Dataset Preview
The full dataset viewer is not available (click to read why). Only showing a preview of the rows.
Not found.
Error code: ResponseNotFound
Need help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
system_prompt string | user_prompt string | ground_truth string |
|---|---|---|
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_a1`
**Tags:** geometry
**Problem:**
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. | 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.
-/
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 ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_a2`
**Tags:** analysis
**Problem:**
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 me... | open MeasureTheory Set
abbrev putnam_1962_a2_solution : Set (β β β) := sorry
-- {f | (β a c : β, 0 β€ a β§ f = (fun x : β β¦ a / (1 - c * x) ^ 2)) β¨ (β a c : β, 0 β€ a β§ 0 < c β§ f = (fun x : β β¦ if x < 1 / c then a / (1 - c * x) ^ 2 else 0)) β¨ (0 β€ f β§ β x : β, 0 < x β f x = 0) β¨ (β e : β, 0 < e β§ f 0 = 0 β§ 0 β€ f β§ β x β ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_a3`
**Tags:** geometry
**Problem:**
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 ... | 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 trian... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_a4`
**Tags:** analysis
**Problem:**
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. | /--
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.
-/
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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_a5`
**Tags:** algebra, combinatorics
**Problem:**
Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \]
**Answer:**
Show that the expression equals $n(n+1)2^{n-2}$. | abbrev putnam_1962_a5_solution : β β β := sorry
-- fun n : β => n * (n + 1) * 2^(n - 2)
/--
Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \]
-/
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 :=
sorry |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_a6`
**Tags:** algebra
**Problem:**
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... | /--
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.... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_b1`
**Tags:** algebra, combinatorics
**Problem:**
Let $x^{(n)} = x(x-1)\cdots(x-n+1)$ for $n$ a positive integer and let $x^{(0)} = 1.$ Prove that \[ (x+y)^{(n)} = \sum_{k=0}^n {n \choose k} x^{(k)} y^{(n-k)}. \] | /--
Let $x^{(n)} = x(x-1)\cdots(x-n+1)$ for $n$ a positive integer and let $x^{(0)} = 1.$ Prove that \[ (x+y)^{(n)} = \sum_{k=0}^n {n \choose k} x^{(k)} y^{(n-k)}. \]
-/
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))
... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_b2`
**Tags:** set_theory
**Problem:**
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 $a < b$. | open MeasureTheory
--Note: The original problem requires a function to be exhibited, but in the official archives the solution depends on an enumeration of the rationals, so we modify the problem to be an existential statement.
/--
Let $\mathbb{S}$ be the set of all subsets of the natural numbers. Prove the existence ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_b3`
**Tags:** analysis
**Problem:**
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... | 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.
-/
theorem putnam_1962_b3
(S : ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_b5`
**Tags:** algebra
**Problem:**
Prove that for every integer $n$ greater than 1: \[ \frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^n + \left(\frac{2}{n} \right)^n + \cdots + \left(\frac{n}{n} \right)^n < 2. \] | open MeasureTheory
/--
Prove that for every integer $n$ greater than 1: \[ \frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^n + \left(\frac{2}{n} \right)^n + \cdots + \left(\frac{n}{n} \right)^n < 2. \]
-/
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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1962_b6`
**Tags:** analysis
**Problem:**
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) =... | 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 $a$.
-/
theorem putn... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1963_a2`
**Tags:** number_theory, algebra
**Problem:**
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 $1$). ... | 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 $n$.
-/
th... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1963_a3`
**Tags:** analysis
**Problem:**
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) = \cd... | open Nat Set Topology Filter
noncomputable abbrev putnam_1963_a3_solution : (β β β) β β β β β β β β := sorry
-- fun (f : β β β) (n : β) (x : β) (t : β) β¦ (x - t) ^ (n - 1) * (f t) / ((n - 1)! * t ^ n)
/--
Find an integral formula (i.e., a function $z$ such that $y(x) = \int_{1}^{x} z(t) dt$) for the solution of the di... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1963_a4`
**Tags:** analysis
**Problem:**
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 sym... | 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{\l... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1963_a6`
**Tags:** geometry
**Problem:**
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 tha... | 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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1963_b1`
**Tags:** algebra
**Problem:**
For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?
