 rubels for the best show the crowd has ever seen, 170 rubels for a mediocre show, 170 rubels for a boring show, and 170 rubels if you are too "sick" to work that day. Clearly, noone would try hard unless they enjoy their job. Also, the difference in wages between a janitor and a surgeon are very slim. This makes working hard in school worth alot less.

As far as I'm concerned: Four legs good, Two legs better.
---------------------------------------------------------------------------------------------------------------------
PenguinIntegral's post sums up why this was split very well.[/quote]

Ok let me answer them:
1) English
2) IMO
3) No idea - do you get credentials for maths?
3) Do you mean the hostel room? Then it is a bit worse. Ourroom had two small beds, two chairs, a desk, some elctric outlets, a ceiling fan, two lights, an attached atrium, and an antechamber with a shower and a toilet seat. No AC, mirror, or microwave.
4) Not really
5) No, but last year a communist student did launch a fight which finally won him a chicken dish one day.

For the second part, I think it is spelled roubles. Now, even putting aside the fact that the performer will work for his fame, (If you are in your school drama troupe, you don't get paid for your work, but you try and out up the best show) answer this simple question: Why does a mother try and cook up a good dinner for the family, when she knows she will live just as comfortably if her cooking skills are revolting?

@orl: The problem is this: not everybody can afford to invest in education. In India, if you are ready to pay 10 dollars per month in school and 10 per month in college, you can get a fairly decent job. Hardly 40% of India's population can afford that.

And yes, nice avatar ragnarok!

Ideology is good. Without it, who needs passion! Bubka, that you possess an ideology is definitely commendable.

Free education for all? Sounds great, if everyone desires it the same way. However, reality does not seem to reflect that. There are always an enterprising few (like yourself, perhaps?) who drive harder and end up at the top in a system where ideally everybody is supposed to remain equal. Maybe that is why revolutions and counter revolutions happen.

[quote="bubka"]... Why does a mother try and cook up a good dinner for the family, when she knows she will live just as comfortably if her cooking skills are revolting? ...[/quote]Does this mother intend to cook for everybody else in her ideal world, and do so around the clock as well? Or does she cook for the love of her family? If the latter is even one of her minor reasons, she is helping to ruin that ideal system :( .

Getting back to the cost of WOOT, it would have been nice for whoever so desires to get into, free of charge of course, MOP or “the world's most reputed (and undoubtedly Asia's best) statistical institute”, etc. Since that is not happening yet, then as a parent, I challenge any family that desires such an education experience to come up with an alternative that is as inexpensive, well done, and well meant.

Bubka, do keep thinking. If nobody thinks about different (and perhaps better) ideas, there is little hope for mankind.

You are a parent? Wow! I did not imagine this discussion would go that far to involve real parents! But maybe some of the administrators are parents anyway.

You have brought up a vital point. This mother must be taught to love the mankind in general. It is not as laughable as it seems. Some mothers love the friends of her children as her own children. Some head mistresses love all the children of the school. Now it is of course impossible that she will know each human being personally, but she must love mankind in general. You should know the thing I am talking about. Like all mothers have a soft spot for children. We shall through conscious evolution give everybody a soft spot for eveybody else. Moreover in the absence of lumpen proletariat, crimes will be practically absent. See what I mean?

[quote="bubka"]... This mother must be taught to love the mankind in general. It is not as laughable as it seems. Some mothers love the friends of her children as her own children. Some head mistresses love all the children of the school. Now it is of course impossible that she will know each human being personally, but she must love mankind in general. You should know the thing I am talking about. Like all mothers have a soft spot for children. We shall through conscious evolution give everybody a soft spot for eveybody else. Moreover in the absence of lumpen proletariat, crimes will be practically absent. See what I mean?[/quote]
In an “all are made equal” system, how probable is the successful co-existence of “must love mankind in general” and “soft spots” while transitioning to “equal soft spot for all”?

Who should be making the rules for such transition? Then are these rule makers still equal in the system to others?

In case not everybody agrees on how the transition should take place, is such transition even going to happen?

Rules? Nobody is going to make any rules. While communism can be established by force, it certainly cannot be run by force. It is necessarily an anarchy. The "equal soft spot" must be brought about through a conscious evolution.

If communism needs to be established by force, then there will be those who disagree with it. These people will rebel, cause riots, and ultimately ruin your civilization.

Communism, if perfectly enacted (which has never been done), causes everybody to have the same power and amount of money. We can divide the people into two groups, the lower class who work minimal time (we can no longer talk about wages) to support the everyday needs themselves and the upper class, and the upper class who seek to better the world through science and knowledge. All of these upper class who would have, at any time in their life, rebelled against an education, would not have gotten one and would be lower class. All of these upper class who were not extremely passionate about their jobs would be lower class (because the jobs are easier for just as good a life). Only a very few intellectuals would be helping the world to develop.

So... great, right? Everybody's needs are accommodated, and maybe a few people aren't happy, but that's their problem. Wrong! Nobody cares about life anymore because there are no more challenges to achieve unless you set them personally. Most people cannot do this, and do not realize that it is necessary. Now everybody is completely bored with life; they hate it! To make things worse, the upper class cannot keep up, because they aren't big enough and they aren't making enough money to support their research. We fall back into the Dark Ages, everything reverts, and now money is back in and so are kings, queens, and extensive corruption.

Now for case two. If imperfectly enacted, communism requires a dictator to manage the people for a "limited" amount of time before "pure communism" kicks in. There will always be somebody unhappy with this. Hence, there will always be a dictator. This is why all communistic governments are dictatorships.

