problem_name stringlengths 14 14 | formal_statement stringlengths 75 1.32k | informal_statement stringlengths 47 898 | informal_solution stringlengths 5 303 | tags listlengths 1 3 | split stringclasses 1
value |
|---|---|---|---|---|---|
putnam_1980_b6 | theorem putnam_1980_b6
(G : ℤ × ℤ → ℚ)
(hG : ∀ d n : ℕ, d ≤ n → (d = 1 → G (d, n) = 1/(n : ℚ)) ∧ (d > 1 → G (d, n) = (d/(n : ℚ))*∑ i ∈ Finset.Icc d n, G ((d : ℤ) - 1, (i : ℤ) - 1)))
: ∀ d p : ℕ, 1 < d ∧ d ≤ p ∧ Prime p → ¬p ∣ (G (d, p)).den :=
sorry | For integers $d, n$ with $1 \le d \le n$, let $G(1, n) = \frac{1}{n}$ and $G(d, n) = \frac{d}{n}\sum_{i=d}^{n}G(d - 1, i - 1)$ for all $d > 1$. If $1 < d \le p$ for some prime $p$, prove that the reduced denominator of $G(d, p)$ is not divisible by $p$. | None. | [
"number_theory",
"algebra"
] | test |
putnam_1981_a1 | noncomputable abbrev putnam_1981_a1_solution : ℝ := sorry
theorem putnam_1981_a1
(P : ℕ → ℕ → Prop)
(hP : ∀ n k, P n k ↔ 5^k ∣ ∏ m ∈ Finset.Icc 1 n, (m^m : ℤ))
(E : ℕ → ℕ)
(hE : ∀ n ∈ Ici 1, P n (E n) ∧ ∀ k : ℕ, P n k → k ≤ E n) :
Tendsto (fun n : ℕ => ((E n) : ℝ)/n^2) atTop (𝓝 putnam_1981_a1_solut... | Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$. | The limit equals $\frac{1}{8}$. | [
"analysis",
"number_theory"
] | test |
putnam_1981_a3 | abbrev putnam_1981_a3_solution : Prop := sorry
theorem putnam_1981_a3
(f : ℝ → ℝ)
(hf : f = fun t : ℝ => Real.exp (-t) * ∫ y in (Ico 0 t), ∫ x in (Ico 0 t), (Real.exp x - Real.exp y) / (x - y))
: (∃ L : ℝ, Tendsto f atTop (𝓝 L)) ↔ putnam_1981_a3_solution :=
sorry | Does the limit $$lim_{t \rightarrow \infty}e^{-t}\int_{0}^{t}\int_{0}^{t}\frac{e^x - e^y}{x - y} dx dy$$exist? | The limit does not exist. | [
"analysis"
] | test |
putnam_1981_a5 | abbrev putnam_1981_a5_solution : Prop := sorry
theorem putnam_1981_a5
(Q : Polynomial ℝ → Polynomial ℝ)
(hQ : Q = fun P : Polynomial ℝ => (X^2 + 1)*P*(derivative P) + X*(P^2 + (derivative P)^2))
(n : Polynomial ℝ → ℝ)
(hn : n = fun P : Polynomial ℝ => ({x ∈ Ioi 1 | P.eval x = 0}.ncard : ℝ))
: (∀ P : Polynomial ℝ, {x : ... | Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$
Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots. | $Q(x)$ must have at least $2n - 1$ distinct real roots. | [
"algebra"
] | test |
putnam_1981_b1 | abbrev putnam_1981_b1_solution : ℝ := sorry
theorem putnam_1981_b1
(f : ℕ → ℝ)
(hf : f = fun n : ℕ => ((1 : ℝ)/n^5) * ∑ h ∈ Finset.Icc 1 n, ∑ k ∈ Finset.Icc 1 n, (5*(h : ℝ)^4 - 18*h^2*k^2 + 5*k^4))
: Tendsto f atTop (𝓝 putnam_1981_b1_solution) :=
sorry | Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$ | The limit equals $-1$. | [
"analysis"
] | test |
putnam_1981_b2 | noncomputable abbrev putnam_1981_b2_solution : ℝ := sorry
theorem putnam_1981_b2
(P : ℝ × ℝ × ℝ → Prop)
(hP : P = fun (r, s, t) => 1 ≤ r ∧ r ≤ s ∧ s ≤ t ∧ t ≤ 4)
(f : ℝ × ℝ × ℝ → ℝ)
(hf : f = fun (r, s, t) => (r - 1)^2 + (s/r - 1)^2 + (t/s - 1)^2 + (4/t - 1)^2) :
IsLeast {y | ∃ r s t, P (r, s, t) ∧ ... | Determine the minimum value attained by $$(r - 1)^2 + (\frac{s}{r} - 1)^2 + (\frac{t}{s} - 1)^2 + (\frac{4}{t} - 1)^2$$ across all choices of real $r$, $s$, and $t$ that satisfy $1 \le r \le s \le t \le 4$. | The minimum is $12 - 8\sqrt{2}$. | [
"algebra"
] | test |
putnam_1981_b3 | theorem putnam_1981_b3
(P : ℕ → Prop)
(hP : ∀ n, P n ↔
∀ p : ℕ, (Nat.Prime p ∧ p ∣ n^2 + 3) →
∃ k : ℕ, (p : ℤ) ∣ (k : ℤ)^2 + 3 ∧ k^2 < n) :
∀ n : ℕ, ∃ m : ℕ, (m : ℤ) > n ∧ P m :=
sorry | Prove that, for infinitely many positive integers $n$, all primes $p$ that divide $n^2 + 3$ also divide $k^2 + 3$ for some integer $k$ such that $k^2 < n$. | None. | [
"number_theory"
] | test |
putnam_1981_b4 | abbrev putnam_1981_b4_solution : Prop := sorry
theorem putnam_1981_b4
(VAB : Set (Matrix (Fin 5) (Fin 7) ℝ) → Prop)
(Vrank : Set (Matrix (Fin 5) (Fin 7) ℝ) → ℕ → Prop)
(hVAB : ∀ V, VAB V = (∀ A ∈ V, ∀ B ∈ V, ∀ r s : ℝ, r • A + s • B ∈ V))
(hVrank : ∀ V k, Vrank V k = ∃ A ∈ V, A.rank = k) :
putnam_19... | Let $V$ be a set of $5$ by $7$ matrices, with real entries and with the property that $rA+sB \in V$ whenever $A,B \in V$ and $r$ and $s$ are scalars (i.e., real numbers). \emph{Prove or disprove} the following assertion: If $V$ contains matrices of ranks $0$, $1$, $2$, $4$, and $5$, then it also contains a matrix of ra... | Show that the assertion is false. | [
"linear_algebra"
] | test |
putnam_1981_b5 | abbrev putnam_1981_b5_solution : Prop := sorry
theorem putnam_1981_b5
(sumbits : List ℕ → ℤ)
(B : ℕ → ℤ)
(hsumbits : ∀ bits : List ℕ, sumbits bits = ∑ i : Fin bits.length, (bits[i] : ℤ))
(hB : ∀ n > 0, B n = sumbits (Nat.digits 2 n))
: (∃ q : ℚ, Real.exp (∑' n : Set.Ici 1, B n / ((n : ℝ) * ((n : ℝ) + 1))) = q) ↔ putnam... | Let $B(n)$ be the number of ones in the base two expression for the positive integer $n$. For example, $B(6)=B(110_2)=2$ and $B(15)=B(1111_2)=4$. Determine whether or not $\exp \left(\sum_{n=1}^\infty \frac{B(n)}{n(n+1)}\right)$ is a rational number. Here $\exp(x)$ denotes $e^x$. | Show that the expression is a rational number. | [
"analysis",
"algebra"
] | test |
putnam_1982_a2 | abbrev putnam_1982_a2_solution : Prop := sorry
theorem putnam_1982_a2
(B : ℕ → ℝ → ℝ)
(hB : B = fun (n : ℕ) (x : ℝ) ↦ ∑ k ∈ Finset.Icc 1 n, (k : ℝ) ^ x)
(f : ℕ → ℝ)
(hf : f = fun n ↦ B n (logb n 2) / (n * logb 2 n) ^ 2)
: (∃ L : ℝ, Tendsto (fun N ↦ ∑ j ∈ Finset.Icc 2 N, f j) atTop (𝓝 L)) ↔ putnam_1982_a2_solution :=
s... | Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge? | Prove that the series converges. | [
"algebra"
] | test |
putnam_1982_a3 | noncomputable abbrev putnam_1982_a3_solution : ℝ := sorry
theorem putnam_1982_a3 :
Tendsto (fun t ↦ ∫ x in (0)..t, (arctan (Real.pi * x) - arctan x) / x) atTop (𝓝 putnam_1982_a3_solution) :=
sorry | Evaluate $\int_0^{\infty} \frac{\tan^{-1}(\pi x) - \tan^{-1} x}{x} \, dx$. | Show that the integral evaluates to $\frac{\pi}{2} \ln \pi$. | [
