PutnamBench: Evaluating Neural Theorem-Provers on the Putnam Mathematical Competition
Paper β’ 2407.11214 β’ Published
problem_name stringlengths 14 14 | formal_statement stringlengths 72 1.32k | header stringlengths 15 103 | informal_statement stringlengths 47 898 | informal_solution stringlengths 5 303 | tags listlengths 1 3 | split stringclasses 1
value | name stringlengths 14 14 | uuid stringlengths 16 16 | goal_state stringlengths 34 1.44k |
|---|---|---|---|---|---|---|---|---|---|
putnam_1962_a1 | theorem putnam_1962_a1
(S : Set (β Γ β))
(hS : S.ncard = 5)
(hnoncol : β s β S, s.ncard = 3 β Β¬Collinear β s)
: β T β S, T.ncard = 4 β§ Β¬β t β T, t β convexHull β (T \ {t}) := by | import Mathlib
open MeasureTheory
| Given five points in a plane, no three of which lie on a straight line, show that some four of these points form the vertices of a convex quadrilateral. | None. | [
"geometry"
] | test | putnam_1962_a1 | bf6873c545286fee | S : Set (β Γ β)
hS : S.ncard = 5
hnoncol : β s β S, s.ncard = 3 β Β¬Collinear β s
β’ β T β S, T.ncard = 4 β§ Β¬β t β T, t β (convexHull β) (T \ {t}) |
putnam_1962_a2 | abbrev putnam_1962_a2_solution : Set (β β β) := sorry
theorem putnam_1962_a2
(P : Set β β (β β β) β Prop)
(P_def : β s f, P s f β 0 β€ f β§ β x β s, β¨ t in Ico 0 x, f t = β(f 0 * f x)) :
(β f,
(P (Ioi 0) f β β g β putnam_1962_a2_solution, EqOn f g (Ici 0)) β§
(β e > 0, P (Ioo 0 e) f β β g β putnam_... | import Mathlib
open MeasureTheory Set
| Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having 0 as a left-hand endpoint, such that for every positive member $x$ of $I$ the average of $f$ over the closed interval $[0, x]$ is equal to the geometric mean of the numbers $f(0)$ and $f(x)$. | Show that \[ f(x) = \frac{a}{(1 - cx)^2} \begin{cases} \text{for } 0 \le x < \frac{1}{c}, & \text{if } c > 0\\ \text{for } 0 \le x < \infty, & \text{if } c \le 0, \end{cases} \] where $a > 0$. | [
"analysis"
] | test | putnam_1962_a2 | 137a5c9069c97387 | P : Set β β (β β β) β Prop
P_def : β (s : Set β) (f : β β β), P s f β 0 β€ f β§ β x β s, β¨ (t : β) in Ico 0 x, f t = β(f 0 * f x)
β’ (β (f : β β β),
(P (Ioi 0) f β β g β putnam_1962_a2_solution, EqOn f g (Ici 0)) β§
β e > 0, P (Ioo 0 e) f β β g β putnam_1962_a2_solution, EqOn f g (Ico 0 e)) β§
β f β putnam... |
putnam_1962_a3 | theorem putnam_1962_a3
(A B C A' B' C' P Q R : EuclideanSpace β (Fin 2))
(k : β)
(hk : k > 0)
(hABC : Β¬Collinear β {A, B, C})
(hA' : A' β segment β B C β§ dist C A' / dist A' B = k)
(hB' : B' β segment β C A β§ dist A B' / dist B' C = k)
(hC' : C' β segment β A B β§ dist B C' / dist C' A = k)
(hP : P β segment β B B' β§ P ... | import Mathlib
open MeasureTheory
| Let $\triangle ABC$ be a triangle in the Euclidean plane, with points $P$, $Q$, and $R$ lying on segments $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively such that $$\frac{AQ}{QC} = \frac{BR}{RA} = \frac{CP}{PB} = k$$ for some positive constant $k$. If $\triangle UVW$ is the triangle formed by parts of s... | None. | [
"geometry"
] | test | putnam_1962_a3 | 792d3aa0973261fc | A B C A' B' C' P Q R : EuclideanSpace β (Fin 2)
k : β
hk : k > 0
hABC : Β¬Collinear β {A, B, C}
hA' : A' β segment β B C β§ dist C A' / dist A' B = k
hB' : B' β segment β C A β§ dist A B' / dist B' C = k
hC' : C' β segment β A B β§ dist B C' / dist C' A = k
hP : P β segment β B B' β§ P β segment β C C'
hQ : Q β segment β C ... |
putnam_1962_a4 | theorem putnam_1962_a4
(f : β β β)
(a b : β)
(hdiff : Differentiable β f β§ (Differentiable β (deriv f)))
(hfabs : β x β Set.Icc a b, |f x| β€ 1)
(hfppabs : β x β Set.Icc a b, |(iteratedDeriv 2 f) x| β€ 1)
(hlen2 : b - a β₯ 2)
: β x β Set.Icc a b, |(iteratedDeriv 1 f) x| β€ 2 := by | import Mathlib
| Assume that $\lvert f(x) \rvert \le 1$ and $\lvert f''(x) \rvert \le 1$ for all $x$ on an interval of length at least 2. Show that $\lvert f'(x) \rvert \le 2$ on the interval. | None. | [
"analysis"
] | test | putnam_1962_a4 | 0c9dd4e43dede165 | f : β β β
a b : β
hdiff : Differentiable β f β§ Differentiable β (deriv f)
hfabs : β x β Set.Icc a b, |f x| β€ 1
hfppabs : β x β Set.Icc a b, |iteratedDeriv 2 f x| β€ 1
hlen2 : b - a β₯ 2
β’ β x β Set.Icc a b, |iteratedDeriv 1 f x| β€ 2 |
putnam_1962_a5 | abbrev putnam_1962_a5_solution : β β β := sorry
theorem putnam_1962_a5
: β n β₯ 2, putnam_1962_a5_solution n = β k β Finset.Icc 1 n, Nat.choose n k * k^2 := by | import Mathlib
| Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \] | Show that the expression equals $n(n+1)2^{n-2}$. | [
"algebra",
"combinatorics"
] | test | putnam_1962_a5 | c6716c02ef83e836 | β’ β n β₯ 2, putnam_1962_a5_solution n = β k β Finset.Icc 1 n, n.choose k * k ^ 2 |
putnam_1962_a6 | theorem putnam_1962_a6
(S : Set β)
(hSadd : β a β S, β b β S, a + b β S)
(hSprod : β a β S, β b β S, a * b β S)
(hScond : β r : β, (r β S β¨ -r β S β¨ r = 0) β§ Β¬(r β S β§ -r β S) β§ Β¬(r β S β§ r = 0) β§ Β¬(-r β S β§ r = 0))
: S = { r : β | r > 0 } := by | import Mathlib
| Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $a+b$ and $ab$, and having the property that for every rational number $r$ exactly one of the following three statements is true: \[ r \in S, -r \in S, r = 0. \] Prove that $S$ is the set of all positive rational numbers. | None. | [
"algebra"
] | test | putnam_1962_a6 | fe63d76f80e87cbe | S : Set β
hSadd : β a β S, β b β S, a + b β S
hSprod : β a β S, β b β S, a * b β S
hScond : β (r : β), (r β S β¨ -r β S β¨ r = 0) β§ Β¬(r β S β§ -r β S) β§ Β¬(r β S β§ r = 0) β§ Β¬(-r β S β§ r = 0)
β’ S = {r | r > 0} |
putnam_1962_b1 | theorem putnam_1962_b1
(p : β β β β β)
(x y : β)
(n : β)
(h0 : p 0 = fun x : β => 1)
(hp : β n > 0, p n = fun x : β => β i β Finset.range n, (x - i))
: p n (x+y) = β k β Finset.range (n+1), Nat.choose n k * (p k x) * (p (n - k) y) := by | import Mathlib
| Let $x^{(n)} = x(x-1)\cdots(x-n+1)$ for $n$ a positive integer and let $x^{(0)} = 1.$ Prove that \[ (x+y)^{(n)} = \sum_{k=0}^n {n \choose k} x^{(k)} y^{(n-k)}. \] | None. | [
"algebra",
"combinatorics"
] | test | putnam_1962_b1 | 3cc77f2339c311a3 | p : β β β β β
x y : β
n : β
h0 : p 0 = fun x => 1
hp : β n > 0, p n = fun x => β i β Finset.range n, (x - βi)
β’ p n (x + y) = β k β Finset.range (n + 1), β(n.choose k) * p k x * p (n - k) y |
putnam_1962_b2 | theorem putnam_1962_b2
: β f : β β Set β+, β a b : β, a < b β f a β f b := by | import Mathlib
open MeasureTheory
| Let $\mathbb{S}$ be the set of all subsets of the natural numbers. Prove the existence of a function $f : \mathbb{R} \to \mathbb{S}$ such that $f(a) \subset f(b)$ whenever $a < b$. | None. | [
"set_theory"
] | test | putnam_1962_b2 | a3e79be7684fc70f | β’ β f, β (a b : β), a < b β f a β f b |
putnam_1962_b3 | theorem putnam_1962_b3
(S : Set (EuclideanSpace β (Fin 2)))
(hS : Convex β S β§ 0 β S)
(htopo : (0 β interior S) β¨ IsClosed S)
(hray : β P : EuclideanSpace β (Fin 2), P β 0 β β Q : EuclideanSpace β (Fin 2), SameRay β P Q β§ Q β S)
: Bornology.IsBounded S := by | import Mathlib
open MeasureTheory
| Let $S$ be a convex region in the Euclidean plane, containing the origin, for which every ray from the origin has at least one point outside $S$. Assuming that either the origin is an interior point of $S$ or $S$ is topologically closed, prove that $S$ is bounded. | None. | [
"analysis"
] | test | putnam_1962_b3 | eaaff803157a0989 | S : Set (EuclideanSpace β (Fin 2))
hS : Convex β S β§ 0 β S
htopo : 0 β interior S β¨ IsClosed S
hray : β (P : EuclideanSpace β (Fin 2)), P β 0 β β Q, SameRay β P Q β§ Q β S
β’ Bornology.IsBounded S |
putnam_1962_b5 | theorem putnam_1962_b5
(n : β€)
(ng1 : n > 1)
: (3 * (n : β) + 1) / (2 * n + 2) < β i : Finset.Icc 1 n, ((i : β) / n) ^ (n : β) β§ β i : Finset.Icc 1 n, ((i : β) / n) ^ (n : β) < 2 := by | import Mathlib
open MeasureTheory
| Prove that for every integer $n$ greater than 1: \[ \frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^n + \left(\frac{2}{n} \right)^n + \cdots + \left(\frac{n}{n} \right)^n < 2. \] | None. | [
"algebra"
] | test | putnam_1962_b5 | 16a609f350314380 | n : β€
ng1 : n > 1
β’ (3 * βn + 1) / (2 * βn + 2) < β i, (ββi / βn) ^ βn β§ β i, (ββi / βn) ^ βn < 2 |
putnam_1962_b6 | theorem putnam_1962_b6
(n : β)
(a b : β β β)
(xs : Set β)
(f : β β β)
(hf : f = fun x : β => β k β Finset.Icc 0 n, ((a k) * Real.sin (k * x) + (b k) * Real.cos (k * x)))
(hf1 : β x β Set.Icc 0 (2 * Ο), |f x| β€ 1)
(hxs : xs.ncard = 2 * n β§ xs β Set.Ico 0 (2 * Ο))
(hfxs : β x β xs, |f x| = 1)
: (Β¬β c : β, f = fun x : β =... | import Mathlib
open Real
| Let \[ f(x) = \sum_{k=0}^n a_k \sin kx + b_k \cos kx, \] where $a_k$ and $b_k$ are constants. Show that, if $\lvert f(x) \rvert \le 1$ for $0 \le x \le 2 \pi$ and $\lvert f(x_i) \rvert = 1$ for $0 \le x_1 < x_2 < \cdots < x_{2n} < 2 \pi$, then $f(x) = \cos (nx + a)$ for some constant $a$. | None. | [
"analysis"
] | test | putnam_1962_b6 | 1b4d58ce7a1995f5 | n : β
a b : β β β
xs : Set β
f : β β β
hf : f = fun x => β k β Finset.Icc 0 n, (a k * sin (βk * x) + b k * cos (βk * x))
hf1 : β x β Set.Icc 0 (2 * Ο), |f x| β€ 1
hxs : xs.ncard = 2 * n β§ xs β Set.Ico 0 (2 * Ο)
hfxs : β x β xs, |f x| = 1
β’ (Β¬β c, f = fun x => c) β β a, f = fun x => cos (βn * x + a) |
putnam_1963_a2 | theorem putnam_1963_a2
(f : β β β)
(hfpos : β n, f n > 0)
(hfinc : StrictMonoOn f (Set.Ici 1))
(hf2 : f 2 = 2)
(hfmn : β m n, m > 0 β n > 0 β IsRelPrime m n β f (m * n) = f m * f n)
: β n > 0, f n = n := by | import Mathlib
open Topology Filter
| Let $\{f(n)\}$ be a strictly increasing sequence of positive integers such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every relatively prime pair of positive integers $m$ and $n$ (the greatest common divisor of $m$ and $n$ is equal to $1$). Prove that $f(n)=n$ for every positive integer $n$. | None. | [
"number_theory",
"algebra"
] | test | putnam_1963_a2 | a991a111d428ec4c | f : β β β
hfpos : β (n : β), f n > 0
hfinc : StrictMonoOn f (Set.Ici 1)
hf2 : f 2 = 2
hfmn : β (m n : β), m > 0 β n > 0 β IsRelPrime m n β f (m * n) = f m * f n
β’ β n > 0, f n = n |
putnam_1963_a3 | noncomputable abbrev putnam_1963_a3_solution : (β β β) β β β β β β β β := sorry
theorem putnam_1963_a3
(P : β β (β β β) β (β β β))
(hP : P 0 = id β§ β i y, P (i + 1) y = P i (fun x β¦ x * deriv y x - i * y x))
(n : β)
(hn : 0 < n)
(f y : β β β)
(hf : ContinuousOn f (Ici 1))
(hy : ContDiffOn β ... | import Mathlib
open Nat Set Topology Filter
| Find an integral formula (i.e., a function $z$ such that $y(x) = \int_{1}^{x} z(t) dt$) for the solution of the differential equation $$\delta (\delta - 1) (\delta - 2) \cdots (\delta - n + 1) y = f(x)$$ with the initial conditions $y(1) = y'(1) = \cdots = y^{(n-1)}(1) = 0$, where $n \in \mathbb{N}$, $f$ is continuous ... | Show that the solution is $$y(x) = \int_{1}^{x} \frac{(x - t)^{n - 1} f(t)}{(n - 1)!t^n} dt$$. | [