Conscious evolution just isn't going to happen when crazy people know perfectly well they can steal, murder, etc. without any consequences. Rules are necessary, as is a government. I don't believe democracy is perfect, especially not here in America :wink: , but I still haven't seen a better option.

Read 1984.

What you said about imperfect enaction is perfect. Sorry, but I support that dictatorship. Equality is more important than freedom.

So a dictator has no more or less priveleges than his subjects? They are equal? Even if not, the people rebelling are the people whose lives have been made worse. They have been brought down to the level of the lower class person, which is definitely a bad thing (you must agree with this in some degree, even if you disagree that it is bad enough to justify not switching to communism). So MOST of the upper/middle class will be in disagreement, which will cause the dictator to dispose of them. The same argument applies. The world collapses back into the dark ages.

What you are talking about is simply a revolution that switches the roles of the classes, keeping the world exactly the same (in essence) except that most of the new upper class will have no interest in working.

I believe that over the past few centuries, people have drifted into the classes that best suit them, based on their desire to work and what they want to do with their lives. Over time, more and more people will drift into the right situations and more people will be happy.

Why in the world are there 3 simultaneous threads about communism???

Well I suppose the average income of people in the US is comparable to the costs.

They average income of Indians is quite less(if converted into US Dollars) but the costs are also low.

The only thing is if you convert currencies  then the cost of living in the US seem shocking to an Indian.

Infact to help developing contries like India, publishers like Wiley have printed special low price editions of their books (the AOPS books can be tried to be made available at a cheaper price in such developing countries).

Why if it isn't ol' Vihang! What a pleasant surprise! I suppose you are referring to books like "fundamentals of Physics" by Hallisay, Resnick, and Walker. That was my favourite physics book. I got it at a tenth of the original US price. But let us get back to the point.

[quote="jb05"]They have been brought down to the level of the lower class person, which is definitely a bad thing (you must agree with this in some degree, even if you disagree that it is bad enough to justify not switching to communism).[/quote]

Do you have any idea what you are talking about??! I should see a real class enemy in you if I take your words fully meant!! You are actually saying that the lower class people should be kept lower class and the higher class should consider themselves much higher! I am not blaming you because you are a child, but it only shows the values of a typical American even so early in his life. (for my definiton of Americans, refer to the thread "sep 11 theory".)

[quote="jb05"]Why in the world are there 3 simultaneous threads about communism???[/quote]

Goes on to show that everything must converge to communism someday or the other, especially with me around!

[quote="bubka"]Equality is more important than freedom.[/quote]
This mentality will, and has, ruin(ed) more lives than any capitalist pigs ever will.

[quote="bubka"]
Ok let me answer them:
1) English
2) IMO
3) No idea - do you get credentials for maths?
3) Do you mean the hostel room? Then it is a bit worse. Ourroom had two small beds, two chairs, a desk, some elctric outlets, a ceiling fan, two lights, an attached atrium, and an antechamber with a shower and a toilet seat. No AC, mirror, or microwave.
4) Not really
5) No, but last year a communist student did launch a fight which finally won him a chicken dish one day.
[/quote]

I think MOP is way better. Especially after #'s 3-5.

[quote="Xantos C. Guin"][quote="bubka"]
OK let me answer them:
1) English
2) IMO
3) No idea - do you get credentials for maths?
3) Do you mean the hostel room? Then it is a bit worse. Our room had two small beds, two chairs, a desk, some electrical outlets, a ceiling fan, two lights, an attached atrium, and an antechamber with a shower and a toilet seat. No AC, mirror, or microwave.
4) Not really
5) No, but last year a communist student did launch a fight which finally won him a chicken dish one day.
[/quote]

I think MOP is way better. Especially after #'s 3-5.[/quote]

It might not be as good as MOP but still the food is edible (ignore bubka;it's not that bad) and well there is no option for non-vegetarian food because of some of the customs in some castes.
Some Indians don't even have vegetarian food in Restaurants that also serve non vegetarian food.


limit
analytic geometry
graphing lines
slope
calculus
calculus computations

[quote="Jumbler"]Let $f(4)=2$, $\lim_{h\rightarrow 0}(\frac{f(4+h)}{h}-\frac{2}{h}) = 12$, $g(x) = x^{2}+3x+f(x)$. Find the slope of the tangent to curve $g$ at $(4,30)$[/quote]

Well, notice that

$\lim_{h\rightarrow 0}(\frac{f(4+h)}{h}-\frac{2}{h}) = \lim_{h\rightarrow 0}\frac{f(4+h)-f(4)}{h}=f^{\prime}(4)=12$

hence we have

$g^{\prime}(x) = 2x+3+f^{\prime}(x)\Rightarrow g^{\prime}(4) = 2(4)+3+f^{\prime}(4)= 11+12=23$

so the answer is 23.


algebra
polynomial
modular arithmetic
number theory proposed
number theory

These are special cases of a general result about factors of values of cyclotimic polynomials.

Let $ x$ be an integer and $ p$ a prime number.

1) Show that if $ p\mid 1\plus{}x\plus{}x^2\plus{}x^3\plus{}x^4$ then $ p\equal{}5$ or $ p\equiv 1\pmod{5}$.

2) Show that if $ p\mid 1\plus{}x^3\plus{}x^6$ then $ p\equal{}3$ or $ p\equiv 1\pmod{9}$.