**Answer:**
Show that $a=2$. | open Topology Filter Polynomial
abbrev putnam_1963_b1_solution : β€ := sorry
-- 2
/--
For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$?
-/
theorem putnam_1963_b1
: β a : β€, (X^2 - X + (C a)) β£ (X ^ 13 + X + (C 90)) β a = putnam_1963_b1_solution :=
sorry |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1963_b2`
**Tags:** analysis
**Problem:**
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 $P$?
**Answer:**
Show that $S$ is dense in $P$. | open Topology Filter Polynomial
abbrev putnam_1963_b2_solution : Prop := sorry
-- True
/--
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 $P$?
-/
theorem putnam_1963_b2
(S : Set β)
(hS : S = {2 ^ m * 3 ^ n | (... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1963_b3`
**Tags:** analysis
**Problem:**
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 $y$.
**Answer:**
Show that the solution is the sets ... | open Topology Filter Polynomial
abbrev putnam_1963_b3_solution : Set (β β β) := sorry
-- {(fun u : β => A * Real.sinh (k * u)) | (A : β) (k : β)} βͺ {(fun u : β => A * u) | A : β} βͺ {(fun u : β => A * Real.sin (k * u)) | (A : β) (k : β)}
/--
Find every twice-differentiable real-valued function $f$ with domain the set o... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1963_b5`
**Tags:** analysis
**Problem:**
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 $\lim_{n \to \infty}na_n=0$. | 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 $\lim_{n \to \infty}na_n=0$.
-/
theorem putnam_1963_b5
(a : β€ β β)
(haine... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1963_b6`
**Tags:** geometry, linear_algebra
**Problem:**
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 ... | 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 $... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_a1`
**Tags:** geometry
**Problem:**
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 $\frac{D}{d} \geq \sqrt 3$. | /--
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 $\frac{D}{d} \geq \sqrt 3$.
-/
theorem putnam_1964_a1
(A : Finset (EuclideanSpace β (Fin 2)))
(hAcard : A.card = 6)
(dists : Set β)
(hdists : dists = {d ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_a2`
**Tags:** analysis, algebra
**Problem:**
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*}
**Ans... | open Set
-- Note: uses (β β β) instead of (Icc 0 1 β β)
abbrev putnam_1964_a2_solution : β β Set (β β β) := sorry
-- fun _ β¦ β
/--
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, \\
\i... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_a3`
**Tags:** analysis, algebra
**Problem:**
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, lengt... | 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_n... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_a4`
**Tags:** analysis
**Problem:**
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 $n$. | 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 $n$.
-/
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 ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_a5`
**Tags:** analysis
**Problem:**
Prove that there exists a constant $k$ such that for any sequence $a_i$ of positive numbers,
\[
\sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 + \dots + a_n} \leq k \sum_{n=1}^{\infty}\frac{1}{a_n}.
\] | open Set Function Filter Topology
/--
Prove that there exists a constant $k$ such that for any sequence $a_i$ of positive numbers,
\[
\sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 + \dots + a_n} \leq k \sum_{n=1}^{\infty}\frac{1}{a_n}.
\]
-/
theorem putnam_1964_a5
(pa : (β β β) β Prop)
(hpa : β a, pa a β (β n : β, a ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_a6`
**Tags:** geometry
**Problem:**
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 ... | 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 dis... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_b1`
**Tags:** analysis
**Problem:**
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 $\lim_{n \to \infty} b_n/n = 0$. | 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 $\lim_{n \to \infty} b_n/n = 0$.
-/
theorem putnam_1964_b1
(a b : β β β)
(h : β n, 0 < a n)
(h'... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_b2`
**Tags:** set_theory, combinatorics
**Problem:**
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... | 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 $S$.
-/
theorem putnam_1964_... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_b3`
**Tags:** analysis
**Problem:**
Suppose $f : \mathbb{R} \to \mathbb{R}$ is continuous and for every $\alpha > 0$, $\lim_{n \to \infty} f(n\alpha) = 0$. Prove that $\lim_{x \to \infty} f(x) = 0$. | 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 $\lim_{x \to \infty} f(x) = 0$.
