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putnam_1990_b5
|
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?
|
Show that the answer is yes, such an infinite sequence exists.
|
[
"algebra",
"analysis"
] |
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.
|
putnam_1991_a3
|
Find all real polynomials $p(x)$ of degree $n \geq 2$ for which there exist real numbers $r_1<r_2<\cdots<r_n$ such that
\begin{enumerate}
\item $p(r_i)=0, \qquad i=1,2,\dots,n$, and
\item $p'(\frac{r_i+r_{i+1}}{2})=0 \qquad i=1,2,\dots,n-1$,
\end{enumerate}
where $p'(x)$ denotes the derivative of $p(x)$.
|
Show that the real polynomials with the required property are exactly those that are of degree $2$ with $2$ distinct real zeros.
|
[
"algebra",
"analysis"
] |
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.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition putnam_1991_a3_solution : set {poly R} := [set P : {poly R} | size P = 3%nat /\ (exists r1 r2 : R, r1 <> r2 /\ P.[r1] = 0 /\ P.[r2] = 0)].
Theorem putnam_1991_a3
(P : {poly R})
(n : nat)
(hn : n = (size P).-1)
(hge : ge n 2)
: P \in putnam_1991_a3_solution <->
(exists (r: nat -> R), (forall i : nat, lt i (n - 1) -> r i < r (i.+1)) /\
(forall i : nat, lt i n -> P.[r i] = 0) /\
(forall i : nat, lt i (n.-1) -> (P^`()).[(r i + r i.+1) / 2] = 0)).
Proof. Admitted.
|
putnam_1991_a5
|
Find the maximum value of $\int_0^y \sqrt{x^4+(y-y^2)^2}\,dx$ for $0 \leq y \leq 1$.
|
Show that the maximum value of the integral is $1/3$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1991_a5_solution := 1/3.
Theorem putnam_1991_a5
(f : R -> R := fun y : R => RInt (fun x => sqrt (pow x 4 + pow (y - pow y 2) 2)) 0 y)
: (exists (y : R), 0 <= y <= 1 /\ putnam_1991_a5_solution = f y) /\ (forall (y : R), 0 <= y <= 1 -> putnam_1991_a5_solution >= f y).
Proof. Admitted.
|
putnam_1991_b1
|
For each integer $n \geq 0$, let $S(n)=n-m^2$, where $m$ is the greatest integer with $m^2 \leq n$. Define a sequence $(a_k)_{k=0}^\infty$ by $a_0=A$ and $a_{k+1}=a_k+S(a_k)$ for $k \geq 0$. For what positive integers $A$ is this sequence eventually constant?
|
Show that this sequence is eventually constant if and only if $A$ is a perfect square.
|
[
"algebra"
] |
Require Import Nat.
Open Scope nat_scope.
Definition putnam_1991_b1_solution (A: nat) := exists (m: nat), A = pow m 2.
Theorem putnam_1991_b1
(eS : nat -> nat := fun n => sub n (pow (sqrt n) 2))
(a_seq := fix a (A k: nat) :=
match k with
| O => A
| S k' => a A k' + eS (a A k')
end)
(A : nat)
(hA : gt A 0)
: (exists K c : nat, forall k : nat, k >= K -> a_seq A k = c) <-> putnam_1991_b1_solution A.
Proof. Admitted.
|
putnam_1991_b2
|
Suppose $f$ and $g$ are non-constant, differentiable, real-valued functions defined on $(-\infty,\infty)$. Furthermore, suppose that for each pair of real numbers $x$ and $y$,
\begin{align*}
f(x+y)&=f(x)f(y)-g(x)g(y), \\
g(x+y)&=f(x)g(y)+g(x)f(y).
\end{align*}
If $f'(0)=0$, prove that $(f(x))^2+(g(x))^2=1$ for all $x$.
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1991_b2
(f g : R -> R)
(fgnconst : ~ exists (c : R), f = (fun _ => c) \/ g = (fun _ => c))
(fgdiff : forall x : R, ex_derive f x /\ ex_derive g x)
(fadd : forall x y : R, f (x + y) = f x * f y - g x * g y)
(gadd : forall x y : R, g (x + y) = f x * g y + g x * f y)
: Derive f 0 = 0 -> forall x : R, pow (f x) 2 + pow (g x) 2 = 1.
Proof. Admitted.
|
putnam_1991_b4
|
Suppose $p$ is an odd prime. Prove that $\sum_{j=0}^p \binom{p}{j}\binom{p+j}{j} \equiv 2^p+1 \pmod{p^2}$.
|
None.
|
[
"number_theory",
"algebra"
] |
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.
|
putnam_1991_b5
|
Let $p$ be an odd prime and let $\mathbb{Z}_p$ denote (the field of) integers modulo $p$. How many elements are in the set $\{x^2:x \in \mathbb{Z}_p\} \cap \{y^2+1:y \in \mathbb{Z}_p\}$?
|
Show that the number of elements in the intersection is $\lceil p/4 \rceil$.
|
[
"number_theory"
] |
Require Import Reals Nat ZArith Znumtheory Ensembles Finite_sets Coquelicot.Coquelicot. From mathcomp Require Import fintype seq ssrbool.
Open Scope R.
Definition putnam_1991_b5_solution (p: nat) : nat := p / 4 + 1.
Theorem putnam_1991_b5
(p: nat)
(hp : odd p = true /\ prime (Z.of_nat p))
(A: Ensemble Z := fun z => exists (m: 'I_p), z = Z.of_nat (pow (nat_of_ord m) 2) mod (Z.of_nat p))
(B: Ensemble Z := fun z => exists (m: 'I_p), z = Z.of_nat (pow (nat_of_ord m) 2 + 1) mod (Z.of_nat p))
(C : Ensemble Z := Intersection Z A B)
: cardinal Z C (putnam_1991_b5_solution p).
Proof. Admitted.
|
putnam_1991_b6
|
Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $a^xb^{1-x} \leq a\frac{\sinh ux}{\sinh u}+b\frac{\sinh u(1-x)}{\sinh u}$ for all $u$ with $0<|u| \leq c$ and for all $x$, $0<x<1$. (Note: $\sinh u=(e^u-e^{-u})/2$.)
|
Show that the largest $c$ for which the inequality holds for $0<|u| \leq c$ is $c=|\ln(a/b)|$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1991_b6_solution (a b : R) := Rabs (ln (a / b)).
Theorem putnam_1991_b6
(a b: R)
(hab : a > 0 /\ b > 0)
(ineq_holds : R -> Prop := fun c => forall (u x: R), 0 < Rabs u <= c /\ 0 < x < 1 -> Rpower a x * Rpower b (1 - x) <= a * sinh (u * x) / sinh u + b * sinh (u * (1 - x)) / sinh u)
(c: R)
(hc : ineq_holds c)
(hcub : forall (x: R), ineq_holds x -> x <= c)
: c = putnam_1991_b6_solution a b.
Proof. Admitted.
|
putnam_1992_a1
|
Prove that $f(n) = 1-n$ is the only integer-valued function defined on the integers that satisfies the following conditions.
\begin{itemize}
\item[(i)] $f(f(n)) = n$, for all integers $n$;
\item[(ii)] $f(f(n+2)+2) = n$ for all integers $n$;
\item[(iii)] $f(0) = 1$.
\end{itemize}
|
None.
|
[
"algebra"
] |
Require Import Basics ZArith.
Open Scope Z_scope.
Theorem putnam_1992_a1
(f : Z -> Z)
: f = (fun n => 1 - n) <-> ((forall n : Z, f (f n) = n) /\ (forall n : Z, f (f (n + 2) + 2) = n) /\ f 0 = 1).
Proof. Admitted.
|
putnam_1992_a2
|
Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x=0$ of $(1 + x)^\alpha$. Evaluate
\[
\int_0^1 \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\,dy.
\]
|
Prove that the integral evaluates to $1992$.
|
[
"analysis",
"algebra"
] |
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.
|
putnam_1992_a3
|
For a given positive integer $m$, find all triples $(n, x, y)$ of positive integers, with $n$ relatively prime to $m$, which satisfy
\[
(x^2 + y^2)^m = (xy)^n.
\]
|
Prove that if $m$ is odd, there are no solutions, and if $m$ is even, the only solution is
$(n, x, y) = (m + 1, 2 ^ {m/2}, 2 ^{m/2})$.
|
[
"algebra",
"number_theory"
] |
Require Import Nat. From mathcomp Require Import div fintype perm ssrbool.
Definition putnam_1992_a3_solution (x y n m: nat) := exists r, x = 2 ^ r /\ y = 2 ^ r /\ n = 2 * r + 1 /\ m = 2 * r.
Theorem putnam_1992_a3
(m : nat)
(hm : m > 0)
(hnxy : nat -> nat -> nat -> Prop := fun n x y => n > 0 /\ x > 0 /\ y > 0 /\ coprime n m /\ pow (pow x 2 + pow y 2) m = pow (x * y) n)
: forall n x y, putnam_1992_a3_solution x y n m <-> hnxy n x y.
Proof. Admitted.
|
putnam_1992_a4
|
Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If
\[
f\left( \frac{1}{n} \right) = \frac{n^2}{n^2 + 1}, \qquad n = 1, 2, 3, \dots,
\]
compute the values of the derivatives $f^{(k)}(0), k = 1, 2, 3, \dots$.
|
Prove that
\[
f^{(k)}(0) =
\begin{cases}
(-1)^{k/2}k! & \text{if $k$ is even;} \\
0 & \text{if $k$ is odd.} \\
\end{cases}
\]
|
[
"analysis"
] |
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 n : nat, gt n 0 -> f (1 / INR n) = (INR n)^2 / ((INR n)^2 + 1))
: forall (k : nat), gt k 0 -> (Derive_n f k) 0 = putnam_1992_a4_solution k.
