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The AI Alliance for Solving Mathematics for All (AI4M) 12

informal statement For all odd $n$ show that $8 \mid n^{2}-1$.formal statement theorem exercise_1_27 {n : ℕ} (hn : odd n) : 8 ∣ (n^2 - 1) :=

informal statement Show that if $X$ is a Hausdorff space that is locally compact at the point $x$, then for each neighborhood $U$ of $x$, there is a neighborhood $V$ of $x$ such that $\bar{V}$ is compact and $\bar{V} \subset U$.formal statement theorem exercise_29_10 {X : Type*} [topological_space X] [t2_space X] (x : X) (hx : ∃ U : set X, x ∈ U ∧ is_open U ∧ (∃ K : set X, U ⊂ K ∧ is_compact K)) (U : set X) (hU : is_open U) (hxU : x ∈ U) : ∃ (V : set X), is_open V ∧ x ∈ V ∧ is_compact (closure V) ∧ closure V ⊆ U :=

informal statement Let $n$ be a positive integer, and let $f_{n}(z)=n+(n-1) z+$ $(n-2) z^{2}+\cdots+z^{n-1}$. Prove that $f_{n}$ has no roots in the closed unit disk $\{z \in \mathbb{C}:|z| \leq 1\}$.formal statement theorem exercise_2018_b2 (n : ℕ) (hn : n > 0) (f : ℕ → ℂ → ℂ) (hf : ∀ n : ℕ, f n = λ z, (∑ (i : fin n), (n-i)* z^(i : ℕ))) : ¬ (∃ z : ℂ, ‖z‖ ≤ 1 ∧ f n z = 0) :=

informal statement Show that a group of order 5 must be abelian.formal statement theorem exercise_2_1_21 (G : Type*) [group G] [fintype G] (hG : card G = 5) : comm_group G :=

informal statement Let $X$ be locally path connected. Show that every connected open set in $X$ is path connected.formal statement theorem exercise_25_4 {X : Type*} [topological_space X] [loc_path_connected_space X] (U : set X) (hU : is_open U) (hcU : is_connected U) : is_path_connected U :=

informal statement If $z$ is a complex number, prove that there exists an $r\geq 0$ and a complex number $w$ with $| w | = 1$ such that $z = rw$.formal statement theorem exercise_1_11a (z : ℂ) : ∃ (r : ℝ) (w : ℂ), abs w = 1 ∧ z = r * w :=

informal statement Suppose $V$ is a complex inner-product space and $T \in \mathcal{L}(V)$ is a normal operator such that $T^{9}=T^{8}$. Prove that $T$ is self-adjoint and $T^{2}=T$.formal statement theorem exercise_7_10 {V : Type*} [inner_product_space ℂ V] [finite_dimensional ℂ V] (T : End ℂ V) (hT : T * T.adjoint = T.adjoint * T) (hT1 : T^9 = T^8) : is_self_adjoint T ∧ T^2 = T :=

informal statement Let $p$ and $q$ be distinct odd primes such that $p-1$ divides $q-1$. If $(n, p q)=1$, show that $n^{q-1} \equiv 1(p q)$.formal statement theorem exercise_3_14 {p q n : ℕ} (hp0 : p.prime ∧ p > 2) (hq0 : q.prime ∧ q > 2) (hpq0 : p ≠ q) (hpq1 : p - 1 ∣ q - 1) (hn : n.gcd (p*q) = 1) : n^(q-1) ≡ 1 [MOD p*q] :=

informal statement Prove that $(x, y)$ is not a principal ideal in $\mathbb{Q}[x, y]$.formal statement theorem exercise_9_3_2 {f g : polynomial ℚ} (i j : ℕ) (hfg : ∀ n : ℕ, ∃ a : ℤ, (f*g).coeff = a) : ∃ a : ℤ, f.coeff i * g.coeff j = a :=

informal statement Let $\left\{A_{n}\right\}$ be a sequence of connected subspaces of $X$, such that $A_{n} \cap A_{n+1} \neq \varnothing$ for all $n$. Show that $\bigcup A_{n}$ is connected.formal statement theorem exercise_23_2 {X : Type*} [topological_space X] {A : ℕ → set X} (hA : ∀ n, is_connected (A n)) (hAn : ∀ n, A n ∩ A (n + 1) ≠ ∅) : is_connected (⋃ n, A n) :=

informal statement If $G$ is a nonabelian group of order 6, prove that $G \simeq S_3$.formal statement theorem exercise_2_5_37 (G : Type*) [group G] [fintype G] (hG : card G = 6) (hG' : is_empty (comm_group G)) : G ≃* equiv.perm (fin 3) :=

informal statement Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable.formal statement theorem exercise_2_27a (k : ℕ) (E P : set (euclidean_space ℝ (fin k))) (hE : E.nonempty ∧ ¬ set.countable E) (hP : P = {x | ∀ U ∈ 𝓝 x, ¬ set.countable (P ∩ E)}) : is_closed P ∧ P = {x | cluster_pt x (𝓟 P)} :=