Here is what I had in mind. Suppose
\[ p\mid 1\plus{}x\plus{}x^2\plus{}x^3\plus{}x^4\]
We cannot have $ x\equiv 0 \pmod{p}$, and if $ x\equiv 1 \pmod{p}$ we quickly deduce $ p\equal{}5$. Suppose therefore that $ x$ is neither 0 nor 1 mod $ p$. Now
\[ p\mid 1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{p\minus{}2}\]
by Fermat's theorem. A long division of the second polynomial by the first, shows that $ 5\mid p\minus{}1$.

The case of a prime power, such as 9 instead of 5, is only slightly subtler.

Yes, and that's essentially the same proof as in the link.


abstract algebra
vector
Ring Theory
superior algebra
superior algebra unsolved

Let $ (R,M)$ be the Noetherian local ring (i.e. M is the only maximal ideal of R,) and let $ Q$ be an $ M$-primary ideal of $ R$. Note that the $ R$-module $ (Q: M)/Q$ is annihilated by $ M$, and so can be regarded as a vector space over $ R/M$ in a natural way. Show that the following statements are equivalent:
(i) $ Q$ is irreducible;
(ii) $ dim_{R/M}(Q: M)/Q \equal{} 1$;
(iii) the set of all ideals of $ R$ which strictly contain $ Q$ admits $ (Q: M)$ as smallest element.

$ (i) \rightarrow (ii)$ We first show that $ dim_{R/M}(Q: M)/Q \ge 1$,this is because by the condition $ \sqrt {Q} = M$,hence there exist minimal positive integer $ d$ such that $ M^d \subset Q$(Because the ring is noetherian),hence we can choose $ x \in M^{d - 1} - Q$,thus $ x \in (Q: M)/Q$ which is not zero,hence $ dim_{R/M}(Q: M)/Q \ge 1$.
For if $ dim_{R/M}(Q: M)/Q \ge 2$ we show that $ Q$ is not irreducibe.Then there exist two element $ x_1,x_2 \in (Q: M)$ such that $ r_1x_1 + r_2x_2 \notin Q$ for $ \any r_1 \notin or r_2 \notin M$,then you can prove that $ Q = (Q,x_1) \cap (Q,x_2)$

$ (ii) \rightarrow (iii)$If $ dim_{R/M}(Q: M)/Q = 1$,and for any ideal $ N$ of $ R$ which strictly contains $ Q$,we want to show $ (Q: M) \subset N$.
(We may assume $ N \subset M$,otherhand it is obvious)Because $ N$ contains $ Q$ strictly,we can first select an element $ y \in N - Q$,and $ \square{N} = M = \sqrt {Q}$,hence there exist an minimal positive integer $ e$ such that $ yM^e \subset Q$,hence we can choose an element in $ yM^{e - 1}$,(say it $ z$),such that $ z \notin Q$,but obviously $ z \in N$ and $ z \in (Q: M)$,hence by the condition $ dim_{R/M}(Q: M)/Q = 1$,we have $ (Q: M) = (Q,z) \subset N$.

$ (iii) \rightarrow (i)$,obviously.


Prove that if $ y \ge y^3 \plus{} x^2 \plus{} |x| \plus{} 1$ then $ x^2 \plus{} y^2 \ge 1.$
Find all pairs of $ (x,y)$ such that the first inequality holds while equality in the second one attains.

[quote="hzbrl"]Prove that if $ y \ge y^3 \plus{} x^2 \plus{} |x| \plus{} 1$ then $ x^2 \plus{} y^2 \ge 1.$[/quote]
if $ y\ge 0$ then $ y^3\ge 0$ and $ y\ge1$ if $ y<0$ then $ y^3 \plus{} x^2 \plus{} |x| \plus{} 1<0$ so $ y\le \minus{}1$


A gardener plans to build a fence to enclose a square garden plot.  The perimeter of the plot is 96 feet, and he sets posts at the corners of the square.  The posts along the sides are set 6 feet apart.  How many posts will he use to fence the entire plot?

click on the document to see my diagram and solution :)
its a little hard to see though.

I don't have time to explain but
[hide]16[/hide]

[hide]If the fence has a perimiter of 96 then each side is 24.  The 4 posts are put at the corners and 3 are put between each side.  That is 12 plus 4 gives you [b]16[/b][/hide]

[hide]
If every 6 feet, there is a post, then you do 96/6=16 posts[/hide]


[quote="mathVNpro"]Let $ a,b,c$ be the positive real numbers such that: $ min(a,b,c)\ge \frac {1}{4}.max(a,b,c)$. Find the largest and the smallest value of:
$ P \equal{} \frac {a \minus{} b}{c} \plus{} \frac {b \minus{} c}{a} \plus{} \frac {c \minus{} a}{b}$[/quote]

It's old problem.
The answer is $ maxP\equal{}\frac{3}4$, $ minP\equal{}\frac{\minus{}3}4$. :)