"analysis"
] | test | putnam_1963_a3 | 2369a166e152dbc5 | P : β β (β β β) β β β β
hP : P 0 = id β§ β (i : β) (y : β β β), P (i + 1) y = P i fun x => x * deriv y x - βi * y x
n : β
hn : 0 < n
f y : β β β
hf : ContinuousOn f (Ici 1)
hy : ContDiffOn β (βn) y (Ici 1)
hy1 : ContDiffAt β (βn) y 1
β’ (β i < n, deriv^[i] y 1 = 0) β§ EqOn (P n y) f (Ici 1) β
β x β₯ 1, y x = β« (t : β) ... |
putnam_1963_a4 | theorem putnam_1963_a4
(T : (β β β) β (β β β))
(T_def : β a n, T a n = n * ((1 + a (n + 1)) / a n - 1))
(P : (β β β) β β β Prop)
(P_def : β a C, P a C β C β€ limsup (T a) atTop β¨ Β¬ BddAbove (range (T a))) :
(β a, (β n, 0 < a n) β P a 1) β§ (β C > 1, β a, (β n, 0 < a n) β§ Β¬ P a C) := by | import Mathlib
open Filter Set
| Let $\{a_n\}$ be a sequence of positive real numbers. Show that $\limsup_{n \to \infty} n\left(\frac{1+a_{n+1}}{a_n}-1\right) \geq 1$. Show that the number $1$ on the right-hand side of this inequality cannot be replaced by any larger number. (The symbol $\limsup$ is sometimes written $\overline{\lim}$.) | None. | [
"analysis"
] | test | putnam_1963_a4 | 6b8d0640f9878a53 | T : (β β β) β β β β
T_def : β (a : β β β) (n : β), T a n = βn * ((1 + a (n + 1)) / a n - 1)
P : (β β β) β β β Prop
P_def : β (a : β β β) (C : β), P a C β C β€ limsup (T a) atTop β¨ Β¬BddAbove (range (T a))
β’ (β (a : β β β), (β (n : β), 0 < a n) β P a 1) β§ β C > 1, β a, (β (n : β), 0 < a n) β§ Β¬P a C |
putnam_1963_a6 | theorem putnam_1963_a6
(F1 F2 U V A B C D P Q : EuclideanSpace β (Fin 2))
(r : β)
(E : Set (EuclideanSpace β (Fin 2)))
(hE : E = {H : EuclideanSpace β (Fin 2) | dist F1 H + dist F2 H = r})
(M : EuclideanSpace β (Fin 2))
(hMuv : M = midpoint β U V)
(hr : r > dist F1 F2)
(hUV : U β E β§ V β E β§ U β V)
(hAB : A β E β§ B β E... | import Mathlib
open Topology Filter
| Let $U$ and $V$ be distinct points on an ellipse, with $M$ the midpoint of chord $\overline{UV}$, and let $\overline{AB}$ and $\overline{CD}$ be any two other chords through $M$. If line $UV$ intersects line $AC$ at $P$ and line $BD$ at $Q$, prove that $M$ is the midpoint of segment $\overline{PQ}$. | None. | [
"geometry"
] | test | putnam_1963_a6 | 75c0f03f123c8162 | F1 F2 U V A B C D P Q : EuclideanSpace β (Fin 2)
r : β
E : Set (EuclideanSpace β (Fin 2))
hE : E = {H | dist F1 H + dist F2 H = r}
M : EuclideanSpace β (Fin 2)
hMuv : M = midpoint β U V
hr : r > dist F1 F2
hUV : U β E β§ V β E β§ U β V
hAB : A β E β§ B β E β§ A β B
hCD : C β E β§ D β E β§ C β D
hAC : A β C
hBD : B β D
hdisti... |
putnam_1963_b1 | abbrev putnam_1963_b1_solution : β€ := sorry
theorem putnam_1963_b1
: β a : β€, (X^2 - X + (C a)) β£ (X ^ 13 + X + (C 90)) β a = putnam_1963_b1_solution := by | import Mathlib
open Topology Filter Polynomial
| For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$? | Show that $a=2$. | [
"algebra"
] | test | putnam_1963_b1 | e32bcfed4f271dcd | β’ β (a : β€), X ^ 2 - X + C a β£ X ^ 13 + X + C 90 β a = putnam_1963_b1_solution |
putnam_1963_b2 | abbrev putnam_1963_b2_solution : Prop := sorry
theorem putnam_1963_b2
(S : Set β)
(hS : S = {2 ^ m * 3 ^ n | (m : β€) (n : β€)})
: closure S β Set.Ioi (0 : β) β putnam_1963_b2_solution := by | import Mathlib
open Topology Filter Polynomial
| Let $S$ be the set of all numbers of the form $2^m3^n$, where $m$ and $n$ are integers, and let $P$ be the set of all positive real numbers. Is $S$ dense in $P$? | Show that $S$ is dense in $P$. | [
"analysis"
] | test | putnam_1963_b2 | aa3f2c45027467a4 | S : Set β
hS : S = {x | β m n, 2 ^ m * 3 ^ n = x}
β’ closure S β Set.Ioi 0 β putnam_1963_b2_solution |
putnam_1963_b3 | abbrev putnam_1963_b3_solution : Set (β β β) := sorry
theorem putnam_1963_b3
(f : β β β) :
f β putnam_1963_b3_solution β
(ContDiff β 1 f β§ Differentiable β (deriv f) β§
β x y : β, (f x) ^ 2 - (f y) ^ 2 = f (x + y) * f (x - y)) := by | import Mathlib
open Topology Filter Polynomial
| Find every twice-differentiable real-valued function $f$ with domain the set of all real numbers and satisfying the functional equation $(f(x))^2-(f(y))^2=f(x+y)f(x-y)$ for all real numbers $x$ and $y$. | Show that the solution is the sets of functions $f(u)=A\sinh ku$, $f(u)=Au$, and $f(u)=A\sin ku$ with $A,k \in \mathbb{R}$. | [
"analysis"
] | test | putnam_1963_b3 | 313b8cb55320f769 | f : β β β
β’ f β putnam_1963_b3_solution β
ContDiff β 1 f β§ Differentiable β (deriv f) β§ β (x y : β), f x ^ 2 - f y ^ 2 = f (x + y) * f (x - y) |
putnam_1963_b5 | theorem putnam_1963_b5
(a : β€ β β)
(haineq : β n β₯ 1, β k : β€, (n β€ k β§ k β€ 2 * n) β (0 β€ a k β§ a k β€ 100 * a n))
(haseries : β S : β, Tendsto (fun N : β => β n : Fin N, a n) atTop (π S))
: Tendsto (fun n : β€ => n * a n) atTop (π 0) := by | import Mathlib
open Topology Filter Polynomial
| Let $\{a_n\}$ be a sequence of real numbers satisfying the inequalities $0 \leq a_k \leq 100a_n$ for $n \leq k \leq 2n$ and $n=1,2,\dots$, and such that the series $\sum_{n=0}^\infty a_n$ converges. Prove that $\lim_{n \to \infty}na_n=0$. | None. | [
"analysis"
] | test | putnam_1963_b5 | cdb55710b964b7c2 | a : β€ β β
haineq : β n β₯ 1, β (k : β€), n β€ k β§ k β€ 2 * n β 0 β€ a k β§ a k β€ 100 * a n
haseries : β S, Tendsto (fun N => β n, a ββn) atTop (π S)
β’ Tendsto (fun n => βn * a n) atTop (π 0) |
putnam_1963_b6 | theorem putnam_1963_b6
(d : β)
(S : Set (Fin d β β) β Set (Fin d β β))
(hS : S = fun A : Set (Fin d β β) => β p β A, β q β A, segment β p q)
(A : β β Set (Fin d β β))
(ddim : 1 β€ d β§ d β€ 3)
(hA0 : Nonempty (A 0))
(hAn : β n β₯ 1, A n = S (A (n - 1)))
: β n β₯ 2, A n = A (n + 1) := by | import Mathlib
open Topology Filter Polynomial
| Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of all points that lie on closed segments joining pairs of points of $A$. For a given nonempty set $A_0$, define $A_n=S(A_{n-1})$ for $n=1,2,\dots$. Prove that $A_2=A_3=\cdots$. (A one-point set sho... | None. | [
"geometry",
"linear_algebra"
] | test | putnam_1963_b6 | 8bdc44bda8281d69 | d : β
S : Set (Fin d β β) β Set (Fin d β β)
hS : S = fun A => β p β A, β q β A, segment β p q
A : β β Set (Fin d β β)
ddim : 1 β€ d β§ d β€ 3
hA0 : Nonempty β(A 0)
hAn : β n β₯ 1, A n = S (A (n - 1))
β’ β n β₯ 2, A n = A (n + 1) |
putnam_1964_a1 | theorem putnam_1964_a1
(A : Finset (EuclideanSpace β (Fin 2)))
(hAcard : A.card = 6)
(dists : Set β)
(hdists : dists = {d : β | β a b : EuclideanSpace β (Fin 2), a β A β§ b β A β§ a β b β§ d = dist a b})
: (sSup dists / sInf dists β₯ Real.sqrt 3) := by | import Mathlib
| Let $A_1, A_2, A_3, A_4, A_5, A_6$ be distinct points in the plane. Let $D$ be the longest distance between any pair, and let $d$ the shortest distance. Show that $\frac{D}{d} \geq \sqrt 3$. | None. | [
"geometry"
] | test | putnam_1964_a1 | dad04ff466e09154 | A : Finset (EuclideanSpace β (Fin 2))
hAcard : A.card = 6
dists : Set β
hdists : dists = {d | β a b, a β A β§ b β A β§ a β b β§ d = dist a b}
β’ sSup dists / sInf dists β₯ β3 |
putnam_1964_a2 | abbrev putnam_1964_a2_solution : β β Set (β β β) := sorry
theorem putnam_1964_a2
(Ξ± : β)
: (putnam_1964_a2_solution Ξ± = {f : β β β | (β x β Icc 0 1, f x > 0) β§ ContinuousOn f (Icc 0 1) β§ β« x in (0)..1, f x = 1 β§ β« x in (0)..1, x * f x = Ξ± β§ β« x in (0)..1, x^2 * f x = Ξ±^2}) := by | import Mathlib
open Set
| Let $\alpha$ be a real number. Find all continuous real-valued functions $f : [0, 1] \to (0, \infty)$ such that
\begin{align*}
\int_0^1 f(x) dx &= 1, \\
\int_0^1 x f(x) dx &= \alpha, \\
\int_0^1 x^2 f(x) dx &= \alpha^2. \\
\end{align*} | Prove that there are no such functions. | [
"analysis",
"algebra"
] | test | putnam_1964_a2 | 730066ced765244e | Ξ± : β
β’ putnam_1964_a2_solution Ξ± =
{f |
(β x β Icc 0 1, f x > 0) β§
ContinuousOn f (Icc 0 1) β§
β« (x : β) in 0..1, f x = 1 β§ β« (x : β) in 0..1, x * f x = Ξ± β§ β« (x : β) in 0..1, x ^ 2 * f x = Ξ± ^ 2} |
putnam_1964_a3 | theorem putnam_1964_a3
(x a b : β β β)
(hxdense : range x β Ioo 0 1 β§ closure (range x) β Ioo 0 1)
(hxinj : Injective x)
(ha : a = fun n β¦ x n - sSup ({0} βͺ {p : β | p < x n β§ β i < n, p = x i}))
(hb : b = fun n β¦ sInf ({1} βͺ {p : β | p > x n β§ β i < n, p = x i}) - x n)
: (β' n : β, a n * b n * (a n + b n) = 1 / 3) := ... | import Mathlib
open Set Function
| The distinct points $x_n$ are dense in the interval $(0, 1)$. For all $n \geq 1$, $x_1, x_2, \dots , x_{n-1}$ divide $(0, 1)$ into $n$ sub-intervals, one of which must contain $x_n$. This part is divided by $x_n$ into two sub-intervals, lengths $a_n$ and $b_n$. Prove that $\sum_{n=1}^{\infty} a_nb_n(a_n + b_n) = \frac{... | None. | [
"analysis",
"algebra"
] | test | putnam_1964_a3 | 1334aff9985bc27d | x a b : β β β
hxdense : range x β Ioo 0 1 β§ closure (range x) β Ioo 0 1
hxinj : Injective x
ha : a = fun n => x n - sSup ({0} βͺ {p | p < x n β§ β i < n, p = x i})
hb : b = fun n => sInf ({1} βͺ {p | p > x n β§ β i < n, p = x i}) - x n
β’ β' (n : β), a n * b n * (a n + b n) = 1 / 3 |
putnam_1964_a4 | theorem putnam_1964_a4
(u : β β β€)
(boundedu : β B T : β€, β n : β, B β€ u n β§ u n β€ T)
(hu : β n β₯ 4, u n = ((u (n - 1) + u (n - 2) + u (n - 3) * u (n - 4)) : β) / (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) β§ (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) β 0)
: (β N c : β, c > 0 β§ β n β₯ N, u (n + c) = u n) := by | import Mathlib
open Set Function
| The sequence of integers $u_n$ is bounded and satisfies
\[
u_n = \frac{u_{n-1} + u_{n-2} + u_{n-3}u_{n-4}}{u_{n-1}u_{n-2} + u_{n-3} + u_{n-4}}.
\]
Show that it is periodic for sufficiently large $n$. | None. | [
"analysis"
] | test | putnam_1964_a4 | c5b20a41134533f7 | u : β β β€
boundedu : β B T, β (n : β), B β€ u n β§ u n β€ T
hu :
β n β₯ 4,
β(u n) =
(β(u (n - 1)) + β(u (n - 2)) + β(u (n - 3)) * β(u (n - 4))) /
(β(u (n - 1)) * β(u (n - 2)) + β(u (n - 3)) + β(u (n - 4))) β§
u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4) β 0
β’ β N, β c > 0, β n β₯ N, u (n + c) =... |
putnam_1964_a5 | theorem putnam_1964_a5
(pa : (β β β) β Prop)
(hpa : β a, pa a β (β n : β, a n > 0) β§ β L : β, Tendsto (fun N β¦ β n β Finset.range N, 1 / a n) atTop (π L)) :
β k : β, β a : β β β, pa a β
β' n : β, (n + 1) / (β i β Finset.range (n + 1), a i) β€ k * β' n : β, 1 / a n := by | import Mathlib
open Set Function Filter Topology
| Prove that there exists a constant $k$ such that for any sequence $a_i$ of positive numbers,
\[
\sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 + \dots + a_n} \leq k \sum_{n=1}^{\infty}\frac{1}{a_n}.