informal statement Suppose $E\subset\mathbb{R}^k$ is uncountable, and let $P$ be the set of condensation points of $E$. Prove that $P$ is perfect.formal statement theorem exercise_2_27a (k : ℕ) (E P : set (euclidean_space ℝ (fin k))) (hE : E.nonempty ∧ ¬ set.countable E) (hP : P = {x | ∀ U ∈ 𝓝 x, ¬ set.countable (P ∩ E)}) : is_closed P ∧ P = {x | cluster_pt x (𝓟 P)} :=

informal statement Let $V$ be a vector space which is spanned by a countably infinite set. Prove that every linearly independent subset of $V$ is finite or countably infinite.formal statement theorem exercise_3_5_6 {K V : Type*} [field K] [add_comm_group V] [module K V] {S : set V} (hS : set.countable S) (hS1 : span K S = ⊤) {ι : Type*} (R : ι → V) (hR : linear_independent K R) : countable ι :=

informal statement Show that the rationals $\mathbb{Q}$ are not locally compact.formal statement theorem exercise_29_10 {X : Type*} [topological_space X] [t2_space X] (x : X) (hx : ∃ U : set X, x ∈ U ∧ is_open U ∧ (∃ K : set X, U ⊂ K ∧ is_compact K)) (U : set X) (hU : is_open U) (hxU : x ∈ U) : ∃ (V : set X), is_open V ∧ x ∈ V ∧ is_compact (closure V) ∧ closure V ⊆ U :=

informal statement Prove that if $G$ is an abelian simple group then $G \cong Z_{p}$ for some prime $p$ (do not assume $G$ is a finite group).formal statement theorem exercise_3_4_1 (G : Type*) [comm_group G] [is_simple_group G] : is_cyclic G ∧ ∃ G_fin : fintype G, nat.prime (@card G G_fin) :=

informal statement Prove that the power series $\sum zn/n^2$ converges at every point of the unit circle.formal statement theorem exercise_1_19b (z : ℂ) (hz : abs z = 1) (s : ℕ → ℂ) (h : s = (λ n, ∑ i in (finset.range n), i * z / i ^ 2)) : ∃ y, tendsto s at_top (𝓝 y) :=

/-- `try_solve t` either solves the goal using `t` or does not change the proof state. -/ macro "try_solve " t:tacticSeq : tactic => `(tactic| first | (solve | $t) | skip) -- example : True ∨ False := by (try_solve apply Or.inr); trivial /-- `lax_exact h` completes the goal using `h`, adding a subgoal to rewrite the goal to the type of `h`. It tries to automatically prove this subgoal using `trivial`. The disadvantage is that no inference is possible at all in `h`. -/ macro "lax_exact " t:term : tactic => /- `Eq.subst (Eq.symm ?_) $t` seems to get the motive wrong, so use `cast` and `congr` afterwards instead. -/ `(tactic| focus refine cast (Eq.symm ?_) $t; congr; try_solve trivial) /-- `lax_apply h` is essentially the same as `lax_exact (h ..)`. As a result, it is currently useless (see end of docstring of `lax_exact`). -/ -- In fact, currently, it _is_ the same. macro "lax_apply" t:term : tactic => `(tactic| lax_exact $t ..) /- example (a b : Nat) : a + b - b ≤ a + b := by lax_exact Nat.le_add_right a b apply Nat.add_sub_cancel -/

informal statement Show that 1729 is the smallest positive integer expressible as the sum of two different integral cubes in two ways.formal statement theorem exercise_18_4 {n : ℕ} (hn : ∃ x y z w : ℤ, x^3 + y^3 = n ∧ z^3 + w^3 = n ∧ x ≠ z ∧ x ≠ w ∧ y ≠ z ∧ y ≠ w) : n ≥ 1729 :=

informal statement Prove that if $|G|=1365$ then $G$ is not simple.formal statement theorem exercise_4_5_20 {G : Type*} [fintype G] [group G] (hG : card G = 1365) : ¬ is_simple_group G :=

informal statement Define $f_{n}:[0,1] \rightarrow \mathbb{R}$ by the equation $f_{n}(x)=x^{n}$. Show that the sequence $\left(f_{n}(x)\right)$ converges for each $x \in[0,1]$.formal statement theorem exercise_21_6a (f : ℕ → I → ℝ ) (h : ∀ x n, f n x = x ^ n) : ∀ x, ∃ y, tendsto (λ n, f n x) at_top (𝓝 y) :=

informal statement Let $|G|=p q r$, where $p, q$ and $r$ are primes with $p<q<r$. Prove that $G$ has a normal Sylow subgroup for either $p, q$ or $r$.formal statement theorem exercise_4_5_18 {G : Type*} [fintype G] [group G] (hG : card G = 200) : ∃ N : sylow 5 G, N.normal :=

informal statement Prove that if $|G|=132$ then $G$ is not simple.formal statement theorem exercise_4_5_22 {G : Type*} [fintype G] [group G] (hG : card G = 132) : ¬ is_simple_group G :=

informal statement Let $R$ be the ring of $2 \times 2$ matrices over the real numbers; suppose that $I$ is an ideal of $R$. Show that $I = (0)$ or $I = R$.formal statement theorem exercise_4_3_25 (I : ideal (matrix (fin 2) (fin 2) ℝ)) : I = ⊥ ∨ I = ⊤ :=