Proof. Admitted.
|
putnam_1992_a5
|
For each positive integer $n$, let $a_n = 0$ (or $1$) if the number of $1$'s in the binary representation of $n$ is even (or odd), respectively. Show that there do not exist positive integers $k$ and $m$ such that
\[
a_{k+j} = a_{k+m+j} = a_{k+2m+j},
\]
for $0 \leq j \leq m-1$.
|
None.
|
[
"algebra"
] |
Require Import BinPos Nat ZArith.
Theorem putnam_1992_a5
(k := fix count_ones (n : positive) : nat :=
match n with
| xH => 1
| xO n' => count_ones n'
| xI n' => 1 + count_ones n'
end)
(a : positive -> nat := fun n => (k n) mod 2)
: ~ exists (k m: nat), gt k 0 /\ gt m 0 /\ forall (j: nat), j <= m - 1 -> a (Pos.of_nat (k + j)) = a (Pos.of_nat (k + m + j)) /\ a (Pos.of_nat (k + m + j)) = a (Pos.of_nat (k + 2 * m + j)).
Proof. Admitted.
|
putnam_1992_b1
|
Let $S$ be a set of $n$ distinct real numbers. Let $A_S$ be the set of numbers that occur as averages of two distinct elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_S$?
|
Show that the answer is $2n - 3$.
|
[
"algebra"
] |
Require Import Nat Ensembles Finite_sets Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1992_b1_solution (n: nat) := sub (mul 2 n) 3.
Theorem putnam_1992_b1
(n : nat)
(nge2 : ge n 2)
(A : Ensemble R -> Ensemble R := fun E : Ensemble R => fun x : R => exists a b : R, E a /\ E b /\ a <> b /\ (a + b) / 2 = x)
(min : nat)
(hmineq : exists E : Ensemble R, cardinal R E n /\ cardinal R (A E) min)
(hminlb : forall E : Ensemble R, cardinal R E n -> exists m : nat, le m min /\ cardinal R (A E) m)
: min = putnam_1992_b1_solution n.
Proof. Admitted.
|
putnam_1993_a2
|
Let $(x_n)_{n \geq 0}$ be a sequence of nonzero real numbers such that $x_n^2-x_{n-1}x_{n+1}=1$ for $n=1,2,3,\dots$. Prove there exists a real number $a$ such that $x_{n+1}=ax_n-x_{n-1}$ for all $n \geq 1$.
|
None.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1993_a2
(x : nat -> R)
(xnonzero : forall n : nat, x n <> 0)
(hx : forall n : nat, ge n 1 -> pow (x n) 2 - x (pred n) * x (S n) = 1)
: exists a : R, forall n : nat, ge n 1 -> x (S n) = a * x n - x (pred n).
Proof. Admitted.
|
putnam_1993_a4
|
Let $x_1,x_2,\dots,x_{19}$ be positive integers each of which is less than or equal to $93$. Let $y_1,y_2,\dots,y_{93}$ be positive integers each of which is less than or equal to $19$. Prove that there exists a (nonempty) sum of some $x_i$'s equal to a sum of some $y_j$'s.
|
None.
|
[
"algebra"
] |
Require Import List Bool Reals Peano_dec Coquelicot.Coquelicot.
Open Scope nat_scope.
Fixpoint s_sum (a : nat -> nat) (k : nat) (s : nat -> bool) : nat :=
match k with
| O => a 0
| S k' => (if s k then a k else 0) + s_sum a k' s
end.
Theorem putnam_1993_a4
(x y : nat -> nat)
(hx : forall i : nat, i < 19 -> 1 <= x i <= 93)
(hy : forall j : nat, j < 93 -> 1 <= y j <= 19)
: exists (is js : nat -> bool), (exists i : nat, i < 19 /\ is i = true) /\ s_sum x 18 is = s_sum y 92 js.
Proof. Admitted.
|
putnam_1993_a5
|
Show that $\int_{-100}^{-10} (\frac{x^2-x}{x^3-3x+1})^2\,dx+\int_{\frac{1}{101}}^{\frac{1}{11}} (\frac{x^2-x}{x^3-3x+1})^2\,dx+\int_{\frac{101}{100}}^{\frac{11}{10}} (\frac{x^2-x}{x^3-3x+1})^2\,dx$ is a rational number.
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1993_a5
(f : R -> R := fun x => (pow x 2 - x) / (pow x 3 - 3 * x + 1))
: exists p q : Z,
RInt (fun x => pow (f x) 2) (-100) (-10) +
RInt (fun x => pow (f x) 2) (1/101) (1/11) +
RInt (fun x => pow (f x) 2) (101/100) (11/10) =
IZR p /IZR q.
Proof. Admitted.
|
putnam_1993_b1
|
Find the smallest positive integer $n$ such that for every integer $m$ with $0<m<1993$, there exists an integer $k$ for which $\frac{m}{1993}<\frac{k}{n}<\frac{m+1}{1994}$.
|
Show that the smallest positive integer $n$ satisfying the condition is $n=3987$.
|
[
"algebra"
] |
Require Import Reals.
Open Scope R.
Definition putnam_1993_b1_solution : nat := 3987.
Theorem putnam_1993_b1
(cond : nat -> Prop := fun n => gt n 0 /\ forall (m: nat), and (lt 0 m) (lt m 1993) -> exists (k: nat), INR m / 1993 < INR k / INR n < INR (m + 1) / 1994)
: cond putnam_1993_b1_solution /\ forall (n: nat), cond n -> ge n putnam_1993_b1_solution.
Proof. Admitted.
|
putnam_1993_b4
|
The function $K(x,y)$ is positive and continuous for $0 \leq x \leq 1,0 \leq y \leq 1$, and the functions $f(x)$ and $g(x)$ are positive and continuous for $0 \leq x \leq 1$. Suppose that for all $x$, $0 \leq x \leq 1$, $\int_0^1 f(y)K(x,y)\,dy=g(x)$ and $\int_0^1 g(y)K(x,y)\,dy=f(x)$. Show that $f(x)=g(x)$ for $0 \leq x \leq 1$.
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1993_b4
(K : (R * R) -> R)
(f g : R -> R)
(hK : forall x y : R, 0 <= x <= 1 /\ 0 <= y <= 1 -> K (x, y) > 0 /\ continuous K (x, y))
(hfg : forall x : R, 0 <= x <= 1 -> f x > 0 /\ g x > 0 /\ continuous f x /\ continuous g x /\
RInt (fun y => f y * K (x, y)) 0 1 = g x /\ RInt (fun y => g y * K (x, y)) 0 1 = f x)
: forall x : R, 0 <= x <= 1 -> f x = g x.
Proof. Admitted.
|
putnam_1993_b5
|
Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers.
|
None.
|
[
"geometry",
"number_theory",
"linear_algebra"
] |
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.
|
putnam_1994_a1
|
Suppose that a sequence $a_1,a_2,a_3,\dots$ satisfies $0<a_n \leq a_{2n}+a_{2n+1}$ for all $n \geq 1$. Prove that the series $\sum_{n=1}^\infty a_n$ diverges.
|
None.
|
[
"analysis"
] |
Require Import Nat Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1994_a1
(a : nat -> R)
(ha : forall n : nat, gt n 0 -> 0 < a n <= a (mul 2 n) + a (add (mul 2 n) 1))
: ~ ex_finite_lim_seq (fun n => sum_n (fun m => a m) n).
Proof. Admitted.
|
putnam_1994_b1
|
Find all positive integers $n$ that are within $250$ of exactly $15$ perfect squares.
|
Show that an integer $n$ is within $250$ of exactly $15$ perfect squares if and only if either $315 \leq n \leq 325$ or $332 \leq n \leq 350$.
|
[
"algebra"
] |
Require Import Ensembles Finite_sets ZArith.
Open Scope Z.
Definition putnam_1994_b1_solution (n: Z) := 315 <= n <= 325 \/ 332 <= n <= 350.
Theorem putnam_1994_b1
(n : Z)
(nwithin : Prop := cardinal nat (fun m => Z.abs (n - (Z.of_nat m) ^ 2) <= 250) 15)
: (n > 0 /\ nwithin) <-> putnam_1994_b1_solution n.
Proof. Admitted.
|
putnam_1994_b2
|
For which real numbers $c$ is there a straight line that intersects the curve $x^4+9x^3+cx^2+9x+4$ in four distinct points?
|
Show that there exists such a line if and only if $c<243/8$.
|
[
"geometry",
"algebra"
] |
Require Import Ensembles Reals Finite_sets Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1994_b2_solution (c: R) := c < 243 / 8.
Theorem putnam_1994_b2
(hintersect : R -> Prop := fun c => exists(m b: R), cardinal R (fun x => m * x + b = pow x 4 + 9 * pow x 3 + c * pow x 2 + 9 * x + 4) 4)
: forall (c: R), hintersect c <-> putnam_1994_b2_solution c.
Proof. Admitted.
|
putnam_1994_b3
|
Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f'(x)>f(x)$ for all $x$, there is some number $N$ such that $f(x)>e^{kx}$ for all $x>N$.
|
Show that the desired set is $(-\infty,1)$.
|
[
"analysis"
] |
From mathcomp Require Import all_ssreflect ssralg ssrnum.
From mathcomp Require Import reals 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_1993_b3_solution : set R := [set k | k < 1].