-/
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))
: (T... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_b4`
**Tags:** geometry
**Problem:**
$n$ 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?
**Answer:**
n^2 - n + 2 | open Classical
open scoped InnerProductSpace
abbrev putnam_1964_b4_solution : β β β := sorry
--fun n => n^2 - n + 2
/--
$n$ 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?
-/
theorem pu... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_b5`
**Tags:** analysis, number_theory
**Problem:**
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. | 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.
-/
theorem putnam_1964_b5
(a b : β β β)
(ha : StrictMono a β§ β n : β, a n > 0)
(hb : b 0... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1964_b6`
**Tags:** geometry
**Problem:**
Let $D$ be the unit disk in the plane. Show that we cannot find congruent sets $A, B$ with $A \cap B = \emptyset$ and $A \cup B = D$. | 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 $A \cup B = D$.
-/
theorem putnam_1964_b6
(D : Set (EuclideanSpace β (Fin 2)))
(hD : D = {v : EuclideanSpace β (Fin 2) | dist 0 v β€ 1})
(cong : Set (Eu... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_a1`
**Tags:** geometry
**Problem:**
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... | open EuclideanGeometry Real
noncomputable abbrev putnam_1965_a1_solution : β := sorry
-- Real.pi / 15
/--
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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_a2`
**Tags:** algebra
**Problem:**
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 $n$. | 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 $n$.
-/
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.... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_a3`
**Tags:** analysis
**Problem:**
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 $$\lim_{n \to \infty} \frac{\sum_{k = 1}^{n} e^{ia_{k^2}}}{n^2} = \alpha.$$ | 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 $$\lim_{n \to \infty} \frac{\sum_{k = 1}^{n} e^{ia_{k^2}}}{n^2} = \alpha.$$
-/
theorem putnam_1965_a3
(a : β β β)
(Ξ± : β... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_a4`
**Tags:** combinatorics
**Problem:**
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 $g$ do... | 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 $c$.... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_a5`
**Tags:** combinatorics
**Problem:**
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 $1$?
**Answer:**
There are $2^{n-1}$ such ordering... | open EuclideanGeometry Topology Filter Complex
abbrev putnam_1965_a5_solution : β β β := sorry
-- fun n => 2^(n - 1)
/--
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 $1$... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_a6`
**Tags:** geometry
**Problem:**
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 $m^{-1} + n^{-1} = 1$. | 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 $m^{-1} + n^{-1} = 1$.
-/
theorem putnam_1965_a6
(u v m : β)
(hu : 0 < u)
... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_b1`
**Tags:** analysis
**Problem:**
Find $$\lim_{n \to \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos^2\left(\frac{\pi}{2n}(x_1 + x_2 + \cdots + x_n)\right) dx_1 dx_2 \cdots dx_n.$$
**Answer:**
Show that the limit is $\frac{1}{2}$. | open EuclideanGeometry Topology Filter Complex
noncomputable abbrev putnam_1965_b1_solution : β := sorry
-- 1 / 2
/--
Find $$\lim_{n \to \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos^2\left(\frac{\pi}{2n}(x_1 + x_2 + \cdots + x_n)\right) dx_1 dx_2 \cdots dx_n.$$
-/
theorem putnam_1965_b1
: Tendsto (fun n ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_b2`
**Tags:** combinatorics
**Problem:**
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, re... | 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 $P_r$, ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_b3`
**Tags:** algebra, geometry
**Problem:**
Prove that there are exactly three right triangles (up to orientation and translation) with integer side lengths and area equal to twice their perimeter. | 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.
-/
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 ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_b4`
**Tags:** algebra, analysis
**Problem:**
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 inv... | open EuclideanGeometry Topology Filter Complex
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
-- ((fun h : β β β => h + (fun x : β => x), fun h : β β β => h + (fun _ : β => 1)), ({x : β | x β₯ 0}, Real.sqrt))
/--
Let $$f(x, n) = \frac{{n \choo... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_b5`
**Tags:** combinatorics
**Problem:**
Prove that, if $4E \le V^2$, there exists a graph with $E$ edges and $V$ vertices with no triangles (cycles of length $3$). | 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$).