informal statement Show that if $X$ has a countable dense subset, every collection of disjoint open sets in $X$ is countable.formal statement theorem exercise_30_13 {X : Type*} [topological_space X] (h : ∃ (s : set X), countable s ∧ dense s) (U : set (set X)) (hU : ∀ (x y : set X), x ∈ U → y ∈ U → x ≠ y → x ∩ y = ∅) : countable U :=

informal statement Let $X$ be completely regular. Show that $X$ is connected if and only if the Stone-Čech compactification of $X$ is connected.formal statement theorem exercise_38_6 {X : Type*} (X : Type*) [topological_space X] [regular_space X] (h : ∀ x A, is_closed A ∧ ¬ x ∈ A → ∃ (f : X → I), continuous f ∧ f x = (1 : I) ∧ f '' A = {0}) : is_connected (univ : set X) ↔ is_connected (univ : set (stone_cech X)) :=

informal statement Consider a prime $p$ of the form $4 t+3$. Show that $a$ is a primitive root modulo $p$ iff $-a$ has order $(p-1) / 2$.formal statement theorem exercise_4_5 {p t : ℕ} (hp0 : p.prime) (hp1 : p = 4*t + 3) (a : zmod p) : is_primitive_root a p ↔ ((-a) ^ ((p-1)/2) = 1 ∧ ∀ (k : ℕ), k < (p-1)/2 → (-a)^k ≠ 1) :=

informal statement Suppose that $|G| = pm$, where $p \nmid m$ and $p$ is a prime. If $H$ is a normal subgroup of order $p$ in $G$, prove that $H$ is characteristic.formal statement theorem exercise_2_5_30 {G : Type*} [group G] [fintype G] {p m : ℕ} (hp : nat.prime p) (hp1 : ¬ p ∣ m) (hG : card G = p*m) {H : subgroup G} [fintype H] [H.normal] (hH : card H = p): characteristic H :=

informal statement Suppose that $|G| = pm$, where $p \nmid m$ and $p$ is a prime. If $H$ is a normal subgroup of order $p$ in $G$, prove that $H$ is characteristic.formal statement theorem exercise_2_5_37 (G : Type*) [group G] [fintype G] (hG : card G = 6) (hG' : is_empty (comm_group G)) : G ≃* equiv.perm (fin 3) :=

informal statement Prove that $\frac{(x+2)^p-2^p}{x}$, where $p$ is an odd prime, is irreducible in $\mathbb{Z}[x]$.formal statement theorem exercise_9_4_2d {p : ℕ} (hp : p.prime ∧ p > 2) {f : polynomial ℤ} (hf : f = (X + 2)^p): irreducible (∑ n in (f.support \ {0}), (f.coeff n) * X ^ (n-1) : polynomial ℤ) :=

informal statement Suppose $a, b \in R^k$. Find $c \in R^k$ and $r > 0$ such that $|x-a|=2|x-b|$ if and only if $| x - c | = r$. Prove that $3c = 4b - a$ and $3r = 2 |b - a|$.formal statement theorem exercise_1_19 (n : ℕ) (a b c x : euclidean_space ℝ (fin n)) (r : ℝ) (h₁ : r > 0) (h₂ : 3 • c = 4 • b - a) (h₃ : 3 * r = 2 * ‖x - b‖) : ‖x - a‖ = 2 * ‖x - b‖ ↔ ‖x - c‖ = r :=

informal statement Prove that $G$ cannot have a subgroup $H$ with $|H|=n-1$, where $n=|G|>2$.formal statement theorem exercise_2_1_5 {G : Type*} [group G] [fintype G] (hG : card G > 2) (H : subgroup G) [fintype H] : card H ≠ card G - 1 :=

informal statement Prove that $-(-v) = v$ for every $v \in V$.formal statement theorem exercise_1_3 {F V : Type*} [add_comm_group V] [field F] [module F V] {v : V} : -(-v) = v :=

informal statement Show that a closed subspace of a normal space is normal.formal statement theorem exercise_32_2b {ι : Type*} {X : ι → Type*} [∀ i, topological_space (X i)] (h : ∀ i, nonempty (X i)) (h2 : regular_space (Π i, X i)) : ∀ i, regular_space (X i) :=

informal statement If $r$ is rational $(r \neq 0)$ and $x$ is irrational, prove that $r+x$ is irrational.formal statement theorem exercise_1_1a (x : ℝ) (y : ℚ) : ( irrational x ) -> irrational ( x + y ) :=

informal statement Show that a closed subspace of a normal space is normal.formal statement theorem exercise_32_1 {X : Type*} [topological_space X] (hX : normal_space X) (A : set X) (hA : is_closed A) : normal_space {x // x ∈ A} :=

informal statement Let $R$ be a ring, with $M$ an ideal of $R$. Suppose that every element of $R$ which is not in $M$ is a unit of $R$. Prove that $M$ is a maximal ideal and that moreover it is the only maximal ideal of $R$.formal statement theorem exercise_10_7_10 {R : Type*} [ring R] (M : ideal R) (hM : ∀ (x : R), x ∉ M → is_unit x) : is_maximal M ∧ ∀ (N : ideal R), is_maximal N → N = M :=

