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putnam_1966_a3
theorem putnam_1966_a3 (x : β„• β†’ ℝ) (hx1 : 0 < x 1 ∧ x 1 < 1) (hxi : βˆ€ n β‰₯ 1, x (n + 1) = (x n) * (1 - (x n))) : Tendsto (fun n : β„• => n * (x n)) atTop (𝓝 1) := sorry
124af5f076ccbe26
putnam_1976_a3
abbrev putnam_1976_a3_solution : Set (β„• Γ— β„• Γ— β„• Γ— β„•) := sorry -- {(3, 2, 2, 3), (2, 3, 3, 2)} /-- 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 $1$. -/ theorem putnam_1976_a3 : {(p, r, q, s) : β„• Γ— β„• Γ— β„• Γ— β„• | Nat.Prime p ∧ Nat.Prime q ∧ r > 1 ∧ s > 1 ∧ |(p^r : β„€) - q^s| = 1} = putnam_1976_a3_solution := sorry
862d0def2897b578
putnam_2016_a2
abbrev putnam_2016_a2_solution : ℝ := sorry -- (3 + √ 5) / 2 /-- Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \[ \binom{m}{n-1} > \binom{m-1}{n}. \] Evaluate \[ \lim_{n \to \infty} \frac{M(n)}{n}. \] -/ theorem putnam_2016_a2 (M : β„• β†’ β„•) (hM : βˆ€ n > 0, IsGreatest {m | 0 < m ∧ (m - 1).choose n < m.choose (n - 1)} (M n)) : Tendsto (fun n ↦ M n / (n : ℝ)) atTop (𝓝 putnam_2016_a2_solution) := sorry
9a429a6325dfcc41
putnam_2024_b6
abbrev putnam_2024_b6_solution : ℝ := sorry -- -1/2 /-- For a real number $a$, let $F_a(x) = \sum_{n\geq 1} n^a e^{2n}x^{n^2}$ for $0 \leq x < 1$. Find a real number $c$ such that $\lim_{x \to 1^-} F_a(x)e^{-1/(1-x)} = 0$ for all $a < c$, and $\lim_{x \to 1^-} F_a(x)e^{-1/(1-x)} = \infty$ for all $a > c$. -/ theorem putnam_2024_b6 (F : ℝ β†’ ℝ β†’ ℝ) (c : ℝ) (hF : βˆ€ a, βˆ€ x ∈ Set.Ico 0 1, HasSum (fun (n : β„•) => (n+1)^a * rexp (2*(n+1)) * x^(n+1)^2) (F a x)) : ((βˆ€ a < c, Filter.Tendsto (fun x => F a x * rexp (- 1 / (1-x))) (𝓝[<] 1) (𝓝 0)) ∧ (βˆ€ a > c, Filter.Tendsto (fun x => F a x * rexp (- 1 / (1-x))) (𝓝[<] 1) Filter.atTop)) ↔ c = putnam_2024_b6_solution := sorry
750f7a8d6e992fe0
putnam_2019_b1
abbrev putnam_2019_b1_solution : β„• β†’ β„• := sorry -- (fun n : β„• => 5 * n + 1) /-- Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n \geq 0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k \leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square. -/ theorem putnam_2019_b1 (n : β„•) (Pn : Set (Fin 2 β†’ β„€)) (pZtoR : (Fin 2 β†’ β„€) β†’ EuclideanSpace ℝ (Fin 2)) (sPnsquare : Finset (Fin 2 β†’ β„€) β†’ Prop) (hPn : Pn = {p | (p 0 = 0 ∧ p 1 = 0) ∨ (βˆƒ k ≀ n, (p 0) ^ 2 + (p 1) ^ 2 = 2 ^ k)}) (hpZtoR : βˆ€ p i, (pZtoR p) i = p i) (sPnsquare_def : βˆ€ sPn : Finset (Fin 2 β†’ β„€), sPnsquare sPn ↔ (sPn.card = 4 ∧ βˆƒ p4 : Fin 4 β†’ (Fin 2 β†’ β„€), Set.range p4 = sPn ∧ (βˆƒ s > 0, βˆ€ i : Fin 4, dist (pZtoR (p4 i) : EuclideanSpace ℝ (Fin 2)) (pZtoR (p4 (i + 1))) = s) ∧ (dist (pZtoR (p4 0)) (pZtoR (p4 2)) = dist (pZtoR (p4 1)) (pZtoR (p4 3))))) : {sPn : Finset (Fin 2 β†’ β„€) | (sPn : Set (Fin 2 β†’ β„€)) βŠ† Pn ∧ sPnsquare sPn}.encard = putnam_2019_b1_solution n := sorry
5c2204ebc4ae6f1e
putnam_2009_b1
theorem putnam_2009_b1 (isquotprodprimefact : β„š β†’ Prop) (hisquotprodprimefact : βˆ€ q, isquotprodprimefact q ↔ (βˆƒ (k m : β„•) (a : Fin k β†’ β„•) (b : Fin m β†’ β„•), (βˆ€ i : Fin k, Nat.Prime (a i)) ∧ (βˆ€ j : Fin m, Nat.Prime (b j)) ∧ (q = (∏ i : Fin k, Nat.factorial (a i))/(∏ j : Fin m, Nat.factorial (b j))))) : βˆ€ q : β„š, q > 0 β†’ isquotprodprimefact q := sorry
006bfea59e7e6b9c
putnam_1993_a2
theorem putnam_1993_a2 (x : β„• β†’ ℝ) (xnonzero : βˆ€ n : β„•, x n β‰  0) (hx : βˆ€ n β‰₯ 1, (x n) ^ 2 - x (n - 1) * x (n + 1) = 1) : βˆƒ a : ℝ, βˆ€ n β‰₯ 1, x (n + 1) = a * x n - x (n - 1) := sorry
b053d947122d018b
putnam_1980_a4
theorem putnam_1980_a4 (abcvals : β„€ β†’ β„€ β†’ β„€ β†’ Prop) (habcvals : βˆ€ a b c : β„€, abcvals a b c ↔ (a = 0 ∧ b = 0 ∧ c = 0) ∧ |a| < 1000000 ∧ |b| < 1000000 ∧ |c| < 1000000) : (βˆƒ a b c : β„€, abcvals a b c ∧ |a + b * Real.sqrt 2 + c * Real.sqrt 3| < 10 ^ (-(11 : ℝ))) ∧ (βˆ€ a b c : β„€, abcvals a b c β†’ |a + b * Real.sqrt 2 + c * Real.sqrt 3| > 10 ^ (-(21 : ℝ))) := sorry
ff08cc69439ee179
putnam_1990_a4
abbrev putnam_1990_a4_solution : β„• := sorry -- 3 /-- Consider a paper punch that can be centered at any point of the plane and that, when operated, removes from the plane precisely those points whose distance from the center is irrational. How many punches are needed to remove every point? -/ theorem putnam_1990_a4 : sInf {n : β„• | βˆƒ S : Set (EuclideanSpace ℝ (Fin 2)), S.encard = n ∧ βˆ€ Q : EuclideanSpace ℝ (Fin 2), βˆƒ P ∈ S, Irrational (dist P Q)} = putnam_1990_a4_solution := sorry
61fa14dd6910530f
putnam_2015_a4
abbrev putnam_2015_a4_solution : ℝ := sorry -- 4 / 7 /-- 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 equal to $z$.) -/ theorem putnam_2015_a4 (S : ℝ β†’ Set β„€) (f : ℝ β†’ ℝ) (p : ℝ β†’ Prop) (hS : S = fun (x : ℝ) ↦ {n : β„€ | n > 0 ∧ Even ⌊n * xβŒ‹}) (hf : f = fun (x : ℝ) ↦ βˆ‘' n : S x, 1 / 2 ^ (n : β„€)) (hp : βˆ€ l, p l ↔ βˆ€ x ∈ Set.Ico 0 1, f x β‰₯ l) : IsGreatest p putnam_2015_a4_solution := sorry
