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putnam_1966_b6
|
From mathcomp Require Import all_ssreflect ssrnum ssralg.
From mathcomp Require Import reals exp sequences derive topology normedtype.
From mathcomp Require Import classical_sets cardinality.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1966_b6
(y : R -> R)
(hy : forall x : R, differentiable y x /\ differentiable (y^`()) x)
(diffeq : forall x : R, y^`(2) x + y x * expR x = 0)
: exists r s N : R, forall x : R, x > N -> r <= y x <= s.
Proof. Admitted.
|
import Mathlib
open Topology Filter
/--
Prove that any solution $y(x)$ to the differential equation $y'' + e^{x}y = 0$ remains bounded as $x$ goes to $+\infty$.
-/
theorem putnam_1966_b6
(y : β β β)
(hy : Differentiable β y β§ Differentiable β (deriv y))
(diffeq : deriv (deriv y) + Real.exp * y = 0)
: β r s N : β, β x > N, r β€ y x β§ y x β€ s :=
sorry
|
putnam_1994_b4
|
Require Import Nat List Reals Coquelicot.Coquelicot.
Import ListNotations.
Theorem putnam_1994_b4
(gcdn := fix gcd_n (args : list nat) : nat :=
match args with
| nil => 0%nat
| h :: args' => gcd h (gcd_n args')
end)
(Mmultn := fix Mmult_n {T : Ring} {n : nat} (A : matrix n n) (p : nat) :=
match p with
| O => mk_matrix n n (fun i j : nat => if Nat.eqb i j then one else zero)
| S p' => @Mmult T n n n A (Mmult_n A p')
end)
(A := mk_matrix 2 2 (fun i j =>
match i, j with
| 0, 0 => 3 | 0, 1 => 2
| 1, 0 => 4 | 1, 1 => 3
| _, _ => 0
end))
(I := mk_matrix 2 2 (fun i j =>
match i, j with
| 0, 0 => 1 | 0, 1 => 0
| 1, 0 => 0 | 1, 1 => 1
| _, _ => 0
end))
(dn_mat := fun n => Mplus (Mmultn A n) (opp I))
(dn := fun n => gcdn [Z.to_nat (floor (coeff_mat 0 (dn_mat n) 0 0));
Z.to_nat (floor (coeff_mat 0 (dn_mat n) 0 1));
Z.to_nat (floor (coeff_mat 0 (dn_mat n) 1 0));
Z.to_nat (floor (coeff_mat 0 (dn_mat n) 1 1))])
: forall k : nat, exists N : nat, forall n : nat, ge n N -> ge (dn n) k.
Proof. Admitted.
|
import Mathlib
open Filter Topology
/--
For $n \geq 1$, let $d_n$ be the greatest common divisor of the entries of $A^n-I$, where $A=\begin{pmatrix} 3 & 2 \\ 4 & 3 \end{pmatrix}$ and $I=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Show that $\lim_{n \to \infty} d_n=\infty$.
-/
theorem putnam_1994_b4
(matgcd : Matrix (Fin 2) (Fin 2) β€ β β€)
(A : Matrix (Fin 2) (Fin 2) β€)
(d : β β β€)
(hmatgcd : β M, matgcd M = Int.gcd (Int.gcd (Int.gcd (M 0 0) (M 0 1)) (M 1 0)) (M 1 1))
(hA : A 0 0 = 3 β§ A 0 1 = 2 β§ A 1 0 = 4 β§ A 1 1 = 3)
(hd : β n β₯ 1, d n = matgcd (A ^ n - 1))
: Tendsto d atTop atTop :=
sorry
|
putnam_1967_a3
|
From mathcomp Require Import all_ssreflect all_algebra.
From mathcomp Require Import reals normedtype.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_1967_a3_solution : nat := 5.
Theorem putnam_1967_a3
(pform pzeros pall : {poly R} -> Prop)
(hpform : forall p, pform p <-> size p = 3%nat /\ forall i : nat, lt i 3 -> p`_i = (floor p`_i)%:~R)
(hpzeros : forall p, pzeros p <-> exists z1 z2 : R, 0 < z1 < 1 /\ 0 < z2 < 1 /\ z1 <> z2 /\ p.[z1] = 0 /\ p.[z2] = 0)
(hpall : forall p, pall p <-> pform p /\ pzeros p /\ (p`_2 > 0)%R)
: (exists p, pall p /\ p`_2 = putnam_1967_a3_solution%:R) /\
(forall p, pall p -> p`_2 >= putnam_1967_a3_solution%:R).
Proof. Admitted.
|
import Mathlib
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 :
IsLeast
{a | β P : Polynomial β€,
P.degree = 2 β§
(β z1 z2 : Set.Ioo (0 : β) 1, z1 β z2 β§ aeval (z1 : β) P = 0 β§ aeval (z2 : β) P = 0) β§
P.coeff 2 = a β§ a > 0}
putnam_1967_a3_solution :=
sorry
|
putnam_1988_b2
|
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1988_b2_solution := True.
Theorem putnam_1988_b2
: (forall (a: R), a >= 0 -> forall (x: R), pow (x + 1) 2 >= a * (a + 1) ->
pow x 2 >= a * (a - 1)) <-> putnam_1988_b2_solution.
Proof. Admitted.
|
import Mathlib
open Set Filter Topology
abbrev putnam_1988_b2_solution : Prop := sorry
-- True
/--
Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$.
-/
theorem putnam_1988_b2
: (β x y : β, (y β₯ 0 β§ y * (y + 1) β€ (x + 1) ^ 2) β (y * (y - 1) β€ x ^ 2)) β putnam_1988_b2_solution :=
sorry
|
putnam_1986_a3
|
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1986_a3_solution := PI / 2.
Theorem putnam_1986_a3
(cot : R -> R)
(fcot : cot = fun t => cos t / sin t)
(arccot : R -> R)
(harccot : forall t : R, t >= 0 -> 0 < arccot t <= PI / 2 /\ cot (arccot t) = t)
: Series (fun n => arccot (pow (INR n) 2 + INR n + 1)) = putnam_1986_a3_solution.
Proof. Admitted.
|
import Mathlib
open Real
noncomputable abbrev putnam_1986_a3_solution : β := sorry
-- Real.pi / 2
/--
Evaluate $\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$.
