name stringlengths 14 14 | lean4_statement stringlengths 75 1.48k ⌀ | informal_statement stringlengths 47 898 ⌀ | informal_solution stringlengths 16 303 ⌀ | tags stringclasses 36
values | coq_statement stringlengths 91 1.51k ⌀ | isabelle_statement stringlengths 160 2.65k ⌀ |
|---|---|---|---|---|---|---|
putnam_1993_b1 | abbrev putnam_1993_b1_solution : ℕ := sorry
-- 3987
theorem putnam_1993_b1
(nallmexk : ℕ → Prop)
(hnallmexk : ∀ n : ℕ, nallmexk n = (n > 0 ∧ ∀ m ∈ Set.Ioo 0 1993, ∃ k : ℤ, (m / 1993 < (k : ℝ) / n) ∧ ((k : ℝ) / n < (m + 1) / 1994)))
: nallmexk putnam_1993_b1_solution ∧ (∀ n : ℕ, nallmexk n → n ≥ putnam_1993_b1_solution)... | 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'] | Section putnam_1993_b1.
Require Import Reals.
Open Scope R.
Definition putnam_1993_b1_solution : nat := 3987.
Theorem putnam_1993_b1:
let cond (n: nat) := forall (m: nat), and (lt 0 m) (lt m 1993) -> exists (k: nat), INR m / 1993 < INR k / INR n < INR (m + 1) / 1994 in
exists (mn: nat), cond mn /\ forall (n: n... | theory putnam_1993_b1 imports Complex_Main
begin
definition putnam_1993_b1_solution::nat where "putnam_1993_b1_solution \<equiv> undefined"
(* 3987 *)
theorem putnam_1993_b1:
fixes nallmexk::"nat\<Rightarrow>bool"
defines "nallmexk \<equiv> \<lambda>n::nat. (n > 0 \<and> (\<forall>m \<in> {0<..<1993}. (\<exists>k:... |
putnam_1993_b4 | theorem putnam_1993_b4
(K : ℝ × ℝ → ℝ)
(f g : ℝ → ℝ)
(Kpos : ∀ x y : Set.Icc (0 : ℝ) 1, K (x, y) > 0)
(Kcont : ContinuousOn K {(x, y) : ℝ × ℝ | x ∈ Set.Icc 0 1 ∧ y ∈ Set.Icc 0 1})
(fgpos : ∀ x : Set.Icc (0 : ℝ) 1, f x > 0 ∧ g x > 0)
(fgcont : ContinuousOn f (Set.Icc 0 1) ∧ ContinuousOn g (Set.Icc 0 1))
(fgint : ∀ x : S... | 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... | null | ['analysis'] | Section putnam_1993_b4.
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1993_b4:
forall (K: (R * R) -> R) (f g: R -> R),
(forall (xy : R * R), let (x, y) := xy in 0 <= x <= 1 /\ 0 <= y <= 1 /\ K xy > 0 /\ continuous K xy) /\
(forall (x: R), 0 <= x <= 1 /\ f x > 0 /\ g x > 0 /\ cont... | theory putnam_1993_b4 imports Complex_Main "HOL-Analysis.Interval_Integral"
begin
theorem putnam_1993_b4:
fixes K::"(real\<times>real) \<Rightarrow> real" and f g::"real\<Rightarrow>real"
assumes Kpos : "\<forall>x::real \<in> {0..1}. \<forall>y::real \<in> {0..1}. K (x, y) > 0"
and Kcont : "continuous_on {(x:... |
putnam_1993_b5 | theorem putnam_1993_b5
(pdists : (Fin 4 → (Fin 2 → ℝ)) → Prop)
(hpdists: ∀ p : Fin 4 → (Fin 2 → ℝ), pdists p = ∀ i j : Fin 4, i ≠ j → (Euclidean.dist (p i) (p j) = round (Euclidean.dist (p i) (p j)) ∧ Odd (round (Euclidean.dist (p i) (p j)))))
: ¬∃ p : Fin 4 → (Fin 2 → ℝ), pdists p :=
sorry | Show there do not exist four points in the Euclidean plane such that the pairwise distances between the points are all odd integers. | null | ['geometry', 'number_theory', 'linear_algebra'] | null | theory putnam_1993_b5 imports Complex_Main "HOL-Analysis.Linear_Algebra"