Here is the full solution for you [b]enndb0x[/b]:
Firstly, we will prove: $ A \equal{} |\frac {a \minus{} b}{c} \plus{} \frac {b \minus{} c}{a} \plus{} \frac {c \minus{} a}{b}|\le \frac{3}{4}$.
WLOG, $ a\ge b\ge c > 0$. Denote $ y \equal{} a \minus{} b\ge 0$, $ x \equal{} b \minus{} c\ge 0$, we get: $ b \equal{} x \plus{} c$, $ a \equal{} c \plus{} x \plus{} y$.
From the condition of the problem we get: $ c\ge \frac {1}{4}(c \plus{} x \plus{} y)$ or $ \frac {1}{3}(x \plus{} y)\le c$.
Now, it is easy to notice that: $ A \equal{} |\frac {(a \minus{} b)(b \minus{} c)(c \minus{} a)}{abc}| \equal{} \frac {xy(x \plus{} y)}{c(c \plus{} x)(c \plus{} x \plus{} y)}$.
We trust need to consider that case: $ x \plus{} y > 0$. Then, due to the fact that $ 0 < \frac {1}{3}(x \plus{} y)\le c$, therefore, $ A \le \frac {xy(x \plus{} y)}{\frac {1}{3}(x \plus{} y)[\frac {1}{3}(x \plus{} y) \plus{} x][\frac {1}{3}(x \plus{} y) \plus{} x \plus{} y]}$.
Hence, $ A\le \frac {27}{4}.\frac {xy}{(4x \plus{} y)(x \plus{} y)} \le \frac {3}{4}$ (By [b]AM-GM[/b]).
Therefore, $ \minus{} \frac {3}{4}\le P\le \frac {3}{4}$.
Then we will get the result :D


number theory
greatest common divisor
analytic geometry
graphing lines
slope

Find all $a,b,c\in\mathbb{N}$ such that \[a^{2}+b^{2}=c^{2}.\]

[hide="Massive hint"] What is the square of $m+ni$?  What length does it have? [/hide]

[quote="vishalarul"]Let
$a=r^{2}-s^{2}$
$b=2rs$
$c=r^{2}+s^{2}$
This will always give a solution if $r$ and $s$ are integers, and $r>s$.[/quote]

This does not give all solutions  :wink:  Consider $(9, 12, 15)$.

$a=r^{2}-s^{2}$
$b=2rs$
$c=r^{2}+s^{2}$
This gives only the primitive triplets(i.e. a,b,c are coprime)
$a=k(r^{2}-s^{2})$
$b=2krs$
$c=k(r^{2}+s^{2})$
This will give all the solutions(I think)

[quote="t0rajir0u"]Indeed.  Can you [b]prove[/b] it?   :D[/quote]

Sure  :lol:
We just did this in interm NT class :lol:
[hide]First, if (x,y,z) is a primitive triple, then it's easy to see that (kx,ky,kz) is another triple, for an integer k.
Now to prove that none of a,b,c share a divisor.
Assume a,b are not coprime. Let the gcd be d. then d divides the LHS so d must divide the RHS. contradiction.
Similarly, we can prove that a,c and b,c are coprime.
now WLOG let a be even and b be odd. it follows that c is odd
Back to the initial equation.
Move the b^2 to the RHS and factor diff of squares and divide by 4
Then
$(\frac{a}{2})^{2}=\frac{(c+b)}{2}\frac{(c-b)}{2}$
Let the gcd of (c+b)/2 and (c-b)/2 be d.
then $d|(c+b)/2$ and $d|(c-b)/2$, so $d|(c+b)/2+(c-b)/2=c$, and $d|(c+b)/2-(c-b)/2=b$
Since b and c are coprime, d must be 1.
Now the LHS is a square, and the two factors on RHS are coprime. Therefore, each of those two factors must be squares.
now substitute $(c+b)/2=r^{2}$ and $(c-b)/2=s^{2}$
solving the system of equation, we get the formulas for primitive triples.
$a=r^{2}-s^{2}$
$b=2rs$
$c=r^{2}+s^{2}$
Since (a,b,c) are triples, so are (ka,kb,kc) for k integers so we get the formulas for all triples
$a=k(r^{2}-s^{2})$
$b=2krs$
$c=k(r^{2}+s^{2})$

is that too long? :|
it seems that any proof in my hands become long :|
[/hide]

Very nice  :)  I was not expecting such an elementary solution.

[hide="This is the one I had in mind"] First we assume WLOG that $gcd(a, b) = 1$.  Then

$(a+bi)(a-bi) = c^{2}$

From which it follows $a+bi, a-bi$ must be squares $\in \mathbb{Z}[i]$ (the complete argument here requires that you prove a few properties of $\mathbb{Z}[i]$ that correspond to properties of $\mathbb{Z}$).  The result follows:

$(a+bi) = (m+ni)^{2}= (m^{2}-n^{2})+2mn i$
$(a-bi) = (m-ni)^{2}= (m^{2}-n^{2})-2mn i$
$c = (m+ni)(m-ni) = m^{2}+n^{2}$[/hide]

There's another neat way to do this problem.

Consider the unit circle $C$, $x^{2}+y^{2}= 1$ and the line, $L$, with slope $t \in \mathbb{Q}$ passing through $(-1,0)$. The second intersection of $L$ and $T$ correspond to a right triangle with verticies $(-1,0),(x,0),(x,y)$ where $(x,y)$ is the point of intersection. The sides of this triangle will be $1+x,y,\sqrt{(1+x)^{2}+y^{2})}$.  You can show that $t$ is rational if and only if all of these lengths are rational, and they can be scaled up to integer side lengths.  Since the set of rationals are countably infinite, we can itterate through all the possible right triangles with rational side lengths, and by simply scaling them up, all the triangles of integer side length.


Billy Bob is a superhero. :spidy: One day in Billy Bob's city, there are 10 robberies. How many ways can Billy Bob stop the 10 different thieves, if thief number one has to be caught first and thief number two has to be caught second?