-/
theorem putnam_1986_a3
(cot : β β β)
(fcot : cot = fun ΞΈ β¦ cos ΞΈ / sin ΞΈ)
(arccot : β β β)
(harccot : β t : β, t β₯ 0 β arccot t β Set.Ioc 0 (Real.pi / 2) β§ cot (arccot t) = t)
: (β' n : β, arccot (n ^ 2 + n + 1) = putnam_1986_a3_solution) :=
sorry
|
putnam_2004_b5
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences topology normedtype exp.
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 Open Scope classical_set_scope.
Variable R : realType.
Definition at_left := fun (x : R) => within (fun y => y < x) (nbhs x).
Definition putnam_2004_b5_solution : R := 2 / expR 1.
Theorem putnam_2004_b5
(xprod : R -> R)
(hxprod : forall x : R, 0 < x < 1 -> (fun N : nat => \prod_(0 <= n < N) (expR (ln ((1 + x ^ (n.+1))/(1 + x ^ n)) * (x ^ n)))) @ \oo --> xprod x)
: xprod @ (at_left 1) --> putnam_2004_b5_solution.
Proof. Admitted.
|
import Mathlib
open Nat Topology Filter
abbrev putnam_2004_b5_solution : β := sorry
-- 2 / Real.exp 1
/--
Evaluate $\lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.
-/
theorem putnam_2004_b5
(xprod : β β β)
(hxprod : β x β Set.Ioo 0 1,
Tendsto (fun N β¦ β n β Finset.range N, ((1 + x ^ (n + 1)) / (1 + x ^ n)) ^ (x ^ n))
atTop (π (xprod x))) :
Tendsto xprod (π[<] 1) (π putnam_2004_b5_solution) :=
sorry
|
putnam_1965_a5
|
Require Import Nat Finite_sets. From mathcomp Require Import fintype perm.
Definition putnam_1965_a5_solution : nat -> nat := (fun n : nat => 2 ^ (n - 1)).
Theorem putnam_1965_a5
: forall n : nat, n > 0 -> cardinal {perm 'I_n} (fun p : {perm 'I_n} => forall m : 'I_n, m > 0 -> exists k : 'I_n, k < m /\ (nat_of_ord (p m) = (p k) + 1 \/ nat_of_ord (p m) = (p k) - 1)) (putnam_1965_a5_solution n).
Proof. Admitted.
|
import Mathlib
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 $1$?
-/
theorem putnam_1965_a5
: β n > 0, {p β permsOfFinset (Finset.Icc 1 n) | β m β Finset.Icc 2 n, β k β Finset.Ico 1 m, p m = p k + 1 β¨ p m = p k - 1}.card = putnam_1965_a5_solution n :=
sorry
|
putnam_1982_b5
|
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1982_b5
(F := fix f (n: nat) (x: R) :=
match n with
| O => exp 1
| S n' => ln x / ln (f n' x)
end)
: forall (x: R), x > Rpower (exp 1) (exp 1) ->
ex_finite_lim_seq (fun n => F n x) /\
let g (x: R) := Lim_seq (fun n => F n x) in
continuity_pt g x.
Proof. Admitted.
|
import Mathlib
open Set Function Filter Topology Polynomial Real
/--
For all $x > e^e$, let $S = u_0, u_1, \dots$ be a recursively defined sequence with $u_0 = e$ and $u_{n+1} = \log_{u_n} x$ for all $n \ge 0$. Prove that $S_x$ converges to some real number $g(x)$ and that this function $g$ is continuous for $x > e^e$.
-/
theorem putnam_1982_b5
(T : Set β)
(hT : T = Ioi (Real.exp (Real.exp 1)))
(S : β β β β β)
(hS : β x β T, S x 0 = (Real.exp 1) β§ β n : β, S x (n + 1) = Real.logb (S x n) x)
(g : β β β)
: β x β T, (β L : β, Tendsto (S x) atTop (π L)) β§
(β x β T, Tendsto (S x) atTop (π (g x))) β ContinuousOn g T :=
sorry
|
putnam_2023_a1
|
From mathcomp Require Import all_ssreflect ssrnum ssralg.
From mathcomp Require Import reals trigo normedtype derive topology sequences.
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 Open Scope classical_set_scope.
Variable R : realType.
Definition putnam_2023_a1_solution : nat := 18.
Theorem putnam_2023_a1
(f : nat -> R -> R := fun n x => \prod_(1 <= i < n.+1) cos (i%:R * x))
: gt putnam_2023_a1_solution 0 /\ `|(f putnam_2023_a1_solution)^`(2) 0| > 2023 /\
forall n : nat, gt n 0 -> lt n putnam_2023_a1_solution -> `|(f n)^`(2) 0| <= 2023.
Proof. Admitted.
|
import Mathlib
open Nat
abbrev putnam_2023_a1_solution : β := sorry
-- 18
/--
For a positive integer $n$, let $f_n(x) = \cos(x) \cos(2x) \cos(3x) \cdots \cos(nx)$. Find the smallest $n$ such that $|f_n''(0)| > 2023$.
-/
theorem putnam_2023_a1
(f : β β β β β)
(hf : β n > 0, f n = fun x : β => β i β Finset.Icc 1 n, Real.cos (i * x)) :
IsLeast {n | 0 < n β§ |iteratedDeriv 2 (f n) 0| > 2023} putnam_2023_a1_solution :=
sorry
|
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 + f y|)) β₯ (β« x in (0 : β)..1, |f x|) :=
sorry
|
putnam_1979_a3
|
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.
Definition putnam_1979_a3_solution : (R*R)%type -> Prop := fun '(a, b) => exists m : int, a = m%:~R /\ b = m%:~R.
Theorem putnam_1979_a3
(x : R ^nat)
(hx : forall n : nat, x n <> 0 /\ (ge n 3 -> x n = (x (n.-2)) * (x (n.-1))/(2 * (x (n.-2)) - (x (n.-1)))))
: (forall m : nat, exists n : nat, gt n m /\ (exists a : int, x n = a%:~R)) <-> putnam_1979_a3_solution (x 1%nat, x 2%nat).