begin
theorem putnam_1993_b5:
fixes pdist::"(real \<times> real) \<Rightarrow> (real \<times> real) \<Rightarrow> (real \<times> real) \<Rightarrow> (real \<times> real) \<Rightarrow> bool"
defines "pdist \<equiv> \<lambda>A. \<lambda>B. \<la... |
putnam_1993_b6 | theorem putnam_1993_b6
(S : Fin 3 → ℕ)
(f : Fin 3 → Fin 3 → (Fin 3 → ℕ) → (Fin 3 → ℕ))
(Spos : ∀ i : Fin 3, S i > 0)
(hf : ∀ i j k : Fin 3, (i ≠ j ∧ i ≠ k ∧ j ≠ k) → ∀ S' : Fin 3 → ℕ, if S' i ≤ S' j then ((f i j S') i = 2 * S' i ∧ (f i j S') j = S' j - S' i ∧ (f i j S') k = S' k) else (f i j S' = S'))
: ∃ (Ss : ℕ → (Fi... | Let $S$ be a set of three, not necessarily distinct, positive integers. Show that one can transform $S$ into a set containing $0$ by a finite number of applications of the following rule: Select two of the three integers, say $x$ and $y$, where $x \leq y$ and replace them with $2x$ and $y-x$. | null | ['algebra'] | null | theory putnam_1993_b6 imports Complex_Main
begin
(* Note: Boosted domain to infinite set *)
theorem putnam_1993_b6:
fixes S::"nat\<Rightarrow>nat" and f::"nat \<Rightarrow> nat \<Rightarrow> (nat\<Rightarrow>nat) \<Rightarrow> (nat\<Rightarrow>nat)"
assumes Spos : "\<forall>i \<in> {0..<3}. S i > 0"
and hf : "... |
putnam_2015_a2 | abbrev putnam_2015_a2_solution : ℕ := sorry
-- 181
theorem putnam_2015_a2
(a : ℕ → ℤ)
(abase : a 0 = 1 ∧ a 1 = 2)
(arec : ∀ n ≥ 2, a n = 4 * a (n - 1) - a (n - 2))
: Odd putnam_2015_a2_solution ∧ putnam_2015_a2_solution.Prime ∧ ((putnam_2015_a2_solution : ℤ) ∣ a 2015) :=
sorry | Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n \geq 2$. Find an odd prime factor of $a_{2015}$. | Show that one possible answer is $181$. | ['number_theory'] | Section putnam_2015_a2.
Require Import Nat Reals Znumtheory. From mathcomp Require Import div.
Definition putnam_2015_a2_solution : nat := 181.
Theorem putnam_2015_a2:
let A :=
fix a (n: nat) : nat :=
match n with
| O => 1%nat
| S O => 2%nat
| S ((S n'') as ... | theory putnam_2015_a2 imports Complex_Main
"HOL-Computational_Algebra.Primes"
begin
(* there are several possible correct solutions; this is the one shown on the solutions document *)
definition putnam_2015_a2_solution :: nat where "putnam_2015_a2_solution \<equiv> undefined"
(* 181 *)
theorem putnam_2015_a2:
fixes ... |
putnam_2015_a3 | abbrev putnam_2015_a3_solution : ℂ := sorry
-- 13725
theorem putnam_2015_a3
: Complex.log (∏ a : Fin 2015, ∏ b : Fin 2015, (1 + Complex.exp (2 * Real.pi * Complex.I * (a.1 + 1) * (b.1 + 1) / 2015))) / Complex.log 2 = putnam_2015_a3_solution :=
sorry | Compute $\log_2 \left( \prod_{a=1}^{2015}\prod_{b=1}^{2015}(1+e^{2\pi iab/2015}) \right)$. Here $i$ is the imaginary unit (that is, $i^2=-1$). | Show that the answer is $13725$. | ['number_theory', 'algebra'] | Section putnam_2015_a3.
Require Import Reals ROrderedType Coquelicot.Coquelicot.
Local Open Scope C.
Definition putnam_2015_a3_solution : C := RtoC 13725.
