name stringlengths 14 14 | rocq_statement stringlengths 139 1.42k | lean_statement stringlengths 199 1.61k |
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putnam_2014_a1 | Require Import Reals Factorial Znumtheory Coquelicot.Derive.
Theorem putnam_2014_a1
(f : R -> R := fun x => (1 - x + x^2) * (exp x))
(hf : forall x : R, forall n : nat, ex_derive_n f n x)
(coeff : nat -> R := fun i => Derive_n f i 0 / INR (fact i))
(n : nat)
(hcoeff : coeff n <> 0)
: exists a b:... | import Mathlib
open Topology Filter
/--
Prove that every nonzero coefficient of the Taylor series of \[(1 - x + x^2)e^x\] about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.
-/
theorem putnam_2014_a1
(f : β β β)
(hf : β x : β, f x = (1 - x + x ^ 2) * Real.exp x)
(hfdiff... |
putnam_1962_a6 | Require Import Ensembles QArith.
Theorem putnam_1962_a6
(A : Ensemble Q)
(hSSadd : forall a b : Q, (A a /\ A b) -> A (a + b))
(hSSprod : forall a b : Q, (A a /\ A b) -> A (a * b))
(hSScond : forall r : Q, (A r \/ A (-r) \/ r = 0) /\ ~(A r \/ A (-r)) /\ ~(A r /\ r = 0) /\ ~(A (-r) /\ r = 0))
: A = (f... | import Mathlib
/--
Let $S$ be a set of rational numbers such that whenever $a$ and $b$ are members of $S$, so are $a+b$ and $ab$, and having the property that for every rational number $r$ exactly one of the following three statements is true: \[ r \in S, -r \in S, r = 0. \] Prove that $S$ is the set of all positive r... |
putnam_1962_b6 | Require Import Reals Ensembles Coquelicot.Hierarchy Finite_sets.
Local Coercion INR : nat >-> R.
Theorem putnam_1962_b6
(n : nat)
(a b : nat -> R)
(xs : Ensemble R)
(f : R -> R := (fun x : R => sum_n (fun k : nat => a k * sin (k * x) + b k * cos (k * x)) n))
(hf1 : forall x : R, (0 <= x /\ x <= 2 ... | import Mathlib
open Real
/--
Let \[ f(x) = \sum_{k=0}^n a_k \sin kx + b_k \cos kx, \] where $a_k$ and $b_k$ are constants. Show that, if $\lvert f(x) \rvert \le 1$ for $0 \le x \le 2 \pi$ and $\lvert f(x_i) \rvert = 1$ for $0 \le x_1 < x_2 < \cdots < x_{2n} < 2 \pi$, then $f(x) = \cos (nx + a)$ for some constant $a$.... |
putnam_1966_a4 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1966_a4
(a : nat -> int)
(ha1 : a 1%nat = 2)
(hai : fo... | import Mathlib
open Topology Filter
/--
Prove that the $n$th item in the ascending list of non-perfect-square positive integers equals $n + \{\sqrt{n}\}$, where $\{m\}$ denotes the closest integer to $m$.
-/
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 = Tr... |
putnam_2005_b2 | Require Import Nat List Reals Coquelicot.Coquelicot.
Import ListNotations.
Definition putnam_2005_b2_solution (n: nat) (k: list nat) := (n, k) = (1%nat, [1%nat]) \/ (n, k) = (3%nat, [2%nat; 3%nat; 6%nat]) \/ (n, k) = (3%nat, [2%nat; 6%nat; 3%nat]) \/ (n, k) = (3%nat, [3%nat; 2%nat; 6%nat]) \/ (n, k) = (3%nat, [3%nat; 6... | import Mathlib
open Nat Set
-- Note: uses β β β instead of Fin n β β
abbrev putnam_2005_b2_solution : Set (β Γ (β β β€)) := sorry
-- {(n, k) : β Γ (β β β€) | (n = 1 β§ k 0 = 1) β¨ (n = 3 β§ (k '' {0, 1, 2} = {2, 3, 6})) β¨ (n = 4 β§ (β i : Fin 4, k i = 4))}
/--
Find all positive integers $n,k_1,\dots,k_n$ such that $k_1+\cd... |
putnam_1970_b5 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences topology normedtype.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
L... | import Mathlib
open Metric Set EuclideanGeometry Filter Topology
/--
Let $u_n$ denote the function $u_n(x) = -n$ if $x \leq -n$, $x$ if $-n < x \leq n$, and $n$ otherwise. Let $F$ be a function on the reals. Show that $F$ is continuous if and only if $u_n \circ F$ is continuous for all natural numbers $n$.
-/
theorem... |
putnam_1972_b6 | Require Import Nat Reals. From Coquelicot Require Import Complex Hierarchy.
(* uses (nat -> nat) instead of ('I_k -> nat) *)
Theorem putnam_1972_b6
(k : nat)
(n : nat -> nat)
(zpoly : C -> C := fun z : C => (1 + sum_n (fun i => Cpow z (n i)) (k - 1))%C)
(hk : (k >= 1)%nat)
(hn : (forall i : nat, i <... | import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
/--
Let $n_1 < n_2 < \dots < n_k$ be a set of positive integers. Prove that the polynomail $1 + z^{n_1} + z^{n_2} + \dots + z^{n_k}$ has not roots inside the circle $|z| < (\frac{\sqrt{5}-1}{2}$.
-/
theorem putnam_1972_b6
(k : β)
(hk : k β₯... |
putnam_2003_b4 | Require Import Reals ZArith Coquelicot.Coquelicot.
Theorem putnam_2003_b4
(a b c d e: Z)
(r1 r2 r3 r4: R)
(ha : ~ Z.eq a 0)
: let a := IZR a in
let b := IZR b in
let c := IZR c in
let d := IZR d in
let e := IZR e in
(forall (z: R), a * z ^ 4 + b * z ^ 3 + c * z ^ 2 + d * z + e =... | import Mathlib
open MvPolynomial Set Nat
/--
Let $f(z)=az^4+bz^3+cz^2+dz+e=a(z-r_1)(z-r_2)(z-r_3)(z-r_4)$ where $a,b,c,d,e$ are integers, $a \neq 0$. Show that if $r_1+r_2$ is a rational number and $r_1+r_2 \neq r_3+r_4$, then $r_1r_2$ is a rational number.
-/
theorem putnam_2003_b4
(f : β β β)
(a b c d e : β€)
... |
putnam_2005_a5 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2005_a5_solution := PI * ln 2 / 8.
Theorem putnam_2005_a5
: RInt (fun x => ln (x + 1) / (x ^ 2 + 1)) 0 1 = putnam_2005_a5_solution.
Proof. Admitted.
| import Mathlib
open Nat Set
noncomputable abbrev putnam_2005_a5_solution : β := sorry
-- Real.pi * (Real.log 2) / 8
/--
Evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1}\,dx$.
-/
theorem putnam_2005_a5 :
β« x in (0:β)..1, (Real.log (x+1))/(x^2 + 1) = putnam_2005_a5_solution :=
sorry
|
putnam_2008_b4 | Require Import Nat ZArith Reals Coquelicot.Coquelicot.
Theorem putnam_2008_b4
(p: nat)
(hp : Znumtheory.prime (Z.of_nat p))
(c: nat -> Z)
(n: nat)
(h : nat -> Z := fun x => floor (sum_n (fun i => IZR (c i) * INR (x ^ i)) n))
(hh : forall (i j: nat), i <> j /\ and (le 0 i) (le i (p ^ 2 - 1)) /\ a... | import Mathlib
open Filter Topology Set Nat
/--
Let $p$ be a prime number. Let $h(x)$ be a polynomial with integer coefficients such that $h(0), h(1), \dots, h(p^2-1)$ are distinct modulo $p^2$. Show that $h(0), h(1), \dots, h(p^3-1)$ are distinct modulo $p^3$.
-/
theorem putnam_2008_b4
(p : β)
(hp : Nat.Prime p)
(h ... |
putnam_1976_b6 | From mathcomp Require Import all_algebra all_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope nat_scope.
Theorem putnam_1976_b6
(sigma : nat -> int := fun N => \sum_(d <- divisors N) d)
: forall N : nat, sigma N = (N.*2 + 1) -> (exists m : nat, odd m = tru... | import Mathlib
open Polynomial Filter Topology ProbabilityTheory MeasureTheory
/--
Let $\sigma(N)$ denote the sum of all positive integer divisors of $N$, including $1$ and $N$. Call a positive integer $N$ \textit{quasiperfect} if $\sigma(N) = 2N + 1$. Prove that every quasiperfect number is the square of an odd inte... |
putnam_2007_b3 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2007_b3_solution := let a := (1 + sqrt 5) / 2 in (2 ^ 2006 / sqrt 5) * (a ^ 3997 - Rpower a (-3997)).