informal statement Let $\mathbf{x}_1, \mathbf{x}_2, \ldots$ be a sequence of the points of the product space $\prod X_\alpha$. Show that this sequence converges to the point $\mathbf{x}$ if and only if the sequence $\pi_\alpha(\mathbf{x}_i)$ converges to $\pi_\alpha(\mathbf{x})$ for each $\alpha$.formal statement theorem exercise_21_6a (f : ℕ → I → ℝ ) (h : ∀ x n, f n x = x ^ n) : ∀ x, ∃ y, tendsto (λ n, f n x) at_top (𝓝 y) :=

informal statement Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $|f|$ is constant, then $f$ is constant.formal statement theorem exercise_1_13c {f : ℂ → ℂ} (Ω : set ℂ) (a b : Ω) (h : is_open Ω) (hf : differentiable_on ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, abs (f z) = c) : f a = f b :=

informal statement If $G$ is an abelian group and if $G$ has an element of order $m$ and one of order $n$, where $m$ and $n$ are relatively prime, prove that $G$ has an element of order $mn$.formal statement theorem exercise_2_6_15 {G : Type*} [comm_group G] {m n : ℕ} (hm : ∃ (g : G), order_of g = m) (hn : ∃ (g : G), order_of g = n) (hmn : m.coprime n) : ∃ (g : G), order_of g = m * n :=

informal statement Prove that if $|G|=1365$ then $G$ is not simple.formal statement theorem exercise_4_5_22 {G : Type*} [fintype G] [group G] (hG : card G = 132) : ¬ is_simple_group G :=

informal statement If $C_{0}+\frac{C_{1}}{2}+\cdots+\frac{C_{n-1}}{n}+\frac{C_{n}}{n+1}=0,$ where $C_{0}, \ldots, C_{n}$ are real constants, prove that the equation $C_{0}+C_{1} x+\cdots+C_{n-1} x^{n-1}+C_{n} x^{n}=0$ has at least one real root between 0 and 1.formal statement theorem exercise_5_4 {n : ℕ} (C : ℕ → ℝ) (hC : ∑ i in (finset.range (n + 1)), (C i) / (i + 1) = 0) : ∃ x, x ∈ (set.Icc (0 : ℝ) 1) ∧ ∑ i in finset.range (n + 1), (C i) * (x^i) = 0 :=

informal statement Show that a connected metric space having more than one point is uncountable.formal statement theorem exercise_27_4 {X : Type*} [metric_space X] [connected_space X] (hX : ∃ x y : X, x ≠ y) : ¬ countable (univ : set X) :=

informal statement Let $G$ be a finite group of composite order $n$ with the property that $G$ has a subgroup of order $k$ for each positive integer $k$ dividing $n$. Prove that $G$ is not simple.formal statement theorem exercise_4_2_14 {G : Type*} [fintype G] [group G] (hG : ¬ (card G).prime) (hG1 : ∀ k ∣ card G, ∃ (H : subgroup G) (fH : fintype H), @card H fH = k) : ¬ is_simple_group G :=

informal statement Show that if $X$ is normal, every pair of disjoint closed sets have neighborhoods whose closures are disjoint.formal statement theorem exercise_31_2 {X : Type*} [topological_space X] [normal_space X] {A B : set X} (hA : is_closed A) (hB : is_closed B) (hAB : disjoint A B) : ∃ (U V : set X), is_open U ∧ is_open V ∧ A ⊆ U ∧ B ⊆ V ∧ closure U ∩ closure V = ∅ :=

informal statement Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $\text{Re}(f)$ is constant, then $f$ is constant.formal statement theorem exercise_1_13a {f : ℂ → ℂ} (Ω : set ℂ) (a b : Ω) (h : is_open Ω) (hf : differentiable_on ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, (f z).re = c) : f a = f b :=

informal statement Show that the lower limit topology $\mathbb{R}_l$ and $K$-topology $\mathbb{R}_K$ are not comparable.formal statement theorem exercise_13_6 : ¬ (∀ U, Rl.is_open U → K_topology.is_open U) ∧ ¬ (∀ U, K_topology.is_open U → Rl.is_open U) :=

informal statement Let $A$ be a proper subset of $X$, and let $B$ be a proper subset of $Y$. If $X$ and $Y$ are connected, show that $(X \times Y)-(A \times B)$ is connected.formal statement theorem exercise_24_2 {f : (metric.sphere 0 1 : set ℝ) → ℝ} (hf : continuous f) : ∃ x, f x = f (-x) :=

informal statement Let $H$ be the subgroup generated by two elements $a, b$ of a group $G$. Prove that if $a b=b a$, then $H$ is an abelian group.formal statement theorem exercise_2_2_9 {G : Type*} [group G] {a b : G} (h : a * b = b * a) : ∀ x y : closure {x | x = a ∨ x = b}, x*y = y*x :=

informal statement Show that every locally compact Hausdorff space is regular.formal statement theorem exercise_32_3 {X : Type*} [topological_space X] (hX : locally_compact_space X) (hX' : t2_space X) : regular_space X :=