Theorem putnam_1993_b3
: [set k | forall f (hf : forall x, differentiable f x /\ 0 < f x < f^`() x),
exists N : R, forall x, N < x -> expR (k * x) < f x] = putnam_1993_b3_solution.
Proof. Admitted.
|
putnam_1994_b4
|
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 $\lim_{n \to \infty} d_n=\infty$.
|
None.
|
[
"linear_algebra",
"number_theory",
"analysis"
] |
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.
|
putnam_1994_b5
|
For any real number $\alpha$, define the function $f_\alpha(x)=\lfloor \alpha x \rfloor$. Let $n$ be a positive integer. Show that there exists an $\alpha$ such that for $1 \leq k \leq n$, $f_\alpha^k(n^2)=n^2-k=f_{\alpha^k}(n^2)$.
|
None.
|
[
"algebra"
] |
Require Import Basics ZArith Zpower Reals Coquelicot.Coquelicot.
Open Scope Z_scope.
Theorem putnam_1994_b5
(composen := fix compose_n {A: Type} (f : A -> A) (n : nat) :=
match n with
| O => fun x => x
| S n' => compose f (compose_n f n')
end)
(fa : R -> Z -> Z := fun a x => floor (a * IZR x))
: forall (n: Z), n > 0 -> exists (a: R), forall (k: Z), 1 <= k <= n -> (composen (fa a) (Z.to_nat k)) n^2 = n^2 - k /\ n^2 - k = fa (Rpower a (IZR k)) n^2.
Proof. Admitted.
|
putnam_1994_b6
|
For any integer $a$, set $n_a=101a-100 \cdot 2^a$. Show that for $0 \leq a,b,c,d \leq 99$, $n_a+n_b \equiv n_c+n_d \pmod{10100}$ implies $\{a,b\}=\{c,d\}$.
|
None.
|
[
"number_theory"
] |
Require Import ZArith.
Open Scope Z_scope.
Theorem putnam_1994_b6
(n : Z -> Z := fun a => 101 * a - 100 * 2^a)
: forall a b c d : Z, 0 <= a <= 99 /\ 0 <= b <= 99 /\ 0 <= c <= 99 /\ 0 <= d <= 99 /\
(n a + n b) mod 10100 = (n c + n d) mod 10100 -> (a = c /\ b = d) \/ (a = d /\ b = c).
Proof. Admitted.
|
putnam_1995_a1
|
Let $S$ be a set of real numbers which is closed under multiplication (that is, if $a$ and $b$ are in $S$, then so is $ab$). Let $T$ and $U$ be disjoint subsets of $S$ whose union is $S$. Given that the product of any {\em three} (not necessarily distinct) elements of $T$ is in $T$ and that the product of any three elements of $U$ is in $U$, show that at least one of the two subsets $T,U$ is closed under multiplication.
|
None.
|
[
"algebra"
] |
Require Import Reals Ensembles Coquelicot.Coquelicot.
Theorem putnam_1995_a1
(E T U : Ensemble R)
(hE : forall a : R, forall b : R, In _ E a /\ In _ E b -> In _ E (a * b))
(hsub : Included _ T E /\ Included _ U E)
(hunion : Same_set _ (Union _ T U) E)
(hdisj : Disjoint _ T U)
(hT3 : forall a b c : R, In _ T a /\ In _ T b /\ In _ T c -> In _ T (a * b * c))
(hU3 : forall a b c : R, In _ U a /\ In _ U b /\ In _ U c -> In _ U (a * b * c))
: (forall a b : R, In _ T a /\ In _ T b -> In _ T (a * b)) \/ (forall a b : R, In _ U a /\ In _ U b -> In _ U (a * b)).
Proof. Admitted.
|
putnam_1995_a2
|
For what pairs $(a,b)$ of positive real numbers does the improper integral \[ \int_{b}^{\infty} \left( \sqrt{\sqrt{x+a}-\sqrt{x}} - \sqrt{\sqrt{x}-\sqrt{x-b}} \right)\,dx \] converge?
|
Show that the solution is those pairs $(a,b)$ where $a = b$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1995_a2_solution : R -> R -> Prop := fun (a b: R) => a = b.
Theorem putnam_1995_a2
: forall (a b: R), a > 0 /\ b > 0 -> ((exists limit : R, filterlim (fun n : R => RInt (fun x => sqrt (sqrt (x + a) - sqrt x) - sqrt (sqrt x - (x - b))) b n) (Rbar_locally p_infty) (locally limit)) <-> putnam_1995_a2_solution a b).
Proof. Admitted.
|
putnam_1995_a4
|
Suppose we have a necklace of $n$ beads. Each bead is labeled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_{1},x\_{2},\dots,x_{n}$ satisfy \[\sum_{i=1}^{k} x_{i} \leq k-1 \qquad \mbox{for} \quad k=1,2,\dots,n.\]
|
None.
|
[
"combinatorics"
] |
Require Import Nat ZArith Coquelicot.Coquelicot.
Open Scope Z_scope.
Fixpoint Z_sum (a : nat -> Z) (k : nat) : Z :=
match k with
| O => a O
| S k' => a k + Z_sum a k'
end.
Theorem putnam_1995_a4
(n : nat)
(hn : gt n 0)
(necklace : nat -> Z)
(hnecklacesum : Z_sum necklace (n - 1) = (Z.of_nat n) - 1)
: exists cut : nat, le cut (n-1) /\ (forall k : nat, le k (n-1) -> (Z_sum (fun i => necklace (Nat.modulo (cut + i) n)) k) <= Z.of_nat k).
Proof. Admitted.
|
putnam_1995_a5
|
Let $x_{1},x_{2},\dots,x_{n}$ be differentiable (real-valued) functions of a single variable $f$ which satisfy \begin{align*} \frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} \ \frac{dx_{2}}{dt} &= a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} \ \vdots && \vdots \ \frac{dx_{n}}{dt} &= a_{n1}x_{1} + a_{n2}x_{2} + \cdots + a_{nn}x_{n} \end{align*} for some constants $a_{ij}>0$. Suppose that for all $i$, $x_{i}(t) \to 0$ as $t \to \infty$. Are the functions $x_{1},x_{2},\dots,x_{n}$ necessarily linearly dependent?
|
Show that the answer is yes, the functions must be linearly dependent.
|
[
"linear_algebra",
"analysis"
] |
Require Import Nat Reals Coquelicot.Coquelicot. From mathcomp Require Import fintype.
Definition putnam_1995_a5_solution : Prop := True.
Theorem putnam_1995_a5
(hdiffx := fun (n : nat) (x : nat -> (R -> R)) => forall (i : nat) (t : R), le i (Nat.sub n 1) -> ex_derive (x i) t)
(ha := fun (n : nat) (a : nat -> nat -> R) => forall i j : nat, le i (Nat.sub n 1) /\ le j (Nat.sub n 1) -> a i j > 0)
(hcomb := fun (n : nat) (x : nat -> (R -> R)) (a : nat -> nat -> R) => (forall t : R, forall i : nat, le i (Nat.sub n 1) -> Derive (x i) t = sum_n (fun j => a i j * ((x j) t)) (n - 1)))
(hxlim := fun (n : nat) (x : nat -> (R -> R)) => (forall i : nat, le i (Nat.sub n 1) -> filterlim (x i) (Rbar_locally p_infty) (locally 0)))
: putnam_1995_a5_solution <-> (forall (n : nat) (x : nat -> (R -> R)) (a : nat -> nat -> R), (gt n 0 /\ hdiffx n x /\ ha n a /\ hcomb n x a /\ hxlim n x) -> ~(forall b : nat -> R, (forall t : R, 0 = sum_n (fun i => (b i) * ((x i) t)) (n - 1)) -> (forall i : nat, le i (Nat.sub n 1) -> b i = 0))).
Proof. Admitted.
|
putnam_1995_b4
|
Evaluate \[ \sqrt[8]{2207 - \frac{1}{2207-\frac{1}{2207-\dots}}}. \] Express your answer in the form $\frac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers.
|
Show that the solution is $(3 + 1*\sqrt{5})/2.
|
[
"algebra"
] |
Require Import Reals ZArith Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1995_b4_solution : Z * Z * Z * Z := (3%Z,1%Z,5%Z,2%Z).
Theorem putnam_1995_b4
(contfrac : R)
(hcontfrac : contfrac = 2207 - 1/contfrac)
(hcontfrac' : 1 < contfrac)
: let (abc, d) := putnam_1995_b4_solution in let (ab, c) := abc in let (a, b) := ab in
pow contfrac (1 / 8) = (IZR a + IZR b * sqrt (IZR c))/IZR d.
Proof. Admitted.
|
putnam_1995_b6
|
For a positive real number $\alpha$, define \[ S(\alpha) = \{ \lfloor n\alpha \rfloor : n = 1,2,3,\dots \}. \] Prove that $\{1,2,3,\dots\}$ cannot be expressed as the disjoint union of three sets $S(\alpha), S(\beta)$ and $S(\gamma)$. [As usual, $\lfloor x \rfloor$ is the greatest integer $\leq x$.]
|
None.
|
[
"algebra",
"number_theory"
] |
Require Import Reals Ensembles Coquelicot.Coquelicot.
Theorem putnam_1995_b6
(E : R -> Ensemble nat := fun alpha => fun x : nat => exists n : nat, ge n 1 /\ Z.of_nat x = floor (INR n * alpha))
: ~ exists alpha beta gamma : R, alpha > 0 /\ beta > 0 /\ gamma > 0 /\ (Disjoint _ (E alpha) (E beta) /\ Disjoint _ (E beta) (E gamma) /\ Disjoint _ (E gamma) (E alpha)) /\ (Same_set _ (fun x : nat => ge x 1) (Union _ (Union _ (E alpha) (E beta)) (E gamma))).
