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putnam_1974_b3
|
Prove that if $\alpha$ is a real number such that $\cos (\pi \alpha) = 1/3$, the $\alpha$ is irrational.
|
None.
|
[
"number_theory"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences exp trigo.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Definition is_rational (x : R) := exists p q : int, q != 0 /\ x = (p%:~R) / (q%:~R).
Theorem putnam_1974_b3
(a : R)
(ha : cos (pi * a) = 1/3)
: ~ is_rational a.
Proof. Admitted.
|
putnam_1974_b5
|
Show that $1 + (n/1!) + (n^2/2!) + \dots + (n^n/n!) > e^n/2$ for every integer $n \geq 0$.
|
None.
|
[
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences exp normedtype signed.
Import Order.TTheory GRing.Theory Num.Def Num.Theory.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1974_b5
(f : nat -> R := fun n => (\sum_(0 <= i < n.+1) (n%:R) ^ i / ((i`!)%:R)))
: forall n : nat, f n > (expR n%:R) / 2%R.
Proof. Admitted.
|
putnam_1974_b6
|
For a set with $1000$ elements, how many subsets are there whose candinality is respectively $\equiv 0 \bmod 3, \equiv 1 \bmod 3, \equiv 2 \bmod 3$?
|
Show that there answer is that there are $(2^1000-1)/3$ subsets of cardinality $\equiv 0 \bmod 3$ and $\equiv 1 \bmod 3$, and $1 + (2^1000-1)/3$ subsets of cardinality $\equiv 2 \bmod 3$.
|
[
"set_theory"
] |
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.
|
putnam_1975_a1
|
If an integer $n$ can be written as the sum of two triangular numbers (that is, $n = \frac{a^2 + a}{2} + \frac{b^2 + b}{2}$ for some integers $a$ and $b$), express $4n + 1$ as the sum of the squares of two integers $x$ and $y$, giving $x$ and $y$ in terms of $a$ and $b$. Also, show that if $4n + 1 = x^2 + y^2$ for some integers $x$ and $y$, then $n$ can be written as the sum of two triangular numbers.
|
$x = a + b + 1$ and $y = a - b$ (or vice versa).
|
[
"algebra",
"number_theory"
] |
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_1975_a1_solution : ((int * int) -> int) * ((int * int) -> int) := (fun '(a, b) => a + b + 1, fun '(a, b) => a - b).
Theorem putnam_1975_a1
(nab : set (int * int * int) := [set nab | let '(n, a, b) := nab in n = (a%:~R ^+ 2 + a%:~R)/2 + (b%:~R ^+ 2 + b%:~R)/2])
(nxy : set (int * int * int) := [set nxy | let '(n, x, y) := nxy in 4 * n + 1 = x ^+ 2 + y ^+ 2])
: (forall n a b : int, nab (n, a, b) <-> nxy (n, putnam_1975_a1_solution.1 (a, b), putnam_1975_a1_solution.2 (a, b))) /\
forall n : int, (exists x y : int, nxy (n, x, y)) -> exists a b : int, nab (n, a, b).
Proof. Admitted.
|
putnam_1975_a2
|
For which ordered pairs $(b, c)$ of real numbers do both roots of $z^2 + bz + c$ lie strictly inside the unit disk (i.e., $\{|z| < 1\}$) in the complex plane?
|
The desired region is the strict interior of the triangle with vertices $(0, -1)$, $(2, 1)$, and $(-2, 1)$.
|
[
"algebra"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals complex.
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 complex_scope.
(* Note: normc is defined in real_closed.complex but cannot be recognized when I try to use it here. I substitute an equivalent (to this problem) definition here.*)
Variable R : realType.
Definition putnam_1975_a2_solution : (R * R) -> Prop := fun '(b, c) => c < 1 /\ c - b > -1 /\ c + b > -1.
Theorem putnam_1975_a2
: forall b c : R, (forall z : R[i], z ^+ 2 + (Complex b 0) * z + (Complex c 0) = 0 -> (Re z) ^+ 2 + (Im z) ^+ 2 < 1) <-> putnam_1975_a2_solution (b, c).
Proof. Admitted.
|
putnam_1975_a3
|
If $a$, $b$, and $c$ are real numbers satisfying $0 < a < b < c$, at what points in the set $$\{(x, y, z) \in \mathbb{R}^3 : x^b + y^b + z^b = 1, x \ge 0, y \ge 0, z \ge 0\}$$ does $f(x, y, z) = x^a + y^b + z^c$ attain its maximum and minimum?
|
$f$ attains its maximum at $\left(x_0, (1 - x_0^b)^{\frac{1}{b}}, 0\right)$ and its minimum at $\left(0, (1 - z_0^b)^{\frac{1}{b}}, z_0\right)$, where $x_0 = \left(\frac{a}{b}\right)^{\frac{1}{b-a}}$ and $z_0 = \left(\frac{b}{c}\right)^{\frac{1}{c-b}}$.
|
[
"algebra"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_1975_a3_solution : ((R * R * R) -> (R * R * R)) * ((R * R * R) -> (R * R * R)) :=
(fun xyz => let '(a, b, c) := xyz in (expR (1/(b-a) * ln (a/b)), expR (1/b * ln (1 - expR ((b/(b-a) * ln (a/b))))), 0),
fun xyz => let '(a, b, c) := xyz in (0, expR ((1/b) * ln (1 - expR (b/(c-b) * ln (b/c)))), (expR ((ln (b/c)) * 1/(c-b))))).
Theorem putnam_1975_a3
(a b c : R)
(hi : 0 < a /\ a < b < c)
(P : (R * R * R) -> Prop := fun xyz: R * R * R => let '(x, y, z) := xyz in (x >= 0 /\ y >= 0 /\ z >= 0 /\ expR (b * ln x) + expR (b * ln y) + expR (b * ln z) = 1))
(f : (R * R * R) -> R := fun xyz: R * R * R => let '(x, y, z) := xyz in expR (a * ln x) + expR (b * ln y) + expR (c * ln z))
: (P ((fst putnam_1975_a3_solution) (a, b, c)) /\ forall x y z : R, P (x, y, z) ->
f (x, y, z) <= f ((fst putnam_1975_a3_solution) (a, b, c))) /\
(P ((snd putnam_1975_a3_solution) (a, b, c)) /\ forall x y z : R, P (x, y, z) ->
f (x, y, z) >= f ((snd putnam_1975_a3_solution) (a, b, c))).
Proof. Admitted.
|
putnam_1975_a5
|
On some interval $I$ of the real line, let $y_1(x), y_2(x)$ be linearly independent solutions of the differential equation \[y'' = f(x)y\], where $f(x)$ is a continuous real-valued function. Suppose that $y_1(x) > 0$ and $y_2(x) > 0$ on $I$. Show that there exists a positive constant $c$ such that, on $I$, the function \[z(x) = c \sqrt{y_1(x)y_2(x)}\] satisfies the equation \[z'' + 1/z^3 = f(x)z.\]
|
None.
|
[
"analysis"
] |
From mathcomp Require Import all_ssreflect ssrnum ssralg.
From mathcomp Require Import reals normedtype derive topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
(* Note: Lacking clean way of defining twice-differentiable functions as in Lean *)
(* Note: Slightly weaker as this assumes a closed interval instead of a more general real interval. The heart of the problem is unchanged. *)
Variable R : realType.
Theorem putnam_1975_a5
(a b : R)
(hab : a < b)
(f y1 y2 : R -> R)
(hfcont : continuous f)
(diff : (R -> R) -> Prop := fun g => (forall x : R, differentiable g x /\ differentiable g^`() x) /\ continuous g^`(2))
(hycdiff : diff y1 /\ diff y2)
(hy1sol : forall x : R, y1^`(2) x = (f x) * (y1 x))
(hy2sol : forall x : R, y2^`(2) x = (f x) * (y2 x))
(hylinindep : forall c1 c2 : R, (forall x : R, c1 * y1 x + c2 * y2 x = 0) -> (c1 = 0 /\ c2 = 0))
(hyI : forall x : R, a <= x <= b -> (y1 x > 0 /\ y2 x > 0))
: exists c : R, c > 0 /\ (let z : R -> R := fun x => c * (@Num.sqrt R ((y1 x) * (y2 x))) in
forall x : R, a <= x <= b -> z^`(2) x + 1/((z x) ^+ 3) = (f x) * (z x)).
Proof. Admitted.
|
putnam_1975_b5
|
Let $f_0(x) = e^x$ and $f_{n+1}(x) = xf'_n(x)$ for all $n \ge 0$. Prove that $$\sum_{n=0}^{\infty} \frac{f_n(1)}{n!} = e^e.$$
|
None.
|
[
"analysis",
"algebra"
] |
From mathcomp Require Import all_ssreflect ssrnum ssralg ring_quotient.
From mathcomp Require Import reals sequences exp topology normedtype 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.
Theorem putnam_1975_b5
(f : nat -> R -> R)
(h0 : f 0%nat = expR)
(hf : forall (n : nat) (x : R), f n.+1 x = x * ((f n)^`() x))
: series (fun n => f n 1 / n`!%:R) @ \oo --> (expR (expR 1)).
Proof. Admitted.
|
putnam_1975_b6
|
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 $n > 2$.
|
None.
|
[
"analysis"
] |
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.
|
putnam_1976_a3
|
Find all integer solutions $(p, r, q, s)$ of the equation $|p^r - q^s| = 1$, where $p$ and $q$ are prime and $r$ and $s$ are greater than $1$.
|
The only solutions are $(p, r, q, s) = (3, 2, 2, 3)$ and $(p, r, q, s) = (2, 3, 3, 2)$.
|
[
"number_theory"
] |
From mathcomp Require Import all_algebra all_ssreflect.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Definition putnam_1976_a3_solution : nat -> nat -> nat -> nat -> Prop := fun p r q s => (p = 3%nat /\ r = 2%nat /\ q = 2%nat /\ s = 3%nat) \/ (p = 2%nat /\ r = 3%nat /\ q = 3%nat /\ s = 2%nat).