\] | None. | [
"analysis"
] | test | putnam_1964_a5 | a9fd76d34a23fdb5 | pa : (β β β) β Prop
hpa : β (a : β β β), pa a β (β (n : β), a n > 0) β§ β L, Tendsto (fun N => β n β Finset.range N, 1 / a n) atTop (π L)
β’ β k, β (a : β β β), pa a β β' (n : β), (βn + 1) / β i β Finset.range (n + 1), a i β€ k * β' (n : β), 1 / a n |
putnam_1964_a6 | theorem putnam_1964_a6
(S : Finset β)
(pairs : Set (β Γ β))
(hpairs : pairs = {(a, b) | (a β S) β§ (b β S) β§ (a < b)})
(distance : β Γ β β β)
(hdistance : distance = fun (a, b) β¦ b - a)
(hrepdist : β p β pairs, (β m β pairs, distance m > distance p) β β q β pairs, q β p β§ distance p = distance q)
: (β p q : pairs, q β p... | import Mathlib
open Set Function Filter Topology
| Let $S$ be a finite set of collinear points. Let $k$ be the maximum distance between any two points of $S$. Given a pair of points of $S$ a distance $d < k$ apart, we can find another pair of points of $S$ also a distance $d$ apart. Prove that if two pairs of points of $S$ are distances $a$ and $b$ apart, then $\frac{a... | None. | [
"geometry"
] | test | putnam_1964_a6 | 374495ab7e5e3f78 | S : Finset β
pairs : Set (β Γ β)
hpairs : pairs = {(a, b) | a β S β§ b β S β§ a < b}
distance : β Γ β β β
hdistance :
distance = fun x =>
match x with
| (a, b) => b - a
hrepdist : β p β pairs, (β m β pairs, distance m > distance p) β β q β pairs, q β p β§ distance p = distance q
β’ β (p q : βpairs), q β p β β r, ... |
putnam_1964_b1 | theorem putnam_1964_b1
(a b : β β β)
(h : β n, 0 < a n)
(h' : Summable fun n β¦ (1 : β) / a n)
(h'' : β n, b n = {k | a k β€ n}.ncard) :
Tendsto (fun n β¦ (b n : β) / n) atTop (π 0) := by | import Mathlib
open Set Function Filter Topology
| Let $a_n$ be a sequence of positive integers such that $\sum_{n=1}^{\infty} 1/a_n$ converges. For all $n$, let $b_n$ be the number of $a_n$ which are at most $n$. Prove that $\lim_{n \to \infty} b_n/n = 0$. | None. | [
"analysis"
] | test | putnam_1964_b1 | 87d71f812f07de03 | a b : β β β
h : β (n : β), 0 < a n
h' : Summable fun n => 1 / β(a n)
h'' : β (n : β), b n = {k | a k β€ n}.ncard
β’ Tendsto (fun n => β(b n) / βn) atTop (π 0) |
putnam_1964_b2 | theorem putnam_1964_b2
(S : Type*) [Fintype S] [Nonempty S]
(P : Finset (Set S))
(hPP : β T β P, β U β P, T β© U β β
)
(hPS : Β¬β T : Set S, T β P β§ (β U β P, T β© U β β
))
: (P.card = 2 ^ (Fintype.card S - 1)) := by | import Mathlib
open Set Function Filter Topology
| Let $S$ be a finite set. A set $P$ of subsets of $S$ has the property that any two members of $P$ have at least one element in common and that $P$ cannot be extended (whilst keeping this property). Prove that $P$ contains exactly half of the subsets of $S$. | None. | [
"set_theory",
"combinatorics"
] | test | putnam_1964_b2 | b1123d636b1381af | S : Type u_1
instβΒΉ : Fintype S
instβ : Nonempty S
P : Finset (Set S)
hPP : β T β P, β U β P, T β© U β β
hPS : Β¬β T β P, β U β P, T β© U β β
β’ P.card = 2 ^ (Fintype.card S - 1) |
putnam_1964_b3 | theorem putnam_1964_b3
(f : β β β)
(hf : Continuous f β§ β Ξ± > 0, Tendsto (fun n : β β¦ f (n * Ξ±)) atTop (π 0))
: (Tendsto f atTop (π 0)) := by | import Mathlib
open Set Function Filter Topology
| Suppose $f : \mathbb{R} \to \mathbb{R}$ is continuous and for every $\alpha > 0$, $\lim_{n \to \infty} f(n\alpha) = 0$. Prove that $\lim_{x \to \infty} f(x) = 0$. | None. | [
"analysis"
] | test | putnam_1964_b3 | 4ec6e467e9efb85b | f : β β β
hf : Continuous f β§ β Ξ± > 0, Tendsto (fun n => f (βn * Ξ±)) atTop (π 0)
β’ Tendsto f atTop (π 0) |
putnam_1964_b4 | abbrev putnam_1964_b4_solution : β β β := sorry
theorem putnam_1964_b4
{n : β} (hn : 0 < n)
-- `C` is a collection of `n` great circles on the sphere, i.e a collection of sets
(C : Fin n β Set (EuclideanSpace β (Fin 3)))
--together with a collection of `n` normal vectors `v`
(v : Fin n β EuclideanS... | import Mathlib
open Classical
open scoped InnerProductSpace
| $n$ great circles on the sphere are in general position (in other words at most two circles pass through any two points on the sphere). How many regions do they divide the sphere into? | n^2 - n + 2 | [
"geometry"
] | test | putnam_1964_b4 | 74afa18e4b2ee0cf | n : β
hn : 0 < n
C : Fin n β Set (EuclideanSpace β (Fin 3))
v : Fin n β EuclideanSpace β (Fin 3)
hv : β (i : Fin n), C i = Metric.sphere 0 1 β© {x | βͺv i, xβ«_β = 0}
hv' : β (i : Fin n), v i β 0
hCinj : Function.Injective C
hTβ : β x β Metric.sphere 0 1, β y β Metric.sphere 0 1, {i | {x, y} β C i}.card β€ 2
IsRegion : Set... |
putnam_1964_b5 | theorem putnam_1964_b5
(a b : β β β)
(ha : StrictMono a β§ β n : β, a n > 0)
(hb : b 0 = a 0 β§ β n : β, b (n + 1) = lcm (b n) (a (n + 1)))
: (β L : β, Tendsto (fun N β¦ β n β Finset.range N, (1 : β) / b n) atTop (π L)) := by | import Mathlib
open Set Function Filter Topology
| Let $a_n$ be a strictly monotonic increasing sequence of positive integers. Let $b_n$ be the least common multiple of $a_1, a_2, \dots , a_n$. Prove that $\sum_{n=1}^{\infty} 1/b_n$ converges. | None. | [
"analysis",
"number_theory"
] | test | putnam_1964_b5 | 5bdfe499780a09a9 | a b : β β β
ha : StrictMono a β§ β (n : β), a n > 0
hb : b 0 = a 0 β§ β (n : β), b (n + 1) = lcm (b n) (a (n + 1))
β’ β L, Tendsto (fun N => β n β Finset.range N, 1 / β(b n)) atTop (π L) |
putnam_1964_b6 | theorem putnam_1964_b6
(D : Set (EuclideanSpace β (Fin 2)))
(hD : D = {v : EuclideanSpace β (Fin 2) | dist 0 v β€ 1})
(cong : Set (EuclideanSpace β (Fin 2)) β Set (EuclideanSpace β (Fin 2)) β Prop)
(hcong : β A B, cong A B β β f : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)), B = f '' A β§ β v ... | import Mathlib
open Set Function Filter Topology
| Let $D$ be the unit disk in the plane. Show that we cannot find congruent sets $A, B$ with $A \cap B = \emptyset$ and $A \cup B = D$. | None. | [
"geometry"
] | test | putnam_1964_b6 | 9f8662ade9542175 | D : Set (EuclideanSpace β (Fin 2))
hD : D = {v | dist 0 v β€ 1}
cong : Set (EuclideanSpace β (Fin 2)) β Set (EuclideanSpace β (Fin 2)) β Prop
hcong :
β (A B : Set (EuclideanSpace β (Fin 2))),
cong A B β β f, B = f '' A β§ β (v w : EuclideanSpace β (Fin 2)), dist v w = dist (f v) (f w)
β’ Β¬β A B, cong A B β§ A β© B = β
... |
putnam_1965_a1 | noncomputable abbrev putnam_1965_a1_solution : β := sorry
theorem putnam_1965_a1
(A B C X Y : EuclideanSpace β (Fin 2))
(hABC : Β¬Collinear β {A, B, C})
(hangles : β C A B < β B C A β§ β B C A < Ο/2 β§ Ο/2 < β A B C)
(hX : Collinear β {X, B, C} β§ β X A B = (Ο - β C A B)/2 β§ dist A X = dist A B)
(hY : Collinear β {Y, C, A}... | import Mathlib
open EuclideanGeometry Real
| Let $\triangle ABC$ satisfy $\angle CAB < \angle BCA < \frac{\pi}{2} < \angle ABC$. If the bisector of the external angle at $A$ meets line $BC$ at $P$, the bisector of the external angle at $B$ meets line $CA$ at $Q$, and $AP = BQ = AB$, find $\angle CAB$. | Show that the solution is $\angle CAB = \frac{\pi}{15}$. | [
"geometry"
] | test | putnam_1965_a1 | a5e2a43798c96e41 | A B C X Y : EuclideanSpace β (Fin 2)
hABC : Β¬Collinear β {A, B, C}
hangles : β C A B < β B C A β§ β B C A < Ο / 2 β§ Ο / 2 < β A B C
hX : Collinear β {X, B, C} β§ β X A B = (Ο - β C A B) / 2 β§ dist A X = dist A B
hY : Collinear β {Y, C, A} β§ β Y B C = (Ο - β A B C) / 2 β§ dist B Y = dist A B
β’ β C A B = putnam_1965_a1_solu... |
putnam_1965_a2 | theorem putnam_1965_a2
: β n > 0, β r β Finset.Icc 0 ((n - 1)/2), ((n - 2*r) * Nat.choose n r / (n : β))^2 = (Nat.choose (2*n - 2) (n - 1))/(n : β) := by | import Mathlib
open EuclideanGeometry
| Prove that $$\sum_{r=0}^{\lfloor\frac{n-1}{2}\rfloor} \left(\frac{n - 2r}{n} {n \choose r}\right)^2 = \frac{1}{n} {{2n - 2} \choose {n - 1}}$$ for every positive integer $n$. | None. | [
"algebra"
] | test | putnam_1965_a2 | 1610cfb5afed081f | β’ β n > 0,
β r β Finset.Icc 0 ((n - 1) / 2), ((βn - 2 * βr) * β(n.choose r) / βn) ^ 2 = β((2 * n - 2).choose (n - 1)) / βn |
putnam_1965_a3 | theorem putnam_1965_a3
(a : β β β)
(Ξ± : β)
: Tendsto (fun n : β => (β k β Finset.Icc 1 n, exp (I * a k))/n) atTop (π Ξ±) β
Tendsto (fun n : β => (β k β Finset.Icc 1 (n^2), exp (I * a k))/n^2) atTop (π Ξ±) := by | import Mathlib
open EuclideanGeometry Topology Filter Complex
| Prove that, for any sequence of real numbers $a_1, a_2, \dots$, $$\lim_{n \to \infty} \frac{\sum_{k = 1}^{n} e^{ia_k}}{n} = \alpha$$ if and only if $$\lim_{n \to \infty} \frac{\sum_{k = 1}^{n} e^{ia_{k^2}}}{n^2} = \alpha.$$ | None. | [
"analysis"
] | test | putnam_1965_a3 | 686f1bdfdd61d9eb | a : β β β
Ξ± : β
β’ Tendsto (fun n => (β k β Finset.Icc 1 n, cexp (I * β(a k))) / βn) atTop (π Ξ±) β
Tendsto (fun n => (β k β Finset.Icc 1 (n ^ 2), cexp (I * β(a k))) / βn ^ 2) atTop (π Ξ±) |
putnam_1965_a4 | theorem putnam_1965_a4
{G B : Type*}
[Fintype G] [Nonempty G]
[Fintype B] [Nonempty B]
(dances : G β B β Prop)
(h : (Β¬β b : B, β g : G, dances g b) β§ β g : G, β b : B, dances g b)
: β g h : G, β b c : B, dances g b β§ dances h c β§ Β¬dances h b β§ Β¬dances g c := by | import Mathlib
open EuclideanGeometry Topology Filter Complex
| At a party, no boy dances with every girl, but each girl dances with at least one boy. Prove that there exist girls $g$ and $h$ and boys $b$ and $c$ such that $g$ dances with $b$ and $h$ dances with $c$, but $h$ does not dance with $b$ and $g$ does not dance with $c$. | None. | [
"combinatorics"
] | test | putnam_1965_a4 | def51bfdbefddda6 | G : Type u_1
B : Type u_2
instβΒ³ : Fintype G
instβΒ² : Nonempty G
instβΒΉ : Fintype B
instβ : Nonempty B
dances : G β B β Prop
h : (Β¬β b, β (g : G), dances g b) β§ β (g : G), β b, dances g b
β’ β g h b c, dances g b β§ dances h c β§ Β¬dances h b β§ Β¬dances g c |
putnam_1965_a5 | abbrev putnam_1965_a5_solution : β β β := sorry
theorem putnam_1965_a5
: β n > 0, {p β permsOfFinset (Finset.Icc 1 n) | β m β Finset.Icc 2 n, β k β Finset.Ico 1 m, p m = p k + 1 β¨ p m = p k - 1}.card = putnam_1965_a5_solution n := by | import Mathlib
open EuclideanGeometry Topology Filter Complex
| How many orderings of the integers from $1$ to $n$ satisfy the condition that, for every integer $i$ except the first, there exists some earlier integer in the ordering which differs from $i$ by $1$? | There are $2^{n-1}$ such orderings. | [