[hide]isn,t it obviously 8! ways as the first 2 have to be kept constant and the rest can be permuted in 8! ways.[/hide]

You both are right I think. Hide your solution MUKUL ATRI

[hide]basically it is $8!$ and $8!=40320$[/hide]

Yes, that is the answer I got too. All of you also used the same way that I did to do this problem. However, since I know that knowing many ways to do a problem leads to mathematical success, does anyone know a different way to do this problem? I sure don't... :D

If there are n robberies and if numbers 1 through x have to be caught respectivey, the formula [hide="is"]$(n-x)!$[/hide]

Err, you just pushed up a topic that was on the second page...don't do that.

It's okay as long as it isn't over a month old...

NUTZ, DONT SPAM OR REVIVE

LIKE U DID ON THE MC FORUM

[hide]8! because 2 are arleady done[/hide]

[quote="Karth"]Actually, it's more than a month old...[/quote]
Aug 29-Sep 10 is 12 days...

[hide]there are 8! List the number of thieves on order 1-10,
abcdefghij. if you have to keep AB as the first two thieves cought you have ABC, ABD, ABD, ABE, ABF. ABG, ABH, ABI, ABJ, 8 different possible premutations for the first three apprehensions. then you have 7 chioces for the 4th, so 8 * 7 (56) different possible permutations
ABCD, ABCE, ABCF, ABCG, ABCH, ABCI, ABCJ
ABDC, ABDE, ABDF, ABDG, ABDH, ABDI, ABDJ
ABEC, ABED, ... , ABJI.
then 6 for the 5th, 5 for the 6th, 4 for 7th, 3 for the 8th, 2 for the 9th, and 1 for the 10th apprehension. so we have 8*7*6*5*4*3*2*1 = 8! = 40320 different ways[/hide]


calculus
integration
calculus computations

what is the integral of (sin x)^3 * cos x dx

give me some hints on this problem

1) Use a substitution.
2) Do your own homework.


I watched the [url=http://video.msn.com/v/us/msnbc.htm?f=00&g=b60573a8-041d-405a-9ad6-c53a8b22f6cc&p=hotvideo_m_edpicks&t=m5&rf=http://www.msnbc.msn.com/id/3032084/&fg=&GT1=9033]msn video[/url]. Wow. I told my mom after I had seen it right on the front page of msn and my mom came rushing downstairs :P !

Well, does it matter now who was father to the baby? (for Anna's sake)

chance of son+mother dying (unless both at birth) within 4 months...$\frac{1}{33!}$

-jorian

I don't really care, it's just you've been hearing so much about her this last 1/2 year and all these contreversies have just ended.

comments?

Why would her death stop all the controversies?

Seriously. If I died, none of the general national public would notice. The only thing she could do was take her clothes off in front of massive crowds

The media makes people feel like they know celebrities personally, and so when one of them dies, it makes people feel like somebody they know personally has died. It has nothing to do with person, it has to do with the person's depiction in the media.

[quote="Treething"][quote="7h3.D3m0n.117"]After the autopsie how did she die? I didn't catch it yet but I was wondering if anyone else knew?[/quote]

Most people die before the autopsy is taken.[/quote]

lol no what i meant what did they find in the autopsy

Due to egregious lack of worthwhile content, this thread is being moved to the G&FF.

It is not THAT egregious, JBL, but having said that, it is quite egregious, I agree.


We have registered but may not be able to go due to a conflict with a mandatory celebration :(

[quote="Ubemaya"]I'm pretty sure the Lynbrook team will take this hands down, with the Hos on their team.[/quote]
along with a compsci nub who got a B- in apcs. A deep team might go just as far.

haha okay so looks like our team will be able to go.  Has anyone been before?

I'll be there :D

But thanks to that mandatory celebration that archimedes1 mentioned, I have to leave Stanford between ProCo and ARML practice :(

@archimedes1: I think this is the first year they're ever doing this.

[quote="hunter34"][quote="Ubemaya"]I'm pretty sure the Lynbrook team will take this hands down, with the Hos on their team.[/quote]

Is it just me or does this sound wrong?[/quote]

Ubemaya is referring to Tony Ho, Johnny Ho from Lynbrook. (USACO participants)
Yes and my name has been used in wrong ways b4... :oops:

Darn Gunn might send like a non-trivial number of people because there are a lot of juniors in AP Comp Sci, and the teacher handed out flyers...

ewwww two hours was pretty short
especially for only having one laptop per team

I couldn't stay for the awards, who won?  What were the prizes?

[quote="Ubemaya"]I actually liked the contest difficulty (first time this year!) because I felt it was accessible enough for APCS people and somewhat challenging for USACO people. I would say the first five problems were trivial-USACO bronze level, second five were USACO bronze/easy silver level and third five were medium silver level. The time constraints were ridiculous. I'm pretty sure our team would have done better if we split up into three (or two) individuals, because we kept having to fight over the laptop.[/quote]

BTW, are you going to usaico or not?

Also, can someone post the problems?

yea he is going...
i'm too lazy to post problems tho...

[quote="Scrambled"][quote="CircleSquared"]grr i should go to this next year

but my school is surprisingly uncaring about things like this[/quote]
That's why you should set up your own team? It's not like our school did anything to make this happen either.

Also BackHo will win the OSL.[/quote]

yeah i should

also osl discussion should be in the osl thread?

[quote="CircleSquared"][quote="pythag011"][quote="moogra"]um first five were too trivial for usaco at all[/quote]

USACO bronze has a lot of very very very very trivial problems.[/quote]

does trivial mean brute forceable or do-in-headable[/quote]

I don't understand what you mean by do-in-headable.