Theorem putnam_2007_b3
(X := fix x (n: nat) :=
match n with
| O => 1
| S n' => 3 * x n' + IZR (floor (x n' * sqrt 5))
end)
... | import Mathlib
open Set Nat Function
noncomputable abbrev putnam_2007_b3_solution : β := sorry
-- (2 ^ 2006 / Real.sqrt 5) * (((1 + Real.sqrt 5) / 2) ^ 3997 - ((1 + Real.sqrt 5) / 2) ^ (-3997 : β€))
/--
Let $x_0 = 1$ and for $n \geq 0$, let $x_{n+1} = 3x_n + \lfloor x_n \sqrt{5} \rfloor$. In particular, $x_1 = 5$, $x_... |
putnam_2004_a3 | Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2004_a3
(u : nat -> R)
(hubase : u O = 1 /\ u (S O) = 1 /\ u (S (S O)) = 1)
(hudet : forall n : nat, u n * u (Nat.add n 3) - u (Nat.add n 1) * u (Nat.add n 2) = INR (fact n))
: forall n : nat, exists m : Z, u n = IZR m.
Proof. Admitted.
| import Mathlib
open Nat Topology Filter
/--
Define a sequence $\{u_n\}_{n=0}^\infty$ by $u_0=u_1=u_2=1$, and thereafter by the condition that $\det \begin{pmatrix}
u_n & u_{n+1} \\
u_{n+2} & u_{n+3}
\end{pmatrix} = n!$ for all $n \geq 0$. Show that $u_n$ is an integer for all $n$. (By convention, $0!=1$.)
-/
theorem ... |
putnam_1990_b3 | Require Import Ensembles Finite_sets Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1990_b3
(E : Ensemble (matrix 2 2))
(hE : forall (A: matrix 2 2), E A ->
forall (i j: nat), and (le 0 i) (lt i 2) /\ and (le 0 j) (lt j 2) ->
(coeff_mat 0 A i j) <= 200 /\ exists (m: nat), coeff_mat ... | import Mathlib
open Filter Topology Nat
/--
Let $S$ be a set of $2 \times 2$ integer matrices whose entries $a_{ij}$ (1) are all squares of integers, and, (2) satisfy $a_{ij} \leq 200$. Show that if $S$ has more than $50387$ ($=15^4-15^2-15+2$) elements, then it has two elements that commute.
-/
theorem putnam_1990_b... |
putnam_1992_a4 | Require Import Nat Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1992_a4_solution (k : nat) := if odd k then 0 else pow (-1) (k/2) * INR (fact k).
Theorem putnam_1992_a4
(f : R -> R)
(hfdiff : forall k : nat, continuity (Derive_n f k) /\ forall x : R, ex_derive (Derive_n f k) x)
(hf : forall ... | import Mathlib
open Topology Filter Nat Function
abbrev putnam_1992_a4_solution : β β β := sorry
-- fun k β¦ ite (Even k) ((-1) ^ (k / 2) * factorial k) 0
/--
Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If
\[
f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1,... |
putnam_1990_b2 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences normedtype topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
L... | import Mathlib
open Filter Topology Nat
/--
Prove that for $|x|<1$, $|z|>1$, $1+\sum_{j=1}^\infty (1+x^j)P_j=0$, where $P_j$ is $\frac{(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3) \cdots (z-x^j)}$.
-/
theorem putnam_1990_b2
(x z : β)
(P : β β β)
(xlt1 : |x| < 1)
(zgt1 : |z| > 1)
(hP : β j β₯ 1, P j = (... |
putnam_2018_b6 | Require Import Nat List Ensembles Finite_sets Reals.
Theorem putnam_2018_b6
(E: Ensemble (list nat) := fun l => length l = 2018 /\
forall (n: nat), (List.In n l) -> (n = 1 \/ n = 2 \/ n = 3 \/ n = 4 \/ n = 5 \/ n = 6 \/ n = 10) /\
fold_left Nat.add l 0 = 3860
)
(n : nat)
: cardinal (lis... | import Mathlib
/--
Let $S$ be the set of sequences of length $2018$ whose terms are in the set $\{1,2,3,4,5,6,10\}$ and sum to $3860$. Prove that the cardinality of $S$ is at most $2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}$.
-/
theorem putnam_2018_b6
(S : Finset (Fin 2018 β β€))
(hS : S = {s : Fin 2018 β... |
putnam_2009_a2 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2009_a2_solution : (R -> R) := fun x => Rpower 2 (-1 / 12) * Rpower (sin (6 * x + PI / 4) / (cos (6 * x + PI / 4)) ^ 2) (1 / 6).
Theorem putnam_2009_a2
(f g h: R -> R)
(a b: R)
(hab : a < 0 < b)
(hdiff : forall (x: R), a < x < b -> ex_derive... | import Mathlib
open Topology MvPolynomial Filter Set
noncomputable abbrev putnam_2009_a2_solution : β β β := sorry
-- fun x β¦ 2 ^ (-(1 : β) / 12) * (Real.sin (6 * x + Real.pi / 4) / (Real.cos (6 * x + Real.pi / 4)) ^ 2) ^ ((1 : β) / 6)
/--
Functions $f,g,h$ are differentiable on some open interval around $0$
and sati... |
putnam_2008_b5 | Require Import Reals QArith Coquelicot.Coquelicot. From mathcomp Require Import div.
Open Scope R.
Definition putnam_2008_b5_solution (f : R -> R) := exists n : Z, f = (fun x => x + IZR n) \/ f = (fun x => -x + IZR n).
Theorem putnam_2008_b5
(f: R -> R)
(hf : (forall x: R, ex_derive f x) /\ continuity (Derive f... | import Mathlib
open Filter Topology Set Nat
abbrev putnam_2008_b5_solution : Set (β β β) := sorry
-- {fun (x : β) => x + n | n : β€} βͺ {fun (x : β) => -x + n | n : β€}
/--
Find all continuously differentiable functions f : \mathbb{R} \to \mathbb{R} such that for every rational number $q$, the number $f(q)$ is rational ... |
putnam_2010_a2 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2010_a2_solution (f: R -> R) := exists (c d: R), f = (fun x => c * x + d).
Theorem putnam_2010_a2
(f: R -> R)
: (forall (x: R) (n: nat), gt n 0 -> ex_derive f x /\ Derive f x = (f (x + (INR n)) - f x) / (INR n)) <-> putnam_2010_a2_solution f.
Proof. ... | import Mathlib
abbrev putnam_2010_a2_solution : Set (β β β) := sorry
-- {f : β β β | β c d : β, β x : β, f x = c*x + d}
/--
Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[
f'(x) = \frac{f(x+n)-f(x)}{n}
\]
for all real numbers $x$ and all positive integers $n$.
-/
theorem putnam_2010_a2
: {... |
putnam_1992_a2 | Require Import Reals Binomial Factorial Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1992_a2_solution := 1992.
Theorem putnam_1992_a2
(C : R -> R := fun a => (Derive_n (fun x => Rpower (1 + x) a) 1992) 0 / INR (fact 1992))
: RInt (fun y => C (-y - 1) * sum_n_m (fun k => 1 / (y + INR k)) 1 1992) 0 1 = ... | import Mathlib
open Topology Filter
abbrev putnam_1992_a2_solution : β := sorry
-- 1992
/--
Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x=0$ of $(1 + x)^\alpha$. Evaluate
\[
\int_0^1 \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\,dy.
\]
-/
theorem putnam_1992_a2
(C : β β... |
putnam_1969_b3 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences topology normedtype trigo.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_sc... | import Mathlib
open Matrix Filter Topology Set Nat
/--
Suppose $T$ is a sequence which satisfies $T_n * T_{n+1} = n$ whenever $n \geq 1$, and also $\lim_{n \to \infty} \frac{T_n}{T_{n+1}} = 1. Show that $\pi * T_1^2 = 2$.
-/
theorem putnam_1969_b3
(T : β β β)
(hT1 : β n : β, n β₯ 1 β (T n) * (T (n + 1)) = n)
(hT2 : Te... |
putnam_2022_a1 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2022_a1_solution : R -> R -> Prop := fun a b => (a = 0 /\ b = 0) \/ (Rabs a >= 1) \/ (0 < Rabs a < 1 /\ (b < ln (1 + ((1 - sqrt (1 - a ^2))/ a) ^ 2) - a * (1 - sqrt (1 - a ^ 2) / a) \/ b > ln (1 + ((1 + sqrt (1 - a ^ 2)) / a) ^ 2) - a * (1 + sqrt (1 - a ^ 2)... | import Mathlib
open Polynomial
abbrev putnam_2022_a1_solution : Set (β Γ β) := sorry
-- {(a, b) | (a = 0 β§ b = 0) β¨ 1 β€ |a| β¨ (0 < |a| β§ |a| < 1 β§ letI rm := (1 - β(1 - a ^ 2)) / a; letI rp := (1 + β(1 - a ^ 2)) / a; (b < Real.log (1 + rm ^ 2) - a * rm β¨ b > Real.log (1 + rp ^ 2) - a * rp))}
/--
Determine all ordered... |
putnam_1985_a1 | Require Import Ensembles Finite_sets Nat.