"analysis"
] | test |
putnam_1982_a4 | theorem putnam_1982_a4
(hdiffeq : (ℝ → ℝ) → (ℝ → ℝ) → Prop)
(hdiffeq_def : ∀ y z,
hdiffeq y z ↔
y 0 = 1 ∧ z 0 = 0 ∧
ContDiff ℝ 1 y ∧ ContDiff ℝ 1 z ∧
(∀ x : ℝ, deriv y x = -1 * (z x)^3 ∧ deriv z x = (y x)^3))
(f g : ℝ → ℝ)
(hfgsat : hdiffeq f g)
(hfgonly : ¬(∃ f' g' : ℝ... | Assume that the system of simultaneous differentiable equations \[y' = -z^3, z' = y^3\] with the initial conditions $y(0) = 1, z(0) = 0$ has a unique solution $y = f(x), z = g(x)$ defined for all real $x$. Prove that there exists a positive constant $L$ such that for all real $x$, \[f(x) + L = f(x), g(x + L) = g(x).\] | None. | [
"analysis"
] | test |
putnam_1982_a5 | theorem putnam_1982_a5
(a b c d : ℤ)
(hpos : a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0)
(hac : a + c ≤ 1982)
(hfrac : (a : ℝ) / b + (c : ℝ) / d < 1)
: (1 - (a : ℝ) / b - (c : ℝ) / d > 1 / 1983 ^ 3) :=
sorry | Let $a, b, c, d$ be positive integers satisfying $a + c \leq 1982$ and $\frac{a}{b} + \frac{c}{d} < 1$. Prove that $1 - \frac{a}{b} - \frac{c}{d} > \frac{1}{1983^3}$. | None. | [
"algebra"
] | test |
putnam_1982_a6 | abbrev putnam_1982_a6_solution : Prop := sorry
theorem putnam_1982_a6 :
(∀ b : ℕ → ℕ,
∀ x : ℕ → ℝ,
BijOn b (Ici 1) (Ici 1) →
StrictAntiOn (fun n : ℕ => |x n|) (Ici 1) →
Tendsto (fun n : ℕ => |b n - (n : ℤ)| * |x n|) atTop (𝓝 0) →
Tendsto (fun n : ℕ => ∑ k ∈ Finset.Icc 1 n, x k) atTop (𝓝 ... | Let $b$ be a bijection from the positive integers to the positive integers. Also, let $x_1, x_2, x_3, \dots$ be an infinite sequence of real numbers with the following properties:
\begin{enumerate}
\item
$|x_n|$ is a strictly decreasing function of $n$;
\item
$\lim_{n \rightarrow \infty} |b(n) - n| \cdot |x_n| = 0$;
\i... | The limit need not equal $1$. | [
"analysis"
] | test |
putnam_1982_b2 | noncomputable abbrev putnam_1982_b2_solution : Polynomial ℝ := sorry
theorem putnam_1982_b2
(A : ℝ × ℝ → ℕ)
(g I : ℝ)
(hA : A = fun (x, y) => {(m, n) : ℤ × ℤ | m^2 + n^2 ≤ x^2 + y^2}.ncard)
(hg : g = ∑' k : ℕ, Real.exp (-k^2))
(hI : I = ∫ y : ℝ, ∫ x : ℝ, A (x, y) * Real.exp (-x^2 - y^2))
: I = putnam_1982_b2_solution.e... | Let $A(x, y)$ denote the number of points $(m, n)$ with integer coordinates $m$ and $n$ where $m^2 + n^2 \le x^2 + y^2$. Also, let $g = \sum_{k = 0}^{\infty} e^{-k^2}$. Express the value $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} A(x, y)e^{-x^2 - y^2} dx dy$$ as a polynomial in $g$. | The desired polynomial is $\pi(2g - 1)^2$. | [
"analysis"
] | test |
putnam_1982_b3 | noncomputable abbrev putnam_1982_b3_solution : ℝ := sorry
theorem putnam_1982_b3
(p : ℕ → ℝ)
(hp : p = fun n : ℕ => ({(c, d) : Finset.Icc 1 n × Finset.Icc 1 n | ∃ m : ℕ, m^2 = c + d}.ncard : ℝ) / n^2)
: Tendsto (fun n : ℕ => p n * Real.sqrt n) atTop (𝓝 putnam_1982_b3_solution) :=
sorry | Let $p_n$ denote the probability that $c + d$ will be a perfect square if $c$ and $d$ are selected independently and uniformly at random from $\{1, 2, 3, \dots, n\}$. Express $\lim_{n \rightarrow \infty} p_n \sqrt{n}$ in the form $r(\sqrt{s} - t)$ for integers $s$ and $t$ and rational $r$. | The limit equals $\frac{4}{3}(\sqrt{2} - 1)$. | [
"analysis",
"number_theory",
"probability"
] | test |
putnam_1982_b4 | abbrev putnam_1982_b4_solution : Prop × Prop := sorry
theorem putnam_1982_b4
(P : Finset ℤ → Prop)
(P_def : ∀ n, P n ↔ n.Nonempty ∧ ∀ k, ∏ i ∈ n, i ∣ ∏ i ∈ n, (i + k)) :
((∀ n, P n → 1 ∈ n ∨ -1 ∈ n) ↔ putnam_1982_b4_solution.1) ∧
((∀ n, P n → (∀ i ∈ n, 0 < i) → n = Finset.Icc (1 : ℤ) n.card) ↔ putnam_19... | Let $n_1, n_2, \dots, n_s$ be distinct integers such that, for every integer $k$, $n_1n_2\cdots n_s$ divides $(n_1 + k)(n_2 + k) \cdots (n_s + k)$. Prove or provide a counterexample to the following claims:
\begin{enumerate}
\item
For some $i$, $|n_i| = 1$.
\item
If all $n_i$ are positive, then $\{n_1, n_2, \dots, n_s\... | Both claims are true. | [
"number_theory"
] | test |
putnam_1982_b5 | theorem putnam_1982_b5
(T : Set ℝ)
(hT : T = Ioi (Real.exp (Real.exp 1)))
(S : ℝ → ℕ → ℝ)
(hS : ∀ x ∈ T, S x 0 = (Real.exp 1) ∧ ∀ n : ℕ, S x (n + 1) = Real.logb (S x n) x)
(g : ℝ → ℝ)
: ∀ x ∈ T, (∃ L : ℝ, Tendsto (S x) atTop (𝓝 L)) ∧
(∀ x ∈ T, Tendsto (S x) atTop (𝓝 (g x))) → ContinuousOn g T :=
sorry | For all $x > e^e$, let $S = u_0, u_1, \dots$ be a recursively defined sequence with $u_0 = e$ and $u_{n+1} = \log_{u_n} x$ for all $n \ge 0$. Prove that $S_x$ converges to some real number $g(x)$ and that this function $g$ is continuous for $x > e^e$. | None. | [
"analysis"
] | test |
putnam_1983_a1 | abbrev putnam_1983_a1_solution : ℕ := sorry
theorem putnam_1983_a1
: {n : ℤ | n > 0 ∧ (n ∣ 10 ^ 40 ∨ n ∣ 20 ^ 30)}.encard = putnam_1983_a1_solution :=
sorry | How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40},20^{30}$? | Show that the desired count is $2301$. | [
"number_theory"
] | test |
putnam_1983_a3 | theorem putnam_1983_a3
(p : ℕ)
(F : ℕ → ℕ)
(poddprime : Odd p ∧ p.Prime)
(hF : ∀ n : ℕ, F n = ∑ i ∈ Finset.range (p - 1), (i + 1) * n ^ i)
: ∀ a ∈ Finset.Icc 1 p, ∀ b ∈ Finset.Icc 1 p, a ≠ b → ¬(F a ≡ F b [MOD p]) :=
sorry | Let $p$ be in the set $\{3,5,7,11,\dots\}$ of odd primes and let $F(n)=1+2n+3n^2+\dots+(p-1)n^{p-2}$. Prove that if $a$ and $b$ are distinct integers in $\{0,1,2,\dots,p-1\}$ then $F(a)$ and $F(b)$ are not congruent modulo $p$, that is, $F(a)-F(b)$ is not exactly divisible by $p$. | None. | [
"number_theory",
"algebra"
] | test |
putnam_1983_a4 | theorem putnam_1983_a4
(k m : ℕ)
(S : ℤ)
(kpos : k > 0)
(hm : m = 6 * k - 1)
(hS : S = ∑ j ∈ Finset.Icc 1 (2 * k - 1), (-1 : ℤ) ^ (j + 1) * choose m (3 * j - 1))
: (S ≠ 0) :=
sorry | Prove that for $m = 5 \pmod 6$,
\[
\binom{m}{2} - \binom{m}{5} + \binom{m}{8} - \binom{m}{11} + ... - \binom{m}{m-6} + \binom{m}{m-3} \neq 0.