-/
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.edge... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1965_b6`
**Tags:** geometry
**Problem:**
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... | 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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_a1`
**Tags:** algebra
**Problem:**
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$ wi... | /--
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)$.
-/
theorem putnam... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_a2`
**Tags:** geometry
**Problem:**
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}.$$ | /--
Let $a$, $b$, and $c$ be the side lengths of a triangle with inradius $r$. If $p = \frac{a + b + c}{2}$, show that $$\frac{1}{(p - a)^2} + \frac{1}{(p - b)^2} + \frac{1}{(p - c)^2} \ge \frac{1}{r^2}.$$
-/
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)... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_a3`
**Tags:** analysis
**Problem:**
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$. | 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$.
-/
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) :=
sorry |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_a4`
**Tags:** analysis
**Problem:**
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$. | 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$.
-/
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 ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_a5`
**Tags:** algebra
**Problem:**
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$.
\it... | 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$ ar... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_a6`
**Tags:** analysis
**Problem:**
Prove that $$\sqrt {1 + 2 \sqrt {1 + 3 \sqrt {1 + 4 \sqrt {1 + 5 \sqrt {\dots}}}}} = 3.$$ | open Topology Filter
/--
Prove that $$\sqrt {1 + 2 \sqrt {1 + 3 \sqrt {1 + 4 \sqrt {1 + 5 \sqrt {\dots}}}}} = 3.$$
-/
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) :=
sorry |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_b1`
**Tags:** geometry
**Problem:**
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$. | 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$.
-/
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 ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_b2`
**Tags:** number_theory
**Problem:**
Prove that, for any ten consecutive integers, at least one is relatively prime to all of the others. | /--
Prove that, for any ten consecutive integers, at least one is relatively prime to all of the others.
-/
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) :=
sorry |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_b3`
**Tags:** analysis
**Problem:**
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}$$ also converges. | open Topology Filter
/--
Let $p_1, p_2, \dots$ be a sequence of positive real numbers. Prove that if $\sum_{n=1}^{\infty} \frac{1}{p_n}$ converges, then $$\sum_{n=1}^{\infty} \frac {n^2 p_n}{(\sum_{i=1}^{n} p_i)^2}$$ also converges.
-/
theorem putnam_1966_b3
(p : β β β)
(hpos : β n : β, p n > 0)
(hconv : β r : β, Tend... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_b4`
**Tags:** number_theory, combinatorics
**Problem:**
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... | 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 g... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_b5`
**Tags:** geometry
**Problem:**
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. | 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.
-/
theorem putnam_1966_b5
(S : Finset (EuclideanSpace β (Fin 2)))
(hcard : S.card β₯ 3)
(hS : β s β S, s.card = 3 ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1966_b6`
**Tags:** analysis
**Problem:**
Prove that any solution $y(x)$ to the differential equation $y'' + e^{x}y = 0$ remains bounded as $x$ goes to $+\infty$. | 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$.
-/
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§ ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_a1`
**Tags:** analysis
**Problem:**
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$. | 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$.
-/
theorem putnam_1967_a1
(n : β) (hn : n > 0)
(a : Set.Icc 1... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_a2`
**Tags:** linear_algebra, analysis
**Problem:**
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,\do... | 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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_a3`
**Tags:** algebra
**Problem:**
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.
**Answer:**
Show that the mini... | open Polynomial
abbrev putnam_1967_a3_solution : β := sorry
-- 5
/--
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.
-/
theorem putnam_1967_a3 :
... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_a4`
**Tags:** analysis
**Problem:**
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$. | 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$.
-/
theorem putnam_1967_a4
(lambda : β)
(hlambda : lambda > 1 / 2)
: Β¬β u : β β β, MeasureTheory.Inte... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_a5`
**Tags:** geometry
**Problem:**
Prove that any convex region in the Euclidean plane with area greater than $\pi/4$ contains a pair of points exactly $1$ unit apart. | 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.