Definition putnam_1985_a1_solution := (10, 10, 0, 0).
Theorem putnam_1985_a1
: let (abc, d) := putnam_1985_a1_solution in let (ab, c) := abc in let (a, b) := ab in
cardinal ((Ensemble nat) * (Ensemble nat) * (Ensemble nat)) (fun A => let (A1A2, A3) := A in let (A1, A2)... | import Mathlib
open Set
abbrev putnam_1985_a1_solution : β Γ β Γ β Γ β := sorry
-- (10, 10, 0, 0)
/--
Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that
\begin{enumerate}
\item[(i)] $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, and
\item[(ii)] $A_1 \cap A... |
putnam_2022_a6 | Require Import Nat Reals Coquelicot.Hierarchy. From mathcomp Require Import div fintype seq ssralg ssrbool ssrnat ssrnum .
Definition putnam_2022_a6_solution := fun n : nat => n.
Theorem putnam_2022_a6
(N : nat)
(M : nat)
(n := mul N 2)
(i0 : 'I_n)
(sumIntervals : ('I_n -> R) -> nat -> R := fun s k ... | import Mathlib
open Set
-- Note: uses (β β β) instead of (Fin (2 * n) β β)
abbrev putnam_2022_a6_solution : β β β := sorry
-- (fun n : β => n)
/--
Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\dots,x_{2n}$ with $-1<x_1<x_... |
putnam_1986_a6 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_1986_a6_solution : (nat -> nat) -> nat -> R := fun b n => (\prod_(1 <= i ... | import Mathlib
open Real Equiv
noncomputable abbrev putnam_1986_a6_solution : (β β β) β β β β := sorry
-- fun b n β¦ (β i : Finset.Icc 1 n, b i) / Nat.factorial n
/--
Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ sat... |
putnam_2004_b1 | Require Import Nat Reals QArith Coquelicot.Coquelicot.
Theorem putnam_2004_b1
(c : nat -> Z)
(n : nat)
(r : Q)
(Preq0 : sum_n (fun i => IZR (c i) * (Q2R r) ^ i) n = 0)
: forall i : nat, lt i n -> exists m : Z, IZR m = sum_n (fun j => IZR (c (sub n j)) * (Q2R r) ^ (i + 1 - j)) i.
Proof. Admitted.
| import Mathlib
open Nat Topology Filter
/--
Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers $c_nr,\,c_nr^2+c_{n-1}r,\,c_nr^3+c_{n-1}r^2+c_{n-2}r,\dots,\,c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$ are int... |
putnam_2013_b2 | Require Import Ensembles Finite_sets Reals Coquelicot.Coquelicot.
Definition putnam_2013_b2_solution : R := 3.
Theorem putnam_2013_b2
(E: Ensemble (R -> R) := fun f => forall (x : R), exists (a : nat -> R) (N : nat), f x = 1 + sum_n_m (fun n => a n * cos (2 * PI * INR n * x)) 1 N /\ f x >= 0 /\
forall (n: nat)... | import Mathlib
open Function Set
abbrev putnam_2013_b2_solution : β := sorry
-- 3
/--
Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)
\]
for which:
\begin{enumerate}
\item[(i)]
$f(x) \geq 0$ for all real $x$, and
... |
putnam_2012_a6 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype topology sequences measure lebesgue_measure lebesgue_integral.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Def... | import Mathlib
open Matrix Function
-- Note: this formalization differs from the original problem wording in only allowing axis-aligned rectangles. The problem is solvable given this weaker hypothesis.
abbrev putnam_2012_a6_solution : Prop := sorry
-- True
/--
Let $f(x,y)$ be a continuous, real-valued function on $\m... |
putnam_1964_a1 | From mathcomp Require Import all_algebra all_ssreflect fintype.
From mathcomp Require Import reals.
From mathcomp Require Import classical_sets cardinality.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Local Open S... | import Mathlib
/--
Let $A_1, A_2, A_3, A_4, A_5, A_6$ be distinct points in the plane. Let $D$ be the longest distance between any pair, and let $d$ the shortest distance. Show that $\frac{D}{d} \geq \sqrt 3$.
-/
theorem putnam_1964_a1
(A : Finset (EuclideanSpace β (Fin 2)))
(hAcard : A.card = 6)
(dists : Set β)
(hdis... |
putnam_2021_a4 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2021_a4_solution := (sqrt 2 / 2) * PI * ln 2 / ln 10.
Theorem putnam_2021_a4
(I : nat -> R := fun r => RInt (fun x => RInt (fun y => (1 + 2 * x ^ 2) / (1 + x ^ 4 + 6 * x ^ 2 * y ^ 2 + y ^ 4) - (1 + y ^ 2) / (2 + x ^ 4 + y ^ 4)) 0 (sqrt (INR r ^ 2 - x ^... | import Mathlib
open Filter Topology Metric
noncomputable abbrev putnam_2021_a4_solution : β := sorry
-- ((Real.sqrt 2) / 2) * Real.pi * Real.log 2
/--
Let
\[
I(R) = \iint_{x^2+y^2 \leq R^2} \left( \frac{1+2x^2}{1+x^4+6x^2y^2+y^4} - \frac{1+y^2}{2+x^4+y^4} \right)\,dx\,dy.
\]
Find
\[
\lim_{R \to \infty} I(R),
\]
or sh... |
putnam_1986_b4 | Require Import Reals Ranalysis Coquelicot.Coquelicot.
Definition putnam_1986_b4_solution := True.
Theorem putnam_1986_b4
(G : R -> R)
(hGeq : forall (r: R), exists (m n: Z), G r = Rabs (r - sqrt (IZR (m ^ 2 + 2 * n ^ 2))))
(hGlb : forall (r: R), forall (m n: Z), G r <= Rabs (r - sqrt (IZR (m ^ 2 + 2 * n ^ 2... | import Mathlib
open Real Equiv Polynomial Filter Topology
abbrev putnam_1986_b4_solution : Prop := sorry
-- True
/--
For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\lim_{r\to \infty}G(r)$ exists and equals... |
putnam_1971_a6 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences topology.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1971_a6
(c : R)
(hc : forall n : int, n > 0 -... | import Mathlib
open Set MvPolynomial
/--
Let $c$ be a real number such that $n^c$ is an integer for every positive integer $n$. Show that $c$ is a non-negative integer.
-/
theorem putnam_1971_a6
(c : β)
(hc : β n : β€, n > 0 β β m : β€, (n : β)^c = m)
: β m : β€, m β₯ 0 β§ c = m :=
sorry
|
putnam_2002_a1 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype sequences topology derive.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope... | import Mathlib
open Nat
abbrev putnam_2002_a1_solution : β β β β β := sorry
-- (fun k n : β => (-k) ^ n * (n)!)
/--
Let $k$ be a fixed positive integer. The $n$-th derivative of $\frac{1}{x^k-1}$ has the form $\frac{P_n(x)}{(x^k-1)^{n+1}}$ where $P_n(x)$ is a polynomial. Find $P_n(1)$.
-/
theorem putnam_2002_a1
(k : ... |
putnam_1969_a4 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences topology normedtype measure lebesgue_measure lebesgue_integral.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Import Order.TTheory GRing.Theory Num.Theory.
Set Implicit Arguments.
... | import Mathlib
open Matrix Filter Topology Set Nat
/--
Show that $\int_0^1 x^x dx = \sum_{n=1}^{\infty} (-1)^{n+1}n^{-n}$.
-/
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
|
putnam_1964_b2 | From mathcomp Require Import all_algebra all_ssreflect fintype.
From mathcomp Require Import classical_sets cardinality.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
Local Open Scope card_scope.
Theorem putnam_1964_b2
(A : finType)
(n... | import Mathlib
open Set Function Filter Topology
/--
Let $S$ be a finite set. A set $P$ of subsets of $S$ has the property that any two members of $P$ have at least one element in common and that $P$ cannot be extended (whilst keeping this property). Prove that $P$ contains exactly half of the subsets of $S$.
-/
theo... |
putnam_2010_b1 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2010_b1_solution := False.
Theorem putnam_2010_b1
: (exists (a: nat -> R), forall (m: nat), gt m 0 -> Series (fun i => (a i) ^ m) = (INR m)) <-> putnam_2010_b1_solution.