\] | None. | [
"algebra"
] | test |
putnam_1983_a5 | abbrev putnam_1983_a5_solution : Prop := sorry
theorem putnam_1983_a5 :
(∃ α : ℝ, α > 0 ∧ ∀ n : ℕ, n > 0 → Even (⌊α ^ n⌋ - n)) ↔ putnam_1983_a5_solution :=
sorry | Prove or disprove that there exists a positive real number $\alpha$ such that $[\alpha_n] - n$ is even for all integers $n > 0$. (Here $[x]$ denotes the greatest integer less than or equal to $x$.) | Prove that such an $\alpha$ exists. | [
"analysis"
] | test |
putnam_1983_a6 | noncomputable abbrev putnam_1983_a6_solution : ℝ := sorry
theorem putnam_1983_a6
(F : ℝ → ℝ)
(hF : F = fun a ↦ (a ^ 4 / exp (a ^ 3)) * ∫ x in (0)..a, ∫ y in (0)..(a - x), exp (x ^ 3 + y ^ 3))
: (Tendsto F atTop (𝓝 putnam_1983_a6_solution)) :=
sorry | Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$. | Show that the integral evaluates to $\frac{2}{9}$. | [
"analysis"
] | test |
putnam_1983_b2 | abbrev putnam_1983_b2_solution : Prop := sorry
theorem putnam_1983_b2
(f : ℕ+ → ℕ)
(hf : f = fun (n : ℕ+) ↦
Set.ncard {M : Multiset ℕ |
(∀ m ∈ M, ∃ k : ℕ, m = (2 ^ k : ℤ)) ∧
(∀ m ∈ M, M.count m ≤ 3) ∧
(M.sum : ℤ) = n}) :
putnam_1983_b2_solution ↔
(∃ p : Polynomial ℝ, ∀ n ... | Let $f(n)$ be the number of ways of representing $n$ as a sum of powers of $2$ with no power being used more than $3$ times. For example, $f(7) = 4$ (the representations are $4 + 2 + 1$, $4 + 1 + 1 + 1$, $2 + 2 + 2 + 1$, $2 + 2 + 1 + 1 + 1$). Can we find a real polynomial $p(x)$ such that $f(n) = [p(n)]$, where $[u]$ d... | Prove that such a polynomial exists. | [
"algebra"
] | test |
putnam_1983_b4 | theorem putnam_1983_b4
(f : ℕ → ℤ)
(a : ℕ → ℕ)
(hf : f = fun (n : ℕ) ↦ n + Int.floor (√n))
(ha0 : a 0 > 0)
(han : ∀ n : ℕ, a (n + 1) = f (a n)) :
(∃ i : ℕ, ∃ s : ℤ, a i = s ^ 2) :=
sorry | Let $f(n) = n + [\sqrt n]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Define the sequence $a_i$ by $a_0 = m$, $a_{n+1} = f(a_n)$. Prove that it contains at least one square. | None. | [
"algebra"
] | test |
putnam_1983_b5 | noncomputable abbrev putnam_1983_b5_solution : ℝ := sorry
theorem putnam_1983_b5
(dist_fun : ℝ → ℝ)
(hdist_fun : dist_fun = fun (x : ℝ) ↦ min (x - ⌊x⌋) (⌈x⌉ - x))
(fact : Tendsto (fun N ↦ ∏ n ∈ Finset.Icc 1 N, (2 * n / (2 * n - 1)) * (2 * n / (2 * n + 1)) : ℕ → ℝ) atTop (𝓝 (Real.pi / 2)))
: (Tendsto (fun n ↦ (1 / n) *... | Define $\left\lVert x \right\rVert$ as the distance from $x$ to the nearest integer. Find $\lim_{n \to \infty} \frac{1}{n} \int_{1}^{n} \left\lVert \frac{n}{x} \right\rVert \, dx$. You may assume that $\prod_{n=1}^{\infty} \frac{2n}{(2n-1)} \cdot \frac{2n}{(2n+1)} = \frac{\pi}{2}$. | Show that the limit equals $\ln \left( \frac{4}{\pi} \right)$. | [
"analysis"
] | test |
putnam_1983_b6 | theorem putnam_1983_b6
(n : ℕ)
(npos : n > 0)
(α : ℂ)
(hα : α ^ (2 ^ n + 1) - 1 = 0 ∧ α ≠ 1)
: (∃ p q : Polynomial ℤ, (aeval α p) ^ 2 + (aeval α q) ^ 2 = -1) :=
sorry | Let $n$ be a positive integer and let $\alpha \neq 1$ be a complex $(2n + 1)\textsuperscript{th}$ root of unity. Prove that there always exist polynomials $p(x)$, $q(x)$ with integer coefficients such that $p(\alpha)^2 + q(\alpha)^2 = -1$. | None. | [
"algebra"
] | test |
putnam_1984_a2 | abbrev putnam_1984_a2_solution : ℚ := sorry
theorem putnam_1984_a2
: ∑' k : Set.Ici 1, (6 ^ (k : ℕ) / ((3 ^ ((k : ℕ) + 1) - 2 ^ ((k : ℕ) + 1)) * (3 ^ (k : ℕ) - 2 ^ (k : ℕ)))) = putnam_1984_a2_solution :=
sorry | Express $\sum_{k=1}^\infty (6^k/(3^{k+1}-2^{k+1})(3^k-2^k))$ as a rational number. | Show that the sum converges to $2$. | [
"analysis"
] | test |
putnam_1984_a3 | noncomputable abbrev putnam_1984_a3_solution : MvPolynomial (Fin 3) ℝ := sorry
theorem putnam_1984_a3
(n : ℕ)
(a b : ℝ)
(Mn : ℝ → Matrix (Fin (2 * n)) (Fin (2 * n)) ℝ)
(polyabn : Fin 3 → ℝ)
(npos : n > 0)
(aneb : a ≠ b)
(hMn : Mn = fun x : ℝ => fun i j : Fin (2 * n) => if i = j then x else if Even (i.1 + j.1) then a el... | Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$, and let $M_n$ denote the $2n \times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by
\[
m_{ij}=\begin{cases}
x & \text{if }i=j, \\
a & \text{if }i \neq j\text{ and }i+j\text{ is even}, \\
b & \text{if }i \neq j\text{ and }i+j\text{ is odd}.... | Show that $\lim_{x \to a} \frac{\det M_n}{(x-a)^{2n-2}}=n^2(a^2-b^2)$. | [
"linear_algebra",
"analysis"
] | test |
putnam_1984_a5 | abbrev putnam_1984_a5_solution : ℕ × ℕ × ℕ × ℕ × ℕ := sorry
theorem putnam_1984_a5
(R : Set (Fin 3 → ℝ))
(w : (Fin 3 → ℝ) → ℝ)
(hR : R = {p | (∀ i : Fin 3, p i ≥ 0) ∧ p 0 + p 1 + p 2 ≤ 1})
(hw : ∀ p, w p = 1 - p 0 - p 1 - p 2) :
let (a, b, c, d, n) := putnam_1984_a5_solution;
a > 0 ∧ b > 0 ∧... | Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z \leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral $\iiint_R x^1y^9z^8w^4\,dx\,dy\,dz$ in the form $a!b!c!d!/n!$, where $a$, $b$, $c$, $d$, and $n$ are positive integers. | Show that the integral we desire is $1!9!8!4!/25!$. | [
"analysis"
] | test |
putnam_1984_a6 | abbrev putnam_1984_a6_solution : ℕ := sorry
theorem putnam_1984_a6
(f : ℕ → ℕ)
(hf : ∀ n, some (f n) = (Nat.digits 10 (n !)).find? (fun d ↦ d ≠ 0))
(IsPeriodicFrom : ℕ → (ℕ → ℕ) → ℕ → Prop)
(IsPeriodicFrom_def : ∀ x f p, IsPeriodicFrom x f p ↔ Periodic (f ∘ (· + x)) p)
(P : ℕ → (ℕ → ℕ) → ℕ → Prop)
... | Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$. For instance, $f(5)=2$.
\begin{enumerate}
\item[(a)] Show that if $a_1,a_2,\dots,a_k$ are \emph{distinct} nonnegative integers, then $f(5^{a_1}+5^{a_2}+\dots+5^{a_k})$ depends only on the sum $a_1+a_2+\dots+a_k... | Show that the least such $p$ is $p=4$. | [
"algebra",
"number_theory"
] | test |
putnam_1984_b1 | noncomputable abbrev putnam_1984_b1_solution : Polynomial ℝ × Polynomial ℝ := sorry
theorem putnam_1984_b1
(f : ℕ → ℤ)
(hf : ∀ n > 0, f n = ∑ i : Set.Icc 1 n, ((i)! : ℤ))
: let (P, Q) := putnam_1984_b1_solution; ∀ n ≥ 1, f (n + 2) = P.eval (n : ℝ) * f (n + 1) + Q.eval (n : ℝ) * f n :=
sorry | Let $n$ be a positive integer, and define $f(n)=1!+2!+\dots+n!$. Find polynomials $P(x)$ and $Q(x)$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$. | Show that we can take $P(x)=x+3$ and $Q(x)=-x-2$. | [
"algebra"
] | test |
putnam_1984_b2 | abbrev putnam_1984_b2_solution : ℝ := sorry
theorem putnam_1984_b2
(f : ℝ → ℝ → ℝ)
(hf : ∀ u v : ℝ, f u v = (u - v) ^ 2 + (Real.sqrt (2 - u ^ 2) - 9 / v) ^ 2) :
IsLeast {y | ∃ᵉ (u : Set.Ioo 0 √2) (v > 0), f u v = y} putnam_1984_b2_solution :=
sorry | Find the minimum value of $(u-v)^2+(\sqrt{2-u^2}-\frac{9}{v})^2$ for $0< u<\sqrt{2}$ and $v>0$. | Show that the minimum value is $8$. | [
"geometry",
"analysis"
] | test |
putnam_1984_b3 | abbrev putnam_1984_b3_solution : Prop := sorry
theorem putnam_1984_b3
: (∀ (F : Type*) (_ : Fintype F), Fintype.card F ≥ 2 → (∃ mul : F → F → F, ∀ x y z : F, (mul x z = mul y z → x = y) ∧ (mul x (mul y z) ≠ mul (mul x y) z))) ↔ putnam_1984_b3_solution :=
sorry | Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on F such that for all $x,y,z$ in $F$,
\begin{enumerate}
\item[(i)] $x*z=y*z$ implies $x=y$ (right cancellation holds), and
\item[(ii)] $x*(y*z) \neq (x*y)*z$ (\emph{no} case of associat... | Show that the statement is true. | [
"abstract_algebra"
] | test |
putnam_1984_b5 | noncomputable abbrev putnam_1984_b5_solution : ℤ × Polynomial ℝ × Polynomial ℕ := sorry
theorem putnam_1984_b5
(m : ℕ) (mpos : m > 0)
(d : ℕ → ℕ)
(sumbits : List ℕ → ℕ)
(hsumbits : ∀ bits : List ℕ, sumbits bits = ∑ i : Fin bits.length, bits[i])
(hd : ∀ k : ℕ, d k = sumbits (Nat.digits 2 k)) :
le... | For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)=2$). Let $m$ be a positive integer. Express $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m$ in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials. | Show that $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m=(-1)^m2^{m(m-1)/2}m!$. | [
"algebra",
"analysis"
] | test |
putnam_1985_a1 | abbrev putnam_1985_a1_solution : ℕ × ℕ × ℕ × ℕ := sorry
theorem putnam_1985_a1 :
let (a, b, c, d) := putnam_1985_a1_solution;
{(A1, A2, A3) : Set ℤ × Set ℤ × Set ℤ | A1 ∪ A2 ∪ A3 = Icc 1 10 ∧ A1 ∩ A2 ∩ A3 = ∅}.ncard = 2 ^ a * 3 ^ b * 5 ^ c * 7 ^ d :=
sorry | Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that
\begin{enumerate}
\item[(i)] $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, and
\item[(ii)] $A_1 \cap A_2 \cap A_3 = \emptyset$.
\end{enumerate}
Express your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,... | Prove that the number of such triples is $2^{10}3^{10}$. | [
"algebra"
] | test |
putnam_1985_a3 | noncomputable abbrev putnam_1985_a3_solution : ℝ → ℝ := sorry
theorem putnam_1985_a3
(d : ℝ)
(a : ℕ → ℕ → ℝ)
(ha0 : ∀ m : ℕ, a m 0 = d / 2 ^ m)
(ha : ∀ m : ℕ, ∀ j : ℕ, a m (j + 1) = (a m j) ^ 2 + 2 * a m j)
: Tendsto (fun n ↦ a n n) atTop (𝓝 (putnam_1985_a3_solution d)) :=
sorry | Let $d$ be a real number. For each integer $m \geq 0$, define a sequence $\{a_m(j)\}$, $j=0,1,2,\dots$ by the condition
\begin{align*}
a_m(0) &= d/2^m, \\
a_m(j+1) &= (a_m(j))^2 + 2a_m(j), \qquad j \geq 0.