Theorem putnam_2015_a3:
let Carg (z : C) : R := if Reqb (Im z) 0 then (if Rlt_dec (Re z) 0 then PI else R0) else atan ((Im z)/(Re z)) in
let Clog (n: nat)... | theory putnam_2015_a3 imports Complex_Main
begin
definition putnam_2015_a3_solution :: complex where "putnam_2015_a3_solution \<equiv> undefined"
(* 13725 *)
theorem putnam_2015_a3:
shows "log 2 (Re (\<Prod>a::nat=1..2015. \<Prod>b::nat=1..2015. (1 + exp (2*pi*\<i>*a*b/2015)))) = putnam_2015_a3_solution"
sorry
en... |
putnam_2015_a4 | abbrev putnam_2015_a4_solution : ℝ := sorry
-- 4 / 7
theorem putnam_2015_a4
(S : ℝ → Set ℕ := fun x ↦ {n : ℕ | n > 0 ∧ Even ⌊n * x⌋})
(f : ℝ → ℝ := fun x ↦ ∑' n : S x, 1 / 2 ^ (n : ℕ))
(p : ℝ → Prop := fun l ↦ ∀ x ∈ Set.Ico 0 1, f x ≥ l)
(L : ℝ)
(hpL : p L)
(hLub : ∀ l : ℝ, p l → l ≤ L)
: (L = putnam_2015_a4_solution) ... | For each real number $x$, let
\[
f(x) = \sum_{n\in S_x} \frac{1}{2^n},
\]
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx \rfloor$ is even. What is the largest real number $L$ such that $f(x) \geq L$ for all $x \in [0,1)$? (As usual, $\lfloor z \rfloor$ denotes the greatest integer less than or eq... | Prove that $L = \frac{4}{7}$. | ['analysis'] | Section putnam_2015_a4.
Require Import Nat List Reals. From Coquelicot Require Import Coquelicot Hierarchy Rcomplements.
Definition putnam_2015_a4_solution : R := 4/7.
Theorem putnam_2015_a4:
let f (x: R) := Series (fun n => if even (Z.to_nat (floor ((INR n)*x))) then 1/(2^n) else R0) in
exists (L: R), fora... | theory putnam_2015_a4 imports Complex_Main
"HOL-Analysis.Infinite_Sum"
begin
definition putnam_2015_a4_solution::real where "putnam_2015_a4_solution \<equiv> undefined"
(* 4/7 *)
theorem putnam_2015_a4:
fixes S::"real\<Rightarrow>(nat set)" and f::"real\<Rightarrow>real" and p::"real\<Rightarrow>bool"
defines "S \... |
putnam_2015_a5 | theorem putnam_2015_a5
(q : ℕ)
(Nq : ℕ)
(qodd : Odd q)
(qpos : q > 0)
(hNq : Nq = {a : ℕ | 0 < a ∧ a < (q : ℝ) / 4 ∧ Nat.gcd a q = 1}.encard)
: Odd Nq ↔ ∃ p k : ℕ, q = p ^ k ∧ k > 0 ∧ p.Prime ∧ (p % 8 = 5 ∨ p % 8 = 7) :=
sorry | Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1$. Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8$. | null | ['number_theory'] | Section putnam_2015_a5.
Require Import Nat Reals Arith Znumtheory Ensembles Finite_sets.
Local Open Scope nat_scope.
Theorem putnam_2015_a5:
forall (q: nat), odd q = true ->
forall (Nq: Ensemble nat),
forall (a: nat),
(
and (Rle R0 (INR a)) (Rle (INR a) (Rdiv (INR q) (INR 4)))
/\ gcd... | theory putnam_2015_a5 imports Complex_Main
"HOL-Computational_Algebra.Primes"
begin
theorem putnam_2015_a5:
fixes q :: nat
and Nq :: nat
assumes qodd: "odd q"
and qpos: "q > 0"
and hNq: "Nq = card {a::nat. 0 < a \<and> a < (real q)/4 \<and> gcd a q = 1}"
shows "odd Nq \<longleftrightarrow> (\<exists>p k::n... |
putnam_2015_a6 | theorem putnam_2015_a6
(n : ℕ)
(A B M : Matrix (Fin n) (Fin n) ℝ)
(npos : n > 0)
(hmul : A * M = M * B)
(hpoly : Matrix.charpoly A = Matrix.charpoly B)
: ∀ X : Matrix (Fin n) (Fin n) ℝ, (A - M * X).det = (B - X * M).det :=
sorry | Let $n$ be a positive integer. Suppose that $A$, $B$, and $M$ are $n \times n$ matrices with real entries such that $AM = MB$, and such that $A$ and $B$ have the same characteristic polynomial. Prove that $\det(A-MX)=\det(B-XM)$ for every $n \times n$ matrix $X$ with real entries. | null | ['linear_algebra'] | Section putnam_2015_a6.