Proof. Admitted.
|
putnam_1996_a3
|
Suppose that each of 20 students has made a choice of anywhere from 0 to 6 courses from a total of 6 courses offered. Prove or disprove: there are 5 students and 2 courses such that all 5 have chosen both courses or all 5 have chosen neither course.
|
Show that the solution is that the statement is false.
|
[
"combinatorics"
] |
From mathcomp Require Import all_algebra all_ssreflect classical_sets cardinality.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
Local Open Scope card_scope.
Definition putnam_1996_a3_solution : Prop := False.
Theorem putnam_1996_a3 :
(forall choices : 'I_20 -> set 'I_6,
exists (students : set 'I_20) (courses : set 'I_6),
students #= [set: 'I_5] /\ courses #= [set: 'I_2] /\
(courses `<=` \bigcap_(s in students) (choices s) \/ courses `<=` \bigcap_(s in students) (~` choices s)))
<-> putnam_1996_a3_solution.
Proof. Admitted.
|
putnam_1996_a4
|
Let $S$ be a set of ordered triples $(a, b, c)$ of distinct elements of a finite set $A$. Suppose that \begin{enumerate} \item $(a,b,c) \in S$ if and only if $(b,c,a) \in S$; \item $(a,b,c) \in S$ if and only if $(c,b,a) \notin S$; \item $(a,b,c)$ and $(c,d,a)$ are both in $S$ if and only if $(b,c,d)$ and $(d,a,b)$ are both in $S$. \end{enumerate} Prove that there exists a one-to-one function $g$ from $A$ to $\R$ such that $g(a) < g(b) < g(c)$ implies $(a,b,c) \in S$.
|
None.
|
[
"algebra"
] |
Require Import Basics FinFun Reals Ensembles Finite_sets. From mathcomp Require Import fintype.
Local Open Scope R_scope.
Theorem putnam_1996_a4
(A : finType)
(SS : Ensemble (A * A * A))
(hSdistinct: forall a b c : A, In _ SS (a, b, c) -> a <> b /\ b <> c /\ c <> a)
(hS1 : forall a b c : A, In _ SS (a, b, c) <-> In _ SS (b, c, a))
(hS2 : forall a b c : A, a <> c -> (In _ SS (a, b, c) <-> ~ (In _ SS (c, b, a))))
(hS3 : forall a b c d : A, (In _ SS (a, b, c) /\ In _ SS (c, d, a)) <-> (In _ SS (b, c, d) /\ In _ SS (d, a, b)))
: exists g : A -> R, Injective g /\ (forall a b c : A, g a < g b /\ g b < g c -> In _ SS (a, b, c)).
Proof. Admitted.
|
putnam_1996_a5
|
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 $p^2$.
|
None.
|
[
"number_theory"
] |
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 p) 2.
Proof. Admitted.
|
putnam_1996_a6
|
Let $c>0$ be a constant. Give a complete description, with proof, of the set of all continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that $f(x)=f(x^2+c)$ for all $x \in \mathbb{R}$.
|
Show that if $c \leq 1/4$ then $f$ must be constant, and if $c>1/4$ then $f$ can be defined on $[0,c]$ as any continuous function with equal values on the endpoints, then extended to $x>c$ by the relation $f(x)=f(x^2+c)$, then extended further to $x<0$ by the relation $f(x)=f(-x)$.
|
[
"analysis",
"algebra"
] |
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 -> f x = f (-x))).
Theorem putnam_1996_a6
(c: R)
(hc : c > 0)
: forall (f: R -> R), (continuity f /\ (forall (x: R), f x = pow x 2 + c)) <-> putnam_1996_a6_solution c f.
Proof. Admitted.
|
putnam_1996_b2
|
Show that for every positive integer $n$, $(\frac{2n-1}{e})^{\frac{2n-1}{2}}<1 \cdot 3 \cdot 5 \cdots (2n-1)<(\frac{2n+1}{e})^{\frac{2n+1}{2}}$.
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot. From mathcomp Require Import bigop.
Open Scope R.
Theorem putnam_1996_b2
(n : nat)
(prododd : R := INR (\prod_(1 <= i < (n + 1)) (2 * i - 1)))
(npos : gt n 0)
: Rpower ((2 * INR n - 1) / exp 1) ((2 * INR n - 1) / 2) < prododd < Rpower ((2 * INR n + 1) / exp 1) ((2 * INR n + 1) / 2).
Proof. Admitted.
|
putnam_1996_b3
|
Given that $\{x_1,x_2,\ldots,x_n\}=\{1,2,\ldots,n\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \geq 2$), of $x_1x_2+x_2x_3+\cdots+x_{n-1}x_n+x_nx_1$.
|
Show that the maximum is $(2n^3+3n^2-11n+18)/6$.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Hierarchy Nat.
Definition putnam_1996_b3_solution : nat -> R := (fun n => (2 * INR n ^ 3 + 3 * INR n ^ 2 - 11 * INR n + 18) / 6).
Theorem putnam_1996_b3
(n : nat)
(xset : (nat -> nat) -> Prop := (fun x : nat -> nat => forall y : nat, (le 1 y /\ le y n) -> exists! i, (le 0 i /\ le i (n - 1)) /\ x i = y))
(xsum : (nat -> nat) -> R := (fun x : nat -> nat => sum_n (fun i : nat => INR (x i) * INR (x ((i + 1) mod n))) (n - 1)))
(nge2 : ge n 2)
: (exists x : nat -> nat, xset x /\ xsum x = putnam_1996_b3_solution n) /\ (forall x : nat -> nat, xset x -> xsum x <= putnam_1996_b3_solution n).
Proof. Admitted.
|
putnam_1996_b4
|
For any square matrix $A$, we can define $\sin A$ by the usual power series: $\sin A=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}$. Prove or disprove: there exists a $2 \times 2$ matrix $A$ with real entries such that $\sin A=\begin{pmatrix} 1 & 1996 \\ 0 & 1 \end{pmatrix}$.
|
Show that there does not exist such a matrix $A$.
|
[
"linear_algebra"
] |
Require Import Factorial Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1996_b4_solution := False.
Theorem putnam_1996_b4
(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)
(scale_c : R -> matrix 2 2 -> matrix 2 2 := fun c A => mk_matrix 2 2 (fun i j => c * coeff_mat 0 A i j))
(sinA_mat : nat -> matrix 2 2 -> matrix 2 2 := fun n A => scale_c ((pow (-1) n) / INR (fact (2 * n + 1))) (Mmultn A (Nat.add (Nat.mul 2 n) 1)))
: (exists (A: matrix 2 2),
Series (fun n => coeff_mat 0 (sinA_mat n A) 0 0) = 1 /\
Series (fun n => coeff_mat 0 (sinA_mat n A) 0 1) = 1996 /\
Series (fun n => coeff_mat 0 (sinA_mat n A) 1 0) = 0 /\
Series (fun n => coeff_mat 0 (sinA_mat n A) 1 1) = 1) <->
putnam_1996_b4_solution.
Proof. Admitted.
|
putnam_1997_a3
|
Evaluate \begin{gather*} \int_0^\infty \left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6}+\cdots\right) \\ \left(1+\frac{x^2}{2^2}+\frac{x^4}{2^2\cdot 4^2}+\frac{x^6}{2^2\cdot 4^2 \cdot 6^2}+\cdots\right)\,dx. \end{gather*}
|
Show that the solution is $\sqrt{e}$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1997_a3_solution : R := sqrt (exp 1).
Theorem putnam_1997_a3
(evenfact := fix even_fact (n : nat) : R :=
match n with
| O => 1
| S n' => (2 * INR n) * even_fact n'
end)
(evenfactsqr := fix even_fact_sqr (n : nat) : R :=
match n with
| O => 1
| S n' => pow (2 * INR n) 2 * even_fact_sqr n'
end)
(f : R -> R := fun x => Series (fun n => pow (-1) n * pow x (2 * n + 1) / evenfact n))
(g : R -> R := fun x => Series (fun n => pow x (2 * n) / evenfactsqr n))
: Lim_seq (fun t : nat => RInt (fun x => f x * g x) 0 (INR t)) = putnam_1997_a3_solution.
Proof. Admitted.
|
putnam_1997_a4
|
Let $G$ be a group with identity $e$ and $\phi:G\rightarrow G$ a function such that \[\phi(g_1)\phi(g_2)\phi(g_3)=\phi(h_1)\phi(h_2)\phi(h_3)\] whenever $g_1g_2g_3=e=h_1h_2h_3$. Prove that there exists an element $a\in G$ such that $\psi(x)=a\phi(x)$ is a homomorphism (i.e. $\psi(xy)=\psi(x)\psi(y)$ for all $x,y\in G$).
|
None.
|
[
"abstract_algebra"
] |
(* Note: This formalization is only a statement finite groups (due to mathcomp), but the idea of the problem does not rely on the cardinal of the group, so we include it*)
Require Import Basics. From mathcomp Require Import fingroup.
Open Scope group_scope.
Variable T : finGroupType.
Theorem putnam_1997_a4
(G : {group T})
(f : T -> T)
(hf : forall g1 g2 g3 h1 h2 h3 : T, ((g1 * g2 * g3) = 1 /\ (h1 * h2 * h3) = 1) -> (f g1 * f g2 * f g3 = f h1 * f h2 * f h3))
: exists a : T, let psi := fun g => a * f g in (forall x y : T, psi (x * y) = psi x * psi y).