\end{align*}
Evaluate $\lim_{n \to \infty} a_n(n)$. | Show that the limit equals $e^d - 1$. | [
"analysis"
] | test |
putnam_1985_a4 | abbrev putnam_1985_a4_solution : Set (Fin 100) := sorry
theorem putnam_1985_a4
(a : ℕ → ℕ)
(ha1 : a 1 = 3)
(ha : ∀ i ≥ 1, a (i + 1) = 3 ^ a i) :
{k : Fin 100 | ∀ N : ℕ, ∃ i ≥ N, a i % 100 = k} = putnam_1985_a4_solution :=
sorry | Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i \geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$? | Prove that the only number that occurs infinitely often is $87$. | [
"number_theory"
] | test |
putnam_1985_a5 | abbrev putnam_1985_a5_solution : Set ℕ := sorry
theorem putnam_1985_a5
(I : ℕ → ℝ)
(hI : I = fun (m : ℕ) ↦ ∫ x in (0)..(2 * Real.pi), ∏ k ∈ Finset.Icc 1 m, cos (k * x)) :
{m ∈ Finset.Icc 1 10 | I m ≠ 0} = putnam_1985_a5_solution :=
sorry | Let $I_m = \int_0^{2\pi} \cos(x)\cos(2x)\cdots \cos(mx)\,dx$. For which integers $m$, $1 \leq m \leq 10$ is $I_m \neq 0$? | Prove that the integers $m$ with $1 \leq m \leq 10$ and $I_m \neq 0$ are $m = 3, 4, 7, 8$. | [
"analysis"
] | test |
putnam_1985_a6 | noncomputable abbrev putnam_1985_a6_solution : Polynomial ℝ := sorry
theorem putnam_1985_a6
(Γ : Polynomial ℝ → ℝ)
(f : Polynomial ℝ)
(hΓ : Γ = fun p ↦ ∑ k ∈ Finset.range (p.natDegree + 1), coeff p k ^ 2)
(hf : f = 3 * X ^ 2 + 7 * X + 2) :
let g := putnam_1985_a6_solution;
g.eval 0 = 1 ∧ ∀ n : ℕ, n ≥ 1 → Γ ... | If $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ is a polynomial with real coefficients $a_i$, then set
\[
\Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2.
\]
Let $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with real coefficients such that
\begin{enumerate}
\item[(i)] $g(0)=1$, and
\item[(ii)] $\Gamma(f(x)^n) = \G... | Show that $g(x) = 6x^2 + 5x + 1$ satisfies the conditions. | [
"algebra"
] | test |
putnam_1985_b1 | abbrev putnam_1985_b1_solution : Fin 5 → ℤ := sorry
theorem putnam_1985_b1
(p : (Fin 5 → ℤ) → (Polynomial ℝ))
(hp : p = fun m ↦ ∏ i : Fin 5, ((X : Polynomial ℝ) - m i))
(numnzcoeff : Polynomial ℝ → ℕ)
(hnumnzcoeff : numnzcoeff = fun p ↦ {j ∈ Finset.range (p.natDegree + 1) | coeff p j ≠ 0}.card)
: (Injective putnam_1985... | Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial
\[
p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)
\]
has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved. | Show that the minimum $k = 3$ is obtained for $\{m_1, m_2, m_3, m_4, m_5\} = \{-2, -1, 0, 1, 2\}$. | [
"algebra"
] | test |
putnam_1985_b2 | abbrev putnam_1985_b2_solution : ℕ → ℕ := sorry
theorem putnam_1985_b2
(f : ℕ -> Polynomial ℕ)
(hf0x : f 0 = 1)
(hfn0 : ∀ n ≥ 1, (f n).eval 0 = 0)
(hfderiv : ∀ n : ℕ, derivative (f (n + 1)) = (n + 1) * (Polynomial.comp (f n) (X + 1)))
: Nat.factorization ((f 100).eval 1) = putnam_1985_b2_solution :=
sorry | Define polynomials $f_n(x)$ for $n \geq 0$ by $f_0(x)=1$, $f_n(0)=0$ for $n \geq 1$, and
\[
\frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1)
\]
for $n \geq 0$. Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes. | Show that $f_{100}(1) = 101^{99}$. | [
"algebra"
] | test |
putnam_1985_b3 | theorem putnam_1985_b3
(a : ℕ → ℕ → ℕ)
(apos : ∀ m n : ℕ, a m n > 0)
(ha : ∀ k : ℕ, k > 0 → {(m, n) : ℕ × ℕ | m > 0 ∧ n > 0 ∧ a m n = k}.encard = 8)
: (∃ m n, m > 0 ∧ n > 0 ∧ a m n > m * n) :=
sorry | Let
\[
\begin{array}{cccc} a_{1,1} & a_{1,2} & a_{1,3} & \dots \\
a_{2,1} & a_{2,2} & a_{2,3} & \dots \\
a_{3,1} & a_{3,2} & a_{3,3} & \dots \\
\vdots & \vdots & \vdots & \ddots
\end{array}
\]
be a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Pr... | None. | [
"algebra"
] | test |
putnam_1985_b5 | noncomputable abbrev putnam_1985_b5_solution : ℝ := sorry
theorem putnam_1985_b5
(fact : ∫ x in univ, exp (- x ^ 2) = sqrt (Real.pi))
: (∫ t in Set.Ioi 0, t ^ (- (1 : ℝ) / 2) * exp (-1985 * (t + t ^ (-(1 : ℝ)))) = putnam_1985_b5_solution) :=
sorry | Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$. | Show that the integral evaluates to $\sqrt{\frac{\pi}{1985}}e^{-3970}$. | [
"analysis"
] | test |
putnam_1985_b6 | theorem putnam_1985_b6
(n : ℕ)
(npos : n > 0)
(G : Finset (Matrix (Fin n) (Fin n) ℝ))
(groupG : (∀ g ∈ G, ∀ h ∈ G, g * h ∈ G) ∧ 1 ∈ G ∧ (∀ g ∈ G, ∃ h ∈ G, g * h = 1))
(hG : ∑ M ∈ G, Matrix.trace M = 0)
: (∑ M ∈ G, M = 0) :=
sorry | Let $G$ be a finite set of real $n\times n$ matrices $\{M_i\}$, $1 \leq i \leq r$, which form a group under matrix
multiplication. Suppose that $\sum_{i=1}^r \mathrm{tr}(M_i)=0$, where $\mathrm{tr}(A)$ denotes the trace of the matrix $A$. Prove that $\sum_{i=1}^r M_i$ is the $n \times n$ zero matrix. | None. | [
"abstract_algebra",
"linear_algebra"
] | test |
putnam_1986_a1 | abbrev putnam_1986_a1_solution : ℝ := sorry
theorem putnam_1986_a1
(S : Set ℝ) (f : ℝ → ℝ)
(hS : S = {x : ℝ | x ^ 4 + 36 ≤ 13 * x ^ 2})
(hf : f = fun x ↦ x ^ 3 - 3 * x) :
IsGreatest
{f x | x ∈ S}
putnam_1986_a1_solution :=
sorry | Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \leq 13x^2$. | Show that the maximum value is $18$. | [
"algebra",
"analysis"
] | test |
putnam_1986_a2 | abbrev putnam_1986_a2_solution : ℕ := sorry
theorem putnam_1986_a2
: (Nat.floor ((10 ^ 20000 : ℝ) / (10 ^ 100 + 3)) % 10 = putnam_1986_a2_solution) :=
sorry | What is the units (i.e., rightmost) digit of
\[
\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?
\] | Show that the answer is $3$. | [
"algebra"
] | test |
putnam_1986_a3 | noncomputable abbrev putnam_1986_a3_solution : ℝ := sorry
theorem putnam_1986_a3
(cot : ℝ → ℝ)
(fcot : cot = fun θ ↦ cos θ / sin θ)
(arccot : ℝ → ℝ)
(harccot : ∀ t : ℝ, t ≥ 0 → arccot t ∈ Set.Ioc 0 (Real.pi / 2) ∧ cot (arccot t) = t)
: (∑' n : ℕ, arccot (n ^ 2 + n + 1) = putnam_1986_a3_solution) :=
sorry | Evaluate $\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$. | Show that the sum equals $\pi/2$. | [
"analysis"
] | test |
putnam_1986_a4 | abbrev putnam_1986_a4_solution : ℚ × ℚ × ℚ × ℚ × ℚ × ℚ × ℚ := sorry
theorem putnam_1986_a4
(f : ℕ → ℕ)
(hf : f = fun n ↦
Set.ncard {A : Matrix (Fin n) (Fin n) ℤ |
(∀ i j : Fin n, A i j ∈ ({-1, 0, 1} : Set ℤ)) ∧
∃ S : ℤ, ∀ ϕ : Perm (Fin n), ∑ i : Fin n, A i (ϕ i) = S}) :
let (a1, b1, a2, b2, a3, b3, a4... | A \emph{transversal} of an $n\times n$ matrix $A$ consists of $n$ entries of $A$, no two in the same row or column. Let $f(n)$ be the number of $n \times n$ matrices $A$ satisfying the following two conditions:
\begin{enumerate}
\item[(a)] Each entry $\alpha_{i,j}$ of $A$ is in the set
$\{-1,0,1\}$.