Theorem putnam_2015_a6: True.
Proof. Admitted.
End putnam_2015_a6. | theory putnam_2015_a6 imports Complex_Main
"HOL-Analysis.Determinants"
"HOL-Computational_Algebra.Polynomial"
begin
theorem putnam_2015_a6:
fixes n :: nat
and A B M :: "real^'n^'n"
and charpoly :: "real^'n^'n \<Rightarrow> (real poly)"
assumes npos: "n > 0"
and pncard: "CARD('n) = n"
and hmul: "A ** M = M ... |
putnam_2015_b1 | theorem putnam_2015_b1
(f : ℝ → ℝ)
(nzeros : (ℝ → ℝ) → ℕ → Prop)
(fdiff : ContDiff ℝ 2 f ∧ Differentiable ℝ (iteratedDeriv 2 f))
(hnzeros : ∀ f' : ℝ → ℝ, ∀ n : ℕ, nzeros f' n = ({x : ℝ | f' x = 0}.encard ≥ n))
(fzeros : nzeros f 5)
: nzeros (f + 6 * deriv f + 12 * iteratedDeriv 2 f + 8 * iteratedDeriv 3 f) 2 :=
sorry | Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Prove that $f+6f'+12f''+8f'''$ has at least two distinct real zeros. | null | ['analysis'] | Section putnam_2015_b1.
Require Import Reals List Coquelicot.Derive.
Local Open Scope R_scope.
Theorem putnam_2015_b1 :
forall (f: R -> R),
continuity (Derive_n f 3) ->
exists (l: list R), length l = 5%nat /\ NoDup l /\ forall x, In x l -> f x = 0 ->
let g (x: R):= f x + 6 * (Derive_n f 1) x + 12 * (De... | theory putnam_2015_b1 imports Complex_Main
"HOL-Analysis.Derivative"
begin
theorem putnam_2015_b1:
fixes f :: "real \<Rightarrow> real"
and nzeros :: "(real \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> bool"
assumes fdiff: "f C1_differentiable_on UNIV \<and> (deriv f) C1_differentiable_on UNIV \<and> ((de... |
putnam_2015_b3 | abbrev putnam_2015_b3_solution : Set (Matrix (Fin 2) (Fin 2) ℝ) := sorry
-- {A : Matrix (Fin 2) (Fin 2) ℝ | (∃ α : ℝ, ∀ i j : Fin 2, A i j = α * 1) ∨ (∃ β : ℝ, A 0 0 = β * -3 ∧ A 0 1 = β * -1 ∧ A 1 0 = β * 1 ∧ A 1 1 = β * 3)}
theorem putnam_2015_b3
(S : Set (Matrix (Fin 2) (Fin 2) ℝ))
(M : Matrix (Fin 2) (Fin 2) ℝ)
(Mi... | Let $S$ be the set of all $2 \times 2$ real matrices $M=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S$. | Show that matrices of the form $\alpha A$ or $\beta B$, where $A=\left(\begin{smallmatrix} 1 & 1 \\ 1 & 1 \end{smallmatrix}\right)$, $B=\left(\begin{smallmatrix} -3 & -1 \\ 1 & 3 \end{smallmatrix}\right)$, and $\alpha,\beta \in \mathbb{R}$, are the only matrices in $S$ that satisfy the given condition. | ['linear_algebra'] | null | theory putnam_2015_b3 imports Complex_Main
"HOL-Analysis.Finite_Cartesian_Product"
begin
definition putnam_2015_b3_solution :: "(real^2^2) set" where "putnam_2015_b3_solution \<equiv> undefined"
(* {M::real^2^2. (\<exists>\<alpha> \<beta>::real. M$1$1 = \<alpha>*1 + \<beta>*-3 \<and> M$1$2 = \<alpha>*1 + \<beta>*-1 \<... |
putnam_2015_b4 | abbrev putnam_2015_b4_solution : ℤ × ℕ := sorry
-- (17, 21)
theorem putnam_2015_b4
(quotientof : ℚ → (ℤ × ℕ))
(hquotientof : ∀ q : ℚ, quotientof q = (q.num, q.den))
: quotientof (∑' t : (Fin 3 → ℤ), if (∀ n : Fin 3, t n > 0) ∧ t 0 < t 1 + t 2 ∧ t 1 < t 2 + t 0 ∧ t 2 < t 0 + t 1
then 2^(t 0)/(3^(t 1)*5^(t 2)) else 0) = ... | Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c$. Express
\[
\sum_{(a,b,c) \in T} \frac{2^a}{3^b 5^c}
\]
as a rational number in lowest terms. | The answer is $17/21$. | ['algebra'] | Section putnam_2015_b4.