5df6d46654f02eed
putnam_2005_a4
theorem putnam_2005_a4 (n : β„•) (H : Matrix (Fin n) (Fin n) ℝ) (a b : β„•) (S : Matrix (Fin a) (Fin b) ℝ) (npos : n β‰₯ 1) (Hentries : βˆ€ i j : Fin n, H i j = 1 ∨ H i j = -1) (Hortho : H.HasOrthogonalRows) (hab : 1 ≀ a ∧ a ≀ n ∧ 1 ≀ b ∧ b ≀ n) (Ssub : βˆƒ (rri : Fin a β†’ Fin n) (cri : Fin b β†’ Fin n), rri.Injective ∧ cri.Injective ∧ S = H.submatrix rri cri) (Sentries : βˆ€ (i : Fin a) (j : Fin b), S i j = 1) : a * b ≀ n := sorry
51a6e7c5fee0f21c
putnam_1972_a1
theorem putnam_1972_a1 (n : β„•) (hn : n > 0) (fourAP : β„€ β†’ β„€ β†’ β„€ β†’ β„€ β†’ Prop) (hfourAP : βˆ€ n1 n2 n3 n4, fourAP n1 n2 n3 n4 ↔ n4-n3 = n3-n2 ∧ n3-n2 = n2-n1) : Β¬ βˆƒ r : β„•, r > 0 ∧ r + 3 ≀ n ∧ fourAP (n.choose r) (n.choose (r+1)) (n.choose (r+2)) (n.choose (r+3)) := sorry
783d78e37629142e
putnam_1962_a1
theorem putnam_1962_a1 (S : Set (ℝ Γ— ℝ)) (hS : S.ncard = 5) (hnoncol : βˆ€ s βŠ† S, s.ncard = 3 β†’ Β¬Collinear ℝ s) : βˆƒ T βŠ† S, T.ncard = 4 ∧ Β¬βˆƒ t ∈ T, t ∈ convexHull ℝ (T \ {t}) := sorry
d3ebc30ff4ca058d
putnam_1975_a5
theorem putnam_1975_a5 (I : Interval ℝ) (f y1 y2 : ℝ β†’ ℝ) (hfcont : Continuous f) (hycdiff : ContDiff ℝ 2 y1 ∧ ContDiff ℝ 2 y2) (hy1sol : βˆ€ x : ℝ, iteratedDeriv 2 y1 x = (f x) * (y1 x)) (hy2sol : βˆ€ x : ℝ, iteratedDeriv 2 y2 x = (f x) * (y2 x)) (hylinindep : βˆ€ c1 c2 : ℝ, (βˆ€ x : ℝ, c1 * y1 x + c2 * y2 x = 0) β†’ (c1 = 0 ∧ c2 = 0)) (hyI : βˆ€ x ∈ I, y1 x > 0 ∧ y2 x > 0) : βˆƒ c : ℝ, c > 0 ∧ (let z : ℝ β†’ ℝ := fun x => c * Real.sqrt ((y1 x) * (y2 x)); βˆ€ x ∈ I, iteratedDeriv 2 z x + 1/(z x)^3 = (f x) * (z x)) := sorry
925a0432945ed286
putnam_1965_a5
abbrev putnam_1965_a5_solution : β„• β†’ β„• := sorry -- fun n => 2^(n - 1) /-- How many orderings of the integers from $1$ to $n$ satisfy the condition that, for every integer $i$ except the first, there exists some earlier integer in the ordering which differs from $i$ by $1$? -/ theorem putnam_1965_a5 : βˆ€ n > 0, {p ∈ permsOfFinset (Finset.Icc 1 n) | βˆ€ m ∈ Finset.Icc 2 n, βˆƒ k ∈ Finset.Ico 1 m, p m = p k + 1 ∨ p m = p k - 1}.card = putnam_1965_a5_solution n := sorry
bea1b81423927a9f
putnam_1988_b3