Proof. Admitted.
|
import Mathlib
abbrev putnam_1979_a3_solution : (β Γ β) β Prop := sorry
-- fun (a, b) => β m : β€, a = m β§ b = m
/--
Let $x_1, x_2, x_3, \dots$ be a sequence of nonzero real numbers such that $$x_n = \frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}}$$ for all $n \ge 3$. For which real values of $x_1$ and $x_2$ does $x_n$ attain integer values for infinitely many $n$?
-/
theorem putnam_1979_a3
(x : β β β)
(hx : β n : β, x n β 0 β§ (n β₯ 3 β x n = (x (n - 2))*(x (n - 1))/(2*(x (n - 2)) - (x (n - 1)))))
: (β m : β, β n : β, n > m β§ β a : β€, a = x n) β putnam_1979_a3_solution (x 1, x 2) :=
sorry
|
putnam_1965_b4
|
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.
Import Order.TTheory GRing.Theory Num.Theory.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition putnam_1965_b4_solution : ((((R -> R) -> (R -> R)) * ((R -> R) -> (R -> R))) * ((set R) * (R -> R))) :=
((fun h : R -> R => (fun x : R => h x + x), fun h : R -> R => (fun x => h x + 1)), ([set x : R | x >= 0], @Num.sqrt R)).
Theorem putnam_1965_b4
(f u v : nat -> R -> R)
(hu : forall n : nat, gt n 0 -> forall x : R, u n x = \sum_(0 <= i < n%/2 .+1) ('C(n, 2 * i)%:R * x^i))
(hv : forall n : nat, gt n 0 -> forall x : R, v n x = \sum_(0 <= i < (n.-1)%/2 .+1) ('C(n, 2 * (i.+1))%:R * x^i))
(hf : forall n : nat, gt n 0 -> forall x : R, f n x = u n x / v n x)
(n : nat)
(hn : gt n 0)
(f_seq : R -> (nat -> R) := fun (x : R) => fun (m : nat) => f m x) :
let '((p, q), (s, g)) := putnam_1965_b4_solution in
(forall x : R, v n x <> 0 -> v (n.+1) x <> 0 -> q (f n) x <> 0 -> f (n.+1) x = p (f n) x / q (f n) x) /\
s = [set x : R | exists l : R, f_seq x @ \oo --> l] /\
(forall x : R, x \in s -> (f_seq x) @ \oo --> g x).
Proof. Admitted.
|
import Mathlib
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 $$f(x, n) = \frac{{n \choose 0} + {n \choose 2}x + {n \choose 4}x^2 + \cdots}{{n \choose 1} + {n \choose 3}x + {n \choose 5}x^2 + \cdots}$$ for all real numbers $x$ and positive integers $n$. Express $f(x, n+1)$ as a rational function involving $f(x, n)$ and $x$, and find $\lim_{n \to \infty} f(x, n)$ for all $x$ for which this limit converges.
-/
theorem putnam_1965_b4
(f u v : β β β β β)
(hu : β n > 0, β x, u n x = β i β Finset.Icc 0 (n / 2), (n.choose (2 * i)) * x ^ i)
(hv : β n > 0, β x, v n x = β i β Finset.Icc 0 ((n - 1) / 2), (n.choose (2 * i + 1)) * x ^ i)
(hf : β n > 0, β x, f n x = u n x / v n x)
(n : β)
(hn : 0 < n) :
let β¨β¨p, qβ©, β¨s, gβ©β© := putnam_1965_b4_solution
(β x, v n x β 0 β v (n + 1) x β 0 β q (f n) x β 0 β f (n + 1) x = p (f n) x / q (f n) x) β§
s = {x | β l, Tendsto (fun n β¦ f n x) atTop (π l)} β§
β x β s, Tendsto (fun n β¦ f n x) atTop (π (g x)) :=
sorry
|
putnam_1986_a1
|
Require Import Reals Ensembles Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1986_a1_solution := 18.
Theorem putnam_1986_a1
(f : R -> R := fun x => pow x 3 - 3 * x)
(T : Ensemble R := fun x => pow x 4 + 36 <= 13 * pow x 2)
: (forall x : R, In R T x -> putnam_1986_a1_solution >= f x) /\
(exists x : R, In R T x /\ putnam_1986_a1_solution = f x).
Proof. Admitted.
|
import Mathlib
abbrev putnam_1986_a1_solution : β := sorry
-- 18
/--
Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \leq 13x^2$.
-/
theorem putnam_1986_a1
(S : Set β) (f : β β β)
(hS : S = {x : β | x ^ 4 + 36 β€ 13 * x ^ 2})
(hf : f = fun x β¦ x ^ 3 - 3 * x) :
IsGreatest
{f x | x β S}
putnam_1986_a1_solution :=
sorry
|
putnam_1966_b2
|
From mathcomp Require Import all_ssreflect all_algebra.
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.
Theorem putnam_1966_b2
(S : int -> set int := fun n => [set i | n <= i <= n + 9])
: forall n : int, n > 0 -> (exists k : int, k \in S n /\ (forall m : int, m \in S n -> k <> m -> gcdz m k = 1)).
Proof. Admitted.
|
import Mathlib
/--
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
|
putnam_1985_a5
|
Require Import Nat Reals List Coquelicot.Coquelicot.
Open Scope nat_scope.
Definition putnam_1985_a5_solution (n : nat) := n = 3 \/ n = 4 \/ n = 7 \/ n = 8.
Theorem putnam_1985_a5
(F : nat -> R -> R := fun n x => fold_right Rmult 1%R (map (fun i : nat => cos (INR i * x)) (seq 1 n)))
: forall (n: nat), 1 <= n <= 10 -> (RInt (F n) 0 (2*PI) <> 0%R <-> putnam_1985_a5_solution n).
Proof. Admitted.
|
import Mathlib
open Set Filter Topology Real
abbrev putnam_1985_a5_solution : Set β := sorry
-- {3, 4, 7, 8}
/--
Let $I_m = \int_0^{2\pi} \cos(x)\cos(2x)\cdots \cos(mx)\,dx$. For which integers $m$, $1 \leq m \leq 10$ is $I_m \neq 0$?