Require Import Nat List Reals Coquelicot.Coquelicot. From mathcomp Require Import div.
Open Scope nat_scope.
Definition putnam_2015_b4_solution := (17, 21).
Theorem putnam_2015_b4:
let fix exprl2 (l : list nat) : R :=
match l with
| a :: b :: c :: _ => Rdiv (2 ^ a) (3 ^ b * 5 ^ c)
... | theory putnam_2015_b4 imports Complex_Main
"HOL-Analysis.Infinite_Sum"
"HOL-Analysis.Finite_Cartesian_Product"
begin
definition putnam_2015_b4_solution :: "int \<times> int" where "putnam_2015_b4_solution \<equiv> undefined"
(* (17,21) *)
theorem putnam_2015_b4:
shows "\<exists>q::rat. (\<Sum>\<^sub>\<infinity>t::na... |
putnam_2015_b5 | abbrev putnam_2015_b5_solution : ℕ := sorry
-- 4
theorem putnam_2015_b5
(P : ℕ → ℕ := fun n ↦ {pi : Finset.Icc 1 n → Finset.Icc 1 n | Bijective pi ∧ ∀ i j : Finset.Icc 1 n, Nat.dist i j = 1 → Nat.dist (pi i) (pi j) ≤ 2}.ncard)
: (∀ n : ℕ, n ≥ 2 → (P (n + 5) : ℤ) - (P (n + 4) : ℤ) - (P (n + 3) : ℤ) + (P n : ℤ) = putnam... | Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that
\[
|i-j| = 1 \mbox{ implies } |\pi(i) -\pi(j)| \leq 2
\]
for all $i,j$ in $\{1,2,\dots,n\}$. Show that for $n \geq 2$, the quantity
\[
P_{n+5} - P_{n+4} - P_{n+3} + P_n
\]
does not depend on $n$, and find its value. | Prove that answer is $4$. | ['algebra'] | Section putnam_2015_b5.
Require Import Reals. From mathcomp Require Import fintype perm ssrbool.
Open Scope nat_scope.
Definition putnam_2015_b5_solution := 4.
Theorem putnam_2015_b5:
let cond (n: nat) (π: {perm 'I_n}) : bool :=
forallb (fun i =>
forallb (fun j =>
if Z.to_nat (Z.abs (... | theory putnam_2015_b5 imports Complex_Main
"HOL-Combinatorics.Permutations"
begin
definition putnam_2015_b5_solution::nat where "putnam_2015_b5_solution \<equiv> undefined"
(* 4 *)
theorem putnam_2015_b5:
fixes P::"nat\<Rightarrow>nat"
defines "P \<equiv> \<lambda>n. card ({pi::nat\<Rightarrow>nat. pi permutes {1.... |
putnam_2015_b6 | abbrev putnam_2015_b6_solution : ℝ := sorry
-- Real.pi ^ 2 / 16
theorem putnam_2015_b6
(A : ℕ → ℕ)
(hA : ∀ k > 0, A k = {j : ℕ | Odd j ∧ j ∣ k ∧ j < Real.sqrt (2 * k)}.encard)
: ∑' k : Set.Ici 1, (-1 : ℝ) ^ ((k : ℝ) - 1) * (A k / (k : ℝ)) = putnam_2015_b6_solution :=
sorry | For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\sqrt{2k})$. Evaluate $\sum_{k=1}^\infty (-1)^{k-1}\frac{A(k)}{k}$. | Show that the sum converges to $\pi^2/16$. | ['analysis', 'number_theory'] | null | theory putnam_2015_b6 imports Complex_Main
"HOL-Analysis.Infinite_Sum"
begin
definition putnam_2015_b6_solution :: real where "putnam_2015_b6_solution \<equiv> undefined"
(* pi^2 / 16 *)
theorem putnam_2015_b6:
fixes A :: "nat \<Rightarrow> nat"
assumes hA: "\<forall>k::nat>0. A k = card {j::nat. odd j \<and> j dv... |
putnam_1996_a1 | null | null | null | [] | Section putnam_1996_a1.