"combinatorics"
] | test | putnam_1965_a5 | 517ec60ace5c9664 | β’ β n > 0,
{p β permsOfFinset (Finset.Icc 1 n) |
β m β Finset.Icc 2 n, β k β Finset.Ico 1 m, p m = p k + 1 β¨ p m = p k - 1}.card =
putnam_1965_a5_solution n |
putnam_1965_a6 | theorem putnam_1965_a6
(u v m : β)
(hu : 0 < u)
(hv : 0 < v)
(hm : 1 < m) :
(βα΅ (x > 0) (y > 0),
u * x + v * y = 1 β§
x ^ m + y ^ m = 1 β§
u = x ^ (m - 1) β§
v = y ^ (m - 1)) β
β n, u ^ n + v ^ n = 1 β§ mβ»ΒΉ + nβ»ΒΉ = 1 := by | import Mathlib
open EuclideanGeometry Topology Filter Complex
| Prove that the line $ux + vy = 1$ (where $u \ge 0$ and $v \ge 0$) will lie tangent to the curve $x^m + y^m = 1$ (where $m > 1$) if and only if $u^n + v^n = 1$ for some $n$ such that $m^{-1} + n^{-1} = 1$. | None. | [
"geometry"
] | test | putnam_1965_a6 | 9d1f39db555d05d9 | u v m : β
hu : 0 < u
hv : 0 < v
hm : 1 < m
β’ (β x > 0, β y > 0, u * x + v * y = 1 β§ x ^ m + y ^ m = 1 β§ u = x ^ (m - 1) β§ v = y ^ (m - 1)) β
β n, u ^ n + v ^ n = 1 β§ mβ»ΒΉ + nβ»ΒΉ = 1 |
putnam_1965_b1 | noncomputable abbrev putnam_1965_b1_solution : β := sorry
theorem putnam_1965_b1
: Tendsto (fun n : β β¦ β« x in {x : Fin (n+1) β β | β k : Fin (n+1), x k β Set.Icc 0 1}, (Real.cos (Real.pi/(2 * (n+1)) * β k : Fin (n+1), x k))^2) atTop (π putnam_1965_b1_solution) := by | import Mathlib
open EuclideanGeometry Topology Filter Complex
| Find $$\lim_{n \to \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos^2\left(\frac{\pi}{2n}(x_1 + x_2 + \cdots + x_n)\right) dx_1 dx_2 \cdots dx_n.$$ | Show that the limit is $\frac{1}{2}$. | [
"analysis"
] | test | putnam_1965_b1 | d78c89a440b5c647 | β’ Tendsto
(fun n =>
β« (x : Fin (n + 1) β β) in {x | β (k : Fin (n + 1)), x k β Set.Icc 0 1},
Real.cos (Real.pi / (2 * (βn + 1)) * β k, x k) ^ 2)
atTop (π putnam_1965_b1_solution) |
putnam_1965_b2 | theorem putnam_1965_b2
(n : β)
(hn : n > 1)
(won : Fin n β Fin n β Bool)
(hirrefl : β i : Fin n, won i i = false)
(hantisymm : β i j : Fin n, i β j β won i j = Β¬won j i)
(w l : Fin n β β€)
(hw : w = fun r : Fin n => β j : Fin n, (if won r j then 1 else 0))
(hl : l = fun r : Fin n => n - 1 - w r)
: β r : Fin n, (w r)^2 =... | import Mathlib
open EuclideanGeometry Topology Filter Complex
| A round-robin tournament has $n > 1$ players $P_1, P_2, \dots, P_n$, who each play one game with each other player. Each game results in a win for one player and a loss for the other. If $w_r$ and $l_r$ denote the number of games won and lost, respectively, by $P_r$, prove that $$\sum_{r=1}^{n} w_r^2 = \sum_{r=1}^{n} l... | None. | [
"combinatorics"
] | test | putnam_1965_b2 | 16ff2dfd12f314ea | n : β
hn : n > 1
won : Fin n β Fin n β Bool
hirrefl : β (i : Fin n), won i i = false
hantisymm : β (i j : Fin n), i β j β (won i j = true) = Β¬won j i = true
w l : Fin n β β€
hw : w = fun r => β j, if won r j = true then 1 else 0
hl : l = fun r => βn - 1 - w r
β’ β r, w r ^ 2 = β r, l r ^ 2 |
putnam_1965_b3 | theorem putnam_1965_b3
: {(a, b, c) : β€ Γ β€ Γ β€ | a > 0 β§ a β€ b β§ c > 0 β§ a^2 + b^2 = c^2 β§ a*b/(2 : β) = 2*(a + b + c)}.ncard = 3 := by | import Mathlib
open EuclideanGeometry Topology Filter Complex
| Prove that there are exactly three right triangles (up to orientation and translation) with integer side lengths and area equal to twice their perimeter. | None. | [
"algebra",
"geometry"
] | test | putnam_1965_b3 | fbeae6f00965613a | β’ {(a, b, c) | a > 0 β§ a β€ b β§ c > 0 β§ a ^ 2 + b ^ 2 = c ^ 2 β§ βa * βb / 2 = 2 * (βa + βb + βc)}.ncard = 3 |
putnam_1965_b4 | noncomputable abbrev putnam_1965_b4_solution : ((((β β β) β (β β β)) Γ ((β β β) β (β β β))) Γ ((Set β) Γ (β β β))) := sorry
theorem putnam_1965_b4
(f u v : β β β β β)
(hu : β n > 0, β x, u n x = β i β Finset.Icc 0 (n / 2), (n.choose (2 * i)) * x ^ i)
(hv : β n > 0, β x, v n x = β i β Finset.Icc 0 ((n - 1) /... | import Mathlib
open EuclideanGeometry Topology Filter Complex
| Let $$f(x, n) = \frac{{n \choose 0} + {n \choose 2}x + {n \choose 4}x^2 + \cdots}{{n \choose 1} + {n \choose 3}x + {n \choose 5}x^2 + \cdots}$$ for all real numbers $x$ and positive integers $n$. Express $f(x, n+1)$ as a rational function involving $f(x, n)$ and $x$, and find $\lim_{n \to \infty} f(x, n)$ for all $x$ f... | We have $$f(x, n+1) = \frac{f(x, n) + x}{f(x, n) + 1};$$ $\lim_{n \to \infty} f(x, n) = \sqrt{x}$ for all $x \ge 0$ and diverges otherwise. | [
"algebra",
"analysis"
] | test | putnam_1965_b4 | 02ecc9232ae72702 | f u v : β β β β β
hu : β n > 0, β (x : β), u n x = β i β Finset.Icc 0 (n / 2), β(n.choose (2 * i)) * x ^ i
hv : β n > 0, β (x : β), v n x = β i β Finset.Icc 0 ((n - 1) / 2), β(n.choose (2 * i + 1)) * x ^ i
hf : β n > 0, β (x : β), f n x = u n x / v n x
n : β
hn : 0 < n
β’ match putnam_1965_b4_solution with
| ((p, q), ... |
putnam_1965_b5 | theorem putnam_1965_b5
{K : Type*}
[Fintype K]
(V E : β)
(hV : V = Nat.card K)
(hE: 4*E β€ V^2)
: β G : SimpleGraph K, G.edgeSet.ncard = E β§ β a : K, β w : G.Walk a a, w.length β 3 := by | import Mathlib
open EuclideanGeometry Topology Filter Complex SimpleGraph.Walk
| Prove that, if $4E \le V^2$, there exists a graph with $E$ edges and $V$ vertices with no triangles (cycles of length $3$). | None. | [
"combinatorics"
] | test | putnam_1965_b5 | 84857cc3da753b61 | K : Type u_1
instβ : Fintype K
V E : β
hV : V = Nat.card K
hE : 4 * E β€ V ^ 2
β’ β G, G.edgeSet.ncard = E β§ β (a : K) (w : G.Walk a a), w.length β 3 |
putnam_1965_b6 | theorem putnam_1965_b6
(A B C D : EuclideanSpace β (Fin 2))
(S : Set (EuclideanSpace β (Fin 2)))
(hS : S = {A, B, C, D})
(hdistinct : S.ncard = 4)
(through : (β Γ (EuclideanSpace β (Fin 2))) β (EuclideanSpace β (Fin 2)) β Prop)
(through_def : through = fun (r, P) => fun Q => dist P Q = r)
(h... | import Mathlib
open EuclideanGeometry Topology Filter Complex SimpleGraph.Walk
| Let $A$, $B$, $C$, and $D$ be four distinct points for which every circle through $A$ and $B$ intersects every circle through $C$ and $D$. Prove that $A$, $B$, $C$ and $D$ are either collinear (all lying on the same line) or cocyclic (all lying on the same circle). | None. | [
"geometry"
] | test | putnam_1965_b6 | 1be9052f1fcab75a | A B C D : EuclideanSpace β (Fin 2)
S : Set (EuclideanSpace β (Fin 2))
hS : S = {A, B, C, D}
hdistinct : S.ncard = 4
through : β Γ EuclideanSpace β (Fin 2) β EuclideanSpace β (Fin 2) β Prop
through_def :
through = fun x =>
match (motive := β Γ EuclideanSpace β (Fin 2) β EuclideanSpace β (Fin 2) β Prop) x with
... |
putnam_1966_a1 | theorem putnam_1966_a1
(f : β€ β β€)
(hf : f = fun n : β€ => β m β Finset.Icc 0 n, (if Even m then m / 2 else (m - 1)/2))
: β x y : β€, x > 0 β§ y > 0 β§ x > y β x * y = f (x + y) - f (x - y) := by | import Mathlib
| Let $a_n$ denote the sequence $0, 1, 1, 2, 2, 3, \dots$, where $a_n = \frac{n}{2}$ if $n$ is even and $\frac{n - 1}{2}$ if n is odd. Furthermore, let $f(n)$ denote the sum of the first $n$ terms of $a_n$. Prove that all positive integers $x$ and $y$ with $x > y$ satisfy $xy = f(x + y) - f(x - y)$. | None. | [
"algebra"
] | test | putnam_1966_a1 | d8a7a56b22e4c9ae | f : β€ β β€
hf : f = fun n => β m β Finset.Icc 0 n, if Even m then m / 2 else (m - 1) / 2
β’ β (x y : β€), x > 0 β§ y > 0 β§ x > y β x * y = f (x + y) - f (x - y) |
putnam_1966_a2 | theorem putnam_1966_a2
(r : β)
(A B C : EuclideanSpace β (Fin 2))
(hABC : Β¬Collinear β {A, B, C})
(a b c p : β)
(ha : a = dist B C)
(hb : b = dist C A)
(hc : c = dist A B)
(hp : p = (dist B C + dist C A + dist A B)/2)
(hr : β I : EuclideanSpace β (Fin 2),
(β! P : EuclideanSpace β (Fin 2), dist I P = r β§ Collinear β {P,... | import Mathlib
| Let $a$, $b$, and $c$ be the side lengths of a triangle with inradius $r$. If $p = \frac{a + b + c}{2}$, show that $$\frac{1}{(p - a)^2} + \frac{1}{(p - b)^2} + \frac{1}{(p - c)^2} \ge \frac{1}{r^2}.$$ | None. | [
"geometry"
] | test | putnam_1966_a2 | 003de4f28c6df342 | r : β
A B C : EuclideanSpace β (Fin 2)
hABC : Β¬Collinear β {A, B, C}
a b c p : β
ha : a = dist B C
hb : b = dist C A
hc : c = dist A B
hp : p = (dist B C + dist C A + dist A B) / 2
hr :
β I,
(β! P, dist I P = r β§ Collinear β {P, B, C}) β§
(β! Q, dist I Q = r β§ Collinear β {Q, C, A}) β§
(β! R, dist I R... |
putnam_1966_a3 | theorem putnam_1966_a3
(x : β β β)
(hx1 : 0 < x 1 β§ x 1 < 1)
(hxi : β n β₯ 1, x (n + 1) = (x n) * (1 - (x n)))
: Tendsto (fun n : β => n * (x n)) atTop (π 1) := by | import Mathlib
open Topology Filter
| If $0 < x_1 < 1$ and $x_{n+1} = x_n(1 - x_n)$ for all $n \ge 1$, prove that $\lim_{n \to \infty} nx_n = 1$. | None. | [
"analysis"
] | test | putnam_1966_a3 | 8667d7f9a598a62f | x : β β β
hx1 : 0 < x 1 β§ x 1 < 1
hxi : β n β₯ 1, x (n + 1) = x n * (1 - x n)
β’ Tendsto (fun n => βn * x n) atTop (π 1) |
putnam_1966_a4 | theorem putnam_1966_a4
(a : β β β€)
(ha1 : a 1 = 2)
(hai : β n β₯ 1, a (n + 1) = (if β m : β€, m^2 = a n + 1 = True then a n + 2 else a n + 1))
: β n β₯ 1, a n = n + round (Real.sqrt n) := by | import Mathlib
open Topology Filter
| Prove that the $n$th item in the ascending list of non-perfect-square positive integers equals $n + \{\sqrt{n}\}$, where $\{m\}$ denotes the closest integer to $m$. | None. | [
"analysis"
] | test | putnam_1966_a4 | ccf53814d45d50cf | a : β β β€
ha1 : a 1 = 2
hai : β n β₯ 1, a (n + 1) = if (β m, m ^ 2 = a n + 1) = True then a n + 2 else a n + 1
β’ β n β₯ 1, a n = βn + round ββn |
putnam_1966_a5 | theorem putnam_1966_a5
(C : Set (β β β))
(hC : C = {f : β β β | Continuous f})
(T : (β β β) β (β β β))
(imageTC : β f β C, T f β C)
(linearT : β a b : β, β f β C, β g β C, T ((fun x => a)*f + (fun x => b)*g) = (fun x => a)*(T f) + (fun x => b)*(T g))
(localT : β r s : β, r β€ s β β f β C, β g β C, (β x β Set.Icc r s, f ... | import Mathlib
open Topology Filter
| Let $C$ be the set of continuous functions $f : \mathbb{R} \to \mathbb{R}$. Let $T : C \to C$ satisfty the following two properties:
\begin{enumerate}
\item Linearity: $T(af + bg) = aT(f) + bT(g)$ for all $a, b \in \mathbb{R}$ and all $f, g \in C$.