informal statement Show that every linear map from a one-dimensional vector space to itself is multiplication by some scalar. More precisely, prove that if $\operatorname{dim} V=1$ and $T \in \mathcal{L}(V, V)$, then there exists $a \in \mathbf{F}$ such that $T v=a v$ for all $v \in V$.formal statement theorem exercise_3_1 {F V : Type*} [add_comm_group V] [field F] [module F V] [finite_dimensional F V] (T : V →ₗ[F] V) (hT : finrank F V = 1) : ∃ c : F, ∀ v : V, T v = c • v:=

informal statement Let $G$ be any group. Prove that the map from $G$ to itself defined by $g \mapsto g^{-1}$ is a homomorphism if and only if $G$ is abelian.formal statement theorem exercise_1_6_17 {G : Type*} [group G] (f : G → G) (hf : f = λ g, g⁻¹) : ∀ x y : G, f x * f y = f (x*y) ↔ ∀ x y : G, x*y = y*x :=

informal statement Prove that the intersection of any collection of subspaces of $V$ is a subspace of $V$.formal statement theorem exercise_1_8 {F V : Type*} [add_comm_group V] [field F] [module F V] {ι : Type*} (u : ι → submodule F V) : ∃ U : submodule F V, (⋂ (i : ι), (u i).carrier) = ↑U :=

informal statement Show that 1729 is the smallest positive integer expressible as the sum of two different integral cubes in two ways.formal statement theorem exercise_1_13a {f : ℂ → ℂ} (Ω : set ℂ) (a b : Ω) (h : is_open Ω) (hf : differentiable_on ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, (f z).re = c) : f a = f b :=

informal statement Show that if $U$ is open in $X$ and $A$ is closed in $X$, then $U-A$ is open in $X$, and $A-U$ is closed in $X$.formal statement theorem exercise_17_4 {X : Type*} [topological_space X] (U A : set X) (hU : is_open U) (hA : is_closed A) : is_open (U \ A) ∧ is_closed (A \ U) :=

informal statement Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable.formal statement theorem exercise_2_28 (X : Type*) [metric_space X] [separable_space X] (A : set X) (hA : is_closed A) : ∃ P₁ P₂ : set X, A = P₁ ∪ P₂ ∧ is_closed P₁ ∧ P₁ = {x | cluster_pt x (𝓟 P₁)} ∧ set.countable P₂ :=

informal statement A uniformly continuous function of a uniformly continuous function is uniformly continuous.formal statement theorem exercise_4_19 {f : ℝ → ℝ} (hf : ∀ a b c, a < b → f a < c → c < f b → ∃ x, a < x ∧ x < b ∧ f x = c) (hg : ∀ r : ℚ, is_closed {x | f x = r}) : continuous f :=

informal statement Let $p$ be an odd prime and let $1 + \frac{1}{2} + ... + \frac{1}{p - 1} = \frac{a}{b}$, where $a, b$ are integers. Show that $p \mid a$.formal statement theorem exercise_4_2_9 {p : ℕ} (hp : nat.prime p) (hp1 : odd p) : ∃ (a b : ℤ), (a / b : ℚ) = ∑ i in finset.range p, 1 / (i + 1) → ↑p ∣ a :=

informal statement For $j \in\{1,2,3,4\}$, let $z_{j}$ be a complex number with $\left|z_{j}\right|=1$ and $z_{j} \neq 1$. Prove that $3-z_{1}-z_{2}-z_{3}-z_{4}+z_{1} z_{2} z_{3} z_{4} \neq 0 .$formal statement theorem exercise_2020_b5 (z : fin 4 → ℂ) (hz0 : ∀ n, ‖z n‖ < 1) (hz1 : ∀ n : fin 4, z n ≠ 1) : 3 - z 0 - z 1 - z 2 - z 3 + (z 0) * (z 1) * (z 2) * (z 3) ≠ 0 :=

informal statement Prove that a group of even order contains an element of order $2 .$formal statement theorem exercise_3_5_6 {K V : Type*} [field K] [add_comm_group V] [module K V] {S : set V} (hS : set.countable S) (hS1 : span K S = ⊤) {ι : Type*} (R : ι → V) (hR : linear_independent K R) : countable ι :=

informal statement Let $G$ be a group in which $(a b)^{3}=a^{3} b^{3}$ and $(a b)^{5}=a^{5} b^{5}$ for all $a, b \in G$. Show that $G$ is abelian.formal statement theorem exercise_2_2_5 {G : Type*} [group G] (h : ∀ (a b : G), (a * b) ^ 3 = a ^ 3 * b ^ 3 ∧ (a * b) ^ 5 = a ^ 5 * b ^ 5) : comm_group G :=

informal statement Let $p$ be a prime integer. Prove that the polynomial $x^n-p$ is irreducible in $\mathbb{Q}[x]$.formal statement theorem exercise_13_4_10 {p : ℕ} {hp : nat.prime p} (h : ∃ r : ℕ, p = 2 ^ r + 1) : ∃ (k : ℕ), p = 2 ^ (2 ^ k) + 1 :=

informal statement Prove that if a prime integer $p$ has the form $2^r+1$, then it actually has the form $2^{2^k}+1$.formal statement theorem exercise_13_4_10 {p : ℕ} {hp : nat.prime p} (h : ∃ r : ℕ, p = 2 ^ r + 1) : ∃ (k : ℕ), p = 2 ^ (2 ^ k) + 1 :=