-/
theorem putnam_1985_a5
(I : β β β)
(hI : I = fun (m : β) β¦ β« x in (0)..(2 * Real.pi), β k β Finset.Icc 1 m, cos (k * x)) :
{m β Finset.Icc 1 10 | I m β 0} = putnam_1985_a5_solution :=
sorry
|
putnam_1983_a6
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_1983_a6_solution := 2 / 9.
Theorem putnam_1983_a6
(F : R -> R := fun a => (a ^ 4 / exp (a ^ 3)) * RInt (fun x => RInt (fun y => exp (x ^ 3 + y ^ 3)) 0 (a - x)) 0 a)
: filterlim F (Rbar_locally p_infty) (locally putnam_1983_a6_solution).
Proof. Admitted.
|
import Mathlib
open Nat Filter Topology Real
noncomputable abbrev putnam_1983_a6_solution : β := sorry
-- 2 / 9
/--
Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$.
-/
theorem putnam_1983_a6
(F : β β β)
(hF : F = fun a β¦ (a ^ 4 / exp (a ^ 3)) * β« x in (0)..a, β« y in (0)..(a - x), exp (x ^ 3 + y ^ 3))
: (Tendsto F atTop (π putnam_1983_a6_solution)) :=
sorry
|
putnam_1997_a5
|
Require Import Nat Ensembles Finite_sets List Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1997_a5_solution := True.
Theorem putnam_1997_a5
(E: Ensemble (list nat) := fun l => length l = 10%nat /\ (forall i : nat, lt i 10 -> gt (nth i l 0%nat) 0) /\ sum_n (fun i => 1/ INR (nth i l 0%nat)) 9 = 1)
(m: nat)
: cardinal (list nat) E m -> (odd m = true <-> putnam_1997_a5_solution).
Proof. Admitted.
|
import Mathlib
open Filter Topology
abbrev putnam_1997_a5_solution : Prop := sorry
-- True
/--
Let $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots + 1/a_n=1$. Determine whether $N_{10}$ is even or odd.
-/
theorem putnam_1997_a5
(N : (n : β+) β Set (Fin n β β+))
(hN : N = fun (n : β+) => {t : Fin n β β+ | (β i j : Fin n, i < j β t i <= t j) β§ (β i : Fin n, (1 : β)/(t i) = 1) })
: Odd (N 10).ncard β putnam_1997_a5_solution :=
sorry
|
putnam_2003_a3
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2003_a3_solution := 2 * sqrt 2 - 1.
Theorem putnam_2003_a3
(f : R -> R := fun x => Rabs (sin x + cos x + tan x + 1 / tan x + 1 / cos x + 1 / sin x))
: (exists x : R, f x = putnam_2003_a3_solution) /\ (forall x : R, f x >= putnam_2003_a3_solution).
Proof. Admitted.
|
import Mathlib
open Set
noncomputable abbrev putnam_2003_a3_solution : β := sorry
-- 2 * Real.sqrt 2 - 1
/--
Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$.
-/
theorem putnam_2003_a3
(f : β β β)
(hf : β x : β, f x = |Real.sin x + Real.cos x + Real.tan x + 1 / Real.tan x + 1 / Real.cos x + 1 / Real.sin x|) :
IsLeast (Set.range f) putnam_2003_a3_solution :=
sorry
|
putnam_1991_b4
|
Require Import Nat Reals ZArith Znumtheory Binomial Coquelicot.Coquelicot.
Theorem putnam_1991_b4
(p: nat)
(hp : odd p = true /\ prime (Z.of_nat p))
(expr : R := sum_n (fun j => Binomial.C p j * Binomial.C (p + j) j) p)
: (floor expr) mod (Z.pow (Z.of_nat p) 2) = Z.add (Z.pow 2 (Z.of_nat p)) 1.
Proof. Admitted.
|
import Mathlib
open Filter Topology
/--
Suppose $p$ is an odd prime. Prove that $\sum_{j=0}^p \binom{p}{j}\binom{p+j}{j} \equiv 2^p+1 \pmod{p^2}$.
-/
theorem putnam_1991_b4
(p : β)
(podd : Odd p)
(pprime : Prime p)
: (β j : Fin (p + 1), (p.choose j) * ((p + j).choose j)) β‘ (2 ^ p + 1) [MOD (p ^ 2)] :=
sorry
|
putnam_1972_b5
|
From GeoCoq Require Import Main.Tarski_dev.Ch16_coordinates_with_functions.
Context `{T3D:Tarski_3D}.
Theorem putnam_1972_b5
(A B C D : Tpoint)
(hnonplanar : ~Coplanar A B C D)
(hangles : Ang A B C = Ang C D A /\ Ang B C D = Ang D A B)
: (Cong A B C D /\ Cong B C D A).
Proof. Admitted.
|
import Mathlib
open EuclideanGeometry Set Metric
/--
Let $ABCD$ be a skew (non-planar) quadrilateral. Prove that if $\angle ABC = \angle CDA$ and $\angle BCD = \angle DAB$, then $AB = CD$ and $AD = BC$.
-/
theorem putnam_1972_b5
(A B C D : EuclideanSpace β (Fin 3))
(hnonplanar : Β¬Coplanar β {A, B, C, D})
(hangles : β A B C = β C D A β§ β B C D = β D A B)
: dist A B = dist C D β§ dist B C = dist D A :=
sorry
|
putnam_2022_b2
|
Require Import Ensembles Finite_sets Reals.
Require Import GeoCoq.Main.Tarski_dev.Ch16_coordinates_with_functions.
Context `{T2D:Tarski_2D} `{TE:@Tarski_euclidean Tn TnEQD}.
Definition vect3:= (F * F * F)%type.
Definition cross_prod (v w : vect3) := let '(v1, v2, v3) := v in let '(w1, w2, w3) := w in
(SubF (MulF v2 w3) (MulF v3 w2),
SubF (MulF v3 w1) (MulF v1 w3),
SubF (MulF v1 w2) (MulF v2 w1)).
Definition putnam_2022_b2_solution : Ensemble nat := fun n => n = 1 \/ n = 7.