But yes, usaco bronze has a lot of either BRUTE FORCE QUESTIONS or really classic questions.

In this context, I think "trivial" means you need little to no planning time before you start to code.

[quote="pythag011"][quote="moogra"]um first five were too trivial for usaco at all[/quote]

USACO bronze has a lot of very very very very trivial problems.[/quote]
why am i not surprised by this statement?
anyways... these problems were so *"trivial"* ANYONE who knew how to code could solve them
hmm seems like a very fatal variable problem screwed us up on number 4.

Problems are up!

http://proco.stanford.edu/PastContests/2009Problems

OK, the first few are just... ridiculous.

The medium ones all could be the easiest problem on bronze usaco.

9.1 I don't understand.

the other 9s could be on silver usaco.

we wasted at least 20 minutes trying to figure out the meaning of 9.1 in the end, we noticed we didn't read the first page (lol)

whatever i did all the 2 pt problems, or at least got them accepted

@miller
not all of them were trivial. I see you didn't solve all of them, including #4, which was "trivial"

@pythag
9.4 was easy

meh i was talking about the first 5 problems. they are definitely trivial.
and.. note the fatal variable problem
EDIT: oh yeah it's impossible to solve all of them btw -- i type too slow

oh yes the first 5 were definitely trivial

[quote="kitsune"][quote="miller4math"]these problems were so *"trivial"* ANYONE who knew how to code could solve them[/quote]

Actually, I'm sure most of the people in my AP Comp Sci class would not be able to solve the first one. My teacher never taught us about escape characters. -_-[/quote]

I am sure though that every usaco gold participant would find these problems very easy.

[quote="kitsune"]Well, they were trying to make it accessible to everyone so it would be easy for some people. Luckily there weren't many USACO gold participants there (I think) or my team wouldn't have done as well as it did.  :)[/quote]

Well there was me (Tony Ho), Johnny Ho, David Wang, Albert Gu, Billy Dorminy who are all gold participants. So there weren't too many ppl.
If I missed anyone, please do say so.


algorithm
ARML
induction
number theory unsolved
number theory

What is the biggest number that can't be written as sums of different squares?

Note: I wrote a computer program to solve this prob. and after [hide]128[/hide] every number I tried had a corresponding sum representation. But, I couldn't prove it.

A sketch of an approach:  use the greedy algorithm, subtracting the largest square less than a number.  This process is guaranteed to produce distinct large squares until $ n$ is less than some constant, after which it's a matter of a finite amount of casework to show that all of the resulting possibilities (except for the numbers less than some other constant) can be written as a sum of sufficiently small distinct squares.  Unfortunately, the computations involved are a little annoying.

I couldn't read the article fully but it seems that the corresponding two theorems does not state that the squares must be different.
Also, I searched the greedy algorithm in wikipedia and, as far as I understand, you say subtracting large squares should guarantee the sum of squares representation. But, I couldn't see how that's possible for every case(I think we need to predict the answer to apply that solution). Maybe I misunderstood you...
[hide="What I mean"]Let's say our prediction about answer is true! Then, we can state that after 128 + 144(the smallest square after 128) = 272, every number has a representation. Let's say number is 500. Just subtract 128 from 500 , 500-128 = 372. Pick the nearest smaller square, that is 324. Add the rest to 128, 128 + (372-324) = 176. But we know that 176 can be written as sums of different squares.
So, I think, we need to find all square representations upto 272 by hand.But, I still believe that this problem should have a smarter solution because the answer is 2^7(maybe related to the idea that every number can be written as sum of different 2^n's.)[/hide][/hide]

You are correct that the two-square and four-square theorems do not apply in this case.  In any case, I see what you mean; I was mistaken.

I can't find the questions, but an ARML power round a few years back gave an elementary proof that every number greater than $ 136$ (or something like that) is the sum of five (or six?) distinct squares.  The argument is very simple (it's essentially casework) but again, I can't find the questions.

The greedy algorithm by itself is too greedy.  For example, if the starting number is $ 1{,}000{,}002$, we don't want to choose one million as a square, because then we are stuck with 2, which can't be written as a sum of distinct squares.

Instead, let's choose a slightly less greedy algorithm.  First, check that all the numbers from 129 through 297 work.

Then, given an integer $ n \ge 298$, let $ k$ be a positive integer such that $ k^2 \plus{} 128 < n < 2k^2$.  (We can show that such a $ k$ exists.)  We will use $ k^2$ as the largest square.  What's left is $ n \minus{} k^2$, which is between $ 128$ and $ k^2$.  By induction, $ n \minus{} k^2$ has a good representation, and all of its squares have to be less than $ k^2$.

This method doesn't give the constant (five or six) number of squares that t0r mentioned.

Edit:  I see now that what I wrote is similar to gunes's hidden content called "What I mean".  Yes, this method does require an annoying check of more than 100 numbers.

Thank you for your ideas :)
I assume that being [hide="Answer"]2^7[/hide] is just a coincidence! :(

[hide="This part includes the answer"]I, now, can conclude that if we find all numbers' square representations up to 272(128+144) and see that between 128 and 273 every number has a valid square representation, after 272 every number can be represented as sums of different squares![/hide]

Edit: After searching "sum of different squares" in google, (second and third links from "JSTOR") a man called "R. Sprague" found the answer to my question but I can't reach the article. If anyone can reach the article, I will be very pleased to hear that guy's method.