informal statement Give an example of a nonempty subset $U$ of $\mathbf{R}^2$ such that $U$ is closed under addition and under taking additive inverses (meaning $-u \in U$ whenever $u \in U$), but $U$ is not a subspace of $\mathbf{R}^2$.formal statement theorem exercise_1_6 : ∃ U : set (ℝ × ℝ), (U ≠ ∅) ∧ (∀ (u v : ℝ × ℝ), u ∈ U ∧ v ∈ U → u + v ∈ U) ∧ (∀ (u : ℝ × ℝ), u ∈ U → -u ∈ U) ∧ (∀ U' : submodule ℝ (ℝ × ℝ), U ≠ ↑U') :=

informal statement Let $V$ be a vector space over an infinite field $F$. Show that $V$ cannot be the set-theoretic union of a finite number of proper subspaces of $V$.formal statement theorem exercise_5_2_20 {F V ι: Type*} [infinite F] [field F] [add_comm_group V] [module F V] {u : ι → submodule F V} (hu : ∀ i : ι, u i ≠ ⊤) : (⋃ i : ι, (u i : set V)) ≠ ⊤ :=

informal statement Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable.formal statement theorem exercise_2_24 {X : Type*} [metric_space X] (hX : ∀ (A : set X), infinite A → ∃ (x : X), x ∈ closure A) : separable_space X :=

informal statement Prove that $\sqrt{n+1}-\sqrt{n} \rightarrow 0$ as $n \rightarrow \infty$.formal statement theorem exercise_3_4 (n : ℕ) : tendsto (λ n, (sqrt (n + 1) - sqrt n)) at_top (𝓝 0) :=

informal statement Let $\|\cdot\|$ be any norm on $\mathbb{R}^{m}$ and let $B=\left\{x \in \mathbb{R}^{m}:\|x\| \leq 1\right\}$. Prove that $B$ is compact.formal statement theorem exercise_2_41 (m : ℕ) {X : Type*} [normed_space ℝ ((fin m) → ℝ)] : is_compact (metric.closed_ball 0 1) :=

informal statement Prove that no group of order $p^2 q$, where $p$ and $q$ are prime, is simple.formal statement theorem exercise_6_4_3 {G : Type*} [group G] [fintype G] {p q : ℕ} (hp : prime p) (hq : prime q) (hG : card G = p^2 *q) : is_simple_group G → false :=

informal statement Assume that $f$ is a continuous real function defined in $(a, b)$ such that $f\left(\frac{x+y}{2}\right) \leq \frac{f(x)+f(y)}{2}$ for all $x, y \in(a, b)$. Prove that $f$ is convex.formal statement theorem exercise_4_24 {f : ℝ → ℝ} (hf : continuous f) (a b : ℝ) (hab : a < b) (h : ∀ x y : ℝ, a < x → x < b → a < y → y < b → f ((x + y) / 2) ≤ (f x + f y) / 2) : convex_on ℝ (set.Ioo a b) f :=

informal statement Show that if $X$ has a countable dense subset, every collection of disjoint open sets in $X$ is countable.formal statement theorem exercise_31_2 {X : Type*} [topological_space X] [normal_space X] {A B : set X} (hA : is_closed A) (hB : is_closed B) (hAB : disjoint A B) : ∃ (U V : set X), is_open U ∧ is_open V ∧ A ⊆ U ∧ B ⊆ V ∧ closure U ∩ closure V = ∅ :=

informal statement Prove that in the ring $\mathbb{Z}[x],(2) \cap(x)=(2 x)$.formal statement theorem exercise_10_2_4 : span ({2} : set $ polynomial ℤ) ⊓ (span {X}) = span ({2 * X} : set $ polynomial ℤ) :=

informal statement Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $\text{Re}(f)$ is constant, then $f$ is constant.formal statement theorem exercise_1_13c {f : ℂ → ℂ} (Ω : set ℂ) (a b : Ω) (h : is_open Ω) (hf : differentiable_on ℂ f Ω) (hc : ∃ (c : ℝ), ∀ z ∈ Ω, abs (f z) = c) : f a = f b :=

informal statement Let $G$ be a transitive permutation group on the finite set $A$ with $|A|>1$. Show that there is some $\sigma \in G$ such that $\sigma(a) \neq a$ for all $a \in A$.formal statement theorem exercise_4_3_26 {α : Type*} [fintype α] (ha : fintype.card α > 1) (h_tran : ∀ a b: α, ∃ σ : equiv.perm α, σ a = b) : ∃ σ : equiv.perm α, ∀ a : α, σ a ≠ a :=

informal statement Let $x$ be an element of $G$. Prove that $x^2=1$ if and only if $|x|$ is either $1$ or $2$.formal statement theorem exercise_1_1_16 {G : Type*} [group G] (x : G) (hx : x ^ 2 = 1) : order_of x = 1 ∨ order_of x = 2 :=

informal statement Prove that the intersection of an arbitrary nonempty collection of normal subgroups of a group is a normal subgroup (do not assume the collection is countable).formal statement theorem exercise_3_1_22b {G : Type*} [group G] (I : Type*) (H : I → subgroup G) (hH : ∀ i : I, subgroup.normal (H i)) : subgroup.normal (⨅ (i : I), H i):=