\item[(b)] The sum ... | Prove that $f(n) = 4^n + 2 \cdot 3^n - 4 \cdot 2^n + 1$. | [
"linear_algebra"
] | test |
putnam_1986_a5 | theorem putnam_1986_a5
(n : ℕ) (hn : 1 ≤ n)
(f : Fin n → ((Fin n → ℝ) → ℝ))
(hf : ∀ i, ContDiff ℝ 2 (f i))
(C : Fin n → Fin n → ℝ)
(hf' : ∀ i j : Fin n, ∀ x : Fin n → ℝ, fderiv ℝ (f i) x (Pi.single j 1) - fderiv ℝ (f j) x (Pi.single i 1) = C i j)
: ∃ g : (Fin n → ℝ) → ℝ, ∀ i : Fin n, IsLinearMap ℝ (λ x ↦ f ... | Suppose $f_1(x),f_2(x),\dots,f_n(x)$ are functions of $n$ real variables $x=(x_1,\dots,x_n)$ with continuous second-order partial derivatives everywhere on $\mathbb{R}^n$. Suppose further that there are constants $c_{ij}$ such that $\frac{\partial f_i}{\partial x_j}-\frac{\partial f_j}{\partial x_i}=c_{ij}$ for all $i$... | None. | [
"analysis",
"linear_algebra"
] | test |
putnam_1986_a6 | noncomputable abbrev putnam_1986_a6_solution : (ℕ → ℕ) → ℕ → ℝ := sorry
theorem putnam_1986_a6
(n : ℕ)
(npos : n > 0)
(a : ℕ → ℝ)
(b : ℕ → ℕ)
(bpos : ∀ i ∈ Finset.Icc 1 n, b i > 0)
(binj : ∀ i ∈ Finset.Icc 1 n, ∀ j ∈ Finset.Icc 1 n, b i = b j → i = j)
(f : Polynomial ℝ)
(hf : ∀ x : ℝ, (1 - x) ^ n * f.eval x = 1 + ∑ i :... | Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity
\[
(1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}.
\]
Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \dots, b_n$ a... | Show that $f(1) = b_1 b_2 \dots b_n / n!$. | [
"algebra"
] | test |
putnam_1986_b1 | noncomputable abbrev putnam_1986_b1_solution : ℝ := sorry
theorem putnam_1986_b1
(b h : ℝ)
(hbh : b > 0 ∧ h > 0 ∧ b ^ 2 + h ^ 2 = 2 ^ 2)
(areaeq : b * h = 0.5 * b * (1 - h / 2))
: h = putnam_1986_b1_solution :=
sorry | Inscribe a rectangle of base $b$ and height $h$ and an isosceles triangle of base $b$ (against a corresponding side of the rectangle and pointed in the other direction) in a circle of radius one. For what value of $h$ do the rectangle and triangle have the same area? | Show that the only such value of $h$ is $2/5$. | [
"geometry",
"algebra"
] | test |
putnam_1986_b2 | noncomputable abbrev putnam_1986_b2_solution : Finset (ℂ × ℂ × ℂ) := sorry
theorem putnam_1986_b2
: ({T : ℂ × ℂ × ℂ | ∃ x y z : ℂ, T = (x - y, y - z, z - x) ∧ x * (x - 1) + 2 * y * z = y * (y - 1) + 2 * z * x ∧ y * (y - 1) + 2 * z * x = z * (z - 1) + 2 * x * y} = putnam_1986_b2_solution) :=
sorry | Prove that there are only a finite number of possibilities for the ordered triple $T=(x-y,y-z,z-x)$, where $x,y,z$ are complex numbers satisfying the simultaneous equations
\[
x(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,
\]
and list all such triples $T$. | Show that the possibilities for $T$ are $(0, 0, 0), \, (0, -1, 1), \, (1, 0, -1), \, (-1, 1, 0)$. | [
"algebra"
] | test |
putnam_1986_b3 | theorem putnam_1986_b3
(n p : ℕ)
(nppos : n > 0 ∧ p > 0)
(pprime : Nat.Prime p)
(cong : Polynomial ℤ → Polynomial ℤ → ℤ → Prop)
(hcong : ∀ f g m, cong f g m ↔ ∀ i : ℕ, m ∣ (f - g).coeff i)
(f g h r s : Polynomial ℤ)
(hcoprime : cong (r * f + s * g) 1 p)
(hprod : cong (f * g) h p)
: (... | Let $\Gamma$ consist of all polynomials in $x$ with integer coefficients. For $f$ and $g$ in $\Gamma$ and $m$ a positive integer, let $f \equiv g \pmod{m}$ mean that every coefficient of $f-g$ is an integral multiple of $m$. Let $n$ and $p$ be positive integers with $p$ prime. Given that $f,g,h,r$ and $s$ are in $\Gamm... | None. | [
"number_theory",
"algebra"
] | test |
putnam_1986_b4 | abbrev putnam_1986_b4_solution : Prop := sorry
theorem putnam_1986_b4
(G : ℝ → ℝ)
(hGeq : ∀ r : ℝ, ∃ m n : ℤ, G r = |r - sqrt (m ^ 2 + 2 * n ^ 2)|)
(hGlb : ∀ r : ℝ, ∀ m n : ℤ, G r ≤ |r - sqrt (m ^ 2 + 2 * n ^ 2)|)
: (Tendsto G atTop (𝓝 0) ↔ putnam_1986_b4_solution) :=
sorry | For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\lim_{r\to \infty}G(r)$ exists and equals $0$. | Show that the limit exists and equals $0$. | [
"analysis"
] | test |
putnam_1986_b5 | abbrev putnam_1986_b5_solution : Prop := sorry
theorem putnam_1986_b5
(f : MvPolynomial (Fin 3) ℝ)
(perms : Set (Set (MvPolynomial (Fin 3) ℝ)))
(hf : f = (X 0) ^ 2 + (X 1) ^ 2 + (X 2) ^ 2 + (X 0) * (X 1) * (X 2))
(hperms : perms = {{X 0, X 1, X 2}, {X 0, -X 1, -X 2}, {-X 0, X 1, -X 2}, {-X 0, -X 1, X 2}... | Let $f(x,y,z) = x^2+y^2+z^2+xyz$. Let $p(x,y,z), q(x,y,z)$, $r(x,y,z)$ be polynomials with real coefficients satisfying
\[
f(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z).
\]
Prove or disprove the assertion that the sequence $p,q,r$ consists of some permutation of $\pm x, \pm y, \pm z$, where the number of minus signs is $0... | Prove that the assertion is false. | [
"algebra"
] | test |
putnam_1986_b6 | theorem putnam_1986_b6
(n : ℕ)
(npos : n > 0)
(F : Type*) [Field F]
(A B C D : Matrix (Fin n) (Fin n) F)
(hsymm : IsSymm (A * Bᵀ) ∧ IsSymm (C * Dᵀ))
(hid : A * Dᵀ - B * Cᵀ = 1)
: (Aᵀ * D - Cᵀ * B = 1) :=
sorry | Suppose $A,B,C,D$ are $n \times n$ matrices with entries in a field $F$, satisfying the conditions that $AB^T$ and $CD^T$ are symmetric and $AD^T - BC^T = I$. Here $I$ is the $n \times n$ identity matrix, and if $M$ is an $n \times n$ matrix, $M^T$ is its transpose. Prove that $A^T D - C^T B = I$. | None. | [
"linear_algebra"
] | test |
putnam_1987_a1 | theorem putnam_1987_a1
(A B C D : Set (ℝ × ℝ))
(hA : A = {(x, y) : ℝ × ℝ | x ^ 2 + y ^ 2 ≠ 0 ∧ x ^ 2 - y ^ 2 = x / (x ^ 2 + y ^ 2)})
(hB : B = {(x, y) : ℝ × ℝ | x ^ 2 + y ^ 2 ≠ 0 ∧ 2 * x * y + y / (x ^ 2 + y ^ 2) = 3})
(hC : C = {(x, y) : ℝ × ℝ | x ^ 3 - 3 * x * y ^ 2 + 3 * y = 1})
(hD : D = {(x, y) : ℝ × ℝ | 3 * x ^ 2... | Curves $A$, $B$, $C$, and $D$ are defined in the plane as follows:
\begin{align*}
A&=\left\{ (x,y):x^2-y^2=\frac{x}{x^2+y^2} \right\}, \\
B&=\left\{ (x,y):2xy+\frac{y}{x^2+y^2}=3 \right\}, \\
C&=\left\{ (x,y):x^3-3xy^2+3y=1 \right\}, \\
D&=\left\{ (x,y):3x^2y-3x-y^3=0 \right\}.
