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putnam_2022_a6
|
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 $-1<x_1<x_2<\cdots<x_{2n}<1$ such that the sum of the lengths of the $n$ intervals $[x_1^{2k-1},x_2^{2k-1}],[x_3^{2k-1},x_4^{2k-1}],\dots,[x_{2n-1}^{2k-1},x_{2n}^{2k-1}]$ is equal to $1$ for all integers $k$ with $1 \leq k \leq m$.
|
Show that the largest such $m$ is $n$.
|
[
"algebra"
] |
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 => sum_n (fun i => (((s (nth i0 (enum 'I_n) (i+1))))^(2*k-1) - ((s (nth i0 (enum 'I_n) i)))^(2*k-1))) (n-1))
(valid : nat -> ('I_n -> R) -> Prop := fun m s => forall (k: nat), and (le 1 k) (le k m) -> sumIntervals s k = 1)
(hvalid : nat -> Prop := fun m => exists (s : 'I_n -> R), (forall (i : 'I_n), (s i < s (ordS i)) /\ s (nth i0 (enum 'I_n) 0) > -1 /\ s (nth i0 (enum 'I_n) (n-1)) < 1) -> valid m s)
(hM : hvalid M)
(hMub : forall m : nat, hvalid m -> le m M)
: M = putnam_2022_a6_solution n.
Proof. Admitted.
|
putnam_2022_b1
|
Suppose that $P(x)=a_1x+a_2x^2+\cdots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\dots$ for all $x$. Prove that $b_k$ is nonzero for all $k \geq 0$.
|
None.
|
[
"analysis",
"algebra"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals derive normedtype sequences topology 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.
Theorem putnam_2022_b1
(n : nat)
(P : {poly R})
(B : R ^nat)
(npos : ge n 1)
(Pconst : P`_(0%nat) = 0)
(Pdegreee : size P = n.+1)
(Pint : forall k : nat, ge k 1 /\ le k n -> exists c : int, (P`_k = c%:~R /\ (k = 1%nat -> (c %% 2)%Z = 1)))
(hB : forall x : R, (fun n : nat => \sum_(0 <= k < n) (B k * x^k)) @ \oo --> expR (P.[x]))
: forall k : nat, B k <> 0.
Proof. Admitted.
|
putnam_2022_b2
|
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 $S=\{v \times w:v,w \in S\}$?
|
Show that the possible values of $n$ are $1$ and $7$.
|
[
"algebra"
] |
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.
|
putnam_2022_b4
|
Find all integers $n$ with $n \geq 4$ for which there exists a sequence of distinct real numbers $x_1,\dots,x_n$ such that each of the sets $\{x_1,x_2,x_3\},\{x_2,x_3,x_4\},\dots,\{x_{n-2},x_{n-1},x_n\},\{x_{n-1},x_n,x_1\}$, and $\{x_n,x_1,x_2\}$ forms a $3$-term arithmetic progression when arranged in increasing order.
|
Show that the values of $n$ in question are the multiples of $3$ starting with $9$.
|
[
"algebra"
] |
From mathcomp Require Import all_ssreflect ssrint ssralg ssrnum.
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.
Definition putnam_2022_b4_solution : set nat := [set n | ge n 9 /\ 3%|n].
Theorem putnam_2022_b4
(n : nat)
(ap3 : R -> R -> R -> Prop)
(xprog : (nat -> R) -> Prop)
(hap3 : forall x0 x1 x2 : R, ap3 x0 x1 x2 <-> forall o0 o1 o2 : R, o0 < o1 < o2 -> [set x : R | x = o0 \/ x = o1 \/ x = o2] = [set x : R | x = x0 \/ x = x1 \/ x = x2] -> (o1 - o0 = o2 - o1))
(hxprog : forall x, xprog x <->
((forall i j : nat, lt i n /\ lt j n /\ i <> j -> x i <> x j) /\
forall i : nat, lt i n -> ap3 (x i) (x ((i.+1) %% n)) (x ((i.+2) %% n))))
: exists x, xprog x <-> n \in putnam_2022_b4_solution.
Proof. Admitted.
|
putnam_2022_b6
|
Find all continuous functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that $f(xf(y))+f(yf(x))=1+f(x+y)$ for all $x,y>0$.
|
Show that the only such functions are the functions $f(x)=\frac{1}{1+cx}$ for some $c \geq 0$.
|
[
"analysis"
] |
Require Import Reals.