Proof. Admitted.
|
putnam_1997_a5
|
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.
|
Show that $N_{10}$ is odd.
|
[
"number_theory"
] |
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.
|
putnam_1997_a6
|
For a positive integer $n$ and any real number $c$, define $x_k$ recursively by $x_0=0$, $x_1=1$, and for $k\geq 0$, \[x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1}.\] Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$. Find $x_k$ in terms of $n$ and $k$, $1\leq k\leq n$.
|
Show that the solution is that $x_k = {n - 1 \choose k - 1}$.
|
[
"algebra"
] |
Require Import Binomial Nat Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1997_a6_solution : nat -> nat -> R := (fun n k : nat => Binomial.C (n - 1) (k - 1)).
Theorem putnam_1997_a6
(n : nat)
(maxc : R)
(X := fix x (c: R) (k: nat) : R :=
match k with
| O => 0
| S O => 1
| S ((S k'') as k') => (c * x c k' - INR (n - k) * x c k'') / INR k'
end)
(npos : gt n 0)
(hmaxc : X maxc (add n 1) = 0 /\ (forall c : R, X c (add n 1) = 0 -> c <= maxc))
: forall (k: nat), (le 1 k /\ le k n) -> X maxc k = putnam_1997_a6_solution n k.
Proof. Admitted.
|
putnam_1997_b1
|
Let $\{x\}$ denote the distance between the real number $x$ and the nearest integer. For each positive integer $n$, evaluate \[F_n=\sum_{m=1}^{6n-1} \min(\{\frac{m}{6n}\},\{\frac{m}{3n}\}).\] (Here $\min(a,b)$ denotes the minimum of $a$ and $b$.)
|
Show that the solution is $n$.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1997_b1_solution : nat -> R := (fun n : nat => INR n).
Theorem putnam_1997_b1
(rnd : R -> R := fun x => Rmin (Rabs (IZR (floor x) - x)) (Rabs (IZR (floor (x + 1)) - x)))
: forall (n: nat), gt n 0 -> sum_n_m (fun m => Rmin (rnd (INR m / (6 * INR n))) (rnd (INR m / (3 * INR n)))) 1 (6 * n - 1) = putnam_1997_b1_solution n.
Proof. Admitted.
|
putnam_1997_b2
|
Let $f$ be a twice-differentiable real-valued function satisfying \[f(x)+f''(x)=-xg(x)f'(x),\] where $g(x)\geq 0$ for all real $x$. Prove that $|f(x)|$ is bounded.
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1997_b2
(f g : R -> R)
(hg : forall x : R, g x >= 0)
(hfdiff : forall x : R, ex_derive f x /\ ex_derive_n f 2 x)
(hfg : forall x : R, f x + Derive_n f 2 x = -x * g x * Derive f x)
: exists M : R, (forall x : R, -M <= abs (f x) <= M).
Proof. Admitted.
|
putnam_1997_b3
|
For each positive integer $n$, write the sum $\sum_{m=1}^n 1/m$ in the form $p_n/q_n$, where $p_n$ and $q_n$ are relatively prime positive integers. Determine all $n$ such that 5 does not divide $q_n$.
|
Show that the solution is the set of natural numbers which are between $1$ and $4$, or between $20$ and $24$, or between $100$ and $104$, or between $120$ and $124$.
|
[
"number_theory"
] |
Require Import Reals Coquelicot.Coquelicot. From mathcomp Require Import div.
Open Scope R.
Definition putnam_1997_b3_solution : nat -> Prop := (fun n : nat => (le 1 n /\ le n 4) \/ (le 20 n /\ le n 24) \/ (le 100 n /\ le n 104) \/ (le 120 n /\ le n 124)).
Theorem putnam_1997_b3
(n : nat)
(p q : nat)
(hn : gt n 0)
(hpq : gt p 0 /\ gt q 0 /\ coprime p q = true /\ sum_n_m (fun m => 1 / INR m) 1 n = INR p / INR q)
: neq (q mod 5) 0 <-> putnam_1997_b3_solution n.
Proof. Admitted.
|
putnam_1997_b5
|
Prove that for $n\geq 2$, \[\overbrace{2^{2^{\cdots^{2}}}}^{\mbox{$n$ terms}} \equiv \overbrace{2^{2^{\cdots^{2}}}}^{\mbox{$n-1$ terms}} \quad \pmod{n}.\]
|
None.
|
[
"number_theory"
] |
Require Import Nat.
Theorem putnam_1997_b5
(tetration := fix tetration' (b m: nat) : nat :=
match m with
| O => 1
| S m' => b ^ (tetration' b m')
end)
(n : nat)
(hn : n >= 2)
: (tetration 2 n) mod n = (tetration 2 (n-1)) mod n.
Proof. Admitted.
|
putnam_1998_a3
|
Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that \[f(a)\cdot f'(a) \cdot f''(a) \cdot f'''(a)\geq 0 .\]
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1998_a3
(f : R -> R)
(hf : (forall x : R, ex_derive f x /\ ex_derive_n f 2 x /\ ex_derive_n f 3 x) /\ continuity (Derive_n f 3))
: exists (a: R), (Derive_n f 0) a * (Derive_n f 1) a * (Derive_n f 2) a * (Derive_n f 3) a >= 0.
Proof. Admitted.
|
putnam_1998_a4
|
Let $A_1=0$ and $A_2=1$. For $n>2$, the number $A_n$ is defined by concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from left to right. For example $A_3=A_2 A_1=10$, $A_4=A_3 A_2 = 101$, $A_5=A_4 A_3 = 10110$, and so forth. Determine all $n$ such that $11$ divides $A_n$.
|
Show that the solution is those n for which n can be written as 6k+1 for some integer k.
|
[
"algebra"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
Definition putnam_1998_a4_solution : set nat := [set n | n = 1 %[mod 6]].
Theorem putnam_1998_a4
(A : nat -> list nat)
(hA1 : A 1 = [:: 0])
(hA2 : A 2 = [:: 1])
(hA : forall n, gt n 0 -> A (n.+2) = A (n.+1) ++ A n)
(of_digits : list nat -> nat := fun L => foldl (fun x y => 10 * y + x) 0 L)
: [set n : nat | ge n 1 /\ 11 %| of_digits (A n)] = putnam_1998_a4_solution.
Proof. Admitted.
|
putnam_1998_b1
|
Find the minimum value of \[\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\] for $x>0$.
|
Show that the minimum value is 6.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1998_b1_solution : R := 6.
Theorem putnam_1998_b1
(f : R -> R := fun x => ((x + 1 / x) ^ 6 - (x ^ 6 + 1 / (x ^ 6)) - 2) / ((x + 1 / x) ^ 3 + (x ^ 3 + 1 / (x ^ 3))))
(m : R)
(hm : exists (x: R), x > 0 /\ f x = m)
(hmub : forall (x: R), x > 0 -> f x >= m)
: m = putnam_1998_b1_solution.
Proof. Admitted.
|
putnam_1998_b4
|
Find necessary and sufficient conditions on positive integers $m$ and $n$ so that \[\sum_{i=0}^{mn-1} (-1)^{\lfloor i/m \rfloor +\lfloor i/n\rfloor}=0.\]
|
Show that the sum is 0 if and only if the largest powers of $2$ dividing $m$ and $n$ are different.
|
[
"number_theory"
] |
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.
|
putnam_1998_b5
|
Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is, \[N=1111\cdots 11.\] Find the thousandth digit after the decimal point of $\sqrt N$.
|
Show that the thousandth digit is 1.
|
[
"number_theory"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1998_b5_solution : nat := 1.
Theorem putnam_1998_b5
(N : R := sum_n (fun i => 10^i) 1997)
: Z.to_nat (floor (10^1000 * sqrt N)) mod 10 = putnam_1998_b5_solution.
Proof. Admitted.
|
putnam_1998_b6
|
Prove that, for any integers $a, b, c$, there exists a positive integer $n$ such that $\sqrt{n^3+an^2+bn+c}$ is not an integer.
|
None.
|
[
"number_theory"
] |
Require Import Reals.
Open Scope Z.
Theorem putnam_1998_b6
(a b c : Z)
: exists (n: Z), n > 0 /\ (forall (m : Z), n * n * n + a * n * n + b * n + c <> m * m).
Proof. Admitted.
|
putnam_1999_a2
|
Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),\dots,f_k(x$) such that \[p(x) = \sum_{j=1}^k (f_j(x))^2.\]
|
None.
|
[
"algebra"
] |
From mathcomp.analysis Require Import reals. From mathcomp Require Import ssrnat ssrnum ssralg fintype poly.
Open Scope ring_scope.
Theorem putnam_1999_a2
(R : realType)
(p : {poly R})
(hpos : forall x, (p.[x] >= 0) = true)
: exists (k : nat) (f : nat -> {poly R}), p = \sum_(i < k) ((f i)*(f i)).
Proof. Admitted.
|
putnam_1999_a3
|
Consider the power series expansion \[\frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n.\] Prove that, for each integer $n\geq 0$, there is an integer $m$ such that \[a_n^2 + a_{n+1}^2 = a_m .\]
|
None.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_1999_a3
(f : R -> R := fun x => 1/(1 - 2 * x - x^2))
(a : nat -> R)
(hf : exists epsilon : R, epsilon > 0 /\ (forall x : R, 0 <= Rabs (x) < epsilon -> filterlim (fun n : nat => sum_n (fun i => a i * x^i) n) eventually (locally (f x))))
: forall n : nat, exists m : nat, (a n)^2 + (a (S n))^2 = a m.