abbrev putnam_1988_b3_solution : ℝ := sorry -- (1 + Real.sqrt 3) / 2 /-- 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 $n \in N$. -/ theorem putnam_1988_b3 (r : β„€ β†’ ℝ) (hr : βˆ€ n β‰₯ 1, (βˆƒ c d : β„€, (c β‰₯ 0 ∧ d β‰₯ 0) ∧ c + d = n ∧ r n = |c - d * Real.sqrt 3|) ∧ (βˆ€ c d : β„€, (c β‰₯ 0 ∧ d β‰₯ 0 ∧ c + d = n) β†’ |c - d * Real.sqrt 3| β‰₯ r n)) : IsLeast {g : ℝ | g > 0 ∧ (βˆ€ n : β„€, n β‰₯ 1 β†’ r n ≀ g)} putnam_1988_b3_solution := sorry
3368febfcde43087
putnam_2020_b4
abbrev putnam_2020_b4_solution : ℝ := sorry -- 1 / 4040 /-- Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v} = (s_0, s_1, \cdots, s_{2n-1}, s_{2n})$ for which $s_0 = s_{2n} = 0$ and $|s_j - s_{j-1}| = 1$ for $j=1,2,\cdots,2n$. Define \[ q(\mathbf{v}) = 1 + \sum_{j=1}^{2n-1} 3^{s_j}, \] and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v} \in V_n$. Evaluate $M(2020)$. -/ theorem putnam_2020_b4 (V : β„• β†’ Set (β„• β†’ β„€)) (q : β„• β†’ (β„• β†’ β„€) β†’ ℝ) (M : β„• β†’ ℝ) (hV : V = fun n ↦ ({s : β„• β†’ β„€ | s 0 = 0 ∧ (βˆ€ j β‰₯ 2 * n, s j = 0) ∧ (βˆ€ j ∈ Icc 1 (2 * n), |s j - s (j - 1)| = 1)})) (hq : q = fun n s ↦ 1 + βˆ‘ j in Finset.Icc 1 (2 * n - 1), 3 ^ (s j)) (hM : M = fun n ↦ (βˆ‘' v : V n, 1 / (q n v)) / (V n).ncard) : (M 2020 = putnam_2020_b4_solution) := sorry
10f1f9763f543308
putnam_2004_a5
theorem putnam_2004_a5 (m n : β„•) (mnpos : Inhabited (Fin m Γ— Fin n)) (adj : (Fin m Γ— Fin n) β†’ (Fin m Γ— Fin n) β†’ Prop) (connected : (Fin m Γ— Fin n β†’ Prop) β†’ (Fin m Γ— Fin n) β†’ (Fin m Γ— Fin n) β†’ Prop) (cmr : (Fin m Γ— Fin n β†’ Prop) β†’ β„•) (hadj : adj = fun (⟨a, _⟩, ⟨b, _⟩) (⟨c, _⟩, ⟨d, _⟩) ↦ a = c ∧ Nat.dist b d = 1 ∨ b = d ∧ Nat.dist a c = 1) (hconnected : connected = fun C P Q ↦ βˆƒ (S : List (Fin m Γ— Fin n)) (hS : S β‰  []), S.head hS = P ∧ S.getLast hS = Q ∧ (βˆƒ p : Prop, βˆ€ i ∈ Finset.range S.length, C S[i]! = p) ∧ (βˆ€ i ∈ Finset.range (S.length - 1), adj S[i]! S[i+1]!)) (hcmr : cmr = fun C ↦ {R : Set (Fin m Γ— Fin n) | βˆƒ P ∈ R, βˆ€ Q, Q ∈ R ↔ connected C P Q}.ncard) : (βˆ‘ C : Fin m Γ— Fin n β†’ Prop, cmr C > 2 ^ (m * n) * (m * n / (8 : β„š))) := sorry
0233cf9172791e68
putnam_2014_a5
theorem putnam_2014_a5 (P : β„• β†’ Polynomial β„‚) (hP : βˆ€ n, P n = βˆ‘ i in Finset.Icc 1 n, i * Polynomial.X ^ (i - 1)) : βˆ€ (j k : β„•), (j > 0 ∧ k > 0) β†’ j β‰  k β†’ IsCoprime (P j) (P k) := sorry
7cf74d57af24c945
putnam_1991_a5