informal statement Let $I, J$ be ideals in a ring $R$. Prove that the residue of any element of $I \cap J$ in $R / I J$ is nilpotent.formal statement theorem exercise_10_4_6 {R : Type*} [comm_ring R] [no_zero_divisors R] {I J : ideal R} (x : I ⊓ J) : is_nilpotent ((ideal.quotient.mk (I*J)) x) :=

informal statement An ideal $N$ is called nilpotent if $N^{n}$ is the zero ideal for some $n \geq 1$. Prove that the ideal $p \mathbb{Z} / p^{m} \mathbb{Z}$ is a nilpotent ideal in the ring $\mathbb{Z} / p^{m} \mathbb{Z}$.formal statement theorem exercise_8_1_12 {N : ℕ} (hN : N > 0) {M M': ℤ} {d : ℕ} (hMN : M.gcd N = 1) (hMd : d.gcd N.totient = 1) (hM' : M' ≡ M^d [ZMOD N]) : ∃ d' : ℕ, d' * d ≡ 1 [ZMOD N.totient] ∧ M ≡ M'^d' [ZMOD N] :=

informal statement If $a > 1$ is an integer, show that $n \mid \varphi(a^n - 1)$, where $\phi$ is the Euler $\varphi$-function.formal statement theorem exercise_2_4_36 {a n : ℕ} (h : a > 1) : n ∣ (a ^ n - 1).totient :=

informal statement Prove that any two nonabelian groups of order 21 are isomorphic.formal statement theorem exercise_2_8_12 {G H : Type*} [fintype G] [fintype H] [group G] [group H] (hG : card G = 21) (hH : card H = 21) (hG1 : is_empty(comm_group G)) (hH1 : is_empty (comm_group H)) : G ≃* H :=

informal statement Prove that $\sum 1/k(\log(k))^p$ diverges when $p \leq 1$.formal statement theorem exercise_3_63b (p : ℝ) (f : ℕ → ℝ) (hp : p ≤ 1) (h : f = λ k, (1 : ℝ) / (k * (log k) ^ p)) : ¬ ∃ l, tendsto f at_top (𝓝 l) :=

informal statement A ring $R$ is called a Boolean ring if $a^{2}=a$ for all $a \in R$. Prove that every Boolean ring is commutative.formal statement theorem exercise_7_1_15 {R : Type*} [ring R] (hR : ∀ a : R, a^2 = a) : comm_ring R :=

informal statement Suppose $U$ is a subspace of $V$. Prove that $U^{\perp}=\{0\}$ if and only if $U=V$formal statement theorem exercise_6_16 {K V : Type*} [is_R_or_C K] [inner_product_space K V] {U : submodule K V} : U.orthogonal = ⊥ ↔ U = ⊤ :=

informal statement Let $(p_n)$ be a sequence and $f:\mathbb{N}\to\mathbb{N}$. The sequence $(q_k)_{k\in\mathbb{N}}$ with $q_k=p_{f(k)}$ is called a rearrangement of $(p_n)$. Show that if $f$ is an injection, the limit of a sequence is unaffected by rearrangement.formal statement theorem exercise_2_12a (f : ℕ → ℕ) (p : ℕ → ℝ) (a : ℝ) (hf : injective f) (hp : tendsto p at_top (𝓝 a)) : tendsto (λ n, p (f n)) at_top (𝓝 a) :=

informal statement Let $p$ be an odd prime and let $1 + \frac{1}{2} + ... + \frac{1}{p - 1} = \frac{a}{b}$, where $a, b$ are integers. Show that $p \mid a$.formal statement theorem exercise_4_3_25 (I : ideal (matrix (fin 2) (fin 2) ℝ)) : I = ⊥ ∨ I = ⊤ :=

informal statement Show that there is an infinite number of integers a such that $f(x) = x^7 + 15x^2 - 30x + a$ is irreducible in $Q[x]$.formal statement theorem exercise_5_2_20 {F V ι: Type*} [infinite F] [field F] [add_comm_group V] [module F V] {u : ι → submodule F V} (hu : ∀ i : ι, u i ≠ ⊤) : (⋃ i : ι, (u i : set V)) ≠ ⊤ :=

informal statement Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable.formal statement theorem exercise_3_1a (f : ℕ → ℝ) (h : ∃ (a : ℝ), tendsto (λ (n : ℕ), f n) at_top (𝓝 a)) : ∃ (a : ℝ), tendsto (λ (n : ℕ), |f n|) at_top (𝓝 a) :=

informal statement If $G_1$ and $G_2$ are cyclic groups of orders $m$ and $n$, respectively, prove that $G_1 \times G_2$ is cyclic if and only if $m$ and $n$ are relatively prime.formal statement theorem exercise_2_9_2 {G H : Type*} [fintype G] [fintype H] [group G] [group H] (hG : is_cyclic G) (hH : is_cyclic H) : is_cyclic (G × H) ↔ (card G).coprime (card H) :=