Proof. Admitted.
| import Mathlib
open Filter Topology Set
abbrev putnam_2010_b1_solution : Prop := sorry
-- False
/--
Is there an infinite sequence of real numbers $a_1, a_2, a_3, \dots$ such that \[ a_1^m + a_2^m + a_3^m + \cdots = m \] for every positive integer $m$?
-/
theorem putnam_2010_b1
: (β a : β β β, β m : β, m > 0 β β' i : ... |
putnam_2009_b5 | Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2009_b5
(f: R -> R)
(hf : forall (x: R), 1 < x -> (ex_derive f x /\ Derive f x = (x ^ 2 - (f x) ^ 2) / (x ^ 2 * ((f x) ^ 2 + 1))))
: is_lim f p_infty p_infty.
Proof. Admitted.
| import Mathlib
open Topology MvPolynomial Filter Set Metric
/--
Let $f: (1, \infty) \to \mathbb{R}$ be a differentiable function such that
\[
f'(x) = \frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)}
\qquad \mbox{for all $x>1$.}
\]
Prove that $\lim_{x \to \infty} f(x) = \infty$.
-/
theorem putnam_2009_b5
(f : β β β)
(hfdiff : D... |
putnam_1971_a3 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1971_a3
(a b c : R * R)
(r : R)
(habclattice : fst a ... | import Mathlib
open Set
/--
The three vertices of a triangle of sides $a,b,c$ are lattice points and lie on a circle of radius $R$. Show that $abc \geq 2R$.
-/
theorem putnam_1971_a3
(a b c : β Γ β)
(R : β)
(habclattice : a.1 = round a.1 β§ a.2 = round a.2 β§ b.1 = round b.1 β§ b.2 = round b.2 β§ c.1 = round ... |
putnam_2009_a1 | Require Import Reals GeoCoq.Main.Tarski_dev.Ch16_coordinates_with_functions.
Context `{T2D:Tarski_2D} `{TE:@Tarski_euclidean Tn TnEQD}.
Open Scope R.
Definition putnam_2009_a1_solution := True.
Theorem putnam_2009_a1
: (forall f: Tpoint -> R,
(forall (A B C D: Tpoint), Square A B C D ->
f A + f B + f C + ... | import Mathlib
open Topology MvPolynomial Filter
abbrev putnam_2009_a1_solution : Prop := sorry
-- True
/--
Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f(P)=0$ for all points $P$ in the plane?
-/
theorem putnam_2009_a1
: ... |
putnam_1971_a4 | Require Import Nat Reals Coquelicot.Coquelicot.
Theorem putnam_1971_a4
(epsilon : R)
(hepsilon : 0 < epsilon < 1)
(P : nat -> (R * R) -> R := fun n '(x, y) => (x + y)^n * (x^2 - (2 - epsilon)*x*y + y^2))
: exists N : nat, forall n : nat, ge n N -> (exists (k : nat) (coeff : (nat * nat) -> R),
(... | import Mathlib
open Set MvPolynomial
/--
Show that for $\epsilon \in (0,1)$, the expression $(x + y)^n (x^2 - 2-\epsilon)xy + y^2)$ is a polynomial with positive coefficients for $n$ sufficiently large, where $n$ is an integer.
-/
theorem putnam_1971_a4
(Ξ΅ : β)
(hΞ΅ : 0 < Ξ΅ β§ Ξ΅ < 1)
(P : β β β β MvPolynomial (Fin 2) β... |
putnam_2010_b4 | Require Import Reals Coquelicot.Coquelicot.
Definition P : (nat -> R) -> nat -> R -> R := fun c n x => sum_n (fun i => c i * x ^ i) n.
Definition putnam_2010_b4_solution (c1 c2: nat -> R) (n m: nat) := exists (a b c d: R), b * c - a * d = 1 /\ P c1 n = (fun x => a * x + b) /\ P c2 m = (fun x => c * x + d).
Theorem putn... | import Mathlib
open Filter Topology Set
abbrev putnam_2010_b4_solution : Set (Polynomial β Γ Polynomial β) := sorry
-- {(p, q) : Polynomial β Γ Polynomial β | p.degree β€ 1 β§ q.degree β€ 1 β§ p.coeff 0 * q.coeff 1 - p.coeff 1 * q.coeff 0 = 1}
/--
Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for... |
putnam_1973_b4 | From mathcomp Require Import all_ssreflect ssrnum ssralg.
From mathcomp Require Import reals sequences topology measure lebesgue_measure lebesgue_integral normedtype derive.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Imp... | import Mathlib
open Nat Set MeasureTheory Topology Filter
-- Note: Boosted domain to β, which is fine because you can extend any such function f from [0,1] to β satisfying the same properties. There may be multiple correct answers.
abbrev putnam_1973_b4_solution : β β β := sorry
-- (fun x => x)
/--
Suppose $f$ is a f... |
putnam_1981_a1 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype sequences topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local... | import Mathlib
open Topology Filter Set Polynomial Function
noncomputable abbrev putnam_1981_a1_solution : β := sorry
-- 1/8
/--
Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$.
-/
theorem putnam_1981_a1
(P : β β β β Prop... |
putnam_2005_b1 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2005_b1_solution := (fun x y => match x, y with | 0, 1 => -1 | 0, 2 => 1 | 1, 0 => 2 | 1, 1 => -4 | 2, 0 => 4 | _, _ => 0 end, (2%nat, 2%nat)).
Theorem putnam_2005_b1
(p : R -> R -> R := fun x y => sum_n (fun i => (sum_n (fun j => (fst putnam_2005_b1_sol... | import Mathlib
open Nat Set
-- Note: There might be multiple possible correct answers.
noncomputable abbrev putnam_2005_b1_solution : MvPolynomial (Fin 2) β := sorry
-- (MvPolynomial.X 1 - 2 * MvPolynomial.X 0) * (MvPolynomial.X 1 - 2 * MvPolynomial.X 0 - 1)
/--
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor... |
putnam_1980_b1 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
Local Open Scope ring_scope.
Variable R : realType... | import Mathlib
open Real
abbrev putnam_1980_b1_solution : Set β := sorry
-- {c : β | c β₯ 1 / 2}
/--
For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?
-/
theorem putnam_1980_b1
(c : β)
: (β x : β, (exp x + exp (-x)) / 2 β€ exp (c * x ^ 2)) β c β putnam_1980_b1_solution :=
sorry
|
putnam_1980_b6 | From mathcomp Require Import all_algebra all_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Theorem putnam_1980_b6
(G : (int * int)%type -> rat)
(hG : forall d n : nat, leq d n -> ((d = 1%nat -> G (d%:Z, n%:Z) = (1%:Q)/(n%:Q)) /\ (gtn d (1%na... | import Mathlib
open Set
/--
For integers $d, n$ with $1 \le d \le n$, let $G(1, n) = \frac{1}{n}$ and $G(d, n) = \frac{d}{n}\sum_{i=d}^{n}G(d - 1, i - 1)$ for all $d > 1$. If $1 < d \le p$ for some prime $p$, prove that the reduced denominator of $G(d, p)$ is not divisible by $p$.
-/
theorem putnam_1980_b6
(G : β€ Γ β€... |
putnam_1962_a2 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype sequences topology derive measure lebesgue_measure lebesgue_integral.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ri... | import Mathlib
open MeasureTheory Set
abbrev putnam_1962_a2_solution : Set (β β β) := sorry
-- {f : β β β | β a c : β, a β₯ 0 β§ f = fun x β¦ a / (1 - c * x) ^ 2}
/--
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 me... |
putnam_2015_a2 | Require Import Nat Reals Znumtheory BinInt. From mathcomp Require Import div.
Definition putnam_2015_a2_solution : nat := 181.
Theorem putnam_2015_a2
(A := fix a (n: nat) : Z :=
match n with
| O => 1%Z
| S O => 2%Z
| S ((S n'') as n') => Z.sub (4*(a n')) (a n'')
end)
: odd p... | import Mathlib
-- Note: this problem admits several possible correct solutions; this is the one shown on the solutions document
abbrev putnam_2015_a2_solution : β := sorry
-- 181
/--
Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$.
-/
theorem putnam_2015_a2
(a : ... |
putnam_1990_a2 | Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1990_a2_solution := True.
Theorem putnam_1990_a2
(numform : R -> Prop := fun x => exists (n m: nat), x = Rpower (INR n) (1/3) - Rpower (INR m) (1/3))
: (exists (s: nat -> R), (forall (i: nat), numform (s i)) /\ is_lim_seq s (sqrt 2)) <-... | import Mathlib
open Filter Topology Nat
abbrev putnam_1990_a2_solution : Prop := sorry
-- True
/--
Is $\sqrt{2}$ the limit of a sequence of numbers of the form $\sqrt[3]{n}-\sqrt[3]{m}$ ($n,m=0,1,2,\dots$)?