\item Locality: If $f \in C$ and $g \in C$ are identical on some interv... | None. | [
"algebra"
] | test | putnam_1966_a5 | 131de7736ecbd7a3 | C : Set (β β β)
hC : C = {f | Continuous f}
T : (β β β) β β β β
imageTC : β f β C, T f β C
linearT :
β (a b : β), β f β C, β g β C, T ((fun x => a) * f + (fun x => b) * g) = (fun x => a) * T f + (fun x => b) * T g
localT : β (r s : β), r β€ s β β f β C, β g β C, (β x β Set.Icc r s, f x = g x) β β x β Set.Icc r s, T f ... |
putnam_1966_a6 | theorem putnam_1966_a6
(a : β β (β β β))
(ha : β n β₯ 1, a n n = n β§ β m β₯ 1, m < n β a n m = m * Real.sqrt (1 + a n (m + 1)))
: Tendsto (fun n => a n 1) atTop (π 3) := by | import Mathlib
open Topology Filter
| Prove that $$\sqrt {1 + 2 \sqrt {1 + 3 \sqrt {1 + 4 \sqrt {1 + 5 \sqrt {\dots}}}}} = 3.$$ | None. | [
"analysis"
] | test | putnam_1966_a6 | 32e9a00c6b2e15cb | a : β β β β β
ha : β n β₯ 1, a n n = βn β§ β m β₯ 1, m < n β a n m = βm * β(1 + a n (m + 1))
β’ Tendsto (fun n => a n 1) atTop (π 3) |
putnam_1966_b1 | theorem putnam_1966_b1
(n : β)
(hn : n β₯ 3)
(L : ZMod n β (EuclideanSpace β (Fin 2)))
(hsq : β i : ZMod n, L i 0 β Set.Icc 0 1 β§ L i 1 β Set.Icc 0 1)
(hnoncol : β i j k : ZMod n, i β j β§ j β k β§ k β i β Β¬Collinear β {L i, L j, L k})
(hconvex : β i : ZMod n, segment β (L i) (L (i + 1)) β© interior (convexHull β {L j | j ... | import Mathlib
open Topology
| If a convex polygon $L$ is contained entirely within a square of side length $1$, prove that the sum of the squares of the side lengths of $L$ is no greater than $4$. | None. | [
"geometry"
] | test | putnam_1966_b1 | eb91cfae34261d99 | n : β
hn : n β₯ 3
L : ZMod n β EuclideanSpace β (Fin 2)
hsq : β (i : ZMod n), (L i).ofLp 0 β Set.Icc 0 1 β§ (L i).ofLp 1 β Set.Icc 0 1
hnoncol : β (i j k : ZMod n), i β j β§ j β k β§ k β i β Β¬Collinear β {L i, L j, L k}
hconvex : β (i : ZMod n), segment β (L i) (L (i + 1)) β© interior ((convexHull β) {x | β j, L j = x}) = β
... |
putnam_1966_b2 | theorem putnam_1966_b2
(S : β€ β Set β€)
(hS : S = fun n : β€ => {n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9})
: β n : β€, n > 0 β (β k β S n, β m β S n, k β m β IsCoprime m k) := by | import Mathlib
| Prove that, for any ten consecutive integers, at least one is relatively prime to all of the others. | None. | [
"number_theory"
] | test | putnam_1966_b2 | 353c73a67df034df | S : β€ β Set β€
hS : S = fun n => {n, n + 1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9}
β’ β n > 0, β k β S n, β m β S n, k β m β IsCoprime m k |
putnam_1966_b3 | theorem putnam_1966_b3
(p : β β β)
(hpos : β n : β, p n > 0)
(hconv : β r : β, Tendsto (fun m : β => β n β Finset.Icc 1 m, 1/(p n)) atTop (π r))
: β r : β, Tendsto (fun m : β => β n β Finset.Icc 1 m, (p n) * n^2/(β i β Finset.Icc 1 n, p i)^2) atTop (π r) := by | import Mathlib
open Topology Filter
| Let $p_1, p_2, \dots$ be a sequence of positive real numbers. Prove that if $\sum_{n=1}^{\infty} \frac{1}{p_n}$ converges, then $$\sum_{n=1}^{\infty} \frac {n^2 p_n}{(\sum_{i=1}^{n} p_i)^2}$$ also converges. | None. | [
"analysis"
] | test | putnam_1966_b3 | 1d52fe1e85598f4b | p : β β β
hpos : β (n : β), p n > 0
hconv : β r, Tendsto (fun m => β n β Finset.Icc 1 m, 1 / p n) atTop (π r)
β’ β r, Tendsto (fun m => β n β Finset.Icc 1 m, p n * βn ^ 2 / (β i β Finset.Icc 1 n, p i) ^ 2) atTop (π r) |
putnam_1966_b4 | theorem putnam_1966_b4
(m n : β)
(S : Finset β)
(hS : (β i β S, i > 0) β§ S.card = m * n + 1)
: β T β S, (T.card = m + 1 β§ β j β T, β i β T, i β j β Β¬(j β£ i)) β¨ (T.card = n + 1 β§ β i β T, β j β T, j < i β j β£ i) := by | import Mathlib
open Topology Filter
| Let $a_1, a_2, ...$ be an increasing sequence of $mn + 1$ positive integers. Prove that there exists either a subset of $m + 1$ $a_i$ such that no element of the subset divides any other, or a subset of $n + 1$ $a_i$ such that each element of the subset (except the greatest) divides the next greatest element. | None. | [
"number_theory",
"combinatorics"
] | test | putnam_1966_b4 | 37384af7788d519d | m n : β
S : Finset β
hS : (β i β S, i > 0) β§ S.card = m * n + 1
β’ β T β S, (T.card = m + 1 β§ β j β T, β i β T, i β j β Β¬j β£ i) β¨ T.card = n + 1 β§ β i β T, β j β T, j < i β j β£ i |
putnam_1966_b5 | theorem putnam_1966_b5
(S : Finset (EuclideanSpace β (Fin 2)))
(hcard : S.card β₯ 3)
(hS : β s β S, s.card = 3 β Β¬Collinear β s.toSet)
: β L : ZMod S.card β (EuclideanSpace β (Fin 2)), (β p β S, β! i : ZMod S.card, p = L i) β§
β i j : ZMod S.card, i β j β (β I : EuclideanSpace β (Fin 2),
(I β segment β (L i) (L (i + 1)) ... | import Mathlib
open Topology Filter
| Prove that any set of $n \ge 3$ distinct points in the Euclidean plane, no three of which are collinear, forms the vertex set of some simple (non-self-intersecting) closed polygon. | None. | [
"geometry"
] | test | putnam_1966_b5 | ce6da46ef3a2f9b1 | S : Finset (EuclideanSpace β (Fin 2))
hcard : S.card β₯ 3
hS : β s β S, s.card = 3 β Β¬Collinear β s.toSet
β’ β L,
(β p β S, β! i, p = L i) β§
β (i j : ZMod S.card),
i β j β
β (I : EuclideanSpace β (Fin 2)),
I β segment β (L i) (L (i + 1)) β§ I β segment β (L j) (L (j + 1)) β
... |
putnam_1966_b6 | theorem putnam_1966_b6
(y : β β β)
(hy : Differentiable β y β§ Differentiable β (deriv y))
(diffeq : deriv (deriv y) + Real.exp * y = 0)
: β r s N : β, β x > N, r β€ y x β§ y x β€ s := by | import Mathlib
open Topology Filter
| Prove that any solution $y(x)$ to the differential equation $y'' + e^{x}y = 0$ remains bounded as $x$ goes to $+\infty$. | None. | [
"analysis"
] | test | putnam_1966_b6 | f187a1b81ec0a67e | y : β β β
hy : Differentiable β y β§ Differentiable β (deriv y)
diffeq : deriv (deriv y) + Real.exp * y = 0
β’ β r s N, β x > N, r β€ y x β§ y x β€ s |
putnam_1967_a1 | theorem putnam_1967_a1
(n : β) (hn : n > 0)
(a : Set.Icc 1 n β β)
(f : β β β)
(hf : f = (fun x : β => β i : Set.Icc 1 n, a i * Real.sin (i * x)))
(flesin : β x : β, abs (f x) β€ abs (Real.sin x))
: abs (β i : Set.Icc 1 n, i * a i) β€ 1 := by | import Mathlib
open Nat Topology Filter
| Let $f(x)=a_1\sin x+a_2\sin 2x+\dots+a_n\sin nx$, where $a_1,a_2,\dots,a_n$ are real numbers and where $n$ is a positive integer. Given that $|f(x)| \leq |\sin x|$ for all real $x$, prove that $|a_1|+|2a_2|+\dots+|na_n| \leq 1$. | None. | [
"analysis"
] | test | putnam_1967_a1 | 6cdecccb922d137d | n : β
hn : n > 0
a : β(Set.Icc 1 n) β β
f : β β β
hf : f = fun x => β i, a i * Real.sin (ββi * x)
flesin : β (x : β), |f x| β€ |Real.sin x|
β’ |β i, ββi * a i| β€ 1 |
putnam_1967_a2 | theorem putnam_1967_a2
(S : β β β€)
(hS0 : S 0 = 1)
(hSn : β n β₯ 1, S n = {A : Matrix (Fin n) (Fin n) β | (β i j, A i j = A j i) β§ (β j, (β i, A i j) = 1)}.ncard) :
(β n β₯ 1, S (n + 1) = S n + n * S (n - 1)) β§
(β x : β, (β' n : β, S n * (x ^ n / (n)!)) = Real.exp (x + x ^ 2 / 2)) := by | import Mathlib
open Nat Topology Filter
| Define $S_0$ to be $1$. For $n \geq 1$, let $S_n$ be the number of $n \times n$ matrices whose elements are nonnegative integers with the property that $a_{ij}=a_{ji}$, ($i,j=1,2,\dots,n$) and where $\sum_{i=1}^n a_{ij}=1$, ($j=1,2,\dots,n$). Prove
\begin{enumerate}
\item[(a)] $S_{n+1}=S_n+nS_{n-1}$
\item[(b)] $\sum_{n... | None. | [
"linear_algebra",
"analysis"
] | test | putnam_1967_a2 | cd63ea20c2bd9be7 | S : β β β€
hS0 : S 0 = 1
hSn : β n β₯ 1, S n = β{A | (β (i j : Fin n), A i j = A j i) β§ β (j : Fin n), β i, A i j = 1}.ncard
β’ (β n β₯ 1, S (n + 1) = S n + βn * S (n - 1)) β§
β (x : β), β' (n : β), β(S n) * (x ^ n / βn !) = Real.exp (x + x ^ 2 / 2) |
putnam_1967_a3 | abbrev putnam_1967_a3_solution : β := sorry
theorem putnam_1967_a3 :
IsLeast
{a | β P : Polynomial β€,
P.degree = 2 β§
(β z1 z2 : Set.Ioo (0 : β) 1, z1 β z2 β§ aeval (z1 : β) P = 0 β§ aeval (z2 : β) P = 0) β§
P.coeff 2 = a β§ a > 0}
putnam_1967_a3_solution := by | import Mathlib
open Polynomial
| Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists. | Show that the minimum possible value for $a$ is $5$. | [
"algebra"
] | test | putnam_1967_a3 | 36901dccb4845a10 | β’ IsLeast {a | β P, P.degree = 2 β§ (β z1 z2, z1 β z2 β§ (aeval βz1) P = 0 β§ (aeval βz2) P = 0) β§ P.coeff 2 = a β§ a > 0}
βputnam_1967_a3_solution |
putnam_1967_a4 | theorem putnam_1967_a4
(lambda : β)
(hlambda : lambda > 1 / 2)
: Β¬β u : β β β, MeasureTheory.IntegrableOn u (Set.Icc 0 1) β§ β x β Set.Icc 0 1, u x = 1 + lambda * (β« y in Set.Ioo x 1, u y * u (y - x)) := by | import Mathlib
open Nat Topology Filter
| Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u$ such that for all $x$ in the closed interval $0 \leq x \leq 1$, $u(x)=1+\lambda\int_x^1 u(y)u(y-x)\,dy$. | None. | [
"analysis"
] | test | putnam_1967_a4 | b327ceabbb63989d | lambda : β
hlambda : lambda > 1 / 2
β’ Β¬β u,
MeasureTheory.IntegrableOn u (Set.Icc 0 1) MeasureTheory.volume β§
β x β Set.Icc 0 1, u x = 1 + lambda * β« (y : β) in Set.Ioo x 1, u y * u (y - x) |
putnam_1967_a5 | theorem putnam_1967_a5
(R : Set (EuclideanSpace β (Fin 2)))
(hR : Convex β R β§ (MeasureTheory.volume R).toReal > Real.pi / 4)
: β P β R, β Q β R, dist P Q = 1 := by | import Mathlib
open Nat Topology Filter
| Prove that any convex region in the Euclidean plane with area greater than $\pi/4$ contains a pair of points exactly $1$ unit apart. | None. | [
"geometry"
] | test | putnam_1967_a5 | 4d08ebb881d3f701 | R : Set (EuclideanSpace β (Fin 2))
hR : Convex β R β§ (MeasureTheory.volume R).toReal > Real.pi / 4
β’ β P β R, β Q β R, Dist.dist P Q = 1 |
putnam_1967_a6 | abbrev putnam_1967_a6_solution : β := sorry
theorem putnam_1967_a6
(abneq0 : (Fin 4 β β) β (Fin 4 β β) β Prop)
(habneq0 : abneq0 = (fun a b : Fin 4 β β => a 0 * b 1 - a 1 * b 0 β 0))
(numtuples : (Fin 4 β β) β (Fin 4 β β) β β)
(hnumtuples : β a b : Fin 4 β β, numtuples a b = {s : Fin 4 β β | β x : Fin 4 β β, (β i : Fin... | import Mathlib
open Nat Topology Filter
| Given real numbers $\{a_i\}$ and $\{b_i\}$, ($i=1,2,3,4$), such that $a_1b_2-a_2b_1 \neq 0$. Consider the set of all solutions $(x_1,x_2,x_3,x_4)$ of the simultaneous equations $a_1x_1+a_2x_2+a_3x_3+a_4x_4=0$ and $b_1x_1+b_2x_2+b_3x_3+b_4x_4=0$, for which no $x_i$ ($i=1,2,3,4$) is zero. Each such solution generates a $... | Show that the maximum number of distinct $4$-tuples is eight. | [
"algebra",
"geometry"
] | test | putnam_1967_a6 | 037a09ade2c3eb5b | abneq0 : (Fin 4 β β) β (Fin 4 β β) β Prop
habneq0 : abneq0 = fun a b => a 0 * b 1 - a 1 * b 0 β 0
numtuples : (Fin 4 β β) β (Fin 4 β β) β β
hnumtuples :
β (a b : Fin 4 β β),
β(numtuples a b) =
{s |
β x,
(β (i : Fin 4), x i β 0) β§ β i, a i * x i = 0 β§ β i, b i * x i = 0 β§ β (i : Fin 4),... |
putnam_1967_b1 | theorem putnam_1967_b1
(r : β)
(L : ZMod 6 β (EuclideanSpace β (Fin 2)))
(P Q R: EuclideanSpace β (Fin 2))
(hP : P = midpoint β (L 1) (L 2))
(hQ : Q = midpoint β (L 3) (L 4))
(hR : R = midpoint β (L 5) (L 0))
(hr : r > 0)
(hcyclic : β (O : EuclideanSpace β (Fin 2)), β i : ZMod 6, dist O (L i) = r)
(horder : β i j : ZMo... | import Mathlib
open Nat Topology Filter
| Let $\hexagon ABCDEF$ be a hexagon inscribed in a circle of radius $r$. If $AB = CD = EF = r$, prove that the midpoints of $\overline{BC}$, $\overline{DE}$, and $\overline{FA}$ form the vertices of an equilateral triangle. | None. | [
"geometry"
] | test | putnam_1967_b1 | f194e7d0c26ac885 | r : β
L : ZMod 6 β EuclideanSpace β (Fin 2)
P Q R : EuclideanSpace β (Fin 2)
hP : P = midpoint β (L 1) (L 2)
hQ : Q = midpoint β (L 3) (L 4)
hR : R = midpoint β (L 5) (L 0)
hr : r > 0
hcyclic : β O, β (i : ZMod 6), Dist.dist O (L i) = r
horder :
β (i j : ZMod 6),
i β j β i + 1 = j β¨ i = j + 1 β¨ segment β (L i) (L... |
putnam_1967_b2 | theorem putnam_1967_b2
(p r A B C Ξ± Ξ² Ξ³ : β)
(prbound : 0 β€ p β§ p β€ 1 β§ 0 β€ r β§ r β€ 1)
(id1 : β x y : β, (p * x + (1 - p) * y) ^ 2 = A * x ^ 2 + B * x * y + C * y ^ 2)
(id2 : β x y : β, (p * x + (1 - p) * y) * (r * x + (1 - r) * y) = Ξ± * x ^ 2 + Ξ² * x * y + Ξ³ * y ^ 2)
: max (max A B) C β₯ 4 / 9 β§ max (max Ξ± Ξ²) Ξ³ β₯ 4 / 9... | import Mathlib
open Nat Topology Filter
| Let $0 \leq p \leq 1$ and $0 \leq r \leq 1$ and consider the identities
\begin{enumerate}
\item[(a)] $(px+(1-p)y)^2=Ax^2+Bxy+Cy^2$,
\item[(b)] $(px+(1-p)y)(rx+(1-r)y)=\alpha x^2+\beta xy+\gamma y^2$.