-/
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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_a6`
**Tags:** algebra, geometry
**Problem:**
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=... | open Nat Topology Filter
abbrev putnam_1967_a6_solution : β := sorry
-- 8
/--
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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_b1`
**Tags:** geometry
**Problem:**
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. | 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.
-/
theorem putnam_1967_b1
(r : β)
(L : ZMod 6 β (EuclideanSpace β (... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_b2`
**Tags:** algebra
**Problem:**
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 $... | 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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_b3`
**Tags:** analysis
**Problem:**
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)$. | 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)$.
-/
theorem putnam_1967_b3
(f g : β β β)
(fgcont : Continuous f β§ Continuous g)
(fgperiod : Function.Periodic f 1 ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_b4`
**Tags:** number_theory
**Problem:**
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 lo... | open Nat Topology Filter
-- Note: uses (β β Set.Icc 1 n β Bool) instead of (Set.Icc 0 n β Set.Icc 1 n β Bool)
/--
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 ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_b5`
**Tags:** algebra
**Problem:**
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}.$ | 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}.$
-/
theorem putnam_1967_b5
(n : β)
(hn : n > 0)
: (1 : β)/2 = β i β Finset.range n, (Nat.choose (n + i - 1) i) * ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1967_b6`
**Tags:** analysis
**Problem:**
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}... | open Nat Topology Filter
-- Note: boosted domain of f to the 2D plane and made it partially differentiable everywhere
/--
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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_a1`
**Tags:** analysis
**Problem:**
Prove that $$\frac{22}{7} - \pi = \int_{0}^{1} \frac{x^4(1 - x)^4}{1 + x^2} dx$$. | /--
Prove that $$\frac{22}{7} - \pi = \int_{0}^{1} \frac{x^4(1 - x)^4}{1 + x^2} dx$$.
-/
theorem putnam_1968_a1
: 22/7 - Real.pi = β« x in (0)..1, x^4 * (1 - x)^4 / (1 + x^2) :=
sorry |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_a2`
**Tags:** analysis
**Problem:**
For all integers $a$, $b$, $c$, $d$, $e$, and $f$ such that $ad \neq bc$ and any real number $\epsilon > 0$, prove that there exist rational numbers $r$ and $s$ such that $$0 < |ra + sb - e| < \varepsilon$$ and $$0 < |rc + sd - f| < \varepsilon.$$ | /--
For all integers $a$, $b$, $c$, $d$, $e$, and $f$ such that $ad \neq bc$ and any real number $\epsilon > 0$, prove that there exist rational numbers $r$ and $s$ such that $$0 < |ra + sb - e| < \varepsilon$$ and $$0 < |rc + sd - f| < \varepsilon.$$
-/
theorem putnam_1968_a2
(a b c d e f : β€)
(Ξ΅ : β)
(hne : a * d β b... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_a3`
**Tags:** combinatorics
**Problem:**
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 elemen... | 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 on... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_a4`
**Tags:** geometry, algebra
**Problem:**
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$. | 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$.
-/
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 t... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_a5`
**Tags:** algebra
**Problem:**
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$.
**Answer:**
The supremum is $8$. | open Finset Polynomial
abbrev putnam_1968_a5_solution : β := sorry
-- 8
/--
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$.
-/
theorem putnam_1968_a5
(V : Set β[X])
(V_def : V = {P : β[X] | P.de... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_a6`
**Tags:** algebra
**Problem:**
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.
**Answer:**
The set of such polynomials is $$\{\pm (x - 1), \pm (x + 1), \pm (x^2 + x - 1), \pm (x^2 - x -... | open Finset Polynomial
abbrev putnam_1968_a6_solution : Set β[X] := sorry
-- {X - 1, -(X - 1), X + 1, -(X + 1), X^2 + X - 1, -(X^2 + X - 1), X^2 - X - 1, -(X^2 - X - 1), X^3 + X^2 - X - 1, -(X^3 + X^2 - X - 1), X^3 - X^2 - X + 1, -(X^3 - X^2 - X + 1)}
/--
Find all polynomials of the form $\sum_{0}^{n} a_{i} x^{n-i}$ w... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_b1`
**Tags:** probability
**Problem:**
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{pro... | open MeasureTheory
open scoped ProbabilityTheory
abbrev putnam_1968_b1_solution : β β β β β β β := sorry
-- fun a b c => a + b - c
/--
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{pr... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_b2`
**Tags:** abstract_algebra
**Problem:**
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 $A$. | 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 $A$.