abbrev putnam_1991_a5_solution : ℝ := sorry -- 1 / 3 /-- Find the maximum value of $\int_0^y \sqrt{x^4+(y-y^2)^2}\,dx$ for $0 \leq y \leq 1$. -/ theorem putnam_1991_a5 (f : Set.Icc (0 : ℝ) 1 β†’ ℝ) (hf : βˆ€ y : Set.Icc 0 1, f y = ∫ x in Set.Ioo 0 y, Real.sqrt (x ^ 4 + (y - y ^ 2) ^ 2)) : IsGreatest (f '' (Set.Icc 0 1)) putnam_1991_a5_solution := sorry
9c01ca3b1e4e1716
putnam_1981_a5
abbrev putnam_1981_a5_solution : Prop := sorry -- True /-- Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$ Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots. -/ theorem putnam_1981_a5 (Q : Polynomial ℝ β†’ Polynomial ℝ) (hQ : Q = fun P : Polynomial ℝ => (X^2 + 1)*P*(derivative P) + X*(P^2 + (derivative P)^2)) (n : Polynomial ℝ β†’ ℝ) (hn : n = fun P : Polynomial ℝ => ({x ∈ Ioi 1 | P.eval x = 0}.ncard : ℝ)) : (βˆ€ P : Polynomial ℝ, {x : ℝ | (Q P).eval x = 0}.ncard β‰₯ 2*(n P) - 1) ↔ putnam_1981_a5_solution := sorry
23e23d3a8c9f69f2
putnam_1986_a1
abbrev putnam_1986_a1_solution : ℝ := sorry -- 18 /-- Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36 \leq 13x^2$. -/ theorem putnam_1986_a1 (S : Set ℝ) (f : ℝ β†’ ℝ) (hS : S = {x : ℝ | x ^ 4 + 36 ≀ 13 * x ^ 2}) (hf : f = fun x ↦ x ^ 3 - 3 * x) : IsGreatest {f x | x ∈ S} putnam_1986_a1_solution := sorry
0f75686146092a9d
putnam_1989_b2
abbrev putnam_1989_b2_solution : Prop := sorry -- True /-- Let $S$ be a non-empty set with an associative operation that is left and right cancellative ($xy=xz$ implies $y=z$, and $yx=zx$ implies $y=z$). Assume that for every $a$ in $S$ the set $\{a^n:\,n=1, 2, 3, \ldots\}$ is finite. Must $S$ be a group? -/ theorem putnam_1989_b2 : (βˆ€ (S : Type) [Nonempty S] [Semigroup S] [IsCancelMul S] (h_fin : βˆ€ a : S, {(a * Β·)^[n] a | n : β„•}.Finite), βˆƒ e : S, βˆ€ x, e * x = x ∧ x * e = x ∧ βˆƒ y, x * y = e ∧ y * x = e) ↔ putnam_1989_b2_solution := sorry
d15a5b0e0cb2f6ff
putnam_2021_b5
theorem putnam_2021_b5 (n : β„•) (npos : n β‰₯ 1) (veryodd : Matrix (Fin n) (Fin n) β„€ β†’ Prop) (hveryodd : βˆ€ A, veryodd A ↔ βˆ€ m ∈ Set.Icc 1 n, βˆ€ reind : Fin m β†’ Fin n, Function.Injective reind β†’ Odd (A.submatrix reind reind).det) : βˆ€ A, veryodd A β†’ (βˆ€ k β‰₯ 1, veryodd (A ^ k)) := sorry
4904d260164ca0fb
putnam_1964_a4
theorem putnam_1964_a4 (u : β„• β†’ β„€) (boundedu : βˆƒ B T : β„€, βˆ€ n : β„•, B ≀ u n ∧ u n ≀ T) (hu : βˆ€ n β‰₯ 4, u n = ((u (n - 1) + u (n - 2) + u (n - 3) * u (n - 4)) : ℝ) / (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) ∧ (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) β‰  0) : (βˆƒ N c : β„•, c > 0 ∧ βˆ€ n β‰₯ N, u (n + c) = u n) := sorry