Theorem putnam_1976_a3
: forall (p r q s: nat), ((prime p /\ prime q /\ gt r 1 /\ gt s 1 /\ `|(p^r)%:Z - (q^s)%:Z| = 1) <-> putnam_1976_a3_solution p r q s).
Proof. Admitted.
|
putnam_1976_a6
|
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 $y$.
|
None.
|
[
"analysis",
"algebra"
] |
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.
|
putnam_1976_b5
|
Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$
|
The sum equals $n!$.
|
[
"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_1976_b5_solution : nat -> R -> R := fun n => (fun x => (n `!)%:R).
Theorem putnam_1976_b5
: forall (n : nat) (x : R), \sum_(0 <= k < n.+1) ((-1) ^+ k * 'C(n, k)%:R * (x - k%:R) ^+ n) = putnam_1976_b5_solution n x.
Proof. Admitted.
|
putnam_1976_b6
|
Let $\sigma(N)$ denote the sum of all positive integer divisors of $N$, including $1$ and $N$. Call a positive integer $N$ \textit{quasiperfect} if $\sigma(N) = 2N + 1$. Prove that every quasiperfect number is the square of an odd integer.
|
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_1976_b6
(sigma : nat -> int := fun N => \sum_(d <- divisors N) d)
: forall N : nat, sigma N = (N.*2 + 1) -> (exists m : nat, odd m = true /\ N = m ^ 2).
Proof. Admitted.
|
putnam_1977_a1
|
Show that if four distinct points of the curve $y = 2x^4 + 7x^3 + 3x - 5$ are collinear, then their average $x$-coordinate is some constant $k$. Find $k$.
|
Prove that $k = -\frac{7}{8}$.
|
[
"algebra"
] |
From mathcomp Require Import all_algebra all_ssreflect fintype.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_1977_a1_solution : R := -7/8.
Theorem putnam_1977_a1
(y : R -> R := fun x => 2 * x ^+ 4 + 7 * x ^+ 3 + 3 * x - 5)
(pts : seq R)
(hpts : size pts = 4%nat /\ uniq pts)
(hcollinear : exists (m b : R), all (fun x => y x == m * x + b) pts)
: (\sum_(x <- pts) x) / 4 = putnam_1977_a1_solution.
Proof. Admitted.
|
putnam_1977_a2
|
Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d}$.
|
Prove that the solutions are $d = a$ and $b = -c$, $d = b$ and $a = -c$, or $d = c$ and $a = -b$, with $a, b, c, d$ nonzero.
|
[
"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_1977_a2_solution : R -> R -> R -> R -> Prop := fun a b c d => (c = -a /\ d = b) \/ (c = -b /\ d = a).
Theorem putnam_1977_a2
: forall a b c d : R, a <> 0 -> b <> 0 -> c <> 0 -> d <> 0 -> ((a + b + c = d /\ 1 / a + 1/b + 1/c = 1/d) <-> putnam_1977_a2_solution a b c d).
Proof. Admitted.
|
putnam_1977_a3
|
Let $f, g, h$ be functions $\mathbb{R} \to \mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \frac{h(x + 4) + h(x - 4)}{2}$.
|
Prove that $h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$ suffices.
|
[
"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_1977_a3_solution : (R -> R) -> (R -> R) -> (R -> R) := fun f g x => g x - f (x - 3) + f (x - 1) + f (x + 1) - f (x + 3).
Theorem putnam_1977_a3
(f g h : R -> R)
(hf : forall x, f x = (h (x + 1) + h (x - 1)) / 2)
(hg : forall x, g x = (h (x + 4) + h (x - 4)) / 2)
: h = putnam_1977_a3_solution f g.
Proof. Admitted.
|
putnam_1977_a4
|
Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$.
|
Prove that the sum equals $\frac{x}{1 - x}$.
|
[
"algebra",
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
(* Note: This formalization encodes rational functions manually. There may be a better way with existing mathcomp terminology. *)
Variable R : realType.
Definition putnam_1977_a4_solution : {poly R}*{poly R} := ('X, 1%:P - 'X).
Theorem putnam_1977_a4
: forall x : R, 0 < x < 1 -> ((fun n : nat => \sum_(0 <= i < n) ((x ^+ (2 ^ n))/(1 - x ^+ (2 ^ (n.+1))))) @ \oo --> putnam_1977_a4_solution.1.[x] / putnam_1977_a4_solution.2.[x]).
Proof. Admitted.
|
putnam_1977_a5
|
Let $p$ be a prime and $m \geq n$ be non-negative integers. Show that $\binom{pm}{pn} = \binom{m}{n} \pmod p$, where $\binom{m}{n}$ is the binomial coefficient.
|
None.
|
[
"algebra",
"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_1977_a5
(p n m : nat)
(hp : prime p)
(hmgen : ge m n)
: 'C(mult p m, mult p n) = 'C(m, n) %[mod p].
Proof. Admitted.
|
putnam_1977_b1
|
Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$.
|
Prove that the product equals $\frac{2}{3}$.
|
[
"algebra",
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype topology sequences boolp.
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_1977_b1_solution : R := 2/3.
Theorem putnam_1977_b1
(f : R ^nat := fun n : nat => \prod_(0 <= i < n) (if (i <= 1)%nat then 1 else (i%:R ^+ 3 - 1)/(i%:R ^+ 3 + 1)))
: f @ \oo --> putnam_1977_b1_solution.
Proof. Admitted.
|
putnam_1977_b5
|
If $a_1, a_2, \dots, a_n$ are real numbers with $n > 1$ and $A$ satisfies $$A + \sum_{i = 1}^{n} a_i^2 < \frac{1}{n-1}\left(\sum_{i=1}^{n}a_i\right)^2,$$ prove that $A < 2a_{i}a_{j}$ for all $i, j$ with $1 \le i < j \le n$.
|
None.
|
[
"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.
Theorem putnam_1977_b5
(a : seq R)
(A : R)
(ha : gt (size a) 1)
(hA : A + \sum_(i <- a) (i ^+ 2) <= 1/((size a)%:R - 1) * (\sum_(i <- a) i) ^+ 2)
: pairwise (fun x y => A < 2 * x * y) a.
Proof. Admitted.
|
putnam_1978_a1
|
Let $S = \{1, 4, 7, 10, 13, 16, \dots , 100\}$. Let $T$ be a subset of $20$ elements of $S$. Show that we can find two distinct elements of $T$ with sum $104$.
|
None.
|
[
"algebra"
] |
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 ring_scope.
Local Open Scope classical_set_scope.
Local Open Scope card_scope.
Theorem putnam_1978_a1
(A B : set int)
(hA : A = [set k : int | exists j : int, 0 <= j /\ j <= 33 /\ k = 3 * j + 1])
(hB : B `<=` A /\ B #= [set : `I_(20)])
: exists m n : int, m \in B /\ n \in B /\ m <> n /\ m + n = 104.
Proof. Admitted.
|
putnam_1978_a3
|
Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let
\[
I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx.
\]
For which $k$ is $I_k$ smallest?
|
Show that $I_k$ is smallest for $k = 2$.
|
[
"analysis",
"algebra"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals measure lebesgue_integral lebesgue_measure topology normedtype exp sequences.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Definition putnam_1978_a3_solution : nat := 2.
Theorem putnam_1978_a3
(p : {poly R} := 2 *: ('X^6 + 1%:P) + 4 *: ('X^5 + 'X) + 3 *: ('X^4 + 'X^2) + 5 *: 'X^3)
(I : nat -> R := fun k => \int[mu]_(x in [set t : R | 0 <= t]) (fun x => x ^ k / p.[x]) x)
: putnam_1978_a3_solution \in index_iota 1 5 /\ (forall k : nat, k \in index_iota 1 5 -> I putnam_1978_a3_solution <= I k).
Proof. Admitted.
|
putnam_1978_a5
|
Let $a_1, a_2, \dots , a_n$ be reals in the interval $(0, \pi)$ with arithmetic mean $\mu$. Show that
\[
\prod_{i=1}^n \left( \frac{\sin a_i}{a_i} \right) \leq \left( \frac{\sin \mu}{\mu} \right)^n.
\]
|
None.
|
[
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals trigo.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1978_a5
(A : seq R)
(hanemp : gt (size A) 0)
(ha : all (fun x => 0 < x < pi) A)
(mu : R := \sum_(a <- A) (a / (size A)%:R))
: \prod_(a <- A) ((sin a)/a) <= (sin mu / mu) ^+ (size A).
Proof. Admitted.
|
putnam_1978_a6
|
Given $n$ distinct points in the plane, prove that fewer than $2n^{3/2}$ pairs of these points are a distance of $1$ apart.
|
None.
|
[
"geometry",
"combinatorics"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences boolp normedtype.
Import Order.TTheory Order.Syntax GRing.Theory Num.Def Num.Theory Monoid Monoid.Theory.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope seq_scope.
Variable R : realType.
Definition sq_dist (p q : (R * R)%type) := (p.1 - q.1) ^ 2 + (p.2 - q.2) ^ 2.
Definition ind_is_unit_dist_apart (p q : (R * R)%type) : R := if sq_dist p q == 1 then 1%R else 0%R.
Theorem putnam_1978_a6
(n : nat)
(npos : gt n 0)
(A : seq (R*R))
(hA : size A = n /\ uniq A)
(unit_count : R)
: \sum_(p <- A) ((\sum_(q <- A) ind_is_unit_dist_apart p q)) <= 2 * (2 * expR (3/2 * ln n%:R)).
Proof. Admitted.
|
putnam_1978_b2
|
Find
\[
\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{i^2j + 2ij + ij^2}.