\end{enumerate}
Show that (with respect to $p$ and $r$)
\begin{enumerate}
\item[(a)] $\max\{A,B,C\} \geq 4/9$,
\item[(b)... | None. | [
"algebra"
] | test | putnam_1967_b2 | 94da212413b0e74e | p r A B C Ξ± Ξ² Ξ³ : β
prbound : 0 β€ p β§ p β€ 1 β§ 0 β€ r β§ r β€ 1
id1 : β (x y : β), (p * x + (1 - p) * y) ^ 2 = A * x ^ 2 + B * x * y + C * y ^ 2
id2 : β (x y : β), (p * x + (1 - p) * y) * (r * x + (1 - r) * y) = Ξ± * x ^ 2 + Ξ² * x * y + Ξ³ * y ^ 2
β’ max (max A B) C β₯ 4 / 9 β§ max (max Ξ± Ξ²) Ξ³ β₯ 4 / 9 |
putnam_1967_b3 | theorem putnam_1967_b3
(f g : β β β)
(fgcont : Continuous f β§ Continuous g)
(fgperiod : Function.Periodic f 1 β§ Function.Periodic g 1)
: Tendsto (fun n : β€ => β« x in Set.Ioo 0 1, f x * g (n * x)) atTop (π ((β« x in Set.Ioo 0 1, f x) * (β« x in Set.Ioo 0 1, g x))) := by | import Mathlib
open Nat Topology Filter
| If $f$ and $g$ are continuous and periodic functions with period $1$ on the real line, then $\lim_{n \to \infty} \int_0^1 f(x)g(nx)\,dx=(\int_0^1 f(x)\,dx)(\int_0^1 g(x)\,dx)$. | None. | [
"analysis"
] | test | putnam_1967_b3 | 45b049396821eb2b | f g : β β β
fgcont : Continuous f β§ Continuous g
fgperiod : Function.Periodic f 1 β§ Function.Periodic g 1
β’ Tendsto (fun n => β« (x : β) in Set.Ioo 0 1, f x * g (βn * x)) atTop
(π ((β« (x : β) in Set.Ioo 0 1, f x) * β« (x : β) in Set.Ioo 0 1, g x)) |
putnam_1967_b4 | theorem putnam_1967_b4
(n : β)
(lockers : β β Set.Icc 1 n β Bool)
(npos : n β₯ 1)
(hlockers0 : β i : Set.Icc 1 n, lockers 0 i = false)
(hlockersk : β k β Set.Icc 1 n, β i : Set.Icc 1 n, lockers k i = if k β£ i then !(lockers (k - 1) i) else (lockers (k - 1) i))
: β i : Set.Icc 1 n, lockers n i β (β j : β€, j ^ 2 = i) := b... | import Mathlib
open Nat Topology Filter
| A certain locker room contains $n$ lockers numbered $1,2,3,\cdots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1,T_2,\cdots,T_n$ whereby with the operation $T_k$, $1 \leq k \leq n$, the condition of being locked or unlocked is changed for all those lockers and only those lockers w... | None. | [
"number_theory"
] | test | putnam_1967_b4 | 738a6a4b2bf0a4b4 | n : β
lockers : β β β(Set.Icc 1 n) β Bool
npos : n β₯ 1
hlockers0 : β (i : β(Set.Icc 1 n)), lockers 0 i = false
hlockersk :
β k β Set.Icc 1 n, β (i : β(Set.Icc 1 n)), lockers k i = if k β£ βi then !lockers (k - 1) i else lockers (k - 1) i
β’ β (i : β(Set.Icc 1 n)), lockers n i = true β β j, j ^ 2 = ββi |
putnam_1967_b5 | theorem putnam_1967_b5
(n : β)
(hn : n > 0)
: (1 : β)/2 = β i β Finset.range n, (Nat.choose (n + i - 1) i) * (2 : β)^(-(n : β€) - i) := by | import Mathlib
open Nat Topology Filter
| For any positive integer $n$, prove that the sum of the first $n$ terms of the bimonial expansion of $(2 - 1)^{-n}$ (starting with the maximal exponent of $2$) is $\frac{1}{2}.$ | None. | [
"algebra"
] | test | putnam_1967_b5 | 40cb561811f3b91a | n : β
hn : n > 0
β’ 1 / 2 = β i β Finset.range n, β((n + i - 1).choose i) * 2 ^ (-βn - βi) |
putnam_1967_b6 | theorem putnam_1967_b6
(f : β β β β β)
(fdiff : (β y : β, Differentiable β (fun x : β => f x y)) β§ (β x : β, Differentiable β (fun y : β => f x y)))
(fcont : ContinuousOn (fun p : β Γ β => f p.1 p.2) {p : β Γ β | p.1 ^ 2 + p.2 ^ 2 β€ 1})
(fbound : β x y : β, (x ^ 2 + y ^ 2 β€ 1) β |f x y| β€ 1)
: β x0 y0 : β, (x0 ^ 2 + y0... | import Mathlib
open Nat Topology Filter
| Let $f$ be a real-valued function having partial derivatives and which is defined for $x^2+y^2 \leq 1$ and is such that $|f(x,y)| \leq 1$. Show that there exists a point $(x_0,y_0)$ in the interior of the unit circle such that $\left(\frac{\partial f}{\partial x} (x_0,y_0)\right)^2+\left(\frac{\partial f}{\partial y} (... | None. | [
"analysis"
] | test | putnam_1967_b6 | 7646d460aab1c793 | f : β β β β β
fdiff : (β (y : β), Differentiable β fun x => f x y) β§ β (x : β), Differentiable β fun y => f x y
fcont : ContinuousOn (fun p => f p.1 p.2) {p | p.1 ^ 2 + p.2 ^ 2 β€ 1}
fbound : β (x y : β), x ^ 2 + y ^ 2 β€ 1 β |f x y| β€ 1
β’ β x0 y0, x0 ^ 2 + y0 ^ 2 < 1 β§ deriv (fun x => f x y0) x0 ^ 2 + deriv (fun y => f ... |
putnam_1968_a1 | theorem putnam_1968_a1
: 22/7 - Real.pi = β« x in (0)..1, x^4 * (1 - x)^4 / (1 + x^2) := by | import Mathlib
| Prove that $$\frac{22}{7} - \pi = \int_{0}^{1} \frac{x^4(1 - x)^4}{1 + x^2} dx$$. | None. | [
"analysis"
] | test | putnam_1968_a1 | 6ddcc2dea11c665a | β’ 22 / 7 - Real.pi = β« (x : β) in 0..1, x ^ 4 * (1 - x) ^ 4 / (1 + x ^ 2) |
putnam_1968_a2 | theorem putnam_1968_a2
(a b c d e f : β€)
(Ξ΅ : β)
(hne : a * d β b * c)
(hΞ΅ : Ξ΅ > 0)
: β r s : β, (|r * a + s * b - e| : β) β Set.Ioo 0 Ξ΅ β§ (|r * c + s * d - f| : β) β Set.Ioo 0 Ξ΅ := by | import Mathlib
| For all integers $a$, $b$, $c$, $d$, $e$, and $f$ such that $ad \neq bc$ and any real number $\epsilon > 0$, prove that there exist rational numbers $r$ and $s$ such that $$0 < |ra + sb - e| < \varepsilon$$ and $$0 < |rc + sd - f| < \varepsilon.$$ | None. | [
"analysis"
] | test | putnam_1968_a2 | 6a3d36f5d5f213d9 | a b c d e f : β€
Ξ΅ : β
hne : a * d β b * c
hΞ΅ : Ξ΅ > 0
β’ β r s, |βr * βa + βs * βb - βe| β Set.Ioo 0 Ξ΅ β§ |βr * βc + βs * βd - βf| β Set.Ioo 0 Ξ΅ |
putnam_1968_a3 | theorem putnam_1968_a3
(Ξ± : Type*) [Finite Ξ±] :
β (n : β) (s : Fin (2 ^ n) β Set Ξ±),
s 0 = β
β§
(β t, β! i, s i = t) β§
(β i, i.1 + 1 < 2 ^ n β (s i β s (i + 1)).ncard = 1) := by | import Mathlib
open Finset symmDiff
| Let $S$ be a finite set. Prove that there exists a list of subsets of $S$ such that
\begin{enumerate}
\item The first element of the list is the empty set,
\item Each subset of $S$ occurs exactly once in the list, and
\item Each successive element in the list is formed by adding or removing one element from the previou... | None. | [
"combinatorics"
] | test | putnam_1968_a3 | 81aed69d33446958 | Ξ± : Type u_1
instβ : Finite Ξ±
β’ β n s, s 0 = β
β§ (β (t : Set Ξ±), β! i, s i = t) β§ β (i : Fin (2 ^ n)), βi + 1 < 2 ^ n β (s i β s (i + 1)).ncard = 1 |
putnam_1968_a4 | theorem putnam_1968_a4
(n : β)
(S : Fin n β (EuclideanSpace β (Fin 3)))
(hS : β i : Fin n, dist 0 (S i) = 1)
: β i : Fin n, β j : Fin n, (if i < j then (dist (S i) (S j))^2 else (0 : β)) β€ n^2 := by | import Mathlib
open Finset
| Prove that the sum of the squares of the distances between any $n$ points on the unit sphere $\{(x, y, z) \mid x^2 + y^2 + z^2 = 1\}$ is at most $n^2$. | None. | [
"geometry",
"algebra"
] | test | putnam_1968_a4 | d93c0f59066c4c1a | n : β
S : Fin n β EuclideanSpace β (Fin 3)
hS : β (i : Fin n), dist 0 (S i) = 1
β’ (β i, β j, if i < j then dist (S i) (S j) ^ 2 else 0) β€ βn ^ 2 |
putnam_1968_a5 | abbrev putnam_1968_a5_solution : β := sorry
theorem putnam_1968_a5
(V : Set β[X])
(V_def : V = {P : β[X] | P.degree = 2 β§ β x β Set.Icc 0 1, |P.eval x| β€ 1})
: sSup {|(derivative P).eval 0| | P β V} = putnam_1968_a5_solution := by | import Mathlib
open Finset Polynomial
| Let $V$ be the set of all quadratic polynomials with real coefficients such that $|P(x)| \le 1$ for all $x \in [0, 1]$. Find the supremum of $|P'(0)|$ across all $P \in V$. | The supremum is $8$. | [
"algebra"
] | test | putnam_1968_a5 | 9266e30da7ac100e | V : Set β[X]
V_def : V = {P | P.degree = 2 β§ β x β Set.Icc 0 1, |eval x P| β€ 1}
β’ sSup {x | β P β V, |eval 0 (derivative P)| = x} = putnam_1968_a5_solution |
putnam_1968_a6 | abbrev putnam_1968_a6_solution : Set β[X] := sorry
theorem putnam_1968_a6
: {P : β[X] | P.natDegree β₯ 1 β§ (β k β Set.Icc 0 P.natDegree, P.coeff k = 1 β¨ P.coeff k = -1) β§
β z : β, P.eval z = 0 β β r : β, r = z} = putnam_1968_a6_solution := by | import Mathlib
open Finset Polynomial
| Find all polynomials of the form $\sum_{0}^{n} a_{i} x^{n-i}$ with $n \ge 1$ and $a_i = \pm 1$ for all $0 \le i \le n$ whose roots are all real. | The set of such polynomials is $$\{\pm (x - 1), \pm (x + 1), \pm (x^2 + x - 1), \pm (x^2 - x - 1), \pm (x^3 + x^2 - x - 1), \pm (x^3 - x^2 - x + 1)\}.$$ | [
"algebra"
] | test | putnam_1968_a6 | 9f9fba64846b5784 | β’ {P |
P.natDegree β₯ 1 β§
(β k β Set.Icc 0 P.natDegree, P.coeff k = 1 β¨ P.coeff k = -1) β§ β (z : β), eval z P = 0 β β r, βr = z} =
putnam_1968_a6_solution |
putnam_1968_b1 | abbrev putnam_1968_b1_solution : β β β β β β β := sorry
theorem putnam_1968_b1
{Ξ© : Type*}
[MeasureSpace Ξ©]
[IsProbabilityMeasure (β : Measure Ξ©)]
(X Y : Ξ© β β€)
(hX : Measurable X)
(hY : Measurable Y)
(hX' : Set.Finite (X '' Set.univ))
(hY' : Set.Finite (Y '' Set.univ))
(k : β€) :
... | import Mathlib
open MeasureTheory
open scoped ProbabilityTheory
| The random variables $X, Y$ can each take a finite number of integer values. They are not necessarily independent. Express $\mathrm{prob}(\min(X, Y) = k)$ in terms of $p_1 = \mathrm{prob}(X = k)$, $p_2 = \mathrm{prob}(Y = k)$ and $p_3 = \mathrm{prob(max(X, Y) = k)$. | $\mathrm{prob}(\min(X, Y) = k) = p_1 + p_2 - p_3.$ | [
"probability"
] | test | putnam_1968_b1 | 0a2dcc7e6007635a | Ξ© : Type u_1
instβΒΉ : MeasureSpace Ξ©
instβ : IsProbabilityMeasure β
X Y : Ξ© β β€
hX : Measurable X
hY : Measurable Y
hX' : (X '' Set.univ).Finite
hY' : (Y '' Set.univ).Finite
k : β€
β’ (β {Ο | min (X Ο) (Y Ο) = k}).toReal =
putnam_1968_b1_solution (β (X β»ΒΉ' {k})).toReal (β (Y β»ΒΉ' {k})).toReal (β {Ο | max (X Ο) (Y Ο) =... |
putnam_1968_b2 | theorem putnam_1968_b2
{G : Type*}
[Group G]
(hG : Finite G)
(A : Set G)
(hA : A.ncard > (Nat.card G : β)/2)
: β g : G, β x β A, β y β A, g = x * y := by | import Mathlib
open Finset Polynomial
| Let $G$ be a finite group (with a multiplicative operation), and $A$ be a subset of $G$ that contains more than half of $G$'s elements. Prove that every element of $G$ can be expressed as the product of two elements of $A$. | None. | [
"abstract_algebra"
] | test | putnam_1968_b2 | 404623094360cbd9 | G : Type u_1
instβ : Group G