e572ebab9b3d44de
putnam_1999_b2
theorem putnam_1999_b2 (P Q : Polynomial β„‚) (hQ : Q.natDegree = 2) (hP : βˆ€ x : β„‚, P.eval x = (Q.eval x) * (Polynomial.derivative^[2] P).eval x) : (βˆƒ x1 x2, x1 β‰  x2 ∧ P.eval x1 = 0 ∧ P.eval x2 = 0) β†’ (βˆƒ f : Fin (P.natDegree) β†’ β„‚, (βˆ€ i j, i β‰  j β†’ f i β‰  f j) ∧ (βˆ€ i, P.eval (f i) = 0)) := sorry
3f25619dab63bc71
putnam_1974_a4
abbrev putnam_1974_a4_solution : β„• β†’ β„š := sorry -- (fun n ↦ (1 : β„š) / (2 ^ (n - 1)) * (n * (n - 1).choose ⌊n / 2βŒ‹β‚Š)) /-- Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$. -/ theorem putnam_1974_a4 (n : β„•) (hn : 0 < n) : (1 : β„š) / (2 ^ (n - 1)) * βˆ‘ k in Finset.Icc 0 ⌊n / 2βŒ‹β‚Š, (n - 2 * k) * (n.choose k) = putnam_1974_a4_solution n := sorry
83f368ab91d21a84
putnam_2003_a1
abbrev putnam_2003_a1_solution : β„• β†’ β„• := sorry -- fun n => n /-- Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, \[ n = a_1 + a_2 + \dots + a_k, \] with $k$ an arbitrary positive integer and $a_1 \leq a_2 \leq \dots \leq a_k \leq a_1 + 1$? For example, with $n = 4$, there are four ways: $4, 2 + 2, 1 + 1 + 2, 1 + 1 + 1 + 1$ -/ theorem putnam_2003_a1 (n : β„•) (hn : n > 0) : Set.encard {a : β„• β†’ β„€ | βˆƒ k > 0, (βˆ‘ i : Fin k, a i = n) ∧ (βˆ€ i : Fin k, a i > 0) ∧ (βˆ€ i : Fin (k - 1), a i ≀ a (i + 1)) ∧ a (k - 1) ≀ a 0 + 1 ∧ (βˆ€ i β‰₯ k, a i = 0)} = putnam_2003_a1_solution n := sorry
ab4b405ebbe55c27
putnam_1977_a2
abbrev putnam_1977_a2_solution : ℝ β†’ ℝ β†’ ℝ β†’ ℝ β†’ Prop := sorry -- fun a b c d ↦ d = a ∧ b = -c ∨ d = b ∧ a = -c ∨ d = c ∧ a = -b /-- Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d}$. -/ theorem putnam_1977_a2 : βˆ€ a b c d : ℝ, putnam_1977_a2_solution a b c d ↔ a β‰  0 β†’ b β‰  0 β†’ c β‰  0 β†’ d β‰  0 β†’ (a + b + c = d ∧ 1 / a + 1 / b + 1 / c = 1 / d) := sorry
33e0539de05c62df
putnam_1967_a2
theorem putnam_1967_a2 (S : β„• β†’ β„€) (hS0 : S 0 = 1) (hSn : βˆ€ n β‰₯ 1, S n = {A : Matrix (Fin n) (Fin n) β„• | (βˆ€ i j, A i j = A j i) ∧ (βˆ€ j, (βˆ‘ i, A i j) = 1)}.ncard) : (βˆ€ n β‰₯ 1, S (n + 1) = S n + n * S (n - 1)) ∧ (βˆ€ x : ℝ, (βˆ‘' n : β„•, S n * (x ^ n / (n)!)) = Real.exp (x + x ^ 2 / 2)) := sorry
e6b864edaccc45b1
putnam_2022_b3