-/
theorem putnam_1990_a2
(numform : β β Prop)
(hnumform : β x : β, numform x β β n m : β, x = n ^ ((1 : β)... |
putnam_2013_b5 | Require Import Basics Reals Ensembles Finite_sets. From mathcomp Require Import fintype.
Theorem putnam_2013_b5
(n : nat)
(composen := fix compose_n (f : 'I_n -> 'I_n) (n : nat) :=
match n with
| O => fun x => x
| S n' => compose f (compose_n f n')
end)
(k : 'I_n)
(npos: ge n... | import Mathlib
open Function Set
/--
Let $X=\{1,2,\dots,n\}$, and let $k \in X$. Show that there are exactly $k \cdot n^{n-1}$ functions $f:X \to X$ such that for every $x \in X$ there is a $j \geq 0$ such that $f^{(j)}(x) \leq k$. [Here $f^{(j)}$ denotes the $j$\textsuperscript{th} iterate of $f$, so that $f^{(0)}(x... |
putnam_2003_b1 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_2003_b1_solution : Prop := False.
Theorem putnam_2003_b1
: (exists a ... | import Mathlib
open MvPolynomial Set
abbrev putnam_2003_b1_solution : Prop := sorry
-- False
/--
Do there exist polynomials $a(x), b(x), c(y), d(y)$ such that \[ 1 + xy + x^2y^2 = a(x)c(y) + b(x)d(y)\] holds identically?
-/
theorem putnam_2003_b1
: (β a b c d : Polynomial β, (β x y : β, 1 + x * y + x ^ 2 * y ^ 2 = a.... |
putnam_1974_a4 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences exp.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_1974_a4_solution : nat -> R := fun n => (n%:R / 2 ^+ (n.-1)... | import Mathlib
open Set Nat
noncomputable abbrev putnam_1974_a4_solution : β β β := sorry
-- (fun n β¦ (1 : β) / (2 ^ (n - 1)) * (n * (n - 1).choose βn / 2ββ))
/--
Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$.
-/
theorem putnam_1974_a4
(n : β)
(hn : 0 < n) :
(1 : β) / (... |
putnam_1985_b3 | Require Import Nat Ensembles Finite_sets.
Theorem putnam_1985_b3
(a : nat -> nat -> nat)
(apos : forall m n : nat, a m n > 0)
(ha : forall k : nat, k > 0 -> cardinal (nat * nat) (fun t => let (m, n) := t in m > 0 /\ n > 0 /\ a m n = k) 8)
: exists m n : nat, m > 0 /\ n > 0 /\ a m n > m * n.
Proof. Admit... | import Mathlib
open Set Filter Topology Real Polynomial Function
/--
Let
\[
\begin{array}{cccc} a_{1,1} & a_{1,2} & a_{1,3} & \dots \\
a_{2,1} & a_{2,2} & a_{2,3} & \dots \\
a_{3,1} & a_{3,2} & a_{3,3} & \dots \\
\vdots & \vdots & \vdots & \ddots
\end{array}
\]
be a doubly infinite array of positive integers, and sup... |
putnam_1996_a6 | Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1996_a6_solution (c: R) (f: R -> R) := if Rle_dec c (1/4) then (exists (d: R), f = (fun _ => d)) else ((forall (x: R), 0 <= x <= c -> continuity_pt f x) /\ f 0 = f c /\ (forall (x: R), x > 0 -> f x = f (pow x 2 + c)) /\ (forall (x: R), x < 0 -... | import Mathlib
open Function
abbrev putnam_1996_a6_solution : β β Set (β β β) := sorry
-- (fun c : β => if c β€ 1 / 4 then {f : β β β | β d : β, β x : β, f x = d} else {f : β β β | ContinuousOn f (Set.Icc 0 c) β§ f 0 = f c β§ (β x > 0, f x = f (x ^ 2 + c)) β§ (β x < 0, f x = f (-x))})
/--
Let $c>0$ be a constant. Give a ... |
putnam_1971_b2 | Require Import Reals Ensembles Coquelicot.Coquelicot.
Definition putnam_1971_b2_solution : Ensemble (R -> R) := fun f => (f = fun x : R => (x^3 - x^2 - 1)/(2 * x * (x-1))).
Theorem putnam_1971_b2
(S : Ensemble R := fun x => x <> 0 /\ x <> 1)
(P : (R -> R) -> Prop := fun F => forall x : R, In _ S x -> F x + F ((... | import Mathlib
open Set MvPolynomial
abbrev putnam_1971_b2_solution : Set (β β β) := sorry
-- {fun x : β => (x^3 - x^2 - 1)/(2 * x * (x - 1))}
/--
Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, ... |
putnam_1996_a5 | Require Import Binomial Reals Znumtheory Coquelicot.Coquelicot. From mathcomp Require Import div.
Open Scope R.
Theorem putnam_1996_a5
(p : nat)
(hp : prime (Z.of_nat p) /\ gt p 3)
(k : nat := Z.to_nat (floor (2 * INR p / 3)))
: exists (m : nat), sum_n_m (fun i => Binomial.C p i) 1 k = INR m * pow (INR... | import Mathlib
open Function
/--
If $p$ is a prime number greater than 3 and $k = \lfloor 2p/3 \rfloor$, prove that the sum \[\binom p1 + \binom p2 + \cdots + \binom pk \] of binomial coefficients is divisible by $p^2$.
-/
theorem putnam_1996_a5
(p : β)
(hpprime : Prime p)
(hpge3 : p > 3)
(k : β)
(hk : k = Nat.floor ... |
putnam_1963_b6 | Require Import Ensembles. From GeoCoq Require Import Main.Tarski_dev.Ch16_coordinates_with_functions.
(* Note: This formalization assumes a 3D space; 1D and 2D spaces can be seen as lines and planes in this larger space. *)
Context `{T3D:Tarski_3D}.
Theorem putnam_1963_b6
(T : Ensemble Tpoint -> Ensemble Tpoint := ... | import Mathlib
open Topology Filter Polynomial
/--
Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of all points that lie on closed segments joining pairs of points of $A$. For a given nonempty set $A_0$, define $A_n=S(A_{n-1})$ for $n=1,2,\dot... |
putnam_1978_a5 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals trigo.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1978_a5
(A : seq R)
(hanemp : gt (size A) 0)
(ha : all (fu... | import Mathlib
open Set Real
/--
Let $a_1, a_2, \dots , a_n$ be reals in the interval $(0, \pi)$ with arithmetic mean $\mu$. Show that
\[
\prod_{i=1}^n \left( \frac{\sin a_i}{a_i} \right) \leq \left( \frac{\sin \mu}{\mu} \right)^n.
\]
-/
theorem putnam_1978_a5
(n : β)
(npos : n > 0)
(a : Fin n β β)
(ha : β i : Fin n,... |
putnam_1972_b3 | From mathcomp Require Import fingroup.
Open Scope group_scope.
(* Note: This formalization is only for finite groups (due to mathcomp), but this is sufficiently general since the group generated by A and B is finite. *)
Variable T : finGroupType.
Theorem putnam_1972_b3
(G : {group T})
(A B : T)
(hab : A * B... | import Mathlib
open EuclideanGeometry Filter Topology Set MeasureTheory Metric
/--
Let $A$ and $B$ be two elements in a group such that $ABA = BA^2B$, $A^3 = 1$, and $B^{2n-1} = 1$ for some positive integer $n$. Prove that $B = 1$.
-/
theorem putnam_1972_b3
(G : Type*) [Group G]
(A B : G)
(hab : A * B * A = B * A^2 *... |
putnam_1964_b3 | From mathcomp Require Import all_ssreflect all_algebra.
From mathcomp Require Import reals sequences normedtype topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local... | import Mathlib
open Set Function Filter Topology
/--
Suppose $f : \mathbb{R} \to \mathbb{R}$ is continuous and for every $\alpha > 0$, $\lim_{n \to \infty} f(n\alpha) = 0$. Prove that $\lim_{x \to \infty} f(x) = 0$.
-/
theorem putnam_1964_b3
(f : β β β)
(hf : Continuous f β§ β Ξ± > 0, Tendsto (fun n : β β¦ f (n * Ξ±)) at... |
putnam_2000_a2 | Require Import Reals.
Open Scope Z.
Theorem putnam_2000_a2
: forall (m: Z), exists (n: Z), n >= m /\
exists (a1 a2 b1 b2 c1 c2: Z), n = a1*a1 + a2*a2 /\ n+1 = b1*b1 + b2*b2 /\ n+2 = c1*c1 + c2*c2.