Hi
    Can you tell me how I can get the following output

********A B C D
******+  E F G H
********-----------
******=  I  J K L
********-----------

Where **** represents spacing before the start of the line and ---- represents an actual line.
The letters A,B,... could represent alphabets or numbers.

Thanks.

\[ \setlength{\tabcolsep}{1mm}
\begin{tabular}{cccccc}
& &    A & B & C & D \\
& \plus{} & E & F & G & H \\
\cline{3\minus{}6}
& \equal{} & I & J & K & L \\
\cline{3\minus{}6}
\end{tabular}\]

If I had to change the number of alphabets in any given column and have everything right aligned, then would I have to change the cline command? How? And what do the \tabcolsep, \begin{tabular} and {cccccc} commands do?

Thank you for the quick reply.  :)

Edit: Figured out how to align for any case. But still am not sure what the commands mean.

Tables are here [[LaTeX:Layout#Tables]]. In addition to what's there
\tabcolsep is the distance between columns which has been made smaller to look better for this case (try it without \setlength{\tabcolsep}{1mm} to see what happens)
\cline{3-6} means draw a line from column 3 to 6 whereas \hline draws a line across all the columns.


AMC
AMC 10
MATHCOUNTS
trigonometry
AIME
USA(J)MO
USAMO

This is probably not in the right section.

Any no, it's probaby not AMC difficulty (I assume you mean AMC10 and 12, anything higher would be insulting). For challenging SAT prep, visit the MathCounts forum.

I have heard of Gelfand's books, I believe his Trigonometry book is considered to be a classic; I'd expect somewhat of the same quality in his Algebra book too.

[quote="Boy Soprano II"]First, unless I am mistaken, those books that you are using have a lot about beating the system.  If you wish to learn mathematics, you might as well purchase a book that doesn't waste so much time, space, and money on the system and has better math content as well.  I personally have a strong dislike for the market of things designed to beat testing systems, instead of having you actually learn things.

Second, do not waste too much time preparing for the SAT.  There are many things (such as the AMC and beyond, and even MathCounts) which are both much more challenging and more worthwhile.  High school is a time for high school, not for preparing for college.  It is a time when we are given opportunity to pick and choose wherever we would wish to excel, and to excel there.  Unless I am mistaken, this will be somewhat diminished in college and will vanish altogether when we finish our schooling.  Now, would you spend these great and precious years in slavery to some test so that you can put a slightly better number on an application for a college, or would you spend them living and learning?  I think that many people lose sight of this.

(In answer to your original question, Phelpedo is completely right.)[/quote]

I agree with this statement. However, don't while focusing on the AMC, don't COMLPETELY ignore the SAT. An SAT score plays a vital role in college admissions. However, as far as the Math part is concerned, just study for the AMC. If you can study for the AMC and do well, you are effectively guaranteed a 770+ on the Math section. However, as most of us would agree, the math section would help us negligibly on the Writing and Critical Reading portions (unless there is a particular passage on the AMCs or mathematicians  :P ). Go and study AoPS 1 and 2. Read these completely. Then start working AIME problems. After you can get about 7 or 8 consistently, start chugging Olympiad problems with Engel's Problem Solving Strategies. WOOT might be a good option to consider as well (though it's fairly expensive.) After this, just keep working on Olympiad problems. After you get pretty good at Olympiad problems, start praying. I find this part to be particularly helpful, as when I pray, the problems seem magically easier on the tests. (Placebo effect, for sure, but definitely makes you think it works  :D )

[quote="Phelpedo"][quote="Karth"] An SAT score plays a vital role in college admissions. [/quote]

It does???  :rotfl:  :rotfl:  :rotfl:[/quote]
It plays, at least nominally, some role in the admissions to most universities, although many are rejecting it.  However, if a string of big, prestigious schools rejected the SAT, the finincial implications would be disasterous.  Hence the SAT stays, even though many studies have shown that it is not a good predictor of college performance, that it is biased culturally and also in favor of males, that it is well-established system that can be worked, and that, for some peculiar reason, the most reliable predictor of success in college is actually GPA.  However, I believe that the main purpose of these measures is to filter out people who shouldn't even be considered for admission and thus lighten the load of admissions officers, and not to determine who will be admitted from serious candidates.

But to make college admissions the purpose of one's life is actually self-defeating.  For if you were an admissions officer, whom would you want attending your school, someone with 2400 SATs who belongs to five honor societies; or someone with 2150 on SATs who spurns most honor societies, but who does a few things very passionately and loves learning?  If I were a professor, I would not want the person who always asks what is necessary to know for a test; I would want the person who is actually interested in the material.  So in brief, I say that if your SAT score is good enough, then stop paying homage to that system.

[quote="Boy Soprano II"][quote="Phelpedo"][quote="Karth"] An SAT score plays a vital role in college admissions. [/quote]

For if you were an admissions officer, whom would you want attending your school, someone with 2400 SATs who belongs to five honor societies; or someone with 2150 on SATs who spurns most honor societies, but who does a few things very passionately and loves learning?  If I were a professor, I would not want the person who always asks what is necessary to know for a test; I would want the person who is actually interested in the material.  [/quote][/quote]

Unfortunately, most unfortunately, I wish that this were the case. I wish that we didn't have to do 9304912834 hours of service and get a 239432 on our SATs to make it into Harvard/Princeton, etc. But these days, admissions officers aren't looking only for USAMO. They wouldn't really know how passionate the applicant is about a certain subject. For instance, somebody may just be preparing for USAMO just because they want to get into a good college. IMO and MOSP are entirely different stories. However, if someone just prepares for this stoicly, then, the professor has no way to tell whether the applicant truly enjoys mathematics. Thus, they mostly pay attention to sports AND academics AND clubs. This is the main reason as to why many practically add the SAT as one of their gods (including monotheistic religions). Of course, my religion being polytheistic, I can append the SAT to my pantheon of over 1000 gods  :D

[quote="kyyuanmathcount"][quote="simulacrum"]ne1 ever read the barron's how to prepare for the sat ii math level iic book? its kinda challeging(much more challegning then the real test) and i wanted to know if it was amc level.  and what about algebra by i.m gelfand?[/quote]

Could you please not use "chatspeak" or "13375p34k"?.