informal statement Show that if $a$ is negative then $p \equiv q(4 a) together with p\not | a$ imply $(a / p)=(a / q)$.formal statement theorem exercise_5_37 {p q : ℕ} [fact(p.prime)] [fact(q.prime)] {a : ℤ} (ha : a < 0) (h0 : p ≡ q [ZMOD 4*a]) (h1 : ¬ ((p : ℤ) ∣ a)) : legendre_sym p a = legendre_sym q a :=

informal statement Consider a prime $p$ of the form $4 t+3$. Show that $a$ is a primitive root modulo $p$ iff $-a$ has order $(p-1) / 2$.formal statement theorem exercise_4_8 {p a : ℕ} (hp : odd p) : is_primitive_root a p ↔ (∀ q ∣ (p-1), q.prime → ¬ a^(p-1) ≡ 1 [MOD p]) :=

informal statement Let $X$ be completely regular, let $A$ and $B$ be disjoint closed subsets of $X$. Show that if $A$ is compact, there is a continuous function $f \colon X \rightarrow [0, 1]$ such that $f(A) = \{0\}$ and $f(B) = \{1\}$.formal statement theorem exercise_38_6 {X : Type*} (X : Type*) [topological_space X] [regular_space X] (h : ∀ x A, is_closed A ∧ ¬ x ∈ A → ∃ (f : X → I), continuous f ∧ f x = (1 : I) ∧ f '' A = {0}) : is_connected (univ : set X) ↔ is_connected (univ : set (stone_cech X)) :=

informal statement Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such thatformal statement theorem exercise_2_12a (f : ℕ → ℕ) (p : ℕ → ℝ) (a : ℝ) (hf : injective f) (hp : tendsto p at_top (𝓝 a)) : tendsto (λ n, p (f n)) at_top (𝓝 a) :=

informal statement Show that $ \int_{-\infty}^{\infty} \frac{\cos x}{x^2 + a^2} dx = \pi \frac{e^{-a}}{a}$ for $a > 0$.formal statement theorem exercise_3_3 (a : ℝ) (ha : 0 < a) : tendsto (λ y, ∫ x in -y..y, real.cos x / (x ^ 2 + a ^ 2)) at_top (𝓝 (real.pi * (real.exp (-a) / a))) :=

informal statement Prove that any two nonabelian groups of order 21 are isomorphic.formal statement theorem exercise_2_9_2 {G H : Type*} [fintype G] [fintype H] [group G] [group H] (hG : is_cyclic G) (hH : is_cyclic H) : is_cyclic (G × H) ↔ (card G).coprime (card H) :=

informal statement Prove that a group of order $p^2$, $p$ a prime, has a normal subgroup of order $p$.formal statement theorem exercise_2_5_44 {G : Type*} [group G] [fintype G] {p : ℕ} (hp : nat.prime p) (hG : card G = p^2) : ∃ (N : subgroup G) (fin : fintype N), @card N fin = p ∧ N.normal :=

informal statement Show that if $X$ is an infinite set, it is connected in the finite complement topology.formal statement theorem exercise_23_9 {X Y : Type*} [topological_space X] [topological_space Y] (A₁ A₂ : set X) (B₁ B₂ : set Y) (hA : A₁ ⊂ A₂) (hB : B₁ ⊂ B₂) (hA : is_connected A₂) (hB : is_connected B₂) : is_connected ({x | ∃ a b, x = (a, b) ∧ a ∈ A₂ ∧ b ∈ B₂} \ {x | ∃ a b, x = (a, b) ∧ a ∈ A₁ ∧ b ∈ B₁}) :=

informal statement Show that the subgroup of all rotations in a dihedral group is a maximal subgroup.formal statement theorem exercise_2_4_16b {n : ℕ} {hn : n ≠ 0} {R : subgroup (dihedral_group n)} (hR : R = subgroup.closure {dihedral_group.r 1}) : R ≠ ⊤ ∧ ∀ K : subgroup (dihedral_group n), R ≤ K → K = R ∨ K = ⊤ :=

informal statement Prove that a group of order 200 has a normal Sylow 5-subgroup.formal statement theorem exercise_4_5_20 {G : Type*} [fintype G] [group G] (hG : card G = 1365) : ¬ is_simple_group G :=

informal statement Suppose that $f$ is holomorphic in an open set $\Omega$. Prove that if $|f|$ is constant, then $f$ is constant.formal statement theorem exercise_1_19b (z : ℂ) (hz : abs z = 1) (s : ℕ → ℂ) (h : s = (λ n, ∑ i in (finset.range n), i * z / i ^ 2)) : ∃ y, tendsto s at_top (𝓝 y) :=

informal statement Suppose $V$ is a complex inner-product space and $T \in \mathcal{L}(V)$ is a normal operator such that $T^{9}=T^{8}$. Prove that $T$ is self-adjoint and $T^{2}=T$.formal statement theorem exercise_7_14 {𝕜 V : Type*} [is_R_or_C 𝕜] [inner_product_space 𝕜 V] [finite_dimensional 𝕜 V] {T : End 𝕜 V} (hT : is_self_adjoint T) {l : 𝕜} {ε : ℝ} (he : ε > 0) : ∃ v : V, ‖v‖= 1 ∧ (‖T v - l • v‖ < ε → (∃ l' : T.eigenvalues, ‖l - l'‖ < ε)) :=