Proof. Admitted.
| import Mathlib
open Topology Filter
/--
Prove that there exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers.
-/
theorem putnam_2000_a2 :
β n : β,
β N : β€,
β i : Fin 6 β β,
N > n β§
N = (i 0)^2 + (i 1)^2 β§
N + 1 = (i 2)^2 + (i 3)^... |
putnam_2001_b3 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2001_b3_solution : R := 3.
Theorem putnam_2001_b3
(closest : nat -> R := (fun n : nat => IZR (floor (sqrt (INR n) + 0.5))))
: Series (fun n : nat => sum_n_m (fun n' : nat => (Rpower 2 (closest n') + Rpower 2 (-closest n')) / (2 ^ n')) 1 n) = putnam_2... | import Mathlib
open Topology Filter Polynomial Set
abbrev putnam_2001_b3_solution : β := sorry
-- 3
/--
For any positive integer $n$, let $\langle n \rangle$ denote the closest integer to $\sqrt{n}$. Evaluate $\sum_{n=1}^\infty \frac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n}$.
-/
theorem putnam_2001_b3
: β' ... |
putnam_1985_a6 | From mathcomp Require Import all_ssreflect all_algebra.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Open Scope ring_scope.
Variable R : realType.
Definition putnam_1985_a6_solution : {poly R} := 6 * 'X ^ 2 + 5 * 'X + 1.
Theorem putnam_1985_a6
... | import Mathlib
open Set Filter Topology Real Polynomial
noncomputable abbrev putnam_1985_a6_solution : Polynomial β := sorry
-- 6 * X ^ 2 + 5 * X + 1
/--
If $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ is a polynomial with real coefficients $a_i$, then set
\[
\Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2.
\]
Let $F(x) = 3x... |
putnam_1966_a5 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals topology normedtype.
From mathcomp Require Import classical_sets.
Import numFieldTopology.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope ... | import Mathlib
open Topology Filter
/--
Let $C$ be the set of continuous functions $f : \mathbb{R} \to \mathbb{R}$. Let $T : C \to C$ satisfty the following two properties:
\begin{enumerate}
\item Linearity: $T(af + bg) = aT(f) + bT(g)$ for all $a, b \in \mathbb{R}$ and all $f, g \in C$.
\item Locality: If $f \in C$ ... |
putnam_2021_b2 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences topology normedtype.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local... | import Mathlib
open Filter Topology
noncomputable abbrev putnam_2021_b2_solution : β := sorry
-- 2 / 3
/--
Determine the maximum value of the sum $S = \sum_{n=1}^\infty \frac{n}{2^n}(a_1a_2 \dots a_n)^{1/n}$ over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying $\sum_{k=1}^\infty a_k=1$.
-/
th... |
putnam_1988_b3 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_1988_b3_solution := (1 + sqrt 3) / 2.
Theorem putnam_1988_b3
(r : Z -> R)
(hr : forall (n: Z), Z.ge n 1 -> (exists c d : Z, Z.ge c 0 /\ Z.ge d 0 /\ (Z.add c d) = n /\ r n = Rabs (IZR c - IZR d * sqrt 3)) /\ (forall c d : Z, (Z.ge c 0 /\ Z.ge d 0 /\ (... | import Mathlib
open Set Filter Topology
noncomputable abbrev putnam_1988_b3_solution : β := sorry
-- (1 + Real.sqrt 3) / 2
/--
For every $n$ in the set $N=\{1,2,\dots\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, th... |
putnam_1987_b2 | Require Import Binomial Reals Coquelicot.Coquelicot.
Theorem putnam_1987_b2
: forall (n r s: nat), ge n (r + s) ->
sum_n (fun i => Binomial.C s i / Binomial.C n (r + i)) s = (INR n + 1)/((INR n + 1 - INR s) * Binomial.C (n - s) r).
Proof. Admitted.
| import Mathlib
open MvPolynomial Real Nat
/--
Let $r, s$ and $t$ be integers with $0 \leq r$, $0 \leq s$ and $r+s \leq t$. Prove that
\[
\frac{\binom s0}{\binom tr}
+ \frac{\binom s1}{\binom{t}{r+1}} + \cdots
+ \frac{\binom ss}{\binom{t}{r+s}}
= \frac{t+1}{(t+1-s)\binom{t-s}{r}}.
\]
-/
theorem putnam_1987_b2
(r s t :... |
putnam_2003_a2 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences normedtype.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_2003_a2
(n : nat)
(hn : gt n 0)
(a b :... | import Mathlib
open MvPolynomial
/--
Let $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_n$ be nonnegative real numbers. Show that $(a_1a_2 \cdots a_n)^{1/n}+(b_1b_2 \cdots b_n)^{1/n} \leq [(a_1+b_1)(a_2+b_2) \cdots (a_n+b_n)]^{1/n}$.
-/
theorem putnam_2003_a2
(n : β)
(hn : 0 < n)
(a b : Fin n β β)
(abnneg :... |
putnam_1980_a5 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals trigo lebesgue_integral lebesgue_measure measure.
From mathcomp Require Import classical_sets cardinality.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
... | import Mathlib
/--
Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations $0=\int_0^xP(t)\sin t\,dt=\int_0^xP(t)\cos t\,dt$ has only finitely many real solutions $x$.
-/
theorem putnam_1980_a5
(P : Polynomial β)
(Pnonconst : P.degree > 0) :
Set.Finite... |
putnam_2003_b6 | Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2003_b6
(f : R -> R)
(hf : continuity f)
: RInt (fun x => RInt (fun y => Rabs (f x + f y)) 0 1) 0 1 >= RInt (fun x => Rabs (f x)) 0 1.
Proof. Admitted.
| import Mathlib
open MvPolynomial Set Nat
/--
Let $f(x)$ be a continuous real-valued function defined on the interval $[0,1]$. Show that \[ \int_0^1 \int_0^1 | f(x) + f(y) |\,dx\,dy \geq \int_0^1 |f(x)|\,dx. \]
-/
theorem putnam_2003_b6
(f : β β β)
(hf : Continuous f)
: (β« x in (0 : β)..1, (β« y in (0 : β)..1, |f x ... |
putnam_1985_a3 | Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1985_a3_solution (x: R) := exp x - 1.
Theorem putnam_1985_a3
(x: R)
(A := fix a (i j: nat) :=
match (i,j) with
| (i, 0) => x/pow 2 i
| (i, S j') => pow (a i j') 2 + 2 * a i j'
end)
: Lim_seq (fun n => A... | import Mathlib
open Set Filter Topology Real
noncomputable abbrev putnam_1985_a3_solution : β β β := sorry
-- fun d β¦ exp d - 1
/--
Let $d$ be a real number. For each integer $m \geq 0$, define a sequence $\{a_m(j)\}$, $j=0,1,2,\dots$ by the condition
\begin{align*}
a_m(0) &= d/2^m, \\
a_m(j+1) &= (a_m(j))^2 + 2a_m(j... |
putnam_2021_a5 | Require Import Nat. From mathcomp Require Import bigop div fintype eqtype seq ssrbool ssrnat.
Variables (I : finType) (P : pred I).
Definition putnam_2021_a5_solution (n: nat) := ~ (n %| 42 \/ n %| 46).
Theorem putnam_2021_a5
(A : pred 'I_2021 := fun n => let m := nat_of_ord n in ((1 <= m <= 2021) && (gcd m 2021 ... | import Mathlib
open Filter Topology
abbrev putnam_2021_a5_solution : Set β := sorry
-- {j : β | Β¬(42 β£ j) β§ Β¬(46 β£ j)}
/--
Let $A$ be the set of all integers $n$ such that $1 \leq n \leq 2021$ and $\gcd(n,2021)=1$. For every nonnegative integer $j$, let $S(j)=\sum_{n \in A}n^j$. Determine all values of $j$ such that ... |
putnam_2019_b2 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2019_b2_solution := 8 / PI ^ 3.
Theorem putnam_2019_b2
(a : nat -> R := fun n => sum_n_m (fun k => let k := INR k in let n := INR n in (sin (2 * (k + 1) * PI / (2 * n))) / ((cos ((k - 1) * PI / (2 * n))) ^ 2 * (cos ((k * PI) / (2 * n))) ^ 2)) 1 (n - 1))... | import Mathlib
open Topology Filter Set
noncomputable abbrev putnam_2019_b2_solution : β := sorry
-- 8/Real.pi^3
/--
For all $n \geq 1$, let
\[
a_n = \sum_{k=1}^{n-1} \frac{\sin \left( \frac{(2k-1)\pi}{2n} \right)}{\cos^2 \left( \frac{(k-1)\pi}{2n} \right) \cos^2 \left( \frac{k\pi}{2n} \right)}.