\end{align*}
Prove that $A \cap B=C \cap ... | None. | [
"algebra"
] | test |
putnam_1987_a2 | abbrev putnam_1987_a2_solution : ℕ := sorry
theorem putnam_1987_a2
(seqind seqsize f : ℕ → ℕ)
(hseqind : seqind 1 = 1 ∧ ∀ i ≥ 2, seqind i = seqind (i - 1) + (Nat.digits 10 (i - 1)).length)
(hseqsize : ∀ i ≥ 1, ∀ j : Fin ((Nat.digits 10 i).length), seqsize (seqind i + j) = (Nat.digits 10 i).length)
(hf :... | The sequence of digits $123456789101112131415161718192021 \dots$ is obtained by writing the positive integers in order. If the $10^n$-th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ because the $100$th digit enters th... | Show that the value of $f(1987)$ is $1984$. | [
"algebra"
] | test |
putnam_1987_a4 | noncomputable abbrev putnam_1987_a4_solution : ℂ := sorry
theorem putnam_1987_a4
(P : MvPolynomial (Fin 3) ℂ)
(hPreal : ∀ i : Fin 3 →₀ ℕ, (coeff i P).im = 0)
(F : ℝ → ℝ → ℝ)
(vars : ℂ → ℂ → ℂ → (Fin 3 → ℂ))
(hvars : vars = fun a b c ↦ fun i ↦ ite (i = 0) a (ite (i = 1) b c))
(h : ∀ x y z u : ℝ, eval (vars (u * x) (u * ... | Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that
\[
P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$},
\]
and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$.... | Prove that $|C - A| = \frac{5}{3}\sqrt{30}$. | [
"algebra"
] | test |
putnam_1987_a5 | abbrev putnam_1987_a5_solution : Prop := sorry
theorem putnam_1987_a5
(curl : ((Fin 3 → ℝ) → (Fin 3 → ℝ)) → ((Fin 3 → ℝ) → (Fin 3 → ℝ)))
(curl_def : ∀ f x, curl f x = ![
fderiv ℝ f x (Pi.single 1 1) 2 - fderiv ℝ f x (Pi.single 2 1) 1,
fderiv ℝ f x (Pi.single 2 1) 0 - fderiv ℝ f x (Pi.single 0 1) 2,
... | Let $\vec{G}(x,y)=\left(\frac{-y}{x^2+4y^2},\frac{x}{x^2+4y^2},0\right)$. Prove or disprove that there is a vector-valued function $\vec{F}(x,y,z)=(M(x,y,z),N(x,y,z),P(x,y,z))$ with the following properties:
\begin{enumerate}
\item[(i)] $M$, $N$, $P$ have continuous partial derivatives for all $(x,y,z) \neq (0,0,0)$;
\... | Show that there is no such $\vec{F}$. | [
"analysis"
] | test |
putnam_1987_a6 | abbrev putnam_1987_a6_solution : Set ℝ := sorry
theorem putnam_1987_a6
(a : ℕ → ℕ)
(ha : a = fun n ↦ {i | (digits 3 n).get i = 0}.ncard)
: ({x : ℝ | x > 0 ∧ Summable (fun n ↦ x ^ (a n) / (n ^ 3))} = putnam_1987_a6_solution) :=
sorry | For each positive integer $n$, let $a(n)$ be the number of zeroes in the base $3$ representation of $n$. For which positive real numbers $x$ does the series
\[
\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}
\]
converge? | Show that for positive $x$, the series converges if and only if $x < 25$. | [
"algebra",
"analysis"
] | test |
putnam_1987_b1 | abbrev putnam_1987_b1_solution : ℝ := sorry
theorem putnam_1987_b1
: (∫ x in (2)..4, sqrt (log (9 - x)) / (sqrt (log (9 - x)) + sqrt (log (x + 3))) = putnam_1987_b1_solution) :=
sorry | Evaluate
\[
\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
\] | Prove that the integral evaluates to $1$. | [
"analysis"
] | test |
putnam_1987_b2 | theorem putnam_1987_b2
(r s t : ℕ)
(hsum : r + s ≤ t)
: (∑ i : Finset.range (s + 1), (choose s i : ℚ) / (choose t (r + i)) = ((t + 1) : ℚ) / ((t + 1 - s) * choose (t - s) r)) :=
sorry | Let $r, s$ and $t$ be integers with $0 \leq r$, $0 \leq s$ and $r+s \leq t$. Prove that
\[
\frac{\binom s0}{\binom tr}
+ \frac{\binom s1}{\binom{t}{r+1}} + \cdots
+ \frac{\binom ss}{\binom{t}{r+s}}
= \frac{t+1}{(t+1-s)\binom{t-s}{r}}.
\] | None. | [
"algebra"
] | test |
putnam_1987_b3 | theorem putnam_1987_b3
(F : Type*) [Field F]
(hF : (1 : F) + 1 ≠ 0)
: {(x, y) : F × F | x ^ 2 + y ^ 2 = 1} = {(1, 0)} ∪ {((r ^ 2 - 1) / (r ^ 2 + 1), (2 * r) / (r ^ 2 + 1)) | r ∈ {r' : F | r' ^ 2 ≠ -1}} :=
sorry | Let $F$ be a field in which $1+1 \neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and $(x,y)=\left(\frac{r^2-1}{r^2+1},\frac{2r}{r^2+1}\right)$, where $r$ runs through the elements of $F$ such that $r^2 \neq -1$. | None. | [
"abstract_algebra"
] | test |
putnam_1987_b4 | abbrev putnam_1987_b4_solution : Prop × ℝ × Prop × ℝ := sorry
theorem putnam_1987_b4
(x y : ℕ → ℝ)
(hxy1 : (x 1, y 1) = (0.8, 0.6))
(hx : ∀ n ≥ 1, x (n + 1) = (x n) * cos (y n) - (y n) * sin (y n))
(hy : ∀ n ≥ 1, y (n + 1) = (x n) * sin (y n) + (y n) * cos (y n)) :
let (existsx, limx, existsy, limy)... | Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \cos y_n - y_n \sin y_n$ and $y_{n+1}= x_n \sin y_n + y_n \cos y_n$ for $n=1,2,3,\dots$. For each of $\lim_{n\to \infty} x_n$ and $\lim_{n \to \infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist. | Show that $\lim_{n \to \infty} x_n = -1$ and $\lim_{n \to \infty} y_n = 0$. | [
"analysis"
] | test |
putnam_1987_b5 | theorem putnam_1987_b5
(n : ℕ)
(npos : n > 0)
(M : Matrix (Fin (2 * n)) (Fin n) ℂ)
(hM : ∀ z : Matrix (Fin 1) (Fin (2 * n)) ℂ, z * M = 0 → (¬∀ i : Fin (2 * n), z 0 i = 0) → ∃ i : Fin (2 * n), (z 0 i).im ≠ 0)
: (∀ r : Matrix (Fin (2 * n)) (Fin 1) ℝ, ∃ w : Matrix (Fin n) (Fin 1) ℂ, ∀ i : (Fin (2 * n)), ((M * w) i 0).re =... | Let $O_n$ be the $n$-dimensional vector $(0,0,\cdots, 0)$. Let $M$ be a $2n \times n$ matrix of complex numbers such that whenever $(z_1, z_2, \dots, z_{2n})M = O_n$, with complex $z_i$, not all zero, then at least one of the $z_i$ is not real. Prove that for arbitrary real numbers $r_1, r_2, \dots, r_{2n}$, there are ... | None. | [
"linear_algebra"
] | test |
putnam_1987_b6 | theorem putnam_1987_b6
(p : ℕ)
(F : Type*) [Field F] [Fintype F]
(S : Set F)
(hp : Odd p ∧ Nat.Prime p)
(Fcard : Fintype.card F = p ^ 2)
(Snz : ∀ x ∈ S, x ≠ 0)
(Scard : S.ncard = ((p : ℤ) ^ 2 - 1) / 2)
(hS : ∀ a : F, a ≠ 0 → Xor' (a ∈ S) (-a ∈ S)) :
(Even ((S ∩ {x | ∃ a ∈ S, x = 2 * ... | Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \cap \{2a: a \in S\}$. Prove that $N$ ... | None. | [
"abstract_algebra"
] | test |
putnam_1988_a1 | abbrev putnam_1988_a1_solution : ℝ := sorry
theorem putnam_1988_a1
(R : Set (Fin 2 → ℝ))
(hR : R = {p | |p 0| - |p 1| ≤ 1 ∧ |p 1| ≤ 1}) :
(volume R).toReal = putnam_1988_a1_solution :=
sorry | Let $R$ be the region consisting of the points $(x,y)$ of the cartesian plane satisfying both $|x|-|y| \leq 1$ and $|y| \leq 1$. Find the area of $R$. | Show that the area of $R$ is $6$. | [
"geometry"
] | test |
putnam_1988_a2 | abbrev putnam_1988_a2_solution : Prop := sorry
theorem putnam_1988_a2
(f : ℝ → ℝ)
(hf : f = fun x ↦ Real.exp (x ^ 2)) :
putnam_1988_a2_solution ↔
(∃ a b : ℝ,
a < b ∧
∃ g : ℝ → ℝ,
(∃ x ∈ Ioo a b, g x ≠ 0) ∧
DifferentiableOn ℝ g (Ioo a b) ∧
∀ x ∈ Ioo a b, deriv (fun y ↦... | A not uncommon calculus mistake is to believe that the product rule for derivatives says that $(fg)' = f'g'$. If $f(x)=e^{x^2}$, determine, with proof, whether there exists an open interval $(a,b)$ and a nonzero function $g$ defined on $(a,b)$ such that this wrong product rule is true for $x$ in $(a,b)$. | Show that such $(a,b)$ and $g$ exist. | [
"analysis"
] | test |
putnam_1988_a3 | abbrev putnam_1988_a3_solution : Set ℝ := sorry
theorem putnam_1988_a3
: {x : ℝ | ∃ L : ℝ, Tendsto (fun t ↦ ∑ n ∈ Finset.Icc (1 : ℕ) t, (((1 / n) / Real.sin (1 / n) - 1) ^ x)) atTop (𝓝 L)} = putnam_1988_a3_solution :=
sorry | Determine, with proof, the set of real numbers $x$ for which
\[
\sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x
\]
converges. | Show that the series converges if and only if $x > \frac{1}{2}$. | [
"analysis"
] | test |
putnam_1988_a4 | abbrev putnam_1988_a4_solution : Prop × Prop := sorry
theorem putnam_1988_a4
(p : ℕ → Prop)
(hp : ∀ n, p n ↔
∀ color : (EuclideanSpace ℝ (Fin 2)) → Fin n,
∃ p q : EuclideanSpace ℝ (Fin 2),
color p = color q ∧ dist p q = 1) :
(let (a, b) := putnam_1988_a4_solution; (p 3 ↔ a) ∧ (p 9 ↔ ... | \begin{enumerate}
\item[(a)] If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart?
\item[(b)] What if ``three'' is replaced by ``nine''?