Proof. Admitted.
|
putnam_1999_a4
|
Sum the series \[\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{m^2 n}{3^m(n3^m+m3^n)}.\]
|
Show that the solution is 9/32.
|
[
"number_theory"
] |
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_1999_a4_solution : R := 9/32.
Theorem putnam_1999_a4:
Series (fun m => Series (fun n => (INR (m + 1) ^ 2 * INR (n + 1)) / (3 ^ (m + 1) * (INR (n + 1) * 3 ^ (m + 1) + INR (m + 1) * 3 ^ (n + 1))))) = putnam_1999_a4_solution.
Proof. Admitted.
|
putnam_1999_a5
|
Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree 1999, then \[|p(0)|\leq C \int_{-1}^1 |p(x)|\,dx.\]
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_1999_a5
(p : (nat -> R) -> R -> R := fun a x => sum_n (fun i => a i * x ^ i) 1999)
: exists (c: R), forall (a: nat -> R), Rabs (p a 0) <= c * RInt (fun x => Rabs (p a x)) (-1) 1.
Proof. Admitted.
|
putnam_1999_a6
|
The sequence $(a_n)_{n\geq 1}$ is defined by $a_1=1, a_2=2, a_3=24,$ and, for $n\geq 4$, \[a_n = \frac{6a_{n-1}^2a_{n-3} - 8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.\] Show that, for all n, $a_n$ is an integer multiple of $n$.
|
None.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_1999_a6
(A : nat -> R := fix a (n: nat) :=
match n with
| O => 1
| S O => 2
| S (S O) => 24
| S (S ((S n'') as n') as n) => (6 * (a n) ^ 2 * a n'' - 8 * a n * (a n') ^ 2) / (a n' * a n'')
end)
: forall (n: nat), exists (k: Z), A n = INR (n + 1) * IZR k.
Proof. Admitted.
|
putnam_1999_b2
|
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P''(x)$, where $Q(x)$ is a quadratic polynomial and $P''(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.
|
None.
|
[
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals complex derive topology normedtype.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1999_b2
(P Q : {poly R[i]})
(hQ : size Q = 3%nat)
(hP : forall x : R[i], P.[x] = Q.[x] * (P^`(2)).[x])
: (exists x1 x2 : R[i], x1 <> x2 /\ P.[x1] = 0 /\ P.[x2] = 0) ->
(exists f : seq R[i], size f = (size P).-1 /\ uniq f /\ all (fun x => P.[x] == 0) f).
Proof. Admitted.
|
putnam_1999_b4
|
Let $f$ be a real function with a continuous third derivative such that $f(x), f'(x), f''(x), f'''(x)$ are positive for all $x$. Suppose that $f'''(x)\leq f(x)$ for all $x$. Show that $f'(x)<2f(x)$ for all $x$.
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Derive.
Open Scope R_scope.
Theorem putnam_1999_b4
(f: R -> R)
(f_cont : (forall n : nat, (le 1 n /\ le n 3) -> (forall x : R, ex_derive_n f n x)) /\ continuity (Derive_n f 3))
(f_pos : forall (x: R), f x > 0 /\ (Derive_n f 1) x > 0 /\ (Derive_n f 2) x > 0 /\ (Derive_n f 3) x > 0)
(hf : forall (x: R), (Derive_n f 3) x <= f x)
: forall (x: R), (Derive_n f 1) x < 2 * f x.
Proof. Admitted.
|
putnam_1999_b6
|
Let $S$ be a finite set of integers, each greater than 1. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.
|
None.
|
[
"number_theory"
] |
Require Import Reals List Znumtheory.
Theorem putnam_1999_b6
(A : list Z)
(Age1 : forall (x: Z), In x A -> x > 1)
(hgcd : forall (n: Z), exists (s: Z), In s A /\ (Zis_gcd s n 1 \/ Zis_gcd s n s))
: exists (s: Z) (t: Z) (p: Z), In s A /\ In t A /\ Zis_gcd s t p /\ prime p.
Proof. Admitted.
|
putnam_2000_a1
|
Let $A$ be a positive real number. What are the possible values of $\sum_{j=0}^\infty x_j^2$, given that $x_0,x_1,\ldots$ are positive numbers for which $\sum_{j=0}^\infty x_j=A$?
|
Show that the possible values comprise the interval $(0,A^2)$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2000_a1_solution : R -> (R -> Prop) := (fun A : R => (fun x : R => 0 < x < A ^ 2)).
Theorem putnam_2000_a1
(A: R)
(hA : A > 0)
: forall SS : R, ((exists x : nat -> R, (forall j : nat, x j > 0) /\ Series x = A /\ Series (fun j => (x j) ^ 2) = SS) <-> (putnam_2000_a1_solution A) SS).
Proof. Admitted.
|
putnam_2000_a2
|
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.
|
None.
|
[
"number_theory"
] |
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.
|
putnam_2000_a4
|
Show that the improper integral $\lim_{B \to \infty} \int_0^B \sin(x)\sin(x^2)\,dx$ converges.
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2000_a4
: ex_finite_lim_seq (fun B : nat => RInt (fun x : R => sin x * sin (x ^ 2)) 0 (INR B)).
Proof. Admitted.
|
putnam_2000_a6
|
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 $a_2=0$.
|
None.
|
[
"algebra"
] |
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.
|
putnam_2000_b1
|
Let $a_j,b_j,c_j$ be integers for $1\leq j\leq N$. Assume for each $j$, at least one of $a_j,b_j,c_j$ is odd. Show that there exist integers $r$, $s$, $t$ such that $ra_j+sb_j+tc_j$ is odd for at least $4N/7$ values of $j$, $1\leq j\leq N$.
|
None.
|
[
"algebra"
] |
Require Import List Nat Reals ZArith Ensembles Finite_sets.
Open Scope Z.
Theorem putnam_2000_b1
(n : nat)
(a b c : nat -> Z)
(SS : Z -> Z -> Z -> Ensemble nat := (fun r s t : Z => (fun j : nat => le 1 j /\ le j n /\ Z.odd (Z.add (Z.add (Z.mul r (a j)) (Z.mul s (b j))) (Z.mul t (c j))) = true)))
(SSsize : Z -> Z -> Z -> nat)
(nge1 : ge n 1)
(habc : forall j : nat, ((le 1 j) /\ (le j n)) -> (Z.odd (a j) = true \/ Z.odd (b j) = true \/ Z.odd (c j) = true))
(hSSsize : forall r s t : Z, cardinal nat (SS r s t) (SSsize r s t))
: exists r s t : Z, Rge (INR (SSsize r s t)) (Rdiv (Rmult 4 (INR n)) 7).
Proof. Admitted.
|
putnam_2000_b2
|
Prove that the expression
\[
\frac{gcd(m,n)}{n}\binom{n}{m}
\]
is an integer for all pairs of integers $n\geq m\geq 1$.
|
None.
|
[
"number_theory",
"algebra"
] |
Require Import Nat Reals.
Open Scope R.
Theorem putnam_2000_b2
: forall (n m: nat), and (ge n m) (ge m 1) -> exists (c: Z), (INR (gcd m n) / INR n) * Binomial.C n m = IZR c.
Proof. Admitted.
|
putnam_2000_b4
|
Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\leq x\leq 1$.
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2000_b4
(f : R -> R)
(hf : forall x : R, f (2*x*x-1) = 2*x*(f x))
(f_cont : continuity f)
: forall x : R, -1 <= x <= 1 -> f x = 0.
Proof. Admitted.
|
putnam_2000_b5
|
Let $S_0$ be a finite set of positive integers. We define finite sets $S_1,S_2,\ldots$ of positive integers as follows: the integer $a$ is in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$ for which $S_N=S_0\cup\{N+a: a\in S_0\}$.
|
None.
|
[
"algebra"
] |
Require Import ZArith Ensembles Finite_sets.
Theorem putnam_2000_b5
(SS : nat -> Ensemble Z)
(hSSfin : forall n : nat, exists m : nat, cardinal _ (SS n) m)
(hSSpos : forall n : nat, forall s : Z, In _ (SS n) s -> Z.gt s 0)
(hSSdef : forall n : nat, forall a : Z, (In _ (SS (n + 1)) a) <-> ((In _ (SS n) (Z.sub a 1) /\ ~ (In _ (SS n) a)) \/ (In _ (SS n) a /\ ~ (In _ (SS n) (Z.sub a 1)))))
: forall n : nat, exists N : nat, N >= n /\ Same_set _ (SS N) (Union _ (SS 0) (fun i : Z => exists a : Z, In _ (SS 0) a /\ i = Z.add a (Z.of_nat N))).
Proof. Admitted.
|
putnam_2001_a1
|
Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\in S$, $a*b\in S$. Assume $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b\in S$.
|
None.
|
[
"abstract_algebra"
] |
Require Import Ensembles RelationClasses.
Theorem putnam_2001_a1
(A : Type)
(op : A->A->A)
(hop : forall (a b: A), op (op a b) a = b)
: forall (a b: A), op a (op b a) = b.
Proof. Admitted.
|
putnam_2001_a3
|
For each integer $m$, consider the polynomial
\[P_m(x)=x^4-(2m+4)x^2+(m-2)^2.\] For what values of $m$ is $P_m(x)$
the product of two non-constant polynomials with integer coefficients?
|
$P_m(x)$ factors into two nonconstant polynomials over
the integers if and only if $m$ is either a square or twice a square.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot Ensembles.
Definition putnam_2001_a3_solution : Ensemble Z := (fun m : Z => exists (n: Z), m = Z.mul n n \/ m = Z.mul 2 (Z.mul n n)).