\]
|
Prove that the sum evaluates to $\frac{7}{4}$.
|
[
"algebra",
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals topology sequences normedtype.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import numFieldNormedType.Exports.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition putnam_1978_b2_solution : rat := 7/4.
Theorem putnam_1978_b2
(f : nat -> R := fun n => \sum_(1 <= i < n.+1) (\sum_(1 <= j < n.+1) (1%R)/(i%:R ^+ 2 * j%:R + 2 * i%:R * j%:R + i%:R * j%:R ^+ 2)))
: (f @ \oo --> ratr putnam_1978_b2_solution).
Proof. Admitted.
|
putnam_1978_b3
|
The polynomials $P_n(x)$ are defined by
\begin{align*}
P_1(x) &= 1 + x, \\
P_2(x) &= 1 + 2x, \\
P_{2n+1}(x) &= P_{2n}(x) + (n + 1) x P_{2n-1}(x), \\
P_{2n+2}(x) &= P_{2n+1}(x) + (n + 1) x P_{2n}(x).
\end{align*}
Let $a_n$ be the largest real root of $P_n(x)$. Prove that $a_n$ is strictly monotonically increasing and tends to zero.
|
None.
|
[
"algebra",
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals topology sequences normedtype.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import numFieldNormedType.Exports.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1978_b3
(P : nat -> {poly R})
(hP1 : P 1%nat = 1%:P + 'X)
(hP2 : P 2%nat = 1%:P + 2 * 'X)
(hPodd : forall n : nat, ge n 1 -> P ((n.*2).+1) = P (n.*2) + (n.+1)%:R * 'X * P ((n.*2).-1))
(hPeven : forall n : nat, ge n 1 -> P ((n.+1).*2) = P ((n.*2).+1) + (n.+1)%:R * 'X * P (n.*2))
(a : nat -> R)
(haroot : forall n : nat, ge n 1 -> (P n).[a n] = 0)
(haub : forall n : nat, ge n 1 -> (forall x : R, (P n).[x] = 0 -> x <= a n))
: (forall i j : nat, ge i 1 -> lt i j -> a i < a j) /\ (a @ \oo --> 0).
Proof. Admitted.
|
putnam_1978_b4
|
Show that we can find integers $a, b, c, d$ such that $a^2 + b^2 + c^2 + d^2 = abc + abd + acd + bcd$, and the smallest of $a, b, c, d$ is arbitrarily large.
|
None.
|
[
"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.
Theorem putnam_1978_b4
: forall N : R, exists a b c d : int, (a%:~R) > N /\ (b%:~R) > N /\ (c%:~R) > N /\ (d%:~R) > N /\ a ^+ 2 + b ^+ 2 + c ^+ 2 + d ^+ 2 = a * b * c + a * b * d + a * c * d + b * c * d.
Proof. Admitted.
|
putnam_1978_b5
|
Find the real polynomial $p(x)$ of degree $4$ with largest possible coefficient of $x^4$ such that $p([-1, 1]) \subseteq [0, 1]$.
|
Prove that $p(x) = 4x^4 - 4x^2 + 1$.
|
[
"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_1978_b5_solution : {poly R} := (4%:P) * 'X^4 - 4%:P * 'X^2 + 1%:P.
Theorem putnam_1978_b5
(S : set {poly R} := [set p : {poly R} | size p = 5%nat /\ (forall x : R, -1 <= x <= 1 -> 0 <= p.[x] <= 1)])
: putnam_1978_b5_solution \in S /\ (forall p : {poly R}, p \in S -> (p`_4) <= (putnam_1978_b5_solution`_4)).
Proof. Admitted.
|
putnam_1978_b6
|
Let $a_{ij}$ be real numbers in $[0, 1]$. Show that
\[
\left( \sum_{i=1}^n \sum_{j=1}^{mi} \frac{a_{ij}}{i} \right) ^2 \leq 2m \sum_{i=1}^n \sum_{j=1}^{mi} a_{ij}.
\]
|
None.
|
[
"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.
Theorem putnam_1978_b6
(a : nat -> nat -> R)
(ha : forall i j : nat, 0 <= a i j <= 1)
(m n : nat)
(mnpos : ge m 0 /\ ge n 0)
: (\sum_(1 <= i < m.+1) (\sum_(1 <= j < (m * i).+1) ((a i j) / i%:R))) ^+ 2 <= 2 * (m%:R) * (\sum_(1 <= i < n.+1) (\sum_(1 <= j < (m*i).+1) a i j)).
Proof. Admitted.
|
putnam_1979_a1
|
For which positive integers $n$ and $a_1, a_2, \dots, a_n$ with $\sum_{i = 1}^{n} a_i = 1979$ does $\prod_{i = 1}^{n} a_i$ attain the greatest value?
|
$n$ equals $660$; all but one of the $a_i$ equal $3$ and the remaining $a_i$ equals $2$.
|
[
"algebra"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import multiset.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope mset_scope.
(* Note: The documentation in finmap/multisets.v appears to contain a typo, `mset n a` is not defined, though `msetn n a` seems to do the intended thing. *)
Definition putnam_1979_a1_solution : {mset nat} := 2 |` (msetn 659 3).
Theorem putnam_1979_a1
(P : {mset nat} -> Prop)
(hP : forall A : {mset nat}, P A <-> ((exists a : nat, a \in A) /\ (forall i : nat, i \in A -> i > 0) /\ (\sum_(j <- A) j = 1979)))
: P putnam_1979_a1_solution /\ (forall A : {mset nat}, P A -> \prod_(j <- putnam_1979_a1_solution) j >= \prod_(j <- A) j).
Proof. Admitted.
|
putnam_1979_a2
|
For which real numbers $k$ does there exist a continuous function $f : \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = kx^9$ for all real $x$?
|
Such a function exists if and only if $k \ge 0$.
|
[
"analysis",
"algebra"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals topology sequences derive normedtype realfun real_interval.
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.
Import numFieldNormedType.Exports.
Variable R : realType.
Definition putnam_1979_a2_solution : set R := [set x : R | x >= 0].
Theorem putnam_1979_a2
: forall k : R, (exists f : R -> R, continuous f /\ (forall x : R, f (f x) = k * x ^+ 9)) <-> k \in putnam_1979_a2_solution.
Proof. Admitted.
|
putnam_1979_a3
|
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 $n$?
|
We must have $x_1 = x_2 = m$ for some integer $m$.
|
[
"algebra"
] |
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.
|
putnam_1979_a5
|
Let $S(x)$ denote the sequence $\lfloor 0 \rfloor, \lfloor x \rfloor, \lfloor 2x \rfloor, \lfloor 3x \rfloor, \dots$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Prove that there exist distinct real roots $\alpha$ and $\beta$ of $x^3 - 10x^2 + 29x - 25$ such that infinitely many positive integers appear in both $S(\alpha)$ and $S(\beta)$.
|
None.
|
[
"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.
Theorem putnam_1979_a5
(S : R -> nat -> int)
(hS : forall (x : R) (n : nat), S x n = floor (n%:R * x))
(P : R -> Prop)
(hP : forall (x : R), P x <-> (x ^+ 3) - 10 * (x ^+ 2) + 29 * x - 25 = 0)
: exists (a b : R), (a <> b) /\ P a /\ P b /\ (forall n : nat, exists m : int, m >= n%:Z /\ (exists c d : nat, S a c = m /\ S b d = m)).
Proof. Admitted.
|
putnam_1979_a6
|
For all $i \in \{0, 1, \dots, n - 1\}$, let $p_i \in [0, 1]$. Prove that there exists some $x \in [0, 1]$ such that $$\sum_{i = 0}^{n - 1} \frac{1}{|x - p_i|} \le 8n\left(\sum_{i = 0}^{n-1} \frac{1}{2i + 1}\right).$$
|
None.
|
[
"algebra"
] |
From mathcomp Require Import all_algebra all_ssreflect fintype.
From mathcomp Require Import reals.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1979_b6
(p : seq R)
(hp : all (fun x => 0 <= x <= 1) p)
: exists x : R, 0 <= x <= 1 /\ (all (fun i => x != i) p) /\ (\sum_(i <- p) 1/`|x - i|) <= 8*(size p)%:R*(\sum_(0 <= i < (size p).+1) (1%R)/(2*(i%:R) + 1)).
Proof. Admitted.
|
putnam_1979_b2
|
If $0 < a < b$, find $$\lim_{t \to 0} \left( \int_{0}^{1}(bx + a(1-x))^t dx \right)^{\frac{1}{t}}$$ in terms of $a$ and $b$.
|
The limit equals $$e^{-1}\left(\frac{b^b}{a^a}\right)^{\frac{1}{b-a}}.$$
|
[
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import boolp reals measure lebesgue_measure lebesgue_integral topology normedtype exp sequences.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import numFieldNormedType.Exports.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Definition putnam_1979_b2_solution : R -> R -> R := fun a b => (expR ((b/(b-a)) * ln b))/(expR 1 * (expR ((a/(b-a)) * ln a))).
Theorem putnam_1979_b2
(a b : R)
(hab : 0 < a < b)
(f : R -> R := fun t : R => expR (1/t * ln (\int[mu]_(x in [set x : R | 0 <= x <= 1]) expR (t * ln (b * x + a * (1 - x))))))
: f @ (dnbhs 0) --> putnam_1979_b2_solution a b.
Proof. Admitted.
|
putnam_1979_b6
|
Let $z_i$ be complex numbers for $i = 1, 2, \dots, n$. Show that
\[
\left \lvert \mathrm{Re} \, [(z_1^2 + z_2^2 + \dots + z_n^2)^{1/2} ] \right \rvert \leq \lvert \mathrm{Re} \, z_1 \rvert + \lvert \mathrm{Re} \, z_2 \rvert + \dots + \lvert \mathrm{Re} \, z_n \rvert.