\end{enumerate} | Prove that the points must exist with three colors, but not necessarily with nine. | [
"geometry",
"combinatorics"
] | test |
putnam_1988_a5 | theorem putnam_1988_a5
: (∃ f : ℝ → ℝ, (∀ x > 0, f (f x) = 6 * x - f x ∧ f x > 0) ∧ (∀ g : ℝ → ℝ, (∀ x > 0, g (g x) = 6 * x - g x ∧ g x > 0) → (∀ x > 0, f x = g x))) :=
sorry | Prove that there exists a \emph{unique} function $f$ from the set $\mathrm{R}^+$ of positive real numbers to $\mathrm{R}^+$ such that
\[
f(f(x)) = 6x-f(x)
\]
and
\[
f(x)>0
\]
for all $x>0$. | None. | [
"analysis"
] | test |
putnam_1988_a6 | abbrev putnam_1988_a6_solution : Prop := sorry
theorem putnam_1988_a6
: (∀ (F V : Type*) (_ : Field F) (_ : AddCommGroup V) (_ : Module F V) (_ : FiniteDimensional F V) (n : ℕ) (A : Module.End F V) (evecs : Set V), (n = Module.finrank F V ∧ evecs ⊆ {v : V | ∃ f : F, A.HasEigenvector f v} ∧ evecs.encard = n + 1 ∧ (∀ sev... | If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer. | Show that the answer is yes, $A$ must be a scalar multiple of the identity. | [
"linear_algebra"
] | test |
putnam_1988_b1 | theorem putnam_1988_b1
: ∀ a ≥ 2, ∀ b ≥ 2, ∃ x y z : ℤ, x > 0 ∧ y > 0 ∧ z > 0 ∧ a * b = x * y + x * z + y * z + 1 :=
sorry | A \emph{composite} (positive integer) is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers. | None. | [
"number_theory",
"algebra"
] | test |
putnam_1988_b2 | abbrev putnam_1988_b2_solution : Prop := sorry
theorem putnam_1988_b2
: (∀ x y : ℝ, (y ≥ 0 ∧ y * (y + 1) ≤ (x + 1) ^ 2) → (y * (y - 1) ≤ x ^ 2)) ↔ putnam_1988_b2_solution :=
sorry | Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$. | Show that this is true. | [
"algebra"
] | test |
putnam_1988_b3 | noncomputable abbrev putnam_1988_b3_solution : ℝ := sorry
theorem putnam_1988_b3
(r : ℤ → ℝ)
(hr : ∀ n ≥ 1,
(∃ c d : ℤ,
(c ≥ 0 ∧ d ≥ 0) ∧
c + d = n ∧ r n = |c - d * Real.sqrt 3|) ∧
(∀ c d : ℤ, (c ≥ 0 ∧ d ≥ 0 ∧ c + d = n) → |c - d * Real.sqrt 3| ≥ r n))
: IsLeast {g : ℝ | g > 0 ... | For every $n$ in the set $N=\{1,2,\dots\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in N$. | Show that the smallest such $g$ is $(1+\sqrt{3})/2$. | [
"algebra"
] | test |
putnam_1988_b4 | theorem putnam_1988_b4
(a : ℕ → ℝ)
(IsPosConv : (ℕ → ℝ) → Prop)
(IsPosConv_def : ∀ a' : ℕ → ℝ, IsPosConv a' ↔
(∀ n ≥ 1, a' n > 0) ∧
(∃ s : ℝ, Tendsto (fun N : ℕ => ∑ n : Set.Icc 1 N, a' n) atTop (𝓝 s))) :
(IsPosConv a) → IsPosConv (fun n : ℕ => (a n) ^ ((n : ℝ) / (n + 1))) :=
sorry | Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty (a_n)^{n/(n+1)}$. | None. | [
"analysis"
] | test |
putnam_1988_b5 | abbrev putnam_1988_b5_solution : ℕ → ℕ := sorry
theorem putnam_1988_b5
(n : ℕ) (hn : n > 0)
(Mn : Matrix (Fin (2 * n + 1)) (Fin (2 * n + 1)) ℝ)
(Mnskewsymm : ∀ i j, Mn i j = -(Mn j i))
(hMn1 : ∀ i j, (1 ≤ (i.1 : ℤ) - j.1 ∧ (i.1 : ℤ) - j.1 ≤ n) → Mn i j = 1)
(hMnn1 : ∀ i j, (i.1 : ℤ) - j.1 > n → Mn i... | For positive integers $n$, let $M_n$ be the $2n+1$ by $2n+1$ skew-symmetric matrix for which each entry in the first $n$ subdiagonals below the main diagonal is $1$ and each of the remaining entries below the main diagonal is $-1$. Find, with proof, the rank of $M_n$. (According to one definition, the rank of a matrix ... | Show that the rank of $M_n$ equals $2n$. | [
"linear_algebra"
] | test |
putnam_1988_b6 | theorem putnam_1988_b6
(trinums : Set ℤ)
(htrinums : trinums = {t : ℤ | ∃ n : ℤ, t ≥ 0 ∧ t = (n * (n + 1)) / 2})
: {(a, b) : ℤ × ℤ | ∀ t > 0, (a * t + b) ∈ trinums ↔ t ∈ trinums}.encard = ⊤ :=
sorry | Prove that there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$, the number $at+b$ is a triangular number if and only if $t$ is a triangular number. (The triangular numbers are the $t_n=n(n+1)/2$ with $n$ in $\{0,1,2,\dots\}$.) | None. | [
"number_theory",
"algebra"
] | test |
putnam_1989_a1 | abbrev putnam_1989_a1_solution : ℕ∞ := sorry
theorem putnam_1989_a1
(pdigalt : List ℕ → Prop)
(hpdigalt : ∀ l, pdigalt l ↔ Odd l.length ∧ (∀ i, l.get i = if Even (i : ℕ) then 1 else 0)) :
{p : ℕ | p.Prime ∧ pdigalt (Nat.digits 10 p)}.encard = putnam_1989_a1_solution :=
sorry | How many primes among the positive integers, written as usual in base $10$, are alternating $1$'s and $0$'s, beginning and ending with $1$? | Show that there is only one such prime. | [
"algebra",
"number_theory"
] | test |
putnam_1989_a2 | noncomputable abbrev putnam_1989_a2_solution : ℝ → ℝ → ℝ := sorry
theorem putnam_1989_a2
(a b : ℝ)
(abpos : a > 0 ∧ b > 0)
: ∫ x in Set.Ioo 0 a, ∫ y in Set.Ioo 0 b, Real.exp (max (b ^ 2 * x ^ 2) (a ^ 2 * y ^ 2)) = putnam_1989_a2_solution a b :=
sorry | Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive. | Show that the value of the integral is $(e^{a^2b^2}-1)/(ab)$. | [
"analysis"
] | test |
putnam_1989_a3 | theorem putnam_1989_a3
(z : ℂ)
(hz : 11 * z ^ 10 + 10 * I * z ^ 9 + 10 * I * z - 11 = 0)
: (‖z‖ = 1) :=
sorry | Prove that if
\[
11z^{10}+10iz^9+10iz-11=0,
\]
then $|z|=1.$ (Here $z$ is a complex number and $i^2=-1$.) | None. | [
"algebra"
] | test |
putnam_1989_a6 | theorem putnam_1989_a6
(F : Type*) [Field F] [Fintype F]
(hF : Fintype.card F = 2)
(α : PowerSeries F)
(hα : ∀ n : ℕ, let bin := [1] ++ (digits 2 n) ++ [1]; PowerSeries.coeff n α = ite (∀ i j : Fin bin.length, i < j → bin.get i = 1 → bin.get j = 1 → (∀ k, i < k → k < j → bin.get k = 0) → Even ((j : ℕ) - (i : ℕ) - 1)) 1... | Let $\alpha=1+a_1x+a_2x^2+\cdots$ be a formal power series with coefficients in the field of two elements. Let
\[
a_n =
\begin{cases}
1 & \parbox{2in}{if every block of zeros in the binary expansion of $n$ has an even number of zeros in the block} \\[.3in]
0 & \text{otherwise.}
\end{cases}
\]
(For example, $a_{36}=1$ b... | None. | [
"algebra",
"abstract_algebra"
] | test |
putnam_1989_b1 | abbrev putnam_1989_b1_solution : ℤ × ℤ × ℤ × ℤ := sorry
theorem putnam_1989_b1
(square Scloser perimeter: Set (EuclideanSpace ℝ (Fin 2)))
(center : EuclideanSpace ℝ (Fin 2))
(square_def : square = {p | ∀ i : Fin 2, p i ∈ Set.Icc 0 1})
(perimeter_def : perimeter = {p ∈ square | p 0 = 0 ∨ p 0 = 1 ∨ p 1 = ... | A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. Express your answer in the form $(a\sqrt{b}+c)/d$, where $a$, $b$, $c$, $d$ are integers and $b$, $d$ ar... | Show that the probability is $(4\sqrt{2}-5)/3$. | [
"probability",
"geometry"
] | test |
putnam_1989_b2 | abbrev putnam_1989_b2_solution : Prop := sorry
theorem putnam_1989_b2 :
(∀ (S : Type) [Nonempty S] [Semigroup S] [IsCancelMul S]
(h_fin : ∀ a : S, {(a * ·)^[n] a | n : ℕ}.Finite),
∃ e : S, ∀ x, e * x = x ∧ x * e = x ∧ ∃ y, x * y = e ∧ y * x = e) ↔
putnam_1989_b2_solution :=
sorry | Let $S$ be a non-empty set with an associative operation that is left and right cancellative ($xy=xz$ implies $y=z$, and $yx=zx$ implies $y=z$). Assume that for every $a$ in $S$ the set $\{a^n:\,n=1, 2, 3, \ldots\}$ is finite. Must $S$ be a group? | Prove that $S$ must be a group. | [
"abstract_algebra"
] | test |
putnam_1989_b3 | noncomputable abbrev putnam_1989_b3_solution : ℕ → ℝ → ℝ := sorry
theorem putnam_1989_b3
(f : ℝ → ℝ)
(hfdiff : Differentiable ℝ f)
(hfderiv : ∀ x > 0, deriv f x = -3 * f x + 6 * f (2 * x))
(hdecay : ∀ x ≥ 0, |f x| ≤ Real.exp (- √x))
(μ : ℕ → ℝ)
(μ_def : ∀ n, μ n = ∫ x in Set.Ioi 0, x ^ n * f x) ... | Let $f$ be a function on $[0,\infty)$, differentiable and satisfying
\[
f'(x)=-3f(x)+6f(2x)
\]