hG : Finite G
A : Set G
hA : βA.ncard > β(Nat.card G) / 2
β’ β (g : G), β x β A, β y β A, g = x * y |
putnam_1968_b4 | theorem putnam_1968_b4
(f : β β β)
(hf : Continuous f β§ β r : β, Tendsto (fun y => β« x in ball 0 y, f x) atTop (π r))
: β r : β, Tendsto (fun y => β« x in (ball 0 y \ ball 0 (1 / y)), f (x - 1/x)) atTop (π r) β§ Tendsto (fun y => β« x in ball 0 y, f x) atTop (π r) := by | import Mathlib
open Finset Polynomial Topology Filter Metric
| Suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous on $(-\infty, \infty)$ and that $\int_{-\infty}^{\infty} f(x) dx$ exists. Prove that $$\int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) dx = \int_{-\infty}^{\infty} f(x) dx.$$ | None. | [
"analysis"
] | test | putnam_1968_b4 | bda68af9107e4229 | f : β β β
hf : Continuous f β§ β r, Tendsto (fun y => β« (x : β) in ball 0 y, f x) atTop (π r)
β’ β r,
Tendsto (fun y => β« (x : β) in ball 0 y \ ball 0 (1 / y), f (x - 1 / x)) atTop (π r) β§
Tendsto (fun y => β« (x : β) in ball 0 y, f x) atTop (π r) |
putnam_1968_b5 | abbrev putnam_1968_b5_solution : β β β := sorry
theorem putnam_1968_b5
(p : β)
(hp : Prime p)
: {M : Matrix (Fin 2) (Fin 2) (ZMod p) | M 0 0 + M 1 1 = 1 β§ M 0 0 * M 1 1 - M 0 1 * M 1 0 = 0}.ncard = putnam_1968_b5_solution p := by | import Mathlib
open Finset Polynomial Topology Filter Metric
| Let $p$ be a prime number. Find the number of distinct $2 \times 2$ matrices $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ such that $a, b, c, d \in \{0, 1, ..., p - 1\}$, $a + d \equiv 1 \pmod p$, and $ad - bc \equiv 0 \pmod p$. | There are $p^2 + p$ such matrices. | [
"linear_algebra",
"number_theory",
"combinatorics"
] | test | putnam_1968_b5 | fa05c57b10a01374 | p : β
hp : Prime p
β’ {M | M 0 0 + M 1 1 = 1 β§ M 0 0 * M 1 1 - M 0 1 * M 1 0 = 0}.ncard = putnam_1968_b5_solution p |
putnam_1968_b6 | theorem putnam_1968_b6
: Β¬β K : β β Set β, (β n : β, IsCompact (K n)) β§ (β S : Set β, IsCompact S β β n : β, S β K n) := by | import Mathlib
open Finset Polynomial Topology Filter Metric
| Prove that no sequence $\{K_n\}_{n=0}^{\infty}$ of compact (closed and bounded) sets of rational numbers has the property that every compact set of rational numbers is contained by at least one $K_n$. | None. | [
"analysis"
] | test | putnam_1968_b6 | 4726d98cae689dd9 | β’ Β¬β K, (β (n : β), IsCompact (K n)) β§ β (S : Set β), IsCompact S β β n, S β K n |
putnam_1969_a1 | abbrev putnam_1969_a1_solution : Set (Set β) := sorry
theorem putnam_1969_a1
: {{z : β | β x : Fin 2 β β, MvPolynomial.eval x f = z} | f : MvPolynomial (Fin 2) β} = putnam_1969_a1_solution := by | import Mathlib
open Matrix Filter Topology Set Nat
| What are the possible ranges (across all real inputs $x$ and $y$) of a polynomial $f(x, y)$ with real coefficients? | Show that the possibles ranges are a single point, any half-open or half-closed semi-infinite interval, or all real numbers. | [
"algebra",
"set_theory"
] | test | putnam_1969_a1 | c6fcdc1f2d88493c | β’ {x | β f, {z | β x, (MvPolynomial.eval x) f = z} = x} = putnam_1969_a1_solution |
putnam_1969_a2 | theorem putnam_1969_a2
(D : (n : β) β Matrix (Fin n) (Fin n) β)
(hD : D = fun (n : β) => Ξ» (i : Fin n) (j : Fin n) => |(i : β) - (j : β)| )
: β n, n β₯ 2 β (D n).det = (-1)^((n : β€)-1) * ((n : β€)-1) * 2^((n : β€)-2) := by | import Mathlib
open Matrix Filter Topology Set Nat
| Let $D_n$ be the determinant of the $n$ by $n$ matrix whose value in the $i$th row and $j$th column is $|i-j|$. Show that $D_n = (-1)^{n-1} * (n-1) * (2^{n-2}).$ | None. | [
"linear_algebra"
] | test | putnam_1969_a2 | 0ebd1c8781e2547c | D : (n : β) β Matrix (Fin n) (Fin n) β
hD : D = fun n i j => |ββi - ββj|
β’ β n β₯ 2, (D n).det = (-1) ^ (βn - 1) * (ββn - 1) * 2 ^ (βn - 2) |
putnam_1969_a4 | theorem putnam_1969_a4
: Tendsto (fun n => β i β Finset.Icc (1 : β€) n, (-1)^(i+1)*(i : β)^(-i)) atTop (π (β« x in Ioo (0 : β) 1, x^x)) := by | import Mathlib
open Matrix Filter Topology Set Nat
| Show that $\int_0^1 x^x dx = \sum_{n=1}^{\infty} (-1)^{n+1}n^{-n}$. | None. | [
"analysis"
] | test | putnam_1969_a4 | 2b05d0fdbf708bed | β’ Tendsto (fun n => β i β Finset.Icc 1 n, (-1) ^ (i + 1) * βi ^ (-i)) atTop (π (β« (x : β) in Ioo 0 1, x ^ x)) |
putnam_1969_a5 | theorem putnam_1969_a5
(x0 y0 t : β)
(ht : 0 < t) :
x0 = y0 β β x y u : β β β,
Differentiable β x β§
Differentiable β y β§
Continuous u β§
deriv x = - 2 β’ y + u β§
deriv y = - 2 β’ x + u β§
x 0 = x0 β§
y 0 = y0 β§
x t = 0 β§
y t = 0 := by | import Mathlib
open Matrix Filter Topology Set Nat
| Consider the system of differential equations $$\frac{dx}{dt} = -2y + u(t), \frac{dy}{dt} = -2x + u(t)$$ for some continuous function $u(t)$. Prove that, if $x(0) \ne y(0)$, the solution will never pass through $(0, 0)$ regardless of the choice of $u(t)$, and if $x(0) = y(0)$, a suitable $u(t)$ can be chosen for any $T... | None. | [
"analysis"
] | test | putnam_1969_a5 | cd64d40ed4d49cee | x0 y0 t : β
ht : 0 < t
β’ x0 = y0 β
β x y u,
Differentiable β x β§
Differentiable β y β§
Continuous u β§ deriv x = -2 β’ y + u β§ deriv y = -2 β’ x + u β§ x 0 = x0 β§ y 0 = y0 β§ x t = 0 β§ y t = 0 |
putnam_1969_a6 | theorem putnam_1969_a6
(x : β β β)
(y : β β β)
(hy1 : β n β₯ 2, y n = x (n-1) + 2 * (x n))
(hy2 : β c : β, Tendsto y atTop (π c))
: β C : β, Tendsto x atTop (π C) := by | import Mathlib
open Matrix Filter Topology Set Nat
| Let $(x_n)$ be a sequence, and let $y_n = x_{n-1} + 2*x_n$ for $n \geq 2$. Suppose that $(y_n)$ converges, then prove that $(x_n)$ converges. | None. | [
"analysis"
] | test | putnam_1969_a6 | 4bba445300272591 | x y : β β β
hy1 : β n β₯ 2, y n = x (n - 1) + 2 * x n
hy2 : β c, Tendsto y atTop (π c)
β’ β C, Tendsto x atTop (π C) |
putnam_1969_b1 | theorem putnam_1969_b1
(n : β)
(hnpos : n > 0)
(hn : 24 β£ n + 1)
: 24 β£ β d β divisors n, d := by | import Mathlib
open Matrix Filter Topology Set Nat
| Let $n$ be a positive integer such that $n+1$ is divisible by $24$. Prove that the sum of all the divisors of $n$ is divisible by $24$. | None. | [
"number_theory"
] | test | putnam_1969_b1 | 72d3bacd75d0aebf | n : β
hnpos : n > 0
hn : 24 β£ n + 1
β’ 24 β£ β d β n.divisors, d |
putnam_1969_b2 | abbrev putnam_1969_b2_solution : Prop := sorry
theorem putnam_1969_b2
(P : β β Prop)
(P_def : β n, P n β β (G : Type) [Group G] [Finite G],
β H : Fin n β Subgroup G, (β i, H i < β€) β β i, (H i : Set G) < β€) :
P 2 β§ (P 3 β putnam_1969_b2_solution) := by | import Mathlib
open Matrix Filter Topology Set Nat
| Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'? | Show that the statement is no longer true if 'two' is replaced by 'three'. | [
"abstract_algebra"
] | test | putnam_1969_b2 | 8790aafb5eca45c1 | P : β β Prop
P_def :
β (n : β),
P n β β (G : Type) [inst : Group G] [Finite G] (H : Fin n β Subgroup G), (β (i : Fin n), H i < β€) β β i, β(H i) < β€
β’ P 2 β§ (P 3 β putnam_1969_b2_solution) |
putnam_1969_b3 | theorem putnam_1969_b3
(T : β β β)
(hT1 : β n : β, n β₯ 1 β (T n) * (T (n + 1)) = n)
(hT2 : Tendsto (fun n => (T n)/(T (n + 1))) atTop (π 1))
: Real.pi * (T 1)^2 = 2 := by | import Mathlib
open Matrix Filter Topology Set Nat
| Suppose $T$ is a sequence which satisfies $T_n * T_{n+1} = n$ whenever $n \geq 1$, and also $\lim_{n \to \infty} \frac{T_n}{T_{n+1}} = 1. Show that $\pi * T_1^2 = 2$. | None. | [
"analysis"
] | test | putnam_1969_b3 | 4b43116facb260ce | T : β β β
hT1 : β n β₯ 1, T n * T (n + 1) = βn
hT2 : Tendsto (fun n => T n / T (n + 1)) atTop (π 1)
β’ Real.pi * T 1 ^ 2 = 2 |
putnam_1969_b4 | theorem putnam_1969_b4
(Ξ : β β EuclideanSpace β (Fin 2))
--Note: the problem doesn't say what regularity conditions we should impose on `Ξ` - hopefully continuity is enough.
(Ξ_cts : ContinuousOn Ξ (Set.Icc 0 1))
(hΞ : eVariationOn Ξ (Set.Icc 0 1) = 1) :
letI : Module.Oriented β (EuclideanSpace β (... | import Mathlib
open scoped Real EuclideanGeometry
| $Ξ$ is a plane curve of length 1. Show that we can find a closed rectangle of area 1/4 which covers $Ξ$. | None. | [
"geometry"
] | test | putnam_1969_b4 | fe72831b6e1fe2b5 | Ξ : β β EuclideanSpace β (Fin 2)
Ξ_cts : ContinuousOn Ξ (Set.Icc 0 1)
hΞ : eVariationOn Ξ (Set.Icc 0 1) = 1
β’ β a b c d,
Ξ '' Set.Icc 0 1 β (convexHull β) {a, b, c, d} β§
dist a b * dist b c = 1 / 4 β§ β‘ a b c = β(Ο / 2) β§ β‘ b c d = β(Ο / 2) β§ β‘ c d a = β(Ο / 2) β§ β‘ d a b = β(Ο / 2) |
putnam_1969_b5 | theorem putnam_1969_b5
(a : β β β)
(ha : StrictMono a β§ (β x : β, a x > 0))
(hinvasum : β C : β, Tendsto (fun n => β i : Fin n, 1/(a i)) atTop (π C))
(k : β β β)
(hk : k = fun x => {n | a n β€ x}.ncard)
: Tendsto (fun t => (k t)/t) atTop (π 0) := by | import Mathlib
open Matrix Filter Topology Set Nat
| Let $a_1 < a_2 < a_3 < \dots$ be an increasing sequence of positive integers. Assume that the sequences $\sum_{i = 1}^{\infty} 1/(a n)$ is convergent. For any number $x$, let $k(x)$ be the number of $a_n$'s which do not exceed $x$. Show that $\lim_{x \to \infty} k(x)/x = 0$. | None. | [
"analysis"
] | test | putnam_1969_b5 | 7c376968b153537b | a : β β β
ha : StrictMono a β§ β (x : β), a x > 0
hinvasum : β C, Tendsto (fun n => β i, 1 / a βi) atTop (π C)
k : β β β
hk : k = fun x => {n | a n β€ x}.ncard
β’ Tendsto (fun t => β(k t) / t) atTop (π 0) |
putnam_1969_b6 | theorem putnam_1969_b6
(A : Matrix (Fin 3) (Fin 2) β)
(B : Matrix (Fin 2) (Fin 3) β)
(p : Fin 3 β Fin 3 β β)
(hp : p 0 0 = 8 β§ p 0 1 = 2 β§ p 0 2 = -2 β§
p 1 0 = 2 β§ p 1 1 = 5 β§ p 1 2 = 4 β§
p 2 0 = -2 β§ p 2 1 = 4 β§ p 2 2 = 5)
(hAB : A * B = Matrix.of p)
: B * A = 9 * (1 : Matrix (Fin 2) (Fin 2) β) := by | import Mathlib
open Matrix Filter Topology Set Nat
| Let $A$ be a $3 \times 2$ matrix and $B$ be a $2 \times 3$ matrix such that $$AB =
\begin{pmatrix}
8 & 2 & -2 \\
2 & 5 & 4 \\
-2 & 4 & 5
\end{pmatrix}.