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1996_a1:
exists (minA: R), forall (A: R),
let packable (n1 n2 a1 a2: R) := (n1 + n2) <= Rmax a1 a2 /\ Rmax n1 n2 <= Rmin a1 a2 in
forall (n1 n2: R), pow n1 2 + pow n2 2 = 1 ->
exists (a1 a2: R), a1 * a2 = ... | null |
putnam_2009_a2 | null | null | null | [] | Section putnam_2009_a2.
Require Import Reals Coquelicot.Coquelicot.
Definition putnam_2009_a2_solution : (R -> R) := fun x => Rpower 2 (-1 / 12) * (sin (6 * x + PI / 4) / (cos (6 * x + PI / 4)) ^ 2) ^ (1 / 6).
Theorem putnam_2009_a2:
forall (f g h: R -> R),
exists (a b: R), a < b /\
forall (x: R), a < x ... | null |
putnam_1975_a5 | null | null | null | [] | Section putnam_1975_a5.
Require Import Basics Factorial Reals Coquelicot.Series.
Open Scope R.
Theorem putnam_1975_a5:
let f0 (x: R) := Rpower (exp 1) x in
let 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 ... | null |
putnam_1975_b6 | null | null | null | [] | Section putnam_1975_b6.
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Theorem putnam_1975_b6
(h : nat -> R := fun n => sum_n (fun r => 1 / (INR r + 1)) n)
: forall (n: nat), gt n 2 -> INR n * INR (n + 1) ^ (1 / n) < h n < INR n - INR (n - 1) * Rpower (INR n) (-R1 / INR (n - 1)) .
Proof. Admitted.
En... | null |
putnam_1997_a1 | null | null | null | [] | Section putnam_1997_a1.
Require Import Reals Rgeom ZArith
GeoCoq.Main.Tarski_dev.Ch16_coordinates_with_functions
GeoCoq.Main.Annexes.midpoint_theorems
GeoCoq.Main.Highschool.circumcenter.
Context `{T2D:Tarski_2D} `{TE:@Tarski_euclidean Tn TnEQD}.
Open Scope R.
Definition putnam_1997_a1_solution := 28.
Theorem putnam_... | null |
putnam_2003_b5 | null | null | null | [] | Section putnam_2003_b5.
Require Import Reals
GeoCoq.Main.Tarski_dev.Ch16_coordinates_with_functions
GeoCoq.Axioms.Definitions
GeoCoq.Main.Highschool.triangles.
Context `{T2D:Tarski_2D} `{TE:@Tarski_euclidean Tn TnEQD}.
Open Scope R.
Definition putnam_2003_b5_solution (pt_to_R : Tpoint -> (R * R)) (dist : Tpoint -> Tpo... | null |
putnam_2014_a4 | null | null | null | [] | Section putnam_2014_a4.
From mathcomp.analysis Require Import probability.
From mathcomp Require Import all_ssreflect.
From mathcomp Require Import ssralg poly ssrnum ssrint interval finmap.
From mathcomp Require Import mathcomp_extra boolp classical_sets functions.
From mathcomp Require Import cardinality fsbigop.
Fro... | null |
putnam_1987_a3 | null | null | null | [] | Section putnam_1987_a3.
Require Import Reals Coquelicot.Coquelicot.
Open Scope R.
Definition putnam_1987_a3_solution1 := False.
Definition putnam_1987_a3_solution2 := True.
Theorem putnam_1987_a3:
exists (f g: R -> R) (x: R),
((Derive_n f 2) x - 2 * (Derive_n f 1) x + f x = 2 * exp x /\ forall (x: R), f x > 0)... | null |
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