\]
|
None.
|
[
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals complex.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Variable R : realType.
Theorem putnam_1979_b6
(l : seq R[i])
: `|Re (sqrtc (\sum_(z <- l) (z ^+ 2)))| <= \sum_(z <- l) `|Re z|.
Proof. Admitted.
|
putnam_1980_a2
|
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $3^r \cdot 7^s=\text{lcm}[a,b,c]=\text{lcm}[a,b,d]=\text{lcm}[a,c,d]=\text{lcm}[b,c,d]$. The answer should be a function of $r$ and $s$. (Note that $\text{lcm}[x,y,z]$ denotes the least common multiple of $x,y,z$.)
|
Show that the number is $(1+4r+6r^2)(1+4s+6s^2)$.
|
[
"number_theory"
] |
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 classical_set_scope.
Local Open Scope card_scope.
Definition putnam_1980_a2_solution : nat -> nat -> nat := fun r s : nat => (1 + 4 * r + 6 * r ^2) * (1 + 4 * s + 6 * s ^ 2).
Theorem putnam_1980_a2
(r s : nat)
(abcdlcm : nat -> nat -> nat -> nat -> Prop)
(rpos : r > 0 /\ s > 0)
(habcdlcm : forall a b c d, abcdlcm a b c d <-> (a > 0 /\ b > 0 /\ c > 0 /\ d > 0 /\ (3 ^ r * 7 ^ s = lcmn (lcmn a b) c) /\ (3 ^ r * 7 ^ s = lcmn (lcmn a b) d) /\ (3 ^ r * 7 ^ s = lcmn (lcmn a c) d) /\ (3 ^ r * 7 ^ s = lcmn (lcmn b c) d)))
: [set t : 4.-tuple nat | abcdlcm (tnth t (inord 0)) (tnth t (inord 1)) (tnth t (inord 2)) (tnth t (inord 3))] #= `I_(putnam_1980_a2_solution r s).
Proof. Admitted.
|
putnam_1980_a3
|
Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$.
|
Show that the integral is $\pi/4$.
|
[
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals lebesgue_integral measure lebesgue_measure trigo sequences exp.
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.
(* Note: Unaware of definition of non-integer powers, workaround is exp log*)
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Definition putnam_1980_a3_solution : R := pi/4.
Theorem putnam_1980_a3
: \int[mu]_(x in [set t : R | 0 < t < pi/2]) (fun (x : R) => 1/(1 + expR (ln (@tan R x) * (@Num.sqrt R 2)))) x = putnam_1980_a3_solution.
Proof. Admitted.
|
putnam_1980_a4
|
\begin{enumerate}
\item[(a)] Prove that there exist integers $a,b,c$, not all zero and each of absolute value less than one million, such that $|a+b\sqrt{2}+c\sqrt{3}|<10^{-11}$.
\item[(b)] Let $a,b,c$ be integers, not all zero and each of absolute value less than one million. Prove that $|a+b\sqrt{2}+c\sqrt{3}|>10^{-21}$.
\end{enumerate}
|
None.
|
[
"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.
Theorem putnam_1980_a4
(abcvals : int -> int -> int -> Prop)
(habcvals : forall (a b c : int), (abcvals a b c <-> (~ (a = 0 /\ b = 0 /\ c = 0) /\ `|a| < 10^6 /\ `|b| < 10^6 /\ `|c| < 10^6)))
: (exists (a b c : int), abcvals a b c /\ `|a%:~R + b%:~R * (@Num.sqrt R 2) + c%:~R * (@Num.sqrt R 3)| < 10^(-11)) /\ (forall (a b c : int), abcvals a b c -> `|a%:~R + b%:~R * (@Num.sqrt R 2) + c%:~R * (@Num.sqrt R 3)| > 10^(-21)).
Proof. Admitted.
|
putnam_1980_a5
|
Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations $0=\int_0^xP(t)\sin t\,dt=\int_0^xP(t)\cos t\,dt$ has only finitely many real solutions $x$.
|
None.
|
[
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals trigo lebesgue_integral lebesgue_measure measure.
From mathcomp Require Import classical_sets cardinality.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
Local Open Scope ring_scope.
Variable R : realType.
Definition mu := [the measure _ _ of @lebesgue_measure R].
Theorem putnam_1980_a5
(P : {poly R})
(Pnonconst : gtn (size P) (1%nat))
: finite_set [set x : R | \int[mu]_(t in [set t : R | 0 <= t <= x]) (fun y => P.[y] * (sin y)) t = 0 /\ \int[mu]_(t in [set t : R | 0 <= t <= x]) (fun y => P.[y] * (cos y)) t = 0].
Proof. Admitted.
|
putnam_1980_b1
|
For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$?
|
Show that the inequality holds if and only if $c \geq 1/2$.
|
[
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals sequences.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_1980_b1_solution : set R := [set x : R | x >= 1/2].
Theorem putnam_1980_b1
(c : R)
: (forall x : R, (expR x + expR (-x)) / 2 <= expR (c * x^2)) <-> c \in putnam_1980_b1_solution.
Proof. Admitted.
|
putnam_1980_b3
|
For which real numbers $a$ does the sequence defined by the initial condition $u_0=a$ and the recursion $u_{n+1}=2u_n-n^2$ have $u_n>0$ for all $n \geq 0$? (Express the answer in the simplest form.)
|
Show that $u_n>0$ for all $n \geq 0$ if and only if $a \geq 3$.
|
[
"algebra"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import sequences reals.
From mathcomp Require Import classical_sets.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope classical_set_scope.
Local Open Scope ring_scope.
Variable R : realType.
Definition putnam_1980_b3_solution : set R := [set x : R | x >= 3].
Theorem putnam_1980_b3
(a : R)
(u : R ^nat)
(hu : u 0%nat = a /\ (forall n : nat , u n.+1 = 2 * u n - ((n%:R) ^ 2)))
: (forall n : nat, u n > 0) <-> a \in putnam_1980_b3_solution.
Proof. Admitted.
|
putnam_1980_b6
|
For integers $d, n$ with $1 \le d \le n$, let $G(1, n) = \frac{1}{n}$ and $G(d, n) = \frac{d}{n}\sum_{i=d}^{n}G(d - 1, i - 1)$ for all $d > 1$. If $1 < d \le p$ for some prime $p$, prove that the reduced denominator of $G(d, p)$ is not divisible by $p$.
|
None.
|
[
"number_theory",
"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_1980_b6
(G : (int * int)%type -> rat)
(hG : forall d n : nat, leq d n -> ((d = 1%nat -> G (d%:Z, n%:Z) = (1%:Q)/(n%:Q)) /\ (gtn d (1%nat) -> G (d%:Z, n%:Z) = ((d%:Z%:Q)/(n%:Z%:Q)) * \sum_(d <= i < n.+1) (G (d%:Z - 1, i%:Z - 1)))))
: forall (d p : nat), (ltn (1%nat) d /\ leq d p /\ prime p) -> ~ (p%:Z %| (denq (G (d%:Z, p%:Z))))%Z.
Proof. Admitted.
|
putnam_1981_a1
|
Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$.
|
The limit equals $\frac{1}{8}$.
|
[
"analysis",
"number_theory"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals normedtype sequences topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Definition putnam_1981_a1_solution : R := 1 / 8.
Theorem putnam_1981_a1
(P : nat -> nat -> Prop := fun n k => (5 ^ k %| (\prod_( 1<= i < n+1) (i%:Z ^+ i)))%Z)
(E : nat -> nat)
(hE : forall n : nat, ge n 1 -> P n (E n) /\ forall k : nat, P n k -> le k (E n))
: (fun n : nat => (E n)%:R / (n%:R ^ 2)) @ \oo --> putnam_1981_a1_solution.
Proof. Admitted.
|
putnam_1981_a3
|
Does the limit $$lim_{t \rightarrow \infty}e^{-t}\int_{0}^{t}\int_{0}^{t}\frac{e^x - e^y}{x - y} dx dy$$exist?
|
The limit does not exist.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_1981_a3_solution := False.
Theorem putnam_1981_a3
: (exists r : R, Lim (fun k => exp (-1*k) * (RInt (fun x => (RInt (fun y => (exp x - exp y) / (x - y)) 0 k)) 0 k)) p_infty = r) <-> putnam_1981_a3_solution.
Proof. Admitted.
|
putnam_1981_b1
|
Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$
|
The limit equals $-1$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_1981_b1_solution := -1.
Theorem putnam_1981_b1
: Lim_seq (fun n => 1/(pow (INR n) 5) * (sum_n (fun r => (sum_n (fun s => 5 * pow (INR r) 4 - 18 * pow (INR r) 2 * pow (INR s) 2 + 5 * pow (INR s) 4) n)) n)) = putnam_1981_b1_solution.
Proof. Admitted.
|
putnam_1981_b2
|
Determine the minimum value attained by $$(r - 1)^2 + (\frac{s}{r} - 1)^2 + (\frac{t}{s} - 1)^2 + (\frac{4}{t} - 1)^2$$ across all choices of real $r$, $s$, and $t$ that satisfy $1 \le r \le s \le t \le 4$.
|
The minimum is $12 - 8\sqrt{2}$.
|
[
"algebra"
] |
Require Import Reals.
Open Scope R.
Definition putnam_1981_b2_solution := 12 - 8 * sqrt 2.
Theorem putnam_1981_b2
(f : R -> R -> R -> R := fun a b c => pow (a-1) 2 + pow (b / a - 1) 2 + pow (c / b - 1) 2 + pow (4 / c - 1) 2)
: (forall (a b c: R), 1 <= a /\ a <= b /\ b <= c /\ c <= 4 -> putnam_1981_b2_solution <= f a b c) /\
(exists (a b c: R), 1 <= a /\ a <= b /\ b <= c /\ c <= 4 -> putnam_1981_b2_solution = f a b c).