Theorem putnam_2022_b2
(p : nat -> Prop := fun n => exists (A: Ensemble vect3), cardinal vect3 A n /\ (A = fun u => exists v w : vect3, In _ A v /\ In _ A w /\ u = cross_prod v w))
: forall (n: nat), (n > 0 /\ p n) <-> In _ putnam_2022_b2_solution n.
Proof. Admitted.
|
import Mathlib
open Polynomial
abbrev putnam_2022_b2_solution : Set β := sorry
-- {1, 7}
/--
Let $\times$ represent the cross product in $\mathbb{R}^3$. For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $S=\{v \times w:v,w \in S\}$?
-/
theorem putnam_2022_b2
(n : β)
(P : Finset (Fin 3 β β) β Prop)
(P_def : β S : Finset (Fin 3 β β), P S β (S = {u : Fin 3 β β | β v w : S, u = crossProduct v w})) :
(0 < n β§ β S : Finset (Fin 3 β β), S.card = n β§ P S) β n β putnam_2022_b2_solution :=
sorry
|
putnam_1962_b5
|
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_1962_b5
(n : nat)
(ng1 : gt n 1)
(sumf : nat -> R := fun N => \sum_(1 <= i < N.+1) ((i%:R / N%:R) ^+ N))
: (3 * (n%:R + 1) + 1) / (2 * n%:R + 2) < sumf n < 2.
Proof. Admitted.
|
import Mathlib
open MeasureTheory
/--
Prove that for every integer $n$ greater than 1: \[ \frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^n + \left(\frac{2}{n} \right)^n + \cdots + \left(\frac{n}{n} \right)^n < 2. \]
-/
theorem putnam_1962_b5
(n : β€)
(ng1 : n > 1)
: (3 * (n : β) + 1) / (2 * n + 2) < β i : Finset.Icc 1 n, ((i : β) / n) ^ (n : β) β§ β i : Finset.Icc 1 n, ((i : β) / n) ^ (n : β) < 2 :=
sorry
|
putnam_1998_b4
|
Require Import Nat ZArith Reals Coquelicot.Coquelicot.
Definition putnam_1998_b4_solution : nat -> nat -> Prop := (fun m n : nat => forall m2 n2 : nat, (m mod (2 ^ m2) = 0%nat /\ m mod (2 ^ (m2 + 1)) <> 0%nat /\ n mod (2 ^ n2) = 0%nat /\ n mod (2 ^ n2 + 1) <> 0%nat) -> m2 <> n2).
Theorem putnam_1998_b4
(hsum : nat -> nat -> R := (fun m n : nat => sum_n (fun i => Rpower (-1) (IZR (floor (INR i / INR m)) + IZR (floor (INR i / INR n)))) (m * n - 1)))
: forall (m n: nat), (gt m 0 /\ gt n 0) -> (hsum m n = 0 <-> putnam_1998_b4_solution m n).
Proof. Admitted.
|
import Mathlib
open Set Function Metric
abbrev putnam_1998_b4_solution : Set (β Γ β) := sorry
-- {nm | let β¨n,mβ© := nm; multiplicity 2 n β multiplicity 2 m}
/--
Find necessary and sufficient conditions on positive integers $m$ and $n$ so that \[\sum_{i=0}^{mn-1} (-1)^{\lfloor i/m \rfloor +\lfloor i/n\rfloor}=0.\]
-/
theorem putnam_1998_b4
(quantity : β β β β β€)
(hquantity : quantity = fun n m => β i β Finset.range (m * n), (-1)^(i/m + i/n))
(n m : β)
(hnm : n > 0 β§ m > 0) :
quantity n m = 0 β β¨n, mβ© β putnam_1998_b4_solution :=
sorry
|
putnam_2011_b6
|
Require Import Nat List Factorial Ensembles Finite_sets Reals Znumtheory ZArith Coquelicot.Coquelicot.
Open Scope nat_scope.
Theorem putnam_2011_b6
(p: nat)
(hp : prime (Z.of_nat p) /\ odd p = true)
: exists (E: Ensemble nat), (forall (n: nat), E n -> lt n p) /\ cardinal nat E ((p + 1) / 2) /\
forall (n: nat), E n -> Z.to_nat (floor (sum_n (fun k => INR (fact k * n ^ k)) (p - 1))) mod p <> 0.
Proof. Admitted.
|
import Mathlib
open Topology Filter Matrix Set
/--
Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\{0,1,2,\dots,p-1\}$,
\[
\sum_{k=0}^{p-1} k! n^k \qquad \mbox{is not divisible by $p$.}
\]
-/
theorem putnam_2011_b6
(p : β)
(hp : Odd p β§ Nat.Prime p)
: {n β Finset.range p | Β¬ p β£ β k : Finset.range p, Nat.factorial k * n^(k : β)}.card β₯ (p + 1)/2 :=
sorry
|
putnam_1974_b6
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import classical_sets cardinality.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope nat_scope.
Local Open Scope classical_set_scope.
Local Open Scope card_scope.
Definition putnam_1974_b6_solution : (nat * nat * nat) := ((2^1000 - 1) %/ 3, (2^1000 - 1) %/ 3, (2^1000 - 1) %/ 3 + 1).
Theorem putnam_1974_b6
(n : nat := 1000)
(count0 count1 count2 : nat)
(hcount0 : [set : 'I_(count0)] #= [set A : set nat | A `<=` [set x : nat | 1 <= x <= n] /\ exists j : nat, A #= [set : 'I_(3*j)]])
(hcount1 : [set : 'I_(count1)] #= [set A : set nat | A `<=` [set x : nat | 1 <= x <= n] /\ exists j : nat, A #= [set : 'I_(3*j+1)]])
(hcount2 : [set : 'I_(count2)] #= [set A : set nat | A `<=` [set x : nat | 1 <= x <= n] /\ exists j : nat, A #= [set : 'I_(3*j+2)]])
: (count0, count1, count2) = putnam_1974_b6_solution.
Proof. Admitted.
|
import Mathlib
open Set Nat Polynomial Filter Topology
abbrev putnam_1974_b6_solution : (β Γ β Γ β) := sorry
-- ((2^1000 - 1)/3, (2^1000 - 1)/3, 1 + (2^1000 - 1)/3)
/--
For a set with $1000$ elements, how many subsets are there whose candinality is respectively $\equiv 0 \bmod 3, \equiv 1 \bmod 3, \equiv 2 \bmod 3$?