-/
theorem putnam_1968_b2
{G : Type*}
[Group G]
(hG : Finite G)
(A :... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_b4`
**Tags:** analysis
**Problem:**
Suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous on $(-\infty, \infty)$ and that $\int_{-\infty}^{\infty} f(x) dx$ exists. Prove that $$\int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) dx = \int_{-\infty}^{\infty} f(x) dx.$$ | 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 $$\int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) dx = \int_{-\infty}^{\infty} f(x) dx.$$
-/
theorem putnam_1968_b4
(f ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_b5`
**Tags:** linear_algebra, number_theory, combinatorics
**Problem:**
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 \... | open Finset Polynomial Topology Filter Metric
abbrev putnam_1968_b5_solution : β β β := sorry
-- fun p => p^2 + p
/--
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 $a, b, c, d \in \{0, 1, ..., p - 1\}$, $a + d \equiv 1 \pmod p$, ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1968_b6`
**Tags:** analysis
**Problem:**
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 $K_n$. | 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 $K_n$.
-/
theorem putnam_1968_b6
: Β¬β K : β β Set β, (β n : β, IsCompact (K... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_a1`
**Tags:** algebra, set_theory
**Problem:**
What are the possible ranges (across all real inputs $x$ and $y$) of a polynomial $f(x, y)$ with real coefficients?
**Answer:**
Show that the possibles ranges are a single point, any half-open or half-closed semi-infinite interval, or all ... | open Matrix Filter Topology Set Nat
abbrev putnam_1969_a1_solution : Set (Set β) := sorry
-- {{x} | x : β} βͺ {Set.Ici x | x : β} βͺ {Set.Iic x | x : β} βͺ {Set.Iio x | x : β} βͺ {Set.Ioi x | x : β} βͺ {Set.univ}
/--
What are the possible ranges (across all real inputs $x$ and $y$) of a polynomial $f(x, y)$ with real coeff... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_a2`
**Tags:** linear_algebra
**Problem:**
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 $D_n = (-1)^{n-1} * (n-1) * (2^{n-2}).$ | 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 $D_n = (-1)^{n-1} * (n-1) * (2^{n-2}).$
-/
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)... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_a4`
**Tags:** analysis
**Problem:**
Show that $\int_0^1 x^x dx = \sum_{n=1}^{\infty} (-1)^{n+1}n^{-n}$. | open Matrix Filter Topology Set Nat
/--
Show that $\int_0^1 x^x dx = \sum_{n=1}^{\infty} (-1)^{n+1}n^{-n}$.
-/
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)) :=
sorry |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_a5`
**Tags:** analysis
**Problem:**
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(... | 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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_a6`
**Tags:** analysis
**Problem:**
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. | 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.
-/
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 ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_b1`
**Tags:** number_theory
**Problem:**
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 $24$. | 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 $24$.
-/
theorem putnam_1969_b1
(n : β)
(hnpos : n > 0)
(hn : 24 β£ n + 1)
: 24 β£ β d β divisors n, d :=
sorry |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_b2`
**Tags:** abstract_algebra
**Problem:**
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'?
**Answer:**
Show that the statement is no longer true if 'two' is replaced by 'three'. | open Matrix Filter Topology Set Nat
abbrev putnam_1969_b2_solution : Prop := sorry
-- False
/--
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'?
-/
theorem putnam_1969_b2
(P : β β Prop)
(P_def : β n, P n β β (G : Type)... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_b3`
**Tags:** analysis
**Problem:**
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 $\pi * T_1^2 = 2$. | 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 $\pi * T_1^2 = 2$.
-/
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 => ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_b4`
**Tags:** geometry
**Problem:**
$Ξ$ is a plane curve of length 1. Show that we can find a closed rectangle of area 1/4 which covers $Ξ$. | 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 $Ξ$.