\]
Determine
\[
\lim_... |
putnam_1993_b5 | Require Import ZArith Reals Coquelicot.Coquelicot. From mathcomp Require Import fintype.
Theorem putnam_1993_b5
(pdists : ('I_4 -> (R * R)) -> Prop)
(hpdists : forall p : 'I_4 -> (R * R), pdists p = (forall i j : 'I_4, i <> j -> (exists k : Z, IZR k = norm (fst (p i) - fst (p j), (snd (p i) - snd (p j))) /\ Z.o... | import Mathlib
/--
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
-/
theorem putnam_1993_b5:
Β¬β p : Fin 4 β (EuclideanSpace β (Fin 2)),
β i j, i β j β
(β n : β€, dist (p i) (p j) = n β§ Odd n) :=
sorry
|
putnam_1967_b2 | From mathcomp Require Import all_ssreflect all_algebra.
From mathcomp Require Import reals normedtype.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1967_b2
(p r A ... | import Mathlib
open Nat Topology Filter
/--
Let $0 \leq p \leq 1$ and $0 \leq r \leq 1$ and consider the identities
\begin{enumerate}
\item[(a)] $(px+(1-p)y)^2=Ax^2+Bxy+Cy^2$,
\item[(b)] $(px+(1-p)y)(rx+(1-r)y)=\alpha x^2+\beta xy+\gamma y^2$.
\end{enumerate}
Show that (with respect to $p$ and $r$)
\begin{enumerate}
... |
putnam_2023_b2 | Require Import BinNums Nat NArith.
Definition putnam_2023_b2_solution := 3.
Theorem putnam_2023_b2
(k := fix count_ones (n : positive) : nat :=
match n with
| xH => 1
| xO n' => count_ones n'
| xI n' => 1 + count_ones n'
end)
: (forall (n: nat), n > 0 -> k (Pos.of_nat (2023*n... | import Mathlib
open Nat
abbrev putnam_2023_b2_solution : β := sorry
-- 3
/--
For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 * n$. What is the minimum value of $k(n)$?
-/
theorem putnam_2023_b2
: sInf {(digits 2 (2023*n)).sum | n > 0} = putnam_2023_b2_solution :=
... |
putnam_1968_b4 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences topology normedtype measure lebesgue_measure lebesgue_integral.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit... | import Mathlib
open Finset Polynomial Topology Filter Metric
/--
Suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous on $(-\infty, \infty)$ and that $\int_{-\infty}^{\infty} f(x) dx$ exists. Prove that $$\int_{-\infty}^{\infty} f\left(x - \frac{1}{x}\right) dx = \int_{-\infty}^{\infty} f(x) dx.$$
-/
theorem pu... |
putnam_2018_b2 | Require Import Reals Coquelicot.Coquelicot. From Coqtail Require Import Cpow.
Open Scope C_scope.
Theorem putnam_2018_b2
(n : nat)
(hn : gt n 0)
(f : nat -> C -> C)
(hf : forall z : C, f n z = sum_n_m (fun i => (((RtoC (INR n)) - (RtoC (INR i))) * z ^ i)) 0 (n-1))
: forall (z : C), Cnorm z <= 1 -> f... | import Mathlib
/--
Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \cdots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \leq 1\}$.
-/
theorem putnam_2018_b2
(n : β)
(hn : n > 0)
(f : β β β β β)
(hf : β z : β, f n z = β i β Finset.range n, (n - i) * ... |
putnam_2001_a5 | Require Import Nat.
Theorem putnam_2001_a5
: exists! (a n: nat), a > 0 /\ n > 0 /\ a ^ (n + 1) - (a + 1) ^ n = 2001.
Proof. Admitted.
| import Mathlib
open Topology Filter Polynomial Set
/--
Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.
-/
theorem putnam_2001_a5
: β! an : β€ Γ β, let (a, n) := an; a > 0 β§ n > 0 β§ a^(n+1) - (a+1)^n = 2001 :=
sorry
|
putnam_2016_b6 | Require Import List Reals Coquelicot.Hierarchy Coquelicot.Series.
Definition putnam_2016_b6_solution := 1.
Theorem putnam_2016_b6:
Series (fun k => (-1)^k/(INR k+1) * Series (fun n => 1/(INR k*(2^n)+1))) = putnam_2016_b6_solution.
Proof. Admitted.
| import Mathlib
open Polynomial Filter Topology Real Set Nat List
abbrev putnam_2016_b6_solution : β := sorry
-- 1
/--
Evaluate $\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n+1}$.
-/
theorem putnam_2016_b6 :
β' k : β, ((-1 : β) ^ ((k + 1 : β€) - 1) / (k + 1 : β)) * β' n : β, (1 : β) / ((k + 1... |
putnam_1974_a6 | From mathcomp Require Import all_algebra all_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Definition putnam_1974_a6_solution : nat := 25%nat.
Theorem putnam_1974_a6
(hdivnallx : {poly int} -> Prop := fun f => (f \is monic) /\ (forall x ... | import Mathlib
open Set Nat Polynomial
abbrev putnam_1974_a6_solution : β := sorry
-- 25
/--
Given $n$, let $k(n)$ be the minimal degree of any monic integral polynomial $f$ such that the value of $f(x)$ is divisible by $n$ for every integer $x$. Find the value of $k(1000000)$.
-/
theorem putnam_1974_a6
(hdivnallx : ... |
putnam_1984_b1 | Require Import Factorial ZArith.
Open Scope Z.
Fixpoint nat_sum (a : nat -> nat) (k : nat) : nat :=
match k with
| O => a O
| S k' => a k + nat_sum a k'
end.
Fixpoint Z_sum (a : nat -> Z) (k : nat) : Z :=
match k with
| O => a O
| S k' => a k + Z_sum a k'
end.
Definition putnam_1984_b1... | import Mathlib
open Topology Filter Nat
-- Note: This problem may have multiple correct answers.
noncomputable abbrev putnam_1984_b1_solution : Polynomial β Γ Polynomial β := sorry
-- (Polynomial.X + 3, -Polynomial.X - 2)
/--
Let $n$ be a positive integer, and define $f(n)=1!+2!+\dots+n!$. Find polynomials $P(x)$ and... |
putnam_2006_b2 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Open Scope ring_scope.
Variable R : realType.
Theorem putnam_2006_b2
(n : nat)
(hn : gt n 0)
(X : seq R)
(hX : uniq X /\ size X... | import Mathlib
/--
Prove that, for every set $X = \{x_1, x_2, \dots, x_n\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that
\[
\left| m + \sum_{s \in S} s \right| \leq \frac{1}{n+1}.
\]
-/
theorem putnam_2006_b2
(n : β)
(npos : n > 0)
(X : Finset β)
(hXcard : X.card = n)
: ... |
putnam_1966_a1 | From mathcomp Require Import all_ssreflect ssrnum ssralg ssrint.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Theorem putnam_1966_a1
(f : nat -> int := fun n => \sum_(0 <= m < n + 1) (if (~~odd m) then (m%:Z)/2 else (m%:Z-1)/2))
: forall x y :... | import Mathlib
/--
Let $a_n$ denote the sequence $0, 1, 1, 2, 2, 3, \dots$, where $a_n = \frac{n}{2}$ if $n$ is even and $\frac{n - 1}{2}$ if n is odd. Furthermore, let $f(n)$ denote the sum of the first $n$ terms of $a_n$. Prove that all positive integers $x$ and $y$ with $x > y$ satisfy $xy = f(x + y) - f(x - y)$.
-... |
putnam_2012_a3 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2012_a3_solution (x: R) := sqrt (1 - x ^ 2).
Theorem putnam_2012_a3
(hf : (R -> R) -> Prop := fun f : R -> R => continuous_on (fun x : R => -1 <= x <= 1) f /\ (forall x : R, -1 <= x <= 1 -> f x = ((2 - x^2)/2)*f (x^2/(2 - x^2))) /\ f 0 = 1 /\ exists (y :... | import Mathlib
open Matrix Function
-- Note: uses (β β β) instead of (Set.Icc (-1 : β) 1 β β)
noncomputable abbrev putnam_2012_a3_solution : β β β := sorry
-- fun x : β => Real.sqrt (1 - x^2)
/--
Let $f: [-1, 1] \to \mathbb{R}$ be a continuous function such that
\begin{itemize}
\item[(i)]
$f(x) = \frac{2-x^2}{2} f \l... |
putnam_1979_a6 | From mathcomp Require Import all_algebra all_ssreflect fintype.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1979_b6
(p : seq R)
(hp : all (fun x => 0 <= x <= 1) p)
... | import Mathlib
open Set
/--
For all $i \in \{0, 1, \dots, n - 1\}$, let $p_i \in [0, 1]$. Prove that there exists some $x \in [0, 1]$ such that $$\sum_{i = 0}^{n - 1} \frac{1}{|x - p_i|} \le 8n\left(\sum_{i = 0}^{n-1} \frac{1}{2i + 1}\right).$$
-/
theorem putnam_1979_a6
(n : β)
(p : β β β)
(hp : β i β Finset.range n,... |
putnam_1978_a3 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals measure lebesgue_integral lebesgue_measure topology normedtype exp sequences.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_... | import Mathlib
open Set Polynomial
abbrev putnam_1978_a3_solution : β := sorry
-- 2
/--
Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let
\[
I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx.