Theorem putnam_2001_a3
(P : Z -> R -> R := fun m x => x ^ 4 - (2 * IZR m + 4) * x ^ 2 + (IZR m - 2) ^ 2)
(p : (nat -> Z) -> R -> nat -> R := fun a x n => sum_n (fun i => IZR (a i) * x ^ i) n)
: forall (m: Z), (exists (a1 a2: nat -> Z) (n1 n2: nat), (exists i : nat, gt i 0 /\ le i n1 /\ a1 i <> 0%Z) /\ (exists i : nat, gt i 0 /\ le i n2 /\ a2 i <> 0%Z) /\
(forall (x: R), P m x = p a1 x n1 * p a2 x n2)) <-> putnam_2001_a3_solution m.
Proof. Admitted.
|
putnam_2001_a5
|
Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.
|
None.
|
[
"number_theory"
] |
Require Import Nat.
Theorem putnam_2001_a5
: exists! (a n: nat), a > 0 /\ n > 0 /\ a ^ (n + 1) - (a + 1) ^ n = 2001.
Proof. Admitted.
|
putnam_2001_b2
|
Find all pairs of real numbers $(x,y)$ satisfying the system of equations
\begin{align*}
\frac{1}{x}+\frac{1}{2y}&=(x^2+3y^2)(3x^2+y^2) \\
\frac{1}{x}-\frac{1}{2y}&=2(y^4-x^4).
\end{align*}
|
Show that $x=(3^{1/5}+1)/2$ and $y=(3^{1/5}-1)/2$ is the unique solution satisfying the given equations.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2001_b2_solution : R -> R -> Prop := (fun x y : R => x = (3 ^ (1 / 5) + 1) / 2 /\ y = (3 ^ (1 / 5) - 1) / 2).
Theorem putnam_2001_b2
(x y : R)
(hx : x <> 0)
(hy : y <> 0)
: (1 / x + 1 / (2 * y) = (x ^ 2 + 3 * y ^ 2) * (3 * x ^ 2 + y ^ 2) /\
1 / x - 1 / (2 * y) = 2 * (y ^ 4 - x ^ 4)) <-> putnam_2001_b2_solution x y.
Proof. Admitted.
|
putnam_2001_b3
|
For any positive integer $n$, let $\langle n \rangle$ denote the closest integer to $\sqrt{n}$. Evaluate $\sum_{n=1}^\infty \frac{2^{\langle n \rangle}+2^{-\langle n \rangle}}{2^n}$.
|
Show that the sum is $3$.
|
[
"analysis"
] |
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_2001_b3_solution.
Proof. Admitted.
|
putnam_2001_b4
|
Let $S$ denote the set of rational numbers different from $\{-1,0,1\}$. Define $f:S\rightarrow S$ by $f(x)=x-1/x$. Prove or disprove that \[\bigcap_{n=1}^\infty f^{(n)}(S) = \emptyset,\] where $f^{(n)}$ denotes $f$ composed with itself $n$ times.
|
None.
|
[
"algebra"
] |
Require Import Basics List QArith. From mathcomp Require Import bigop fintype seq ssrbool ssreflect ssrnat ssrnum ssralg finfun.
Open Scope Q_scope.
Definition image (f: Q -> Q) := fun y => exists (x: Q), (~ In x [:: -1; 0; 1]) /\ f x = y.
Fixpoint compose_n {A : Type} (f : A -> A) (n : nat) :=
match n with
| O => fun x => x
| S n' => compose f (compose_n f n')
end.
Definition putnam_2001_b4_solution : Prop := True.
Theorem putnam_2001_b4
(f : Q -> Q := fun x => x - 1 / x)
: (~exists (x: Q), (~ In x [:: -1; 0; 1]) /\ (forall (n: nat), ge n 1 -> (image (compose_n f n)) x)) <-> putnam_2001_b4_solution.
Proof. Admitted.
|
putnam_2001_b5
|
Let $a$ and $b$ be real numbers in the interval $(0,1/2)$, and let $g$ be a continuous real-valued function such that $g(g(x))=ag(x)+bx$ for all real $x$. Prove that $g(x)=cx$ for some constant $c$.
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2001_b5
(a b: R)
(g: R -> R)
(hab : 0 < a < 1/2 /\ 0 < b < 1/2)
(gcont : continuity g)
(hg : forall (x: R), g (g x) = a * g x + b * x)
: exists c : R, forall x : R, g x = c * x.
Proof. Admitted.
|
putnam_2001_b6
|
Assume that $(a_n)_{n \geq 1}$ is an increasing sequence of positive real numbers such that $\lim a_n/n=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\ldots,n-1$?
|
Show that the answer is yes, there must exist infinitely many such $n$.
|
[
"analysis"
] |
Require Import Nat Reals Coquelicot.Coquelicot.
Definition putnam_2001_b6_solution : Prop := True.
Theorem putnam_2001_b6
(aposinc : (nat -> R) -> Prop := (fun a : nat -> R => forall n : nat, ge n 1 -> (a n > 0 /\ a n < a (add n 1))))
(alim : (nat -> R) -> Prop := (fun a : nat -> R => Lim_seq (fun n : nat => a (add n 1) / INR (add n 1)) = 0))
: (forall a : nat -> R, (aposinc a /\ alim a) -> forall m : nat, exists n : nat, gt n m /\ (forall i : nat, (le 1 i /\ le i (n - 1)) -> a (sub n i) + a (add n i) < 2 * a n)) <-> putnam_2001_b6_solution.
Proof. Admitted.
|
putnam_2002_a1
|
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 $P_n(1)$.
|
Show that $P_n(1)=(-k)^nn!$ for all $n \geq 0$.
|
[
"analysis",
"algebra"
] |
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.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition putnam_2002_a1_solution : nat -> nat -> R := fun k n : nat => (-k%:R) ^ n * n`!%:R.
Theorem putnam_2002_a1
(k : nat)
(P : nat -> {poly R})
(kpos : gt k 0)
(Pderiv : forall n : nat, forall x : R, derive1n n (fun x' : R => 1/(x' ^ k - 1)) x = (P n).[x] / ((x ^ k - 1) ^ (n.+1)))
: forall n : nat, (P n).[1] = putnam_2002_a1_solution k n.
Proof. Admitted.
|
putnam_2002_b3
|
Show that, for all integers $n > 1$,
\[
\frac{1}{2ne} < \frac{1}{e} - \left( 1 - \frac{1}{n} \right)^n
< \frac{1}{ne}.
\]
|
None.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_2002_b3
: forall (n: nat), gt n 1 -> let n := INR n in 1 / (2 * n * exp 1) < 1 / (exp 1) - Rpower (1 - 1 / n) n < 1 / (n * (exp 1)).
Proof. Admitted.
|
putnam_2003_a1
|
Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, \[ n = a_1 + a_2 + \dots + a_k, \] with $k$ an arbitrary positive integer and $a_1 \leq a_2 \leq \dots \leq a_k \leq a_1 + 1$? For example, with $n = 4$, there are four ways: $4, 2 + 2, 1 + 1 + 2, 1 + 1 + 1 + 1$
|
Show that there are $n$ such sums.
|
[
"algebra"
] |
Require Import Nat List Ensembles Finite_sets Coquelicot.Coquelicot.
Definition putnam_2003_a1_solution : nat -> nat := (fun n : nat => n).
Theorem putnam_2003_a1
(n: nat)
(hn : n > 0)
(E: Ensemble (list nat) := fun l => fold_left add l 0 = n /\ (forall i : nat, i < length l -> nth i l 0 > 0) /\ (forall (i j: nat), i < length l /\ j < length l /\ i < j -> nth i l 0 <= nth j l 0) /\ last l 0 <= hd 0 l + 1)
: cardinal (list nat) E (putnam_2003_a1_solution n).
Proof. Admitted.
|
putnam_2003_a2
|
Let $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_n$ be nonnegative real numbers. Show that $(a_1a_2 \cdots a_n)^{1/n}+(b_1b_2 \cdots b_n)^{1/n} \leq [(a_1+b_1)(a_2+b_2) \cdots (a_n+b_n)]^{1/n}$.
|
None.
|
[
"algebra"
] |
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 : 'I_n -> R)
(abnneg : forall i : 'I_n, (a i) >= 0 /\ (b i) >= 0)
: expR (ln (\prod_(i < n) (a i)) * (1 / n%:R)) +
expR (ln (\prod_(i < n) (b i)) * (1 / n%:R)) <=
expR (ln (\prod_(i < n) (a i + b i)) * (1 / n%:R)).
Proof. Admitted.
|
putnam_2003_a3
|
Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$.
|
Show that the minimum is $2\sqrt{2}-1$.
|
[
"analysis"
] |
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.
|
putnam_2003_a4
|
Suppose that $a,b,c,A,B,C$ are real numbers, $a \ne 0$ and $A \ne 0$, such that $|ax^2+bx+c| \leq |Ax^2+Bx+C|$ for all real numbers $x$. Show that $|b^2-4ac| \leq |B^2-4AC|$.
|
None.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2003_a4
(a b c A B C: R)
(ha : a <> 0)
(hA : A <> 0)
(hp : forall (x: R), Rabs (a * x ^ 2 + b * x + c) <= Rabs (A * x ^ 2 + B * x + C))
: Rabs (b ^ 2 - 4 * a * c) <= Rabs (B ^ 2 - 4 * A * C).