$$ Prove that $$BA =
\begin{pmatrix}
9 & 0 \\
0 & 9
\end{pmatrix}.$$ | None. | [
"linear_algebra"
] | test | putnam_1969_b6 | c6d5f5a51fdb6d73 | A : Matrix (Fin 3) (Fin 2) β
B : Matrix (Fin 2) (Fin 3) β
p : Fin 3 β Fin 3 β β
hp : p 0 0 = 8 β§ p 0 1 = 2 β§ p 0 2 = -2 β§ p 1 0 = 2 β§ p 1 1 = 5 β§ p 1 2 = 4 β§ p 2 0 = -2 β§ p 2 1 = 4 β§ p 2 2 = 5
hAB : A * B = of p
β’ B * A = 9 * 1 |
putnam_1970_a1 | theorem putnam_1970_a1
(a b : β)
(ha : a > 0)
(hb : b > 0)
(f : β β β)
(f_def : f = fun x : β => Real.exp (a*x) * Real.cos (b*x))
(p : β β β)
(hp : β c : β, c > 0 β§ β x β ball 0 c, β' n : β, (p n)*x^n = f x)
(S : Set β)
(S_def : S = {n : β | p n = 0})
: S = β
β¨ Β¬Finite S := by | import Mathlib
open Metric Set EuclideanGeometry
| Prove that, for all $a > 0$ and $b > 0$, the power series of $e^{ax} \cos (bx)$ with respect to $x$ has either zero or infinitely many zero coefficients. | None. | [
"analysis"
] | test | putnam_1970_a1 | d484dbd042a0f7e9 | a b : β
ha : a > 0
hb : b > 0
f : β β β
f_def : f = fun x => Real.exp (a * x) * Real.cos (b * x)
p : β β β
hp : β c > 0, β x β ball 0 c, β' (n : β), p n * x ^ n = f x
S : Set β
S_def : S = {n | p n = 0}
β’ S = β
β¨ Β¬Finite βS |
putnam_1970_a2 | theorem putnam_1970_a2
(A B C D E F G : β)
(hle : B^2 - 4*A*C < 0)
: β Ξ΄ > 0, Β¬β x y : β, x^2 + y^2 β Set.Ioo 0 (Ξ΄^2) β§
A*x^2 + B*x*y + C*y^2 + D*x^3 + E*x^2*y + F*x*y^2 + G*y^3 = 0 := by | import Mathlib
open Metric Set EuclideanGeometry
| Let $A$, $B$, $C$, $D$, $E$, $F$, and $G$ be real numbers satisfying $B^2 - 4AC < 0$. Prove that there exists some $\delta > 0$ such that no points $(x, y)$ in the punctured disk $0 < x^2 + y^2 < \delta$ satisfy $$Ax^2 + Bxy + Cy^2 + Dx^3 + Ex^2y + Fxy^2 + Gy^3 = 0.$$ | None. | [
"analysis",
"algebra"
] | test | putnam_1970_a2 | b80d49e6f60022cf | A B C D E F G : β
hle : B ^ 2 - 4 * A * C < 0
β’ β Ξ΄ > 0,
Β¬β x y,
x ^ 2 + y ^ 2 β Ioo 0 (Ξ΄ ^ 2) β§
A * x ^ 2 + B * x * y + C * y ^ 2 + D * x ^ 3 + E * x ^ 2 * y + F * x * y ^ 2 + G * y ^ 3 = 0 |
putnam_1970_a3 | abbrev putnam_1970_a3_solution : β Γ β := sorry
theorem putnam_1970_a3
(L : β β β)
(hL : β n : β, L n β€ (Nat.digits 10 n).length β§
(β k : β, k < L n β (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) β§
(L n β (Nat.digits 10 n).length β (Nat.digits 10 n)[L n]! β (Nat.digits 10 n)[0]!))
: (β n : β, (Nat.digits 10 (n^2))[0]... | import Mathlib
open Metric Set EuclideanGeometry
| Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length. | The maximum attainable length is $3$; the smallest such square is $38^2 = 1444$. | [
"number_theory"
] | test | putnam_1970_a3 | 4b36c8279db48e02 | L : β β β
hL :
β (n : β),
L n β€ (Nat.digits 10 n).length β§
(β k < L n, (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) β§
(L n β (Nat.digits 10 n).length β (Nat.digits 10 n)[L n]! β (Nat.digits 10 n)[0]!)
β’ (β n, (Nat.digits 10 (n ^ 2))[0]! β 0 β§ L (n ^ 2) = putnam_1970_a3_solution.1) β§
(β (n : β)... |
putnam_1970_a4 | theorem putnam_1970_a4
(x : β β β)
(hxlim : Tendsto (fun n => x (n+2) - x n) atTop (π 0))
: Tendsto (fun n => (x (n+1) - x (n))/(n+1)) atTop (π 0) := by | import Mathlib
open Metric Set EuclideanGeometry Filter Topology
| Suppose $(x_n)$ is a sequence such that $\lim_{n \to \infty} (x_n - x_{n-2} = 0$. Prove that $\lim_{n \to \infty} \frac{x_n - x_{n-1}}{n} = 0$. | None. | [
"analysis"
] | test | putnam_1970_a4 | bca3bc7d51027043 | x : β β β
hxlim : Tendsto (fun n => x (n + 2) - x n) atTop (π 0)
β’ Tendsto (fun n => (x (n + 1) - x n) / (βn + 1)) atTop (π 0) |
putnam_1970_b1 | noncomputable abbrev putnam_1970_b1_solution : β := sorry
theorem putnam_1970_b1
: Tendsto (fun n => 1/(n^4) * β i β Finset.Icc (1 : β€) (2*n), ((n^2 + i^2) : β)^((1 : β)/n)) atTop (π putnam_1970_b1_solution) := by | import Mathlib
open Metric Set EuclideanGeometry Filter Topology
| Evaluate the infinite product $\lim_{n \to \infty} \frac{1}{n^4} \prod_{i = 1}^{2n} (n^2 + i^2)^{1/n}$. | Show that the solution is $e^{2 \log(5) - 4 + 2 arctan(2)}$. | [
"analysis"
] | test | putnam_1970_b1 | ec2491a10c2e8862 | β’ Tendsto (fun n => 1 / βn ^ 4 * β i β Finset.Icc 1 (2 * n), (βn ^ 2 + βi ^ 2) ^ (1 / βn)) atTop
(π putnam_1970_b1_solution) |
putnam_1970_b2 | theorem putnam_1970_b2
(T : β)
(H : Polynomial β)
(hT : T > 0)
(hH : H.degree β€ 3)
: (H.eval (-T / Real.sqrt 3) + H.eval (T / Real.sqrt 3))/2 = β¨ t in Set.Icc (-T) T, H.eval t := by | import Mathlib
open Metric Set EuclideanGeometry Filter Topology
| Let $H$ be a polynomial of degree at most $3$ and $T$ be a positive real number. Show that the average value of $H(t)$ over the interval $[-T, T]$ equals the average of $H\left(-\frac{T}{\sqrt{3}}\right)$ and $H\left(\frac{T}{\sqrt{3}}\right)$. | None. | [
"analysis",
"algebra"
] | test | putnam_1970_b2 | 825b1a53c8de5237 | T : β
H : Polynomial β
hT : T > 0
hH : H.degree β€ 3
β’ (Polynomial.eval (-T / β3) H + Polynomial.eval (T / β3) H) / 2 = β¨ (t : β) in Icc (-T) T, Polynomial.eval t H |
putnam_1970_b3 | theorem putnam_1970_b3
(S : Set (β Γ β))
(a b : β)
(hab : a < b)
(hS : β s β S, s.1 β Ioo a b)
(hSclosed : IsClosed S)
: IsClosed {y | β x : β, β¨x,yβ© β S} := by | import Mathlib
open Metric Set EuclideanGeometry Filter Topology
| A closed subset $S$ of $\mathbb{R}^2$ lies in $a < x < b$. Show that its projection on the y-axis is closed. | None. | [
"analysis"
] | test | putnam_1970_b3 | 9eb22dddc14de6b8 | S : Set (β Γ β)
a b : β
hab : a < b
hS : β s β S, s.1 β Ioo a b
hSclosed : IsClosed S
β’ IsClosed {y | β x, (x, y) β S} |
putnam_1970_b4 | theorem putnam_1970_b4
(x : β β β)
(hdiff : DifferentiableOn β x (Set.Icc 0 1) β§ DifferentiableOn β (deriv x) (Set.Icc 0 1))
(hx : x 1 - x 0 = 1)
(hv : deriv x 0 = 0 β§ deriv x 1 = 0)
(hs : β t β Set.Ioo 0 1, |deriv x t| β€ 3/2)
: β t β Set.Icc 0 1, |(deriv (deriv x)) t| β₯ 9/2 := by | import Mathlib
open Metric Set EuclideanGeometry Filter Topology
| Let $x : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function satisfying $x(1) - x(0) = 1$, $x'(0) = x'(1) = 0$, and $|x'(t)| \le \frac{3}{2}$ for all $t \in (0, 1)$. Prove that there exists some $t \in [0, 1]$ such that $|x''(t)| \ge \frac{9}{2}$. | None. | [
"analysis"
] | test | putnam_1970_b4 | b0af7bd1bcfff6ee | x : β β β
hdiff : DifferentiableOn β x (Icc 0 1) β§ DifferentiableOn β (deriv x) (Icc 0 1)
hx : x 1 - x 0 = 1
hv : deriv x 0 = 0 β§ deriv x 1 = 0
hs : β t β Ioo 0 1, |deriv x t| β€ 3 / 2
β’ β t β Icc 0 1, |deriv (deriv x) t| β₯ 9 / 2 |
putnam_1970_b5 | theorem putnam_1970_b5
(ramp : β€ β (β β β))
(ramp_def : ramp = fun (n : β€) => (fun (x : β) => if x β€ -n then (-n : β) else (if -n < x β§ x β€ n then x else (n : β))))
(F : β β β)
: Continuous F β (β n : β, Continuous ((ramp n) β F)) := by | import Mathlib
open Metric Set EuclideanGeometry Filter Topology
| Let $u_n$ denote the function $u_n(x) = -n$ if $x \leq -n$, $x$ if $-n < x \leq n$, and $n$ otherwise. Let $F$ be a function on the reals. Show that $F$ is continuous if and only if $u_n \circ F$ is continuous for all natural numbers $n$. | None. | [
"analysis"
] | test | putnam_1970_b5 | fbdd867e0bdeecfb | ramp : β€ β β β β
ramp_def : ramp = fun n x => if x β€ -βn then -βn else if -βn < x β§ x β€ βn then x else βn
F : β β β
β’ Continuous F β β (n : β), Continuous (ramp βn β F) |
putnam_1970_b6 | theorem putnam_1970_b6
(L : ZMod 4 β (EuclideanSpace β (Fin 2)))
(S : Set (EuclideanSpace β (Fin 2)))
(S_def : S = {L i | i : ZMod 4})
(hSquad : S.ncard = 4 β§ β s β S, s.ncard = 3 β Β¬ Collinear β s)
(hlens : dist (L 0) (L 1) > 0 β§ dist (L 1) (L 2) > 0 β§ dist (L 2) (L 3) > 0 β§ dist (L 3) (L 0) > 0)
(horder : β i : ZMod ... | import Mathlib
open Metric Set EuclideanGeometry Filter Topology
| Prove that if a quadrilateral with side lengths $a$, $b$, $c$, and $d$ and area $\sqrt{abcd}$ can be circumscribed to a circle (i.e., a circle can be inscribed in it), then it must be cyclic (i.e., it can be inscribed in a circle). | None. | [
"geometry"
] | test | putnam_1970_b6 | f114c792ad7c75e8 | L : ZMod 4 β EuclideanSpace β (Fin 2)
S : Set (EuclideanSpace β (Fin 2))
S_def : S = {x | β i, L i = x}
hSquad : S.ncard = 4 β§ β s β S, s.ncard = 3 β Β¬Collinear β s
hlens : dist (L 0) (L 1) > 0 β§ dist (L 1) (L 2) > 0 β§ dist (L 2) (L 3) > 0 β§ dist (L 3) (L 0) > 0
horder : β (i : ZMod 4), segment β (L i) (L (i + 1)) β© in... |
End of preview. Expand in Data Studio
putnambench-satp-v4.27
PutnamBench Lean 4 problems normalized for SATP evaluation, packaged for
Lean 4.27.0 with initial proof goal_state values.
Source
The single source is trishullab/PutnamBench@dc18909
(lean4/src/putnam_*.lean + informal/putnam.json); the upstream
lean4/lean-toolchain is leanprover/lean4:v4.27.0. All 672 Lean problems
are included and every row elaborates with zero errors under the pinned
toolchain.
| Rows | 672 (single train config, split column = "test") |
| Lean toolchain | leanprover/lean4:v4.27.0 |
| Mathlib | a3a10db0e9d66acbebf76c5e6a135066525ac900 |
| Goal-state renderer | Lean.Meta.ppGoal at proof entry, row-local environment |
Differences from upstream
| Field | Upstream | This repo |
|---|---|---|
formal_statement |
proof is sorry (term- or tactic-mode) |
trailing sorry proof removed, normalized to end := by; solution abbrev/def declarations kept verbatim (their := sorry answer placeholders included); docstrings and -- answer comments stripped |
header |
in-file | upstream-verbatim per-file import + open (+ open scoped) + set_option directives |
goal_state |
absent | pretty-printed Lean goal at proof entry (see provenance below) |
uuid |
absent | sha256(canonical(formal_statement))[:16] after normalization |
name |
absent | pinned to problem_name (e.g. putnam_1962_a1) |
problem_name, informal_statement, informal_solution, tags, and
split are passed through verbatim from informal/putnam.json.
Goal-state provenance
goal_state is the exact (β Lean.Meta.ppGoal (β getMainGoal)).pretty
rendering of the statement's initial goal, produced by elaborating
header + formal_statement under the pinned toolchain. Consumers that key
on bytes can rely on this being identical to what a Lean tactic observes at
proof entry in the same environment.
Schema
| Field | Type | Description |
|---|---|---|
name |
str |
Stable identifier = problem_name |
uuid |
str |
sha256(canonical(formal_statement))[:16] |
problem_name |
str |
e.g. putnam_1962_a1 |
formal_statement |
str |
Lean 4 declarations ending in := by (solution abbrevs + theorem) |
header |
str |
imports + opens prepended before the statement |
goal_state |
str |
Pretty-printed Lean goal at proof entry |
informal_statement |
str |
English problem statement |
informal_solution |
str |
Solution sketch or "None." |
tags |
list[str] |
Categories: algebra, analysis, geometry, etc. |
split |
str |
Always "test" |
Related SATP datasets (same Lean 4.27 environment):
NuminaMath-LEAN-satp-v4.27β main training setNuminaMath-LEAN-satp-gaps-v4.27β sketch-hole sub-goal augmentation setminif2f-satp-v4.27β miniF2F validation/test benchmark
Citation
@article{tsoukalas2024putnambench,
title={PutnamBench: Evaluating Neural Theorem-Provers on the Putnam Mathematical Competition},
author={George Tsoukalas and Jasper Lee and John Jennings and Jimmy Xin
and Michelle Ding and Michael Jennings and Amitayush Thakur
and Swarat Chaudhuri},
journal={arXiv preprint arXiv:2407.11214},
year={2024}
}
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