\]
For which $k$ is $I_k$ smallest?
-/
theorem putnam_1978_a3
(p : Polynomial β)
(h... |
putnam_1987_b1 | Require Import Reals Coquelicot.Coquelicot.
Definition putnam_1987_b1_solution := 1.
Theorem putnam_1987_b1
: RInt (fun x => ln (9 - x) ^ (1/2) / ( ln (9 - x) ^ (1/2) + ln (x + 3) ^ (1/2))) 2 4 = putnam_1987_b1_solution.
Proof. Admitted.
| import Mathlib
open MvPolynomial Real Nat
abbrev putnam_1987_b1_solution : β := sorry
-- 1
/--
Evaluate
\[
\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
\]
-/
theorem putnam_1987_b1
: (β« x in (2)..4, sqrt (log (9 - x)) / (sqrt (log (9 - x)) + sqrt (log (x + 3))) = putnam_1987_b1_solution) :=
s... |
putnam_1963_a2 | From mathcomp Require Import all_algebra all_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope nat_scope.
Theorem putnam_1963_a2
(f : nat -> nat)
(hfpos : forall n : nat, 0 < f n)
(hfinc : forall i j : nat, 0 < i -> i < j -> f i < f j)
(hf2 ... | import Mathlib
open Topology Filter
/--
Let $\{f(n)\}$ be a strictly increasing sequence of positive integers such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every relatively prime pair of positive integers $m$ and $n$ (the greatest common divisor of $m$ and $n$ is equal to $1$). Prove that $f(n)=n$ for every positive in... |
putnam_1962_b2 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Open Scope ring_scope.
Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1962_... | import Mathlib
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. Prov... |
putnam_1975_a1 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definiti... | import Mathlib
open Polynomial
abbrev putnam_1975_a1_solution : ((β€ Γ β€) β β€) Γ ((β€ Γ β€) β β€) := sorry
-- (fun (a, b) => a + b + 1, fun (a, b) => a - b)
/--
If an integer $n$ can be written as the sum of two triangular numbers (that is, $n = \frac{a^2 + a}{2} + \frac{b^2 + b}{2}$ for some integers $a$ and $b$), expre... |
putnam_2016_b5 | Require Import Reals Rpower.
Open Scope R.
Definition putnam_2016_b5_solution : (R -> R) -> Prop := fun f => exists (c: R), c > 0 /\ forall (x: R), x > 1 -> f x = Rpower x c .
Theorem putnam_2016_b5
(f : R -> R)
(fle : Prop := (forall x : R, x > 1 -> f x > 1) /\ (forall x y : R, (x > 1 /\ y > 1 /\ x*x <= y <= x... | import Mathlib
open Polynomial Filter Topology Real Set Nat List
abbrev putnam_2016_b5_solution : Set (Set.Ioi (1 : β) β Set.Ioi (1 : β)) := sorry
-- {f : Set.Ioi (1 : β) β Set.Ioi (1 : β) | β c : β, c > 0 β§ β x : Set.Ioi (1 : β), (f x : β) = x ^ c}
/--
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\in... |
putnam_1964_b5 | From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences normedtype topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local... | import Mathlib
open Set Function Filter Topology
/--
Let $a_n$ be a strictly monotonic increasing sequence of positive integers. Let $b_n$ be the least common multiple of $a_1, a_2, \dots , a_n$. Prove that $\sum_{n=1}^{\infty} 1/b_n$ converges.
-/
theorem putnam_1964_b5
(a b : β β β)
(ha : StrictMono a β§ β n : β, a ... |
putnam_2010_a6 | Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2010_a6
(f: R -> R)
(hf : (forall (x: R), x >= 0 -> (forall (y: R), x < y -> f x > f y) /\ continuity_pt f x) /\ is_lim f p_infty 0)
: ~ ex_finite_lim (fun nInc => RInt (fun x => (f x - f (x + 1)) / f x) 0 nInc) p_infty.
Proof. Admitted.
| import Mathlib
open Filter Topology Set
-- Note: uses (β β β) instead of (Ici 0 β β)
/--
Let $f:[0,\infty)\to \mathbb{R}$ be a strictly decreasing continuous function
such that $\lim_{x\to\infty} f(x) = 0$. Prove that
$\int_0^\infty \frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.
-/
theorem putnam_2010_a6
(f : β β β)
(hf : (... |
putnam_1989_a2 | Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1989_a2_solution (a b: R) := (exp (pow (a*b) 2) - 1)/(a * b).
Theorem putnam_1989_a2
(a b: R)
(abpos : a > 0 /\ b > 0)
(f : R -> R -> R := fun x y => Rmax (pow (b*x) 2) (pow (a*y) 2))
: RInt (fun x => (RInt (fun y => exp (f x y)... | import Mathlib
noncomputable abbrev putnam_1989_a2_solution : β β β β β := sorry
-- (fun a b : β => (Real.exp (a ^ 2 * b ^ 2) - 1) / (a * b))
/--
Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive.
-/
theorem putnam_1989_a2
(a b : β)
(abpos : a > 0 β§ b > 0)
: β« x in Set.Ioo 0... |
putnam_2015_a5 | Require Import Nat Reals Arith Znumtheory Ensembles Finite_sets.
Open Scope nat_scope.
Theorem putnam_2015_a5
(q : nat)
(Nq : nat)
(qodd : odd q = true )
(qpos : q > 0)
(hNq : cardinal nat (fun a : nat => and (Rle R0 (INR a)) (Rle (INR a) (Rdiv (INR q) (INR 4))) /\ gcd a q = 1) Nq)
: odd Nq = t... | import Mathlib
/--
Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1$. Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8$.
-/
theorem putnam_2015_a5
(q : β)
... |
putnam_2014_a4 | From mathcomp.analysis Require Import probability.
From mathcomp Require Import all_ssreflect.
From mathcomp Require Import ssralg poly ssrnum ssrint interval finmap.
From mathcomp Require Import mathcomp_extra boolp classical_sets functions.
From mathcomp Require Import cardinality fsbigop.
From HB Require Import stru... | import Mathlib
open Topology Filter Nat
noncomputable abbrev putnam_2014_a4_solution : β := sorry
-- 1 / 3
/--
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1$, $E\left[X^2\right]=2$, and $E\left[X^3\right]=5$. (Here $E\left[Y\right]$ denotes the expectation of ... |
putnam_1971_b1 | Require Import Ensembles RelationClasses.
Theorem putnam_1971_b1
(S : Type)
(op : S -> S -> S)
(hself : forall x : S, op x x = x)
(h2 : forall x y z : S, op (op x y) z = op (op y z) x)
: (forall x y z : S, op (op x y) z = op x (op y z)) /\ (forall x y : S, op x y = op y x).
Proof. Admitted.
| import Mathlib
open Set MvPolynomial
/--
Let $S$ be a set and let $\cdot$ be a binary operation on $S$ satisfying the two following laws: (1) for all $x$ in $S$, $x = x \cdot x$, (2) for all $x,y,z$ in $S$, $(x \cdot y) \cdot z) = (y \cdot z) \cdot x$. Show that $\cdot$ is associative and commutative.
-/
theorem putn... |
putnam_1982_a5 | Require Import Reals.
Open Scope R.
Theorem putnam_1982_a5
(a b c d: nat)
(hpos : Nat.lt 0 a /\ Nat.lt 0 b /\ Nat.lt 0 c /\ Nat.lt 0 d)
(habcd : Nat.le (Nat.add a c) 1982 /\ INR a / INR b + INR c / INR d < 1)
: 1 - INR a / INR b - INR c / INR d > 1/pow 1983 3.
Proof. Admitted.
| import Mathlib
/--
Let $a, b, c, d$ be positive integers satisfying $a + c \leq 1982$ and $\frac{a}{b} + \frac{c}{d} < 1$. Prove that $1 - \frac{a}{b} - \frac{c}{d} > \frac{1}{1983^3}$.
-/
theorem putnam_1982_a5
(a b c d : β€)
(hpos : a > 0 β§ b > 0 β§ c > 0 β§ d > 0)
(hac : a + c β€ 1982)
(hfrac : (a : β) / b + (c : β) / ... |
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