I thought the SAT's for math were nearly trivialized after studying AoPS book 1.[/quote]
It is... Other than a couple of Alg. 2 questions here and there...

[quote="Karth"]
Unfortunately, most unfortunately, I wish that this were the case. I wish that we didn't have to do 9304912834 hours of service and get a 239432 on our SATs to make it into Harvard/Princeton, etc. But these days, admissions officers aren't looking only for USAMO. They wouldn't really know how passionate the applicant is about a certain subject. For instance, somebody may just be preparing for USAMO just because they want to get into a good college. IMO and MOSP are entirely different stories.[/quote]

That's why you write essays on the things you really are passionate about; then colleges can tell if you're doing it because you love it or because it looks good.

Try to stay on topic; the discussion about weight of certain things on admissions to college should belong in the.... College forum

Keep things AMC related

[quote="Boy Soprano II"][quote="Karth"]Unfortunately, most unfortunately, I wish that this were the case. I wish that we didn't have to do 9304912834 hours of service and get a 239432 on our SATs to make it into Harvard/Princeton, etc. But these days, admissions officers aren't looking only for USAMO. They wouldn't really know how passionate the applicant is about a certain subject. For instance, somebody may just be preparing for USAMO just because they want to get into a good college. IMO and MOP are entirely different stories. However, if someone just prepares for this stoicly, then, the professor has no way to tell whether the applicant truly enjoys mathematics. Thus, they mostly pay attention to sports AND academics AND clubs. This is the main reason as to why many practically add the SAT as one of their gods (including monotheistic religions). Of course, my religion being polytheistic, I can append the SAT to my pantheon of over 1000 gods  :D[/quote]

Ah, but this is why they invented college interviews, my good friend. :wink: There is much more than just the application.  If you do not believe me, see [url=http://www.artofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=71]here[/url].  After all, wouldn't it be to a college's advantage to be able to tell if a prospective student is passionate about something?

Also, remember that high school is in many ways an end in itself.  It is first and foremost a time truly to live, not a time to prepare to live at some later date.  Frankly, I don't view the SAT as truly living, because if you're into the stuff that appears on the SAT, you'll soon move on to more challenging things :wink: .[/quote]

Yes, this is very true indeed. However, while interviews are important, most of the time, they come after the college believes that you are applicable or if the college is trying to select you over another student. They will take the better interview (which obviously, we must express our passion for mathematics) and the student will get into the college. I wasn't trying to say that SAT scores are pretty much life, but then again, one musn't take them very lightly. After all, SATs could be what makes you get over the hump to get into super-selective schools.

isn't the AMC a standardized test as well?  The vast majority of the people in my school who make AIME don't do it because they love math, they only do it because its a nice addition to your college application.

I agree with a lot of the things that both Karth and Boysoprano have said. Basically, the SAT is at least somewhat important in college admissions, so it would be in your interest to take the test and acquire a fairly good score. However, it would probably be a bad use of your time to spend lots and lots of time working SAT math problems just to up your score a little bit, if you don't make an 800 the first time.
Really you can think of it in economic terms: there is increasing opportunity cost as you study more and more for the SAT.

[quote="tokenadult"][quote]Hence the SAT stays, even though many studies have shown that it is not a good predictor of college performance[/quote]

The actual finding is a bit more nuanced than that. The most selective colleges in the United States regard SAT scores because those scores DO have some meaning in showing who is ready to do "college-level" work at their level. The [url=http://www.cpec.ca.gov/Commission/Eligibility.asp]University of California studies[/url] of admission test validity showed that SAT II scores are even more helpful than SAT I scores in showing who will do well in (lower-division) college-level work in the UC system. The SAT I and SAT II are here to stay at the top level of college selectivity.[/quote]

I had a question about the math part (SAT II Math B or something?). Is it helpful? What levels does it go until? Any calculus? Is it worth taking (if it isn't, I don't really want to pay $\$$40 for absolutely nothing)?

I think it's helpful. There are many people that get an 800 on SAT 1 math, but getting an 800 on SAT II math indicates to colleges that you really have a good handle of precalculus. people that are not into math competitions (or studying math in general) will miss a lot of things on the SAT II, simply because they have not taken the time to review and learn more advanced topics in the precalculus curriculum.

it doesn't get too hard (easier than AMC). I think that almost anything that you find in a pre-calculus book could be on there. harder topics would be conics, some of the more obscure parts of geometry, and some elementary number theory.

[quote="tokenadult"]SAT II tests are a mandatory requirement for a complete admissions file at some of the better colleges in the United States. The SAT II Math Level 2 (that's the correct name) is a useful test to take, required at at least one college, and not too hard for AoPSers.[/quote]

that's true. MIT and Caltech require the math and at least one other science if I can remember correctly.