Proof. Admitted.
|
putnam_1981_b3
|
Prove that, for infinitely many positive integers $n$, all primes $p$ that divide $n^2 + 3$ also divide $k^2 + 3$ for some integer $k$ such that $k^2 < n$.
|
None.
|
[
"number_theory"
] |
Require Import Nat ZArith Znumtheory. From mathcomp Require Import div.
Open Scope nat_scope.
Theorem putnam_1981_b3
: ~ exists (N: nat),
forall (n: nat),
(forall (p: nat), prime (Z.of_nat p) /\ p %| pow n 2 + 3 = true -> exists (m: nat), p %| pow m 2 + 3 = true /\ pow m 2 < n) ->
n < N.
Proof. Admitted.
|
putnam_1981_b5
|
Let $B(n)$ be the number of ones in the base two expression for the positive integer $n$. For example, $B(6)=B(110_2)=2$ and $B(15)=B(1111_2)=4$. Determine whether or not $\exp \left(\sum_{n=1}^\infty \frac{B(n)}{n(n+1)}\right)$ is a rational number. Here $\exp(x)$ denotes $e^x$.
|
Show that the expression is a rational number.
|
[
"analysis",
"algebra"
] |
Require Import BinNums Nat NArith Coquelicot.Coquelicot.
Local Coercion Raxioms.INR : nat >-> Rdefinitions.R.
Definition putnam_1981_b5_solution := True.
Theorem putnam_1981_b5
(f := fix count_ones (n : positive) : nat :=
match n with
| xH => 1
| xO n' => count_ones n'
| xI n' => 1 + count_ones n'
end)
(k := Series (fun n => Rdefinitions.Rdiv (f (Pos.of_nat n)) (n + pow n 2)))
: exists (a b: nat), Rtrigo_def.exp k = Rdefinitions.Rdiv a b <-> putnam_1981_b5_solution.
Proof. Admitted.
|
putnam_1982_a2
|
Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge?
|
Prove that the series converges.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1982_a2_solution := True.
Theorem putnam_1982_a2
(B : nat -> R -> R := fun n x => sum_n (fun m => Rpower (INR m) x) n)
(f : nat -> R := fun n => B n (ln 2 / ln (INR n)) / ((INR n) * Rpower (ln 2 / ln (INR n)) 2))
: ex_series (fun n => if (lt_dec n 2) then 0 else f n) <-> putnam_1982_a2_solution.
Proof. Admitted.
|
putnam_1982_a3
|
Evaluate $\int_0^{\infty} \frac{\tan^{-1}(\pi x) - \tan^{-1} x}{x} \, dx$.
|
Show that the integral evaluates to $\frac{\pi}{2} \ln \pi$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1982_a3_solution := PI / 2 * ln PI.
Theorem putnam_1982_a3
(f : R -> R := fun x => (atan (PI * x) - atan x) / x)
: Lim (fun n => RInt f 0 n) p_infty = putnam_1982_a3_solution.
Proof. Admitted.
|
putnam_1982_a5
|
Let $a, b, c, d$ be positive integers satisfying $a + c \leq 1982$ and $\frac{a}{b} + \frac{c}{d} < 1$. Prove that $1 - \frac{a}{b} - \frac{c}{d} > \frac{1}{1983^3}$.
|
None.
|
[
"algebra"
] |
Require Import Reals.
Open Scope R.
Theorem putnam_1982_a5
(a b c d: nat)
(hpos : Nat.lt 0 a /\ Nat.lt 0 b /\ Nat.lt 0 c /\ Nat.lt 0 d)
(habcd : Nat.le (Nat.add a c) 1982 /\ INR a / INR b + INR c / INR d < 1)
: 1 - INR a / INR b - INR c / INR d > 1/pow 1983 3.
Proof. Admitted.
|
putnam_1982_a6
|
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 $$\lim_{n \rightarrow \infty} \sum_{k = 1}^{n} x_{b(k)} = 1.$$
|
The limit need not equal $1$.
|
[
"analysis"
] |
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.
|
putnam_1982_b5
|
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 $x > e^e$.
|
None.
|
[
"analysis"
] |
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.
|
putnam_1983_a1
|
How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40},20^{30}$?
|
Show that the desired count is $2301$.
|
[
"number_theory"
] |
Require Import Nat Ensembles Finite_sets. From mathcomp Require Import div.
Definition putnam_1983_a1_solution := 2301.
Theorem putnam_1983_a1
(E: Ensemble nat := fun n => n %| 10^(40) = true \/ n %| 20^(30) = true)
: cardinal nat E putnam_1983_a1_solution.
Proof. Admitted.
|
putnam_1983_a3
|
Let $p$ be in the set $\{3,5,7,11,\dots\}$ of odd primes and let $F(n)=1+2n+3n^2+\dots+(p-1)n^{p-2}$. Prove that if $a$ and $b$ are distinct integers in $\{0,1,2,\dots,p-1\}$ then $F(a)$ and $F(b)$ are not congruent modulo $p$, that is, $F(a)-F(b)$ is not exactly divisible by $p$.
|
None.
|
[
"number_theory",
"algebra"
] |
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_1983_a3
(p : nat)
(F : nat -> nat)
(poddprime : odd p = true /\ prime p)
(hF : forall n : nat, F n = \sum_(0 <= i < p-1) ((i.+1) * n ^ i))
: forall a b : nat, 1 <= a <= p /\ 1 <= b <= p /\ a <> b -> ~ (F a = F b %[mod p]).
Proof. Admitted.
|
putnam_1983_a4
|
Prove that for $m = 5 \pmod 6$,
\[
\binom{m}{2} - \binom{m}{5} + \binom{m}{8} - \binom{m}{11} + ... - \binom{m}{m-6} + \binom{m}{m-3} \neq 0.
\]
|
None.
|
[
"algebra"
] |
Require Import Binomial Reals Coquelicot.Coquelicot.
Open Scope nat_scope.
Theorem putnam_1983_a4
(k : nat)
(kpos : k > 0)
(m : nat := 6 * k - 1)
(s : R := sum_n_m (fun j => Rmult ((-1) ^ (j + 1)) (Binomial.C m (3 * j - 1))) 1 (2*k - 1))
: s <> R0.
Proof. Admitted.
|
putnam_1983_a5
|
Prove or disprove that there exists a positive real number $\alpha$ such that $[\alpha_n] - n$ is even for all integers $n > 0$. (Here $[x]$ denotes the greatest integer less than or equal to $x$.)
|
Prove that such an $\alpha$ exists.
|
[
"analysis"
] |
Require Import Nat ZArith Reals Coquelicot.Coquelicot.
Definition putnam_1983_a5_solution := True.
Theorem putnam_1983_a5
: (exists a : R, a > 0 /\ forall n : nat, gt n 0 -> Z.Even (floor (a^n) - Z.of_nat n)) <-> putnam_1983_a5_solution.
Proof. Admitted.
|
putnam_1983_a6
|
Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$.
|
Show that the integral evaluates to $\frac{2}{9}$.
|
[
"analysis"
] |
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.
|
putnam_1983_b4
|
Let $f(n) = n + [\sqrt n]$, where $[x]$ denotes the greatest integer less than or equal to $x$. Define the sequence $a_i$ by $a_0 = m$, $a_{n+1} = f(a_n)$. Prove that it contains at least one square.
|
None.
|
[
"algebra"
] |
Require Import ZArith Reals Coquelicot.Coquelicot.
Open Scope Z_scope.
Theorem putnam_1983_b4
(m : Z)
(hm : m > 0)
(f : Z -> Z := fun n => n + floor (sqrt (IZR n)))
(A := fix a (n: nat) :=
match n with
| O => m
| S n' => f (a n')
end)
: exists (i : nat) (s : Z), A i = s^2.
Proof. Admitted.
|
putnam_1983_b5
|
Define $\left\lVert x \right\rVert$ as the distance from $x$ to the nearest integer. Find $\lim_{n \to \infty} \frac{1}{n} \int_{1}^{n} \left\lVert \frac{n}{x} \right\rVert \, dx$. You may assume that $\prod_{n=1}^{\infty} \frac{2n}{(2n-1)} \cdot \frac{2n}{(2n+1)} = \frac{\pi}{2}$.
|
Show that the limit equals $\ln \left( \frac{4}{\pi} \right)$.
|
[
"analysis"
] |
Require Import Nat Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1983_b5_solution := ln (4/PI).
Theorem putnam_1983_b5
(mindist : R -> R := fun x => Rmin (x - IZR (floor x)) (IZR (floor (x+1)) - x))
: Lim_seq (fun n => (1/INR n) * (RInt (fun x => mindist (INR n/x)) 1 (INR n))) = putnam_1983_b5_solution.
Proof. Admitted.
|
putnam_1984_a2
|
Express $\sum_{k=1}^\infty (6^k/(3^{k+1}-2^{k+1})(3^k-2^k))$ as a rational number.
|
Show that the sum converges to $2$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope nat_scope.
(* Uses division by 0 in the first term of the series. *)
Definition putnam_1984_a2_solution := 2%R.
Theorem putnam_1984_a2
(f : nat -> R := fun n => Rdiv (pow 6 n) ((pow 3 (n+1) - pow 2 (n+1)) * (pow 3 n - pow 2 n)))
: Series f = putnam_1984_a2_solution.
Proof. Admitted.
|
putnam_1984_a5
|
Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z \leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral $\iiint_R x^1y^9z^8w^4\,dx\,dy\,dz$ in the form $a!b!c!d!/n!$, where $a$, $b$, $c$, $d$, and $n$ are positive integers.
|
Show that the integral we desire is $1!9!8!4!/25!$.
|
[
"analysis"
] |
Require Import Reals Factorial Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1984_a5_solution := INR (fact 9 * fact 8 * fact 4) / INR (fact 25).