Open Scope R.
Definition putnam_2022_b6_solution := fun (f : R -> R) =>
exists (c : R), c >= 0 /\ forall (x : R), x >= 0 /\ f x >= 0 -> f x = 1 / (1 + c * x).
Theorem putnam_2022_b6
(p : (R -> R) -> Prop := fun f : R -> R => forall (x y: R), (x > 0 /\ y > 0 /\ f x > 0 /\ f y > 0) ->
(continuity_pt f x /\ f (x * f y) + f (y * f x) = 1 + f (x + y)))
: forall f : R -> R, p f <-> putnam_2022_b6_solution f.
Proof. Admitted.
|
putnam_2023_a1
|
For a positive integer $n$, let $f_n(x) = \cos(x) \cos(2x) \cos(3x) \cdots \cos(nx)$. Find the smallest $n$ such that $|f_n''(0)| > 2023$.
|
Show that the solution is $n = 18$.
|
[
"analysis"
] |
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.
|
putnam_2023_a2
|
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$.
|
Show that the other real numbers satisfying $p(1/x) = x^2$ are $\pm \frac{1}{n!}.$
|
[
"algebra"
] |
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.
Definition putnam_2023_a2_solution : nat -> set R := fun n => [set x | x = -1 / (n`!)%:R \/ x = 1 / (n`!)%:R].
Theorem putnam_2023_a2
(n : nat)
(hn0 : gt n 0 /\ ~~ odd n)
(p : {poly R})
(hp : p \is monic /\ size p = (n.*2).+1)
(S : set R := [set x | exists k : int, x = k%:~R /\ 1 <= `|k| <= n])
(hpinv : forall k : int, k%:~R \in S -> p.[1 / k%:~R] = k%:~R ^+ 2)
: [set x | (p.[1 / x] == x ^+ 2) && (x \notin S)] = putnam_2023_a2_solution n.
Proof. Admitted.
|
putnam_2023_a3
|
Determine the smallest positive real number $r$ such that there exist differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ and $g\colon \mathbb{R} \to \mathbb{R}$ satisfying
\begin{enumerate}
\item[(a)] $f(0) > 0$,
\item[(b)] $g(0) = 0$,
\item[(c)] $|f'(x)| \leq |g(x)|$ for all $x$,
\item[(d)] $|g'(x)| \leq |f(x)|$ for all $x$, and
\item[(e)] $f(r) = 0$. \end{enumerate}
|
Show that the solution is $r = \pi/2$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2023_a3_solution := PI / 2.
Theorem putnam_2023_a3
(p : R -> Prop := fun t => exists (f g : R -> R), f 0 > 0 /\ g 0 = 0 /\ (forall (x : R), ex_derive f x /\ ex_derive g x) /\ (forall (x: R), (Rabs (Derive f x) <= Rabs (g x) /\ Rabs (Derive g x) <= Rabs (f x))) /\ f t = 0)
: (putnam_2023_a3_solution > 0 /\ p putnam_2023_a3_solution) /\ (forall (t: R), t > 0 -> t < putnam_2023_a3_solution -> ~ (p t)).
Proof. Admitted.
|
putnam_2023_b2
|
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 $k(n)$?
|
Show that the minimum value is 3.
|
[
"number_theory"
] |
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)) >= putnam_2023_b2_solution) /\
(exists (n: nat), n > 0 /\ k (Pos.of_nat (2023*n)) = putnam_2023_b2_solution).
Proof. Admitted.
|
putnam_2023_b5
|
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2,\dots,n\} \to \{1,2,\dots,n\}$ such that $\pi(\pi(k)) \equiv mk \pmod{n}$ for all $k \in \{1,2,\dots,n\}$.
|
Show that the desired property holds if and only if $n=1$ or $n \equiv 2 \pmod{4}$.
|
[
"number_theory"
] |
Require Import PeanoNat. From mathcomp Require Import div fintype perm ssrbool.
Definition putnam_2023_b5_solution (n : nat) := n = 1 \/ n mod 4 = 2.
Theorem putnam_2023_b5
(p : nat -> Prop := fun n => forall m : nat, coprime m n -> exists (π: {perm 'I_n}), forall (k: 'I_n), nat_of_ord (π (π k)) = (m*(nat_of_ord k))%%n)
: forall n : nat, (n > 0 /\ p n) <-> putnam_2023_b5_solution n.
Proof. Admitted.
|
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