-/
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 ... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_b5`
**Tags:** analysis
**Problem:**
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 ... | 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 $\lim_{x \to \infty} k(x)/x = 0$.
-/
... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1969_b6`
**Tags:** linear_algebra
**Problem:**
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 $$BA =
\begin{pmatrix}
9 & 0 \\
0 & 9
\end{pmatrix}.$$ | 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 $$BA =
\begin{pmatrix}
9 & 0 \\
0 & 9
\end{pmatrix}.$$
-/
theorem putnam_1969_b6
(A : Matrix (Fin 3) (Fin 2) β)
(... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1970_a1`
**Tags:** analysis
**Problem:**
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. | 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.
-/
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 (... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1970_a2`
**Tags:** analysis, algebra
**Problem:**
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 $$Ax^2 + Bxy + Cy^2 + Dx^3 + E... | open Metric Set EuclideanGeometry
/--
Let $A$, $B$, $C$, $D$, $E$, $F$, and $G$ be real numbers satisfying $B^2 - 4AC < 0$. Prove that there exists some $\delta > 0$ such that no points $(x, y)$ in the punctured disk $0 < x^2 + y^2 < \delta$ satisfy $$Ax^2 + Bxy + Cy^2 + Dx^3 + Ex^2y + Fxy^2 + Gy^3 = 0.$$
-/
theorem p... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1970_a3`
**Tags:** number_theory
**Problem:**
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.
**Answer:**
The maximum attainable length is $3$; the smalles... | open Metric Set EuclideanGeometry
abbrev putnam_1970_a3_solution : β Γ β := sorry
-- (3, 1444)
/--
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.
-/
theorem putnam_1970_a3
(L : β β β... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1970_a4`
**Tags:** analysis
**Problem:**
Suppose $(x_n)$ is a sequence such that $\lim_{n \to \infty} (x_n - x_{n-2} = 0$. Prove that $\lim_{n \to \infty} \frac{x_n - x_{n-1}}{n} = 0$. | 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 $\lim_{n \to \infty} \frac{x_n - x_{n-1}}{n} = 0$.
-/
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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1970_b1`
**Tags:** analysis
**Problem:**
Evaluate the infinite product $\lim_{n \to \infty} \frac{1}{n^4} \prod_{i = 1}^{2n} (n^2 + i^2)^{1/n}$.
**Answer:**
Show that the solution is $e^{2 \log(5) - 4 + 2 arctan(2)}$. | open Metric Set EuclideanGeometry Filter Topology
noncomputable abbrev putnam_1970_b1_solution : β := sorry
-- Real.exp (2 * Real.log 5 - 4 + 2 * Real.arctan 2)
/--
Evaluate the infinite product $\lim_{n \to \infty} \frac{1}{n^4} \prod_{i = 1}^{2n} (n^2 + i^2)^{1/n}$.
-/
theorem putnam_1970_b1
: Tendsto (fun n => 1/(n... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1970_b2`
**Tags:** analysis, algebra
**Problem:**
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)... | 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 $H\left(\frac{T}{\sqrt{3}}\right)$.
-/
theorem putnam_19... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1970_b3`
**Tags:** analysis
**Problem:**
A closed subset $S$ of $\mathbb{R}^2$ lies in $a < x < b$. Show that its projection on the y-axis is closed. | 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.
-/
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... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1970_b4`
**Tags:** analysis
**Problem:**
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... | 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 $|x''(t)| \ge \frac{9}{2}$.
-/
theo... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1970_b5`
**Tags:** analysis
**Problem:**
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 $n$. | 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 $n$.
-/
theorem putnam_1970_b5
... |
You are an assistant that translates natural-language math statements into Lean 4 theorems that compile with Mathlib. The preamble import Mathlib and open BigOperators Real Nat Topology Rat is already provided. Given a math problem or statement, output only a single Lean 4 theorem whose type states the claim, ending th... | **Theorem name:** `putnam_1970_b6`
**Tags:** geometry
**Problem:**
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). | 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).
-/
theorem putnam_1970_b6
(L : ZM... |
End of preview.
README.md exists but content is empty.
- Downloads last month
- 32