Theorem putnam_1984_a5
: RInt (fun z => RInt (fun y => RInt (fun x => x * pow y 9 * pow z 8 * pow (1 - x - y - z) 4) 0 (1 - y - z)) 0 (1 - z)) 0 1 = putnam_1984_a5_solution.
Proof. Admitted.
|
putnam_1984_b1
|
Let $n$ be a positive integer, and define $f(n)=1!+2!+\dots+n!$. Find polynomials $P(x)$ and $Q(x)$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$.
|
Show that we can take $P(x)=x+3$ and $Q(x)=-x-2$.
|
[
"algebra"
] |
Require Import Factorial ZArith.
Open Scope Z.
Fixpoint nat_sum (a : nat -> nat) (k : nat) : nat :=
match k with
| O => a O
| S k' => a k + nat_sum a k'
end.
Fixpoint Z_sum (a : nat -> Z) (k : nat) : Z :=
match k with
| O => a O
| S k' => a k + Z_sum a k'
end.
Definition putnam_1984_b1_solution : (nat -> Z) * (nat -> Z) := (fun x => match x with | O => 3 | S O => 1 | _ => 0 end, fun x => match x with | O => -2 | S O => -1 | _ => 0 end).
Theorem putnam_1984_b1
(f : nat -> Z)
(poly : nat -> (nat -> Z) -> nat -> Z := fun k p n => Z_sum (fun i => (p i) * (Z.of_nat (n ^ i))) k)
(deg : (nat -> Z) -> nat)
(finpoly : (nat -> Z) -> Prop := fun p => (exists k : nat, p k <> 0) /\ (exists m : nat, forall n : nat, gt n m -> p n = 0))
(hdeg : forall p : nat -> Z, finpoly p -> (p (deg p) <> 0 /\ forall n : nat, gt n (deg p) -> p n = 0))
(hf : forall n : nat, gt n 0 -> f n = (Z_sum (fun i => Z.of_nat (fact i)) n) - Z.of_nat (fact 0))
: finpoly (fst putnam_1984_b1_solution) /\ finpoly (snd putnam_1984_b1_solution) /\
forall n : nat, ge n 1 -> f (Nat.add n 2) = poly (deg (fst putnam_1984_b1_solution)) (fst putnam_1984_b1_solution) n * f (Nat.add n 1) + poly (deg (snd putnam_1984_b1_solution)) (snd putnam_1984_b1_solution) n * f n.
Proof. Admitted.
|
putnam_1984_b2
|
Find the minimum value of $(u-v)^2+(\sqrt{2-u^2}-\frac{9}{v})^2$ for $0<u<\sqrt{2}$ and $v>0$.
|
Show that the minimum value is $8$.
|
[
"geometry",
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1984_b2_solution := 8.
Theorem putnam_1984_b2
(f : R -> R -> R := fun x y => pow (x - y) 2 + pow (sqrt (2 - pow x 2) - 9 / y) 2)
: (forall x y : R, 0 < x < sqrt 2 /\ y > 0 -> f x y >= putnam_1984_b2_solution) /\
(exists x y : R, 0 < x < sqrt 2 /\ y > 0 /\ f x y = putnam_1984_b2_solution).
Proof. Admitted.
|
putnam_1984_b5
|
For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)=2$). Let $m$ be a positive integer. Express $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m$ in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials.
|
Show that $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m=(-1)^m2^{m(m-1)/2}m!$.
|
[
"algebra",
"analysis"
] |
Require Import Nat Reals Coquelicot.Coquelicot.
Definition putnam_1984_b5_solution : Z * (nat -> R) * (nat -> nat) := (2%Z, fun x => INR x * (INR x - 1) / 2, fun x => x).
Theorem putnam_1984_b5
(d : nat -> nat)
(hd : forall n : nat, d n = if eq_nat_dec n 0 then O else if eq_nat_dec (n mod 2) 0 then d (div n 2) else S (d (div n 2)))
(m : nat)
(mpos : gt m 0)
: let (af, g) := putnam_1984_b5_solution in let (a, f) := af in sum_n (fun k => (-1)^(d k) * INR (Nat.pow k m)) (2^m - 1) = (-1) ^ m * Rpower (IZR a) (f m) * INR (fact (g m)).
Proof. Admitted.
|
putnam_1985_a1
|
Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that
\begin{enumerate}
\item[(i)] $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, and
\item[(ii)] $A_1 \cap A_2 \cap A_3 = \emptyset$.
\end{enumerate}
Express your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$ are nonnegative integers.
|
Prove that the number of such triples is $2^{10}3^{10}$.
|
[
"algebra"
] |
Require Import Ensembles Finite_sets Nat.
Definition putnam_1985_a1_solution := (10, 10, 0, 0).
Theorem putnam_1985_a1
: let (abc, d) := putnam_1985_a1_solution in let (ab, c) := abc in let (a, b) := ab in
cardinal ((Ensemble nat) * (Ensemble nat) * (Ensemble nat)) (fun A => let (A1A2, A3) := A in let (A1, A2) := A1A2 in
Union nat (Union nat A1 A2) A3 = fun n => 1 <= n <= 10 /\ Intersection nat (Intersection nat A1 A2) A3 = Empty_set nat)
(2 ^ a * 3 ^ b * 5 ^ c * 7 ^ d).
Proof. Admitted.
|
putnam_1985_a3
|
Let $d$ be a real number. For each integer $m \geq 0$, define a sequence $\{a_m(j)\}$, $j=0,1,2,\dots$ by the condition
\begin{align*}
a_m(0) &= d/2^m, \\
a_m(j+1) &= (a_m(j))^2 + 2a_m(j), \qquad j \geq 0.
\end{align*}
Evaluate $\lim_{n \to \infty} a_n(n)$.
|
Show that the limit equals $e^d - 1$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1985_a3_solution (x: R) := exp x - 1.
Theorem putnam_1985_a3
(x: R)
(A := fix a (i j: nat) :=
match (i,j) with
| (i, 0) => x/pow 2 i
| (i, S j') => pow (a i j') 2 + 2 * a i j'
end)
: Lim_seq (fun n => A n n) = putnam_1985_a3_solution x.
Proof. Admitted.
|
putnam_1985_a4
|
Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i \geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$?
|
Prove that the only number that occurs infinitely often is $87$.
|
[
"number_theory"
] |
Require Import Ensembles Nat Coquelicot.Coquelicot.
Definition putnam_1985_a4_solution := fun k => k = 87.
Theorem putnam_1985_a4
(a : nat -> nat := fix a (n: nat) :=
match n with
| O => 3
| S n' => 3 ^ (a n')
end)
: let E: Ensemble nat := fun k => k < 100 /\ forall (n: nat), exists (i: nat), ge i n /\ (a i) mod 100 = k in
E = putnam_1985_a4_solution.
Proof. Admitted.
|
putnam_1985_a5
|
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 $I_m \neq 0$?
|
Prove that the integers $m$ with $1 \leq m \leq 10$ and $I_m \neq 0$ are $m = 3, 4, 7, 8$.
|
[
"analysis"
] |
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.
|
putnam_1985_a6
|
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 $n \geq 1$.
|
Show that $g(x) = 6x^2 + 5x + 1$ satisfies the conditions.
|
[
"algebra"
] |
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.
|
putnam_1985_b2
|
Define polynomials $f_n(x)$ for $n \geq 0$ by $f_0(x)=1$, $f_n(0)=0$ for $n \geq 1$, and
\[
\frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1)
\]
for $n \geq 0$. Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.
|
Show that $f_{100}(1) = 101^{99}$.
|
[
"algebra"
] |
Require Import Nat List Reals Coquelicot.Coquelicot ZArith Znumtheory.
Open Scope R.
Definition putnam_1985_b2_solution := repeat 101%nat 99.
Theorem putnam_1985_b2
(f : nat -> R -> R)
(polyf : forall n : nat, exists p : nat -> R, exists deg : nat, f n = fun x : R => sum_n (fun m : nat => (p m) * x ^ m) deg)
(hf0x : f O = fun x => 1)
(hfn0 : forall n : nat, ge n 1 -> f n 0 = 0)
(hfderiv : forall n : nat, forall x : R, Derive (f (S n)) x = (INR n + 1) * f n (x + 1))
: (forall n : nat, In n putnam_1985_b2_solution -> prime (Z.of_nat n)) /\ exists a : nat, INR a = f 100%nat 1 /\ fold_left mul putnam_1985_b2_solution 1%nat = a.
Proof. Admitted.
|
putnam_1985_b3
|
Let
\[
\begin{array}{cccc} a_{1,1} & a_{1,2} & a_{1,3} & \dots \\
a_{2,1} & a_{2,2} & a_{2,3} & \dots \\
a_{3,1} & a_{3,2} & a_{3,3} & \dots \\
\vdots & \vdots & \vdots & \ddots
\end{array}
\]
be a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that $a_{m,n} > mn$ for some pair of positive integers $(m,n)$.
|
None.
|
[
"algebra"
] |
Require Import Nat Ensembles Finite_sets.
Theorem putnam_1985_b3
(a : nat -> nat -> nat)
(apos : forall m n : nat, a m n > 0)
(ha : forall k : nat, k > 0 -> cardinal (nat * nat) (fun t => let (m, n) := t in m > 0 /\ n > 0 /\ a m n = k) 8)
: exists m n : nat, m > 0 /\ n > 0 /\ a m n > m * n.
Proof. Admitted.
|
putnam_1985_b5
|
Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$.
|
Show that the integral evaluates to $\sqrt{\frac{\pi}{1985}}e^{-3970}$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1985_b5_solution := sqrt (PI / 1985) * exp (-3970).
Theorem putnam_1985_b5
: filterlim (fun n : R => RInt (fun x => Rpower x (-1/2) * exp (-1985 * (x + 1/x))) 0 n) (Rbar_locally p_infty) (locally putnam_1985_b5_solution).