abbrev putnam_2022_b3_solution : Prop := sorry -- True /-- Assign to each positive real number a color, either red or blue. Let $D$ be the set of all distances $d>0$ such that there are two points of the same color at distance $d$ apart. Recolor the positive reals so that the numbers in $D$ are red and the numbers not in $D$ are blue. If we iterate this recoloring process, will we always end up with all the numbers red after a finite number of steps? -/ theorem putnam_2022_b3 (recolor : (ℝ β†’ Prop) β†’ (ℝ β†’ Prop)) (hrecolor : βˆ€ isred d, recolor isred d ↔ βˆƒ p q : ℝ, p > 0 ∧ q > 0 ∧ p < q ∧ isred p = isred q ∧ q - p = d) : (βˆ€ isred, (βˆƒ k, βˆ€ p > 0, (recolor^[k] isred) p)) ↔ putnam_2022_b3_solution := sorry
23831b3af829652a
putnam_1968_b1
theorem putnam_1968_b1 {Ξ© : Type*} [MeasureSpace Ξ©] [IsProbabilityMeasure (β„™ : Measure Ξ©)] (X Y : Ξ© β†’ β„€) (hX : Measurable X) (hY : Measurable Y) (hX' : Set.Finite (X '' Set.univ)) (hY' : Set.Finite (Y '' Set.univ)) (k : β„€) : (β„™ {Ο‰ : Ξ© | min (X Ο‰) (Y Ο‰) = k}).toReal = putnam_1968_b1_solution (β„™ (X⁻¹' {k})).toReal (β„™ (Y⁻¹' {k})).toReal (β„™ {Ο‰ : Ξ© | max (X Ο‰) (Y Ο‰) = k}).toReal := sorry
bb090240111a30b7
putnam_1982_a3
abbrev putnam_1982_a3_solution : ℝ := sorry -- (Real.pi / 2) * log Real.pi /-- Evaluate $\int_0^{\infty} \frac{\tan^{-1}(\pi x) - \tan^{-1} x}{x} \, dx$. -/ theorem putnam_1982_a3 : Tendsto (fun t ↦ ∫ x in (0)..t, (arctan (Real.pi * x) - arctan x) / x) atTop (𝓝 putnam_1982_a3_solution) := sorry
5b093423bfb3ae0c
putnam_1992_a3
abbrev putnam_1992_a3_solution : β„• β†’ Set (β„• Γ— β„• Γ— β„•) := sorry -- fun m ↦ ite (Odd m) βˆ… {(m + 1, 2 ^ (m / 2), 2 ^ (m / 2))} /-- For a given positive integer $m$, find all triples $(n, x, y)$ of positive integers, with $n$ relatively prime to $m$, which satisfy \[ (x^2 + y^2)^m = (xy)^n. \] -/ theorem putnam_1992_a3 (m : β„•) (mpos : m > 0) (S : Set (β„• Γ— β„• Γ— β„•)) (hS : βˆ€ n x y : β„•, (n, x, y) ∈ S ↔ n > 0 ∧ x > 0 ∧ y > 0 ∧ Coprime n m ∧ (x ^ 2 + y ^ 2) ^ m = (x * y) ^ n) : (S = putnam_1992_a3_solution m) := sorry
e1d5bd6cc7f4e84a
putnam_2017_a3
theorem putnam_2017_a3 (a b : ℝ) (f g : ℝ β†’ ℝ) (I : β„• β†’ ℝ) (altb : a < b) (fgcont : ContinuousOn f (Set.Icc a b) ∧ ContinuousOn g (Set.Icc a b)) (fgimg : f '' (Set.Icc a b) βŠ† Set.Ioi 0 ∧ g '' (Set.Icc a b) βŠ† Set.Ioi 0) (fgint : ∫ x in Set.Ioo a b, f x = ∫ x in Set.Ioo a b, g x) (fneg : βˆƒ x : Set.Icc a b, f x β‰  g x) (hI : βˆ€ n > 0, I n = ∫ x in Set.Ioo a b, ((f x) ^ (n + 