Proof. Admitted.
|
putnam_2003_b1
|
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?
|
Show that no such polynomials exist.
|
[
"linear_algebra",
"algebra"
] |
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 b c d : {poly R}, forall x y : R, 1 + x * y + x ^ 2 * y ^ 2 = a.[x] * c.[y] + b.[x] * d.[y]) <-> putnam_2003_b1_solution.
Proof. Admitted.
|
putnam_2003_b3
|
Show that for each positive integer $n$, $n!=\prod_{i=1}^n \text{lcm}\{1,2,\dots,\lfloor n/i \rfloor\}$. (Here lcm denotes the least common multiple, and $\lfloor x \rfloor$ denotes the greatest integer $\leq x$.)
|
None.
|
[
"number_theory"
] |
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_2003_b3
(n : nat)
: n `! = \prod_(1 <= i < n.+1) (foldl (fun x y => lcmn x y) 1%nat (iota 1 (n%/i))).
Proof. Admitted.
|
putnam_2003_b4
|
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.
|
None.
|
[
"number_theory",
"algebra"
] |
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 = a * (z - r1) * (z - r2) * (z - r3) * (z - r4)) ->
((exists (p q: Z), r1 + r2 = IZR p / IZR q) /\ r1 + r2 <> r3 + r4) -> (exists (p q: Z), r1 * r2 = IZR p / IZR q).
Proof. Admitted.
|
putnam_2003_b6
|
Let $f(x)$ be a continuous real-valued function defined on the interval $[0,1]$. Show that \[ \int_0^1 \int_0^1 | f(x) + f(y) |\,dx\,dy \geq \int_0^1 |f(x)|\,dx. \]
|
None.
|
[
"analysis"
] |
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.
|
putnam_2004_a3
|
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, $0!=1$.)
|
None.
|
[
"linear_algebra"
] |
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.
|
putnam_2004_a4
|
Show that for any positive integer $n$ there is an integer $N$ such that the product $x_1x_2 \cdots x_n$ can be expressed identically in the form $x_1x_2 \cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots+a_{in}x_n)^n$ where the $c_i$ are rational numbers and each $a_{ij}$ is one of the numbers $-1,0,1$.
|
None.
|
[
"algebra"
] |
Require Import List Reals QArith Coquelicot.Coquelicot.
Theorem putnam_2004_a4
(n : nat)
(x : list R)
(avals : nat -> (nat -> nat -> R) -> Prop := fun (N : nat) (a : nat -> nat -> R) => forall i j : nat, lt i N /\ lt j n -> a i j = -1 \/ a i j = 0 \/ a i j = 1)
(npos : gt n 0)
(hx : length x = n)
: exists (N : nat) (c : nat -> Q) (a : nat -> nat -> R), avals N a /\ fold_left Rmult x 1 = sum_n (fun i : nat => Q2R (c i) * (sum_n (fun j : nat => a i j * nth j x 0) (n - 1)) ^ n) (N - 1).
Proof. Admitted.
|
putnam_2004_a6
|
Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0 \leq x \leq 1,0 \leq y \leq 1$. Show that $\int_0^1 \left(\int_0^1 f(x,y)dx\right)^2dy+\int_0^1 \left(\int_0^1 f(x,y)dy\right)^2dx \leq \left(\int_0^1 \int_0^1 f(x,y)dx\,dy\right)^2+\int_0^1 \int_0^1 [f(x,y)]^2dx\,dy$.
|
None.
|
[
"analysis"
] |
Require Import Basics Reals Coquelicot.Coquelicot.
Theorem putnam_2004_a6
(f: R -> R -> R)
(hf : forall x y : R, (0 <= x <= 1 /\ 0 <= y <= 1) -> continuity_2d_pt f x y)
: RInt (fun y => (RInt (fun x => f x y) 0 1)^2) 0 1 + RInt (fun x => (RInt (fun y => f x y) 0 1)^2) 0 1 <=
(RInt (fun x => RInt (fun y => f x y) 0 1) 0 1) ^ 2 + RInt (fun x => RInt (fun y => (f x y) ^ 2) 0 1) 0 1.
Proof. Admitted.
|
putnam_2004_b1
|
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 integers.
|
None.
|
[
"algebra"
] |
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.
|
putnam_2004_b2
|
Let $m$ and $n$ be positive integers. Show that $\frac{(m+n)!}{(m+n)^{m+n}}<\frac{m!}{m^m}\frac{n!}{n^n}$.
|
None.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Theorem putnam_2004_b2
(m n : nat)
(mnpos : gt m 0 /\ gt n 0)
: INR (fact (m + n)) / INR (m + n) ^ (m + n) < INR (fact m) / INR m ^ m * INR (fact n) / INR n ^ n.
Proof. Admitted.
|
putnam_2004_b5
|
Evaluate $\lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.
|
Show that the desired limit is $2/e$.
|
[
"analysis"
] |
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.
|
putnam_2005_a1
|
Show that every positive integer is a sum of one or more numbers of the form $2^r 3^s$, where $r$ and $s$ are nonnegative integers and no summand divides another.
|
None.
|
[
"number_theory"
] |
Require Import Nat List Coquelicot.Coquelicot.
Fixpoint nat_sum (a : nat -> nat) (k : nat) : nat :=
match k with
| O => a 0
| S k' => a k + nat_sum a k'
end.
Theorem putnam_2005_a1
: forall n : nat, n > 0 -> exists k : nat, exists a : nat -> (nat * nat), n = nat_sum (fun i => 2^(fst (a i)) * 3^(snd(a i))) k /\ forall i j : nat, (i <= k /\ j <= k /\ i <> j) -> ~exists m : nat, (2^(fst (a i))*3^(snd (a i)) * m = 2^(fst (a j))*3^(snd (a j))).
Proof. Admitted.
|
putnam_2005_a3
|
Let $p(z)$ be a polynomial of degree $n$ all of whose zeros have absolute value $1$ in the complex plane. Put $g(z)=p(z)/z^{n/2}$. Show that all zeros of $g'(z)=0$ have absolute value $1$.
|
None.
|
[
"analysis",
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot. From Coqtail Require Import Cpow.
Theorem putnam_2005_a3
(csqrt : C -> C)
(c : nat -> C)
(n : nat)
(hn : gt n 0)
(p : C -> C := fun z : C => sum_n (fun i => c i * z^i) n)
(g : C -> C := fun z : C => p z / csqrt (z^n))
(pzeros : forall z : C, p z = 0 -> norm z = 1%R)
(hcsqrt : forall z : C, (csqrt z)^2 = z /\ Re (csqrt z) >= 0 /\ (Re (csqrt z) = 0%R -> Im (csqrt z) >= 0%R))
: forall z : C, z <> 0 -> C_derive g z = 0 -> norm z = 1%R.
Proof. Admitted.
|
putnam_2005_a5
|
Evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1}\,dx$.
|
Show that the solution is $\pi / 8 * \log 2$.
|
[
"analysis"
] |
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.
|
putnam_2005_b1
|
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a \rfloor,\lfloor 2a \rfloor)=0$ for all real numbers $a$. (Note: $\lfloor \nu \rfloor$ is the greatest integer less than or equal to $\nu$.)
|
Show that $P(x,y)=(y-2x)(y-2x-1)$ works.
|
[
"algebra"
] |
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_solution) i j * x ^ i * y ^ j) (fst (snd putnam_2005_b1_solution)))) (snd (snd putnam_2005_b1_solution)))
: (~forall x y : R, p x y = 0) /\ forall (a : R), p (IZR (floor a)) (IZR (floor (2 * a))) = 0.
Proof. Admitted.
|
putnam_2005_b2
|
Find all positive integers $n,k_1,\dots,k_n$ such that $k_1+\cdots+k_n=5n-4$ and $\frac{1}{k_1}+\cdots+\frac{1}{k_n}=1$.
|
Show that the solutions are $n=1$ and $k_1=1$, $n=3$ and $(k_1,k_2,k_3)$ is a permutation of $(2,3,6)$, and $n=4$ and $(k_1,k_2,k_3,k_4)=(4,4,4,4)$.
|
[
"algebra"
] |
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%nat; 2%nat]) \/ (n, k) = (3%nat, [6%nat; 2%nat; 3%nat]) \/ (n, k) = (3%nat, [6%nat; 3%nat; 2%nat]) \/ (n, k) = (4%nat, [4%nat; 4%nat; 4%nat; 4%nat]).
Theorem putnam_2005_b2
: forall (n: nat) (k: list nat), length k = n -> ((gt n 0 /\ (forall i : nat, In i k -> gt i 0) /\ fold_left add k 0%nat = sub (mul 5 n) 4 /\ sum_n (fun m => 1 / INR (nth m k 0%nat)) (n - 1) = 1) <-> putnam_2005_b2_solution n k).
Proof. Admitted.
|
putnam_2005_b3
|
Find all differentiable functions $f:(0,\infty) \to (0,\infty)$ for which there is a positive real number $a$ such that $f'(\frac{a}{x})=\frac{x}{f(x)}$ for all $x>0$.
|
Show that the functions are precisely $f(x)=cx^d$ for $c,d>0$ arbitrary except that we must take $c=1$ in case $d=1$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2005_b3_solution (f : R -> R) := exists c d : R, c > 0 /\ d > 0 /\ (d = 1 -> c = 1) /\ forall x : R, x > 0 -> f x = c * Rpower x d.
Theorem putnam_2005_b3
(f : R -> R)
(hf : forall x : R, x > 0 -> f x > 0)
(hf' : forall x : R, x > 0 -> ex_derive f x)
: (exists a : R, a > 0 /\ forall x : R, x > 0 -> Derive f (a / x) = x / f x) <-> putnam_2005_b3_solution f.
Proof. Admitted.
|
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