-/
theorem putnam_1974_b6
(n : β€)
(hn : n = 1000)
(count0 count1 count2 : β)
(hcount0 : count0 = {S | S β Finset.Icc 1 n β§ S.card β‘ 0 [MOD 3]}.ncard)
(hcount1 : count1 = {S | S β Finset.Icc 1 n β§ S.card β‘ 1 [MOD 3]}.ncard)
(hcount2 : count2 = {S | S β Finset.Icc 1 n β§ S.card β‘ 2 [MOD 3]}.ncard)
: (count0, count1, count2) = putnam_1974_b6_solution :=
sorry
|
putnam_1967_a6
|
From mathcomp Require Import all_ssreflect all_algebra fintype seq ssrbool.
From mathcomp Require Import reals normedtype.
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 Scope card_scope.
Variable R : realType.
Definition putnam_1967_a6_solution : nat := 8.
Theorem putnam_1967_a6
(i0 : 'I_4) (idx : nat -> 'I_4 := fun i => nth i0 (ord_enum 4) i)
(abneq0 : ('I_4 -> R) -> ('I_4 -> R) -> Prop := fun a b => a (idx 0%nat) * b (idx 1%nat) - a (idx 1%nat) * b (idx 0%nat) <> 0)
(numtuples : ('I_4 -> R) -> ('I_4 -> R) -> nat)
(hnumtuples : forall a b, [set s : 'I_4 -> R | exists x : 'I_4 -> R,
(forall i, x i <> 0)
/\ (\sum_(i : 'I_4) ((a i) * (x i)) = 0)
/\ (\sum_(i : 'I_4) ((b i) * (x i)) = 0)
/\ (forall i, s i = @Num.sg R (x i))] #= [set: 'I_(numtuples a b)])
: ((exists a b, abneq0 a b /\ numtuples a b = putnam_1967_a6_solution) /\
(forall a b, abneq0 a b -> (numtuples a b <= putnam_1967_a6_solution)%nat)).
Proof. Admitted.
|
import Mathlib
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 $b_1x_1+b_2x_2+b_3x_3+b_4x_4=0$, for which no $x_i$ ($i=1,2,3,4$) is zero. Each such solution generates a $4$-tuple of plus and minus signs $(\text{signum }x_1,\text{signum }x_2,\text{signum }x_3,\text{signum }x_4)$. Determine, with a proof, the maximum number of distinct $4$-tuples possible.
-/
theorem putnam_1967_a6
(abneq0 : (Fin 4 β β) β (Fin 4 β β) β Prop)
(habneq0 : abneq0 = (fun a b : Fin 4 β β => a 0 * b 1 - a 1 * b 0 β 0))
(numtuples : (Fin 4 β β) β (Fin 4 β β) β β)
(hnumtuples : β a b : Fin 4 β β, numtuples a b = {s : Fin 4 β β | β x : Fin 4 β β, (β i : Fin 4, x i β 0) β§ (β i : Fin 4, a i * x i) = 0 β§ (β i : Fin 4, b i * x i) = 0 β§ (β i : Fin 4, s i = Real.sign (x i))}.encard)
: (β a b : Fin 4 β β, abneq0 a b β§ numtuples a b = putnam_1967_a6_solution) β§ (β a b : Fin 4 β β, abneq0 a b β numtuples a b β€ putnam_1967_a6_solution) :=
sorry
|
putnam_1986_a2
|
Require Import Nat.
Definition putnam_1986_a2_solution := 3.
Theorem putnam_1986_a2
: (10 ^ (20000) / (10 ^ (100) + 3)) mod 10 = putnam_1986_a2_solution.
Proof. Admitted.
|
import Mathlib
abbrev putnam_1986_a2_solution : β := sorry
-- 3
/--
What is the units (i.e., rightmost) digit of
\[
\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?
\]
-/
theorem putnam_1986_a2
: (Nat.floor ((10 ^ 20000 : β) / (10 ^ 100 + 3)) % 10 = putnam_1986_a2_solution) :=
sorry
|
putnam_2010_b5
|
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2010_b5_solution := False.
Theorem putnam_2010_b5
: (exists (f: R -> R), forall (x y: R), (x < y -> f x < f y) /\ ex_derive f x /\ Derive f x = f (f x)) <-> putnam_2010_b5_solution.
Proof. Admitted.
|
import Mathlib
open Filter Topology Set
abbrev putnam_2010_b5_solution : Prop := sorry
-- False
/--
Is there a strictly increasing function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x) = f(f(x))$ for all $x$?
-/
theorem putnam_2010_b5 :
(β f : β β β, StrictMono f β§ Differentiable β f β§ (β x : β, deriv f x = f (f x))) β putnam_2010_b5_solution :=
sorry
|
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 : f 2 = 2)
(hfmn : forall m n : nat, 0 < m -> 0 < n -> coprime m n -> f (m * n) = f m * f n)
: forall n : nat, 0 < n -> f n = n.
Proof. Admitted.
|
import Mathlib
open Topology Filter
/--
Let $\{f(n)\}$ be a strictly increasing sequence of positive integers such that $f(2)=2$ and $f(mn)=f(m)f(n)$ for every relatively prime pair of positive integers $m$ and $n$ (the greatest common divisor of $m$ and $n$ is equal to $1$). Prove that $f(n)=n$ for every positive integer $n$.
-/
theorem putnam_1963_a2
(f : β β β)
(hfpos : β n, f n > 0)
(hfinc : StrictMonoOn f (Set.Ici 1))
(hf2 : f 2 = 2)
(hfmn : β m n, m > 0 β n > 0 β IsRelPrime m n β f (m * n) = f m * f n)
: β n > 0, f n = n :=
sorry
|
putnam_2009_b6
|
Require Import List ZArith Coquelicot.Coquelicot.
Open Scope Z.