Proof. Admitted.
|
putnam_1986_a1
|
Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \leq 13x^2$.
|
Show that the maximum value is $18$.
|
[
"algebra",
"analysis"
] |
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.
|
putnam_1986_a2
|
What is the units (i.e., rightmost) digit of
\[
\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?
\]
|
Show that the answer is $3$.
|
[
"algebra"
] |
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.
|
putnam_1986_a3
|
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 $\cot \theta = t$.
|
Show that the sum equals $\pi/2$.
|
[
"analysis"
] |
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.
|
putnam_1986_a6
|
Let $a_1, a_2, \dots, a_n$ be real numbers, and let $b_1, b_2, \dots, b_n$ be distinct positive integers. Suppose that there is a polynomial $f(x)$ satisfying the identity
\[
(1-x)^n f(x) = 1 + \sum_{i=1}^n a_i x^{b_i}.
\]
Find a simple expression (not involving any sums) for $f(1)$ in terms of $b_1, b_2, \dots, b_n$ and $n$ (but independent of $a_1, a_2, \dots, a_n$).
|
Show that $f(1) = b_1 b_2 \dots b_n / n!$.
|
[
"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_1986_a6_solution : (nat -> nat) -> nat -> R := fun b n => (\prod_(1 <= i < n.+1) (b i)%:R) / n`!%:R.
Theorem putnam_1986_a6
(n : nat)
(npos : gt n 0)
(a : nat -> R)
(b : nat -> nat)
(bpos : forall i : nat, lt i n /\ gt n 0 -> gt (b i) 0)
(binj : forall i j : nat, lt i n /\ lt j n /\ gt i 0 /\ gt j 0 -> (b i = b j -> i = j))
(f : {poly R})
(hf : forall x : R, (1 - x) ^ n * f.[x] = 1 + \sum_(1 <= i < n.+1) ((a i) * x ^ (b i)))
: f.[1] = putnam_1986_a6_solution b n.
Proof. Admitted.
|
putnam_1986_b2
|
Prove that there are only a finite number of possibilities for the ordered triple $T=(x-y,y-z,z-x)$, where $x,y,z$ are complex numbers satisfying the simultaneous equations
\[
x(x-1)+2yz = y(y-1)+2zx = z(z-1)+2xy,
\]
and list all such triples $T$.
|
Show that the possibilities for $T$ are $(0, 0, 0), \, (0, -1, 1), \, (1, 0, -1), \, (-1, 1, 0)$.
|
[
"algebra"
] |
Require Import Reals Ensembles Finite_sets Coquelicot.Coquelicot.
Open Scope C.
Definition putnam_1986_b2_solution (xyz : C*C*C) := xyz = (RtoC 0, RtoC 0, RtoC 0) \/ xyz = (RtoC 1, RtoC 0, RtoC (-1)) \/ xyz = (RtoC (-1), RtoC 1, RtoC 0) \/ xyz = (RtoC 0, RtoC (-1), RtoC 1).
Theorem putnam_1986_b2
: forall T : C * C * C, putnam_1986_b2_solution T <-> exists '(x, y, z), T = (x - y, y - z, z - x) /\ x * (x - 1) + 2 * y * z = y * (y - 1) + 2 * z * x /\ y * (y - 1) + 2 * z * x = z * (z - 1) + 2 * x * y.
Proof. Admitted.
|
putnam_1986_b4
|
For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\lim_{r\to \infty}G(r)$ exists and equals $0$.
|
Show that the limit exists and equals $0$.
|
[
"analysis"
] |
Require Import Reals Ranalysis Coquelicot.Coquelicot.
Definition putnam_1986_b4_solution := True.
Theorem putnam_1986_b4
(G : R -> R)
(hGeq : forall (r: R), exists (m n: Z), G r = Rabs (r - sqrt (IZR (m ^ 2 + 2 * n ^ 2))))
(hGlb : forall (r: R), forall (m n: Z), G r <= Rabs (r - sqrt (IZR (m ^ 2 + 2 * n ^ 2))))
: filterlim G (Rbar_locally p_infty) (locally 0) <-> putnam_1986_b4_solution.
Proof. Admitted.
|
putnam_1987_b1
|
Evaluate
\[
\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.
\]
|
Prove that the integral evaluates to $1$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_1987_b1_solution := 1.
Theorem putnam_1987_b1
: RInt (fun x => ln (9 - x) ^ (1/2) / ( ln (9 - x) ^ (1/2) + ln (x + 3) ^ (1/2))) 2 4 = putnam_1987_b1_solution.
Proof. Admitted.
|
putnam_1987_b2
|
Let $r, s$ and $t$ be integers with $0 \leq r$, $0 \leq s$ and $r+s \leq t$. Prove that
\[
\frac{\binom s0}{\binom tr}
+ \frac{\binom s1}{\binom{t}{r+1}} + \cdots
+ \frac{\binom ss}{\binom{t}{r+s}}
= \frac{t+1}{(t+1-s)\binom{t-s}{r}}.
\]
|
None.
|
[
"algebra"
] |
Require Import Binomial Reals Coquelicot.Coquelicot.
Theorem putnam_1987_b2
: forall (n r s: nat), ge n (r + s) ->
sum_n (fun i => Binomial.C s i / Binomial.C n (r + i)) s = (INR n + 1)/((INR n + 1 - INR s) * Binomial.C (n - s) r).
Proof. Admitted.
|
putnam_1987_b4
|
Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \cos y_n - y_n \sin y_n$ and $y_{n+1}= x_n \sin y_n + y_n \cos y_n$ for $n=1,2,3,\dots$. For each of $\lim_{n\to \infty} x_n$ and $\lim_{n \to \infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist.
|
Show that $\lim_{n \to \infty} x_n = -1$ and $\lim_{n \to \infty} y_n = 0$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1987_b4_solution := (True, -1, True, 0).
Theorem putnam_1987_b4
(A := fix a (n: nat) : (R * R) :=
match n with
| O => (0.8, 0.6)
| (S n') =>
let (xn, yn) := a n' in
(xn * cos yn - yn * sin yn, xn * sin yn + yn * cos yn)
end)
: let '(existsx, limx, existsy, limy) := putnam_1987_b4_solution in
(ex_lim_seq (fun n => fst (A n)) <-> existsx) /\
(existsx -> Lim_seq (fun n => fst (A n)) = limx) /\
(ex_lim_seq (fun n => snd (A n)) <-> existsy) /\
(existsy -> Lim_seq (fun n => snd (A n)) = limy).
Proof. Admitted.
|
putnam_1988_a2
|
A not uncommon calculus mistake is to believe that the product rule for derivatives says that $(fg)' = f'g'$. If $f(x)=e^{x^2}$, determine, with proof, whether there exists an open interval $(a,b)$ and a nonzero function $g$ defined on $(a,b)$ such that this wrong product rule is true for $x$ in $(a,b)$.
|
Show that such $(a,b)$ and $g$ exist.
|
[
"analysis"
] |
(* Note: omitted the second part of the proof - an example of existence - due to lack of full solution description *)
Require Import Basics Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1988_a2_solution := True.
Theorem putnam_1988_a2
(f : R -> R := fun x => exp (pow x 2))
: (exists (a b: R) (g: R -> R), a < b /\
(exists x: R, a < x < b /\ g x <> 0) /\
forall (x: R), a < x < b -> ex_derive g x /\ Derive (fun y => f y * g y) x = (Derive f x) * (Derive g x))
<-> putnam_1988_a2_solution.
Proof. Admitted.
|
putnam_1988_a3
|
Determine, with proof, the set of real numbers $x$ for which
\[
\sum_{n=1}^\infty \left( \frac{1}{n} \csc \frac{1}{n} - 1 \right)^x
\]
converges.
|
Show that the series converges if and only if $x > \frac{1}{2}$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1988_a3_solution (a: R) := a > 1/2.
Theorem putnam_1988_a3
: forall (a: R), ex_finite_lim_seq (fun m => sum_n_m (fun n => Rpower (1/INR n * (1 / sin (1/INR n)) - 1) a) 1 m) <-> putnam_1988_a3_solution a.
Proof. Admitted.
|
putnam_1988_a4
|
\begin{enumerate}
\item[(a)] If every point of the plane is painted one of three colors, do there necessarily exist two points of the same color exactly one inch apart?
\item[(b)] What if ``three'' is replaced by ``nine''?
\end{enumerate}
|
Prove that the points must exist with three colors, but not necessarily with nine.
|
[
"geometry",
"combinatorics"
] |
Require Import Reals Coquelicot.Coquelicot. From mathcomp Require Import fintype.
Definition putnam_1988_a4_solution : Prop * Prop := (True, False).
Theorem putnam_1988_a4
(p : nat -> Prop := fun n : nat => (forall color : (R * R) -> 'I_n, exists p q : R * R, color p = color q /\ norm (fst p - fst q, snd p - snd q) = 1))
: let (a, b) := putnam_1988_a4_solution in ((p 3%nat) <-> a) /\ ((p 9%nat) <-> b).
Proof. Admitted.
|
putnam_1988_a5
|
Prove that there exists a \emph{unique} function $f$ from the set $\mathrm{R}^+$ of positive real numbers to $\mathrm{R}^+$ such that
\[
f(f(x)) = 6x-f(x)
\]
and
\[
f(x)>0
\]
for all $x>0$.
|
None.
|
[
"analysis"
] |
Require Import Basics Reals.
Open Scope R.
Theorem putnam_1988_a5
: exists! (f: R -> R),
(forall (x: R), x <= 0 -> f x = 0) /\
forall (x: R), x > 0 ->
(f x > 0 /\ (compose f f) x = 6 * x - f x).
Proof. Admitted.
|
putnam_1988_b1
|
A \emph{composite} (positive integer) is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.
|
None.
|
[
"number_theory",
"algebra"
] |
Require Import ZArith Znumtheory.