for $x>0$. Assume that $|f(x)|\le e^{-\sqrt{x}}$ for $x\ge 0$ (so that $f(x)$ tends rapidly to $0$ as $x$ increases). For $n$ a non-negative integer, define
\[
\mu_n=\int_0^\infty x^n f(x)\,dx
\]
(sometimes called the $n$th m... | Show that for each $n \geq 0$, $\mu_n = \frac{n!}{3^n} \left( \prod_{m=1}^{n}(1 - 2^{-m}) \right)^{-1} \mu_0$. | [
"analysis"
] | test |
putnam_1989_b4 | abbrev putnam_1989_b4_solution : Prop := sorry
theorem putnam_1989_b4 :
(∃ S : Type,
Countable S ∧ Infinite S ∧
∃ C : Set (Set S),
¬Countable C ∧
(∀ R ∈ C, R ≠ ∅) ∧
(∀ A ∈ C, ∀ B ∈ C, A ≠ B → (A ∩ B).Finite)
) ↔ putnam_1989_b4_solution :=
sorry | Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite? | Prove that such a collection exists. | [
"set_theory"
] | test |
putnam_1989_b6 | theorem putnam_1989_b6
(n : ℕ) [NeZero n]
(I : (Fin n → ℝ) → Fin (n + 2) → ℝ)
(I_def : ∀ x, I x = Fin.cons 0 (Fin.snoc x 1))
(X : Set (Fin n → ℝ))
(X_def : ∀ x, x ∈ X ↔ 0 < x 0 ∧ x (-1) < 1 ∧ StrictMono x)
(S : (ℝ → ℝ) → (Fin (n + 2) → ℝ) → ℝ)
(S_def : ∀ f x, S f x = ∑ i : Fin n.succ, (x i.s... | Let $(x_1,x_2,\dots,x_n)$ be a point chosen at random from the $n$-dimensional region defined by $0< x_1< x_2<\dots< x_n<1$. Let $f$ be a continuous function on $[0,1]$ with $f(1)=0$. Set $x_0=0$ and $x_{n+1}=1$. Show that the expected value of the Riemann sum $\sum_{i=0}^n (x_{i+1}-x_i)f(x_{i+1})$ is $\int_0^1 f(t)P(t... | None. | [
"probability",
"analysis",
"algebra"
] | test |
putnam_1990_a1 | abbrev putnam_1990_a1_solution : (ℕ → ℤ) × (ℕ → ℤ) := sorry
theorem putnam_1990_a1
(T : ℕ → ℤ)
(hT012 : T 0 = 2 ∧ T 1 = 3 ∧ T 2 = 6)
(hTn : ∀ n, T (n + 3) = (n + 7) * T (n + 2) - 4 * (n + 3) * T (n + 1) + (4 * n + 4) * T n) :
T = putnam_1990_a1_solution.1 + putnam_1990_a1_solution.2 :=
sorry | Let $T_0=2,T_1=3,T_2=6$, and for $n \geq 3$, $T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}$. The first few terms are $2,3,6,14,40,152,784,5168,40576$. Find, with proof, a formula for $T_n$ of the form $T_n=A_n+B_n$, where $\{A_n\}$ and $\{B_n\}$ are well-known sequences. | Show that we have $T_n=n!+2^n$. | [
"algebra"
] | test |
putnam_1990_a2 | abbrev putnam_1990_a2_solution : Prop := sorry
theorem putnam_1990_a2
(numform : ℝ → Prop)
(hnumform : ∀ x : ℝ, numform x ↔ ∃ n m : ℕ, x = n ^ ((1 : ℝ) / 3) - m ^ ((1 : ℝ) / 3)) :
putnam_1990_a2_solution ↔
(∃ s : ℕ → ℝ,
(∀ i : ℕ, numform (s i)) ∧
Tendsto s atTop (𝓝 (Real.sqrt 2))) :=
sorry | Is $\sqrt{2}$ the limit of a sequence of numbers of the form $\sqrt[3]{n}-\sqrt[3]{m}$ ($n,m=0,1,2,\dots$)? | Show that the answer is yes. | [
"analysis"
] | test |
putnam_1990_a4 | abbrev putnam_1990_a4_solution : ℕ := sorry
theorem putnam_1990_a4
: sInf {n : ℕ | ∃ S : Set (EuclideanSpace ℝ (Fin 2)), S.encard = n ∧ ∀ Q : EuclideanSpace ℝ (Fin 2), ∃ P ∈ S, Irrational (dist P Q)} = putnam_1990_a4_solution :=
sorry | Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point? | Show that three punches are needed. | [
"set_theory",
"number_theory"
] | test |
putnam_1990_a5 | abbrev putnam_1990_a5_solution : Prop := sorry
theorem putnam_1990_a5 :
putnam_1990_a5_solution ↔
(∀ n ≥ 1, ∀ A B : Matrix (Fin n) (Fin n) ℝ,
A * B * A * B = 0 → B * A * B * A = 0) :=
sorry | If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$? | Show that the answer is no. | [
"linear_algebra"
] | test |
putnam_1990_a6 | abbrev putnam_1990_a6_solution : ℕ := sorry
theorem putnam_1990_a6 :
((Finset.univ : Finset <| Finset (Set.Icc 1 10) × Finset (Set.Icc 1 10)).filter
fun ⟨S, T⟩ ↦ (∀ s ∈ S, T.card < s) ∧ (∀ t ∈ T, S.card < t)).card =
putnam_1990_a6_solution :=
sorry | If $X$ is a finite set, let $|X|$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $\{1,2,\dots,n\}$ \emph{admissible} if $s>|T|$ for each $s \in S$, and $t>|S|$ for each $t \in T$. How many admissible ordered pairs of subsets of $\{1,2,\dots,10\}$ are there? Prove your answer. | Show that the number of admissible ordered pairs of subsets of $\{1,2,\dots,10\}$ equals the $22$nd Fibonacci number $F_{22}=17711$. | [
"algebra"
] | test |
putnam_1990_b1 | abbrev putnam_1990_b1_solution : Set (ℝ → ℝ) := sorry
theorem putnam_1990_b1
(P : (ℝ → ℝ) → Prop)
(P_def : ∀ f, P f ↔ ∀ x,
(f x) ^ 2 = (∫ t in (0 : ℝ)..x, (f t) ^ 2 + (deriv f t) ^ 2) + 1990)
(f : ℝ → ℝ) :
(ContDiff ℝ 1 f ∧ P f) ↔ f ∈ putnam_1990_b1_solution :=
sorry | Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$. | Show that there are two such functions, namely $f(x)=\sqrt{1990}e^x$, and $f(x)=-\sqrt{1990}e^x$. | [
"analysis"
] | test |
putnam_1990_b2 | theorem putnam_1990_b2
(x z : ℝ)
(P : ℕ → ℝ)
(xlt1 : |x| < 1)
(zgt1 : |z| > 1)
(hP : ∀ j ≥ 1, P j = (∏ i : Fin j, (1 - z * x ^ (i : ℕ))) / (∏ i : Set.Icc 1 j, (z - x ^ (i : ℕ))))
: 1 + (∑' j : Set.Ici 1, (1 + x ^ (j : ℕ)) * P j) = 0 :=
sorry | Prove that for $|x|<1$, $|z|>1$, $1+\sum_{j=1}^\infty (1+x^j)P_j=0$, where $P_j$ is $\frac{(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3) \cdots (z-x^j)}$. | None. | [
"analysis"
] | test |
putnam_1990_b3 | theorem putnam_1990_b3
(S : Set (Matrix (Fin 2) (Fin 2) ℕ))
(hS : ∀ A ∈ S, ∀ i j : Fin 2, (∃ x : ℤ, A i j = x ^ 2) ∧ A i j ≤ 200)
: (S.encard > 50387) → (∃ A ∈ S, ∃ B ∈ S, A ≠ B ∧ A * B = B * A) :=
sorry | Let $S$ be a set of $2 \times 2$ integer matrices whose entries $a_{ij}$ (1) are all squares of integers, and, (2) satisfy $a_{ij} \leq 200$. Show that if $S$ has more than $50387$ ($=15^4-15^2-15+2$) elements, then it has two elements that commute. | None. | [
"linear_algebra"
] | test |
putnam_1990_b4 | abbrev putnam_1990_b4_solution : Prop := sorry
theorem putnam_1990_b4
: (∀ (G : Type*) (_ : Fintype G) (_ : Group G) (n : ℕ) (a b : G), (n = Fintype.card G ∧ G = Subgroup.closure {a, b} ∧ G ≠ Subgroup.closure {a} ∧ G ≠ Subgroup.closure {b}) → (∃ g : ℕ → G, (∀ x : G, {i : Fin (2 * n) | g i = x}.encard = 2)
∧ (∀ i : Fi... | Let $G$ be a finite group of order $n$ generated by $a$ and $b$. Prove or disprove: there is a sequence $g_1,g_2,g_3,\dots,g_{2n}$ such that
\begin{itemize}
\item[(1)] every element of $G$ occurs exactly twice, and
\item[(2)] $g_{i+1}$ equals $g_ia$ or $g_ib$ for $i=1,2,\dots,2n$. (Interpret $g_{2n+1}$ as $g_1$.)
\end{... | Show that such a sequence does exist. | [
"abstract_algebra"
] | test |
putnam_1990_b5 | abbrev putnam_1990_b5_solution : Prop := sorry
theorem putnam_1990_b5 :
(∃ a : ℕ → ℝ, (∀ i, a i ≠ 0) ∧
(∀ n ≥ 1, (∑ i ∈ Finset.Iic n, a i • X ^ i : Polynomial ℝ).roots.toFinset.card = n)) ↔
putnam_1990_b5_solution :=
sorry | Is there an infinite sequence $a_0,a_1,a_2,\dots$ of nonzero real numbers such that for $n=1,2,3,\dots$ the polynomial $p_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ has exactly $n$ distinct real roots? | Show that the answer is yes, such an infinite sequence exists. | [
"algebra",
"analysis"
] | test |
putnam_1991_a2 | abbrev putnam_1991_a2_solution : Prop := sorry
theorem putnam_1991_a2
(n : ℕ) (hn : 1 ≤ n) :
putnam_1991_a2_solution ↔ (∃ A B : Matrix (Fin n) (Fin n) ℝ,
A ≠ B ∧ A ^ 3 = B ^ 3 ∧
A ^ 2 * B = B ^ 2 * A ∧
Nonempty (Invertible (A ^ 2 + B ^ 2))) :=
sorry | Let $\mathbf{A}$ and $\mathbf{B}$ be different $n \times n$ matrices with real entries. If $\mathbf{A}^3=\mathbf{B}^3$ and $\mathbf{A}^2\mathbf{B}=\mathbf{B}^2\mathbf{A}$, can $\mathbf{A}^2+\mathbf{B}^2$ be invertible? | Show that the answer is no. | [
"linear_algebra"
] | test |
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