Theorem putnam_2009_b6
: forall (n: Z), n > 0 -> exists (a: list Z), length a = 2010%nat /\ nth 0 a 0 = 0 /\ nth 2009 a 0 = n /\
forall (i: nat), and (le 1 i) (le i 2009) -> (exists (j: nat), lt j i /\ (exists (k: Z), k >= 0 /\ nth i a 0 = nth j a 0 + 2 ^ k)) \/ exists (b c: nat), lt b i /\ lt c i /\ nth b a 0 > 0 /\ nth c a 0 > 0 /\ nth i a 0 = (nth b a 0) mod (nth c a 0).
Proof. Admitted.
|
import Mathlib
open Topology MvPolynomial Filter Set Metric
/--
Prove that for every positive integer $n$, there is a sequence of integers $a_0, a_1, \dots, a_{2009}$ with $a_0 = 0$ and $a_{2009} = n$ such that each term after $a_0$ is either an earlier term plus $2^k$ for some nonnegative integer $k$, or of the form $b\,\mathrm{mod}\,c$ for some earlier positive terms $b$ and $c$. [Here $b\,\mathrm{mod}\,c$ denotes the remainder when $b$ is divided by $c$, so $0 \leq (b\,\mathrm{mod}\,c) < c$.]
-/
theorem putnam_2009_b6
(n : β) (npos : n > 0) :
(β a : β β β€,
a 0 = 0 β§ a 2009 = n β§
β i : Icc 1 2009,
((β j k : β, j < i β§ a i = a j + 2 ^ k) β¨
β b c : β, b < i β§ c < i β§ a b > 0 β§ a c > 0 β§ a i = (a b) % (a c))) :=
sorry
|
putnam_1982_a6
|
Require Import Nat Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1982_a6_solution := False.
Theorem putnam_1982_a6
(a: nat -> R)
: ((Series a = 1 /\ forall (i j: nat), le i j -> Rabs (a i) > Rabs (a j)) /\
forall (f: nat -> nat), Lim_seq (fun i => Rabs (INR (f i - i)) * Rabs (a i)) = 0 -> exists f', forall x, f' (f x) = x /\ f (f' x) = x ->
Series (fun i => a (f i)) = 1) <-> putnam_1982_a6_solution.
Proof. Admitted.
|
import Mathlib
open Set Function Filter Topology Polynomial Real
abbrev putnam_1982_a6_solution : Prop := sorry
-- False
/--
Let $b$ be a bijection from the positive integers to the positive integers. Also, let $x_1, x_2, x_3, \dots$ be an infinite sequence of real numbers with the following properties:
\begin{enumerate}
\item
$|x_n|$ is a strictly decreasing function of $n$;
\item
$\lim_{n \rightarrow \infty} |b(n) - n| \cdot |x_n| = 0$;
\item
$\lim_{n \rightarrow \infty}\sum_{k = 1}^{n} x_k = 1$.
\end{enumerate}
Prove or disprove: these conditions imply that $$\lim_{n \rightarrow \infty} \sum_{k = 1}^{n} x_{b(k)} = 1.$$
-/
theorem putnam_1982_a6 :
(β b : β β β,
β x : β β β,
BijOn b (Ici 1) (Ici 1) β
StrictAntiOn (fun n : β => |x n|) (Ici 1) β
Tendsto (fun n : β => |b n - (n : β€)| * |x n|) atTop (π 0) β
Tendsto (fun n : β => β k β Finset.Icc 1 n, x k) atTop (π 1) β
Tendsto (fun n : β => β k β Finset.Icc 1 n, x (b k)) atTop (π 1))
β putnam_1982_a6_solution :=
sorry
|
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
(Gamma : {poly R} -> R := fun p => \sum_(i <- p) (i ^+ 2))
(f : {poly R} := 3 * 'X ^ 2 + 7 * 'X + 2)
: let g := putnam_1985_a6_solution in
g.[0] = 1 /\ forall n : nat, ge n 1 -> Gamma (f ^ n) = Gamma (g ^ n).
Proof. Admitted.
|
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 $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with real coefficients such that
\begin{enumerate}
\item[(i)] $g(0)=1$, and
\item[(ii)] $\Gamma(f(x)^n) = \Gamma(g(x)^n)$
\end{enumerate}
for every integer $n \geq 1$.
-/
theorem putnam_1985_a6
(Ξ : Polynomial β β β)
(f : Polynomial β)
(hΞ : Ξ = fun p β¦ β k β Finset.range (p.natDegree + 1), coeff p k ^ 2)
(hf : f = 3 * X ^ 2 + 7 * X + 2) :
let g := putnam_1985_a6_solution;
g.eval 0 = 1 β§ β n : β, n β₯ 1 β Ξ (f ^ n) = Ξ (g ^ n) :=
sorry
|
putnam_2020_b6
|
Require Import Reals. From Coquelicot Require Import Coquelicot Hierarchy Rcomplements.
Open Scope R.
Theorem putnam_2020_b6
(A : nat -> R := fun k => (-1)^(Z.to_nat (floor (INR k * (sqrt 2 - 1)))))
(B : nat -> R := fun n => sum_n_m A 1 n)
: forall (n: nat), B n >= 0.
Proof. Admitted.
|
import Mathlib
open Filter Topology Set
/--
Let $n$ be a positive integer. Prove that $\sum_{k=1}^n(-1)^{\lfloor k(\sqrt{2}-1) \rfloor} \geq 0$.
-/
theorem putnam_2020_b6
(n : β)
(npos : n > 0)
: β k β Finset.Icc 1 n, ((-1) ^ Int.floor (k * (Real.sqrt 2 - 1)) : β) β₯ 0 :=
sorry
|
putnam_1976_a6
|
From mathcomp Require Import all_ssreflect ssrnum ssralg.
From mathcomp Require Import reals derive classical_sets normedtype topology boolp.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1976_a6
(f : R -> R)
(hfdiff : (forall x : R, differentiable f x /\ differentiable f^`() x) /\ continuous f^`(2))
(hfbd : forall x : R, `|f x| <= 1)
(hf0 : (f 0) ^+ 2 + (f^`() 0) ^+ 2 = 4)
: exists y : R, f y + f^`(2) y = 0.
Proof. Admitted.
|
import Mathlib
open Polynomial
/--
Suppose that $f : \mathbb{R} \to \mathbb{R}$ is a twice continuously differentiable function such that $|f(x)| \le 1$ for all real $x$ and $(f(0))^2 + (f'(0))^2 = 4$. Prove that $f(y) + f''(y) = 0$ for some real number $y$.