Open Scope Z.
Theorem putnam_1988_b1
: forall (n: Z), n > 3 /\ ~ prime n -> exists (a b c: Z), a > 0 /\ b > 0 /\ c > 0 /\ n = a * b + b * c + c * a + 1.
Proof. Admitted.
|
putnam_1988_b2
|
Prove or disprove: If $x$ and $y$ are real numbers with $y \geq 0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1) \leq x^2$.
|
Show that this is true.
|
[
"algebra"
] |
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.
|
putnam_1988_b3
|
For every $n$ in the set $N=\{1,2,\dots\}$ of positive integers, let $r_n$ be the minimum value of $|c-d \sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in N$.
|
Show that the smallest such $g$ is $(1+\sqrt{3})/2$.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_1988_b3_solution := (1 + sqrt 3) / 2.
Theorem putnam_1988_b3
(r : Z -> R)
(hr : forall (n: Z), Z.ge n 1 -> (exists c d : Z, Z.ge c 0 /\ Z.ge d 0 /\ (Z.add c d) = n /\ r n = Rabs (IZR c - IZR d * sqrt 3)) /\ (forall c d : Z, (Z.ge c 0 /\ Z.ge d 0 /\ (Z.add c d) = n) -> Rabs (IZR c - IZR d * sqrt 3) >= r n))
: putnam_1988_b3_solution > 0 /\ (forall n : Z, Z.ge n 1 -> r n <= putnam_1988_b3_solution) /\ (forall (g: R), g > 0 -> (forall (n: Z), Z.ge n 1 -> r n <= g) -> g >= putnam_1988_b3_solution).
Proof. Admitted.
|
putnam_1988_b4
|
Prove that if $\sum_{n=1}^\infty a_n$ is a convergent series of positive real numbers, then so is $\sum_{n=1}^\infty (a_n)^{n/(n+1)}$.
|
None.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1988_b4
(a : nat -> R)
(appos : (nat -> R) -> Prop := fun a' : nat -> R => forall n : nat, ge n 1 -> a' n > 0)
(apconv : (nat -> R) -> Prop := fun a' : nat -> R => ex_finite_lim_seq (fun N => sum_n_m a' 1 N))
(apposconv : (nat -> R) -> Prop := fun a' => appos a' /\ apconv a')
: apposconv a -> apposconv (fun n => Rpower (a n) (INR n/(INR n + 1))).
Proof. Admitted.
|
putnam_1988_b6
|
Prove that there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$, the number $at+b$ is a triangular number if and only if $t$ is a triangular number. (The triangular numbers are the $t_n=n(n+1)/2$ with $n$ in $\{0,1,2,\dots\}$.)
|
None.
|
[
"number_theory",
"algebra"
] |
Require Import Ensembles Finite_sets ZArith.
Open Scope Z.
Theorem putnam_1988_b6
(triangular : Z -> Prop := fun n => exists (m: Z), m >= 0 /\ n = (m * (m + 1)) / 2)
(E: Ensemble (Z*Z) := fun '(a, b) => forall (m: Z), m > 0 -> (triangular m <-> triangular (Z.mul a m + b)))
: ~ exists (n: nat), cardinal (Z*Z) E n.
Proof. Admitted.
|
putnam_1989_a2
|
Evaluate $\int_0^a \int_0^b e^{\max\{b^2x^2,a^2y^2\}}\,dy\,dx$ where $a$ and $b$ are positive.
|
Show that the value of the integral is $(e^{a^2b^2}-1)/(ab)$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1989_a2_solution (a b: R) := (exp (pow (a*b) 2) - 1)/(a * b).
Theorem putnam_1989_a2
(a b: R)
(abpos : a > 0 /\ b > 0)
(f : R -> R -> R := fun x y => Rmax (pow (b*x) 2) (pow (a*y) 2))
: RInt (fun x => (RInt (fun y => exp (f x y)) 0 b)) 0 a = putnam_1989_a2_solution a b.
Proof. Admitted.
|
putnam_1989_a3
|
Prove that if
\[
11z^{10}+10iz^9+10iz-11=0,
\]
then $|z|=1.$ (Here $z$ is a complex number and $i^2=-1$.)
|
None.
|
[
"algebra"
] |
Require Import Reals Coquelicot.Coquelicot. From Coqtail Require Import Cpow.
Open Scope C.
Theorem putnam_1989_a3
(f : C -> C := fun z => 11 * Cpow z 10 + 10 * Ci * Cpow z 9 + 10 * Ci * z - 11)
: forall (x: C), f x = 0 -> Cmod x = R1.
Proof. Admitted.
|
putnam_1990_a1
|
Let $T_0=2,T_1=3,T_2=6$, and for $n \geq 3$, $T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}$. The first few terms are $2,3,6,14,40,152,784,5168,40576$. Find, with proof, a formula for $T_n$ of the form $T_n=A_n+B_n$, where $\{A_n\}$ and $\{B_n\}$ are well-known sequences.
|
Show that we have $T_n=n!+2^n$.
|
[
"algebra"
] |
Require Import Nat Factorial Coquelicot.Coquelicot.
Definition putnam_1990_a1_solution := (fun n => fact n, fun n => pow 2 n).
Theorem putnam_1990_a1
(A := fix a (n: nat) : nat :=
match n with
| O => 2
| S O => 3
| S (S O) => 6
| S (S (S n''' as n'') as n') => (n + 4) * a n' - 4 * n * a n'' + (4 * n - 8) * a n'''
end)
: let (b,c) := putnam_1990_a1_solution in forall (n: nat), A n = b n + c n.
Proof. Admitted.
|
putnam_1990_a2
|
Is $\sqrt{2}$ the limit of a sequence of numbers of the form $\sqrt[3]{n}-\sqrt[3]{m}$ ($n,m=0,1,2,\dots$)?
|
Show that the answer is yes.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1990_a2_solution := True.
Theorem putnam_1990_a2
(numform : R -> Prop := fun x => exists (n m: nat), x = Rpower (INR n) (1/3) - Rpower (INR m) (1/3))
: (exists (s: nat -> R), (forall (i: nat), numform (s i)) /\ is_lim_seq s (sqrt 2)) <-> putnam_1990_a2_solution.
Proof. Admitted.
|
putnam_1990_a5
|
If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$?
|
Show that the answer is no.
|
[
"linear_algebra"
] |
From mathcomp Require Import matrix ssralg.
Open Scope ring_scope.
Definition putnam_1990_a5_solution := False.
Theorem putnam_1990_a5
: (forall (R : ringType) (n: nat) (A B : 'M[R]_n), n >= 1 -> mulmx (mulmx (mulmx A B) A) B = 0 -> mulmx (mulmx (mulmx B A) B) A = 0) <-> putnam_1990_a5_solution.
Proof. Admitted.
|
putnam_1990_b1
|
Find all real-valued continuously differentiable functions $f$ on the real line such that for all $x$, $(f(x))^2=\int_0^x [(f(t))^2+(f'(t))^2]\,dt+1990$.
|
Show that there are two such functions, namely $f(x)=\sqrt{1990}e^x$, and $f(x)=-\sqrt{1990}e^x$.
|
[
"analysis"
] |
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1990_b1_solution (f: R -> R) := f = (fun x => sqrt 1990 * exp x) \/ f = (fun x => -sqrt 1990 * exp x).
Theorem putnam_1990_b1
(f : R -> R)
(fderiv : (forall x : R, ex_derive f x) /\ continuity (Derive f))
: (forall x, pow (f x) 2 = RInt (fun t => pow (f t) 2 + pow ((Derive f) t) 2) 0 x + 1990) <->
putnam_1990_b1_solution f.
Proof. Admitted.
|
putnam_1990_b2
|
Prove that for $|x|<1$, $|z|>1$, $1+\sum_{j=1}^\infty (1+x^j)P_j=0$, where $P_j$ is $\frac{(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3) \cdots (z-x^j)}$.
|
None.
|
[
"analysis"
] |
From mathcomp Require Import all_algebra all_ssreflect.
From mathcomp Require Import reals exp sequences normedtype topology.
From mathcomp Require Import classical_sets.
Import numFieldNormedType.Exports.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Local Open Scope ring_scope.
Local Open Scope classical_set_scope.
Variable R : realType.
Theorem putnam_1990_b2
(x z : R)
(P : nat -> R)
(xlt1 : `| x | < 1)
(zgt1 : `| z | > 1)
(hP : forall j : nat, ge j 1 -> P j = (\prod_(0 <= i < j) (1 - z * x ^ i)) / (\prod_(1 <= i < j.+1) (z - x ^ i)))
: (fun n : nat => 1 + \sum_(1 <= j < n) ((1 + x ^ j) * P j)) @ \oo --> 0.
Proof. Admitted.
|
putnam_1990_b3
|
Let $S$ be a set of $2 \times 2$ integer matrices whose entries $a_{ij}$ (1) are all squares of integers, and, (2) satisfy $a_{ij} \leq 200$. Show that if $S$ has more than $50387$ ($=15^4-15^2-15+2$) elements, then it has two elements that commute.
|
None.
|
[
"linear_algebra"
] |
Require Import Ensembles Finite_sets Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1990_b3
(E : Ensemble (matrix 2 2))
(hE : forall (A: matrix 2 2), E A ->
forall (i j: nat), and (le 0 i) (lt i 2) /\ and (le 0 j) (lt j 2) ->
(coeff_mat 0 A i j) <= 200 /\ exists (m: nat), coeff_mat 0 A i j = INR m ^ 2)
: (exists (sz : nat), gt sz 50387 /\ cardinal (matrix 2 2) E sz) -> exists (A B: matrix 2 2), E A /\ E B /\ A <> B /\ Mmult A B = Mmult B A.
Proof. Admitted.
|
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