-/
theorem putnam_1976_a6
(f : β β β)
(hfdiff : ContDiff β 2 f)
(hfbd : β x : β, |f x| β€ 1)
(hf0 : (f 0)^2 + (deriv f 0)^2 = 4)
: β y : β, (f y) + (iteratedDeriv 2 f y) = 0 :=
sorry
|
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.odd k = true)))
: ~ (exists p : 'I_4 -> (R * R), pdists p).
Proof. Admitted.
|
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_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 = putnam_1992_a2_solution.
Proof. Admitted.
|
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 β)
(hC : C = fun Ξ± β¦ taylorCoeffWithin (fun x β¦ (1 + x) ^ Ξ±) 1992 Set.univ 0)
: (β« y in (0)..1, C (-y - 1) * β k β Finset.Icc (1 : β) 1992, 1 / (y + k) = putnam_1992_a2_solution) :=
sorry
|
putnam_2014_a5
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals.
From mathcomp Require Import complex.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_2014_a5
(P : nat -> {poly R[i]} := fun n => \sum_(1 <= i < n.+1) i%:R *: 'X^(i.-1))
: forall j k : nat, (gt j 0 /\ gt k 0) -> j <> k -> gcdp_rec (P j) (P k) = 1.
Proof. Admitted.
|
import Mathlib
open Topology Filter Nat
/--
Let \[ P_n(x) = 1 + 2 x + 3 x^2 + \cdots + n x^{n-1}.\] Prove that the polynomials $P_j(x)$ and $P_k(x)$ are relatively prime for all positive integers $j$ and $k$ with $j \neq k$.
-/
theorem putnam_2014_a5
(P : β β Polynomial β)
(hP : β n, P n = β i β Finset.Icc 1 n, i * Polynomial.X ^ (i - 1))
: β (j k : β), (j > 0 β§ k > 0) β j β k β IsCoprime (P j) (P k) :=
sorry
|
putnam_2012_a2
|
Require Import ssreflect.
Theorem putnam_2012_a2
(S : Type)
(op : S -> S -> S)
(is_comm : (S -> S -> S) -> Prop := fun op => forall (x y : S), op x y = op y x)
(is_assc : (S -> S -> S) -> Prop := fun op => forall (x y z : S), op x (op y z) = op (op x y) z)
(hop : is_comm op /\ is_assc op)
(hS : forall x y : S, exists z : S, op x z = y)
(a b c : S)
(habc : op a c = op b c)
: a = b.
Proof. Admitted.
|
import Mathlib
open Matrix
/--
Let $*$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x*z=y$. (This $z$ may depend on $x$ and $y$.) Show that if $a,b,c$ are in $S$ and $a*c=b*c$, then $a=b$.
-/
theorem putnam_2012_a2
(S : Type*) [CommSemigroup S]
(a b c : S)
(hS : β x y : S, β z : S, x * z = y)
(habc : a * c = b * c)
: a = b :=
sorry
|
putnam_1975_b6
|
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.
Theorem putnam_1975_b6
(s : R ^nat := fun n => \sum_(1 <= i < n.+1) (1 / i%:R))
: (forall n : nat, gt n 1 -> n%:R * (expR (ln ((n + 1)%:R) * 1/n%:R)) < n%:R + s n)
/\ (forall n : nat, gt n 2 -> (n%:R - 1) * (expR ((ln n%:R) * -1/(n%:R-1))) < n%:R - s n).
Proof. Admitted.
|
import Mathlib
open Polynomial Real Complex Matrix Filter Topology Multiset
/--
Show that if $s_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + 1/n, then $n(n+1)^{1/n} < n + s_n$ whenever $n > 1$ and $(n-1)n^{-1/(n-1)} < n - s_n$ whenever $n > 2$.
-/
theorem putnam_1975_b6
(s : β β β)
(hs : s = fun (n : β) => β i β Finset.Icc 1 n, 1/(i : β))
: (β n : β, n > 1 β n * (n+1 : β)^(1/(n : β)) < n + s n) β§ (β n : β, n > 2 β ((n : β) - 1)*((n : β)^(-1/(n-1 : β))) < n - s n) :=
sorry
|
putnam_1990_b5
|
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences topology normedtype.
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 Scope card_scope.
Variable R : realType.
Definition putnam_1990_b5_solution : Prop := True.
Theorem putnam_1990_b5 :
(exists a : nat -> R, (forall i : nat, a i != 0) /\
(forall n : nat, ge n 1 -> (exists roots : seq R, uniq roots /\ size roots = n /\ all (fun x => 0 == \sum_(0 <= i < n.+1) (a i) * (x) ^ i) roots))).
Proof. Admitted.
|
import Mathlib
open Filter Polynomial Topology Nat
abbrev putnam_1990_b5_solution : Prop := sorry
-- True
/--
Is there an infinite sequence $a_0,a_1,a_2,\dots$ of nonzero real numbers such that for $n=1,2,3,\dots$ the polynomial $p_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$ has exactly $n$ distinct real roots?
-/
theorem putnam_1990_b5 :
(β a : β β β, (β i, a i β 0) β§
(β n β₯ 1, (β i β Finset.Iic n, a i β’ X ^ i : Polynomial β).roots.toFinset.card = n)) β
putnam_1990_b5_solution :=
sorry
|
putnam_2000_a6
|
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_2000_a6
(f : {poly int})
(a : nat -> int)
(ha0 : a 0%nat = 0)
(ha : forall n : nat, a (n.+1) = f.[a n])
: (exists m : nat, gt m 0 /\ a m = 0) -> (a 1%nat = 0 \/ a 2%nat = 0).
Proof. Admitted.
|
import Mathlib
open Topology Filter
/--
Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0,a_1,\ldots$ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n\geq 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.
-/
theorem putnam_2000_a6
(f : Polynomial β€)
(a : β β β€)
(ha0 : a 0 = 0)
(ha : β n : β, a (n + 1) = f.eval (a n))
: ((β m > 0, a m = 0) β (a 1 = 0 β¨ a 2 = 0)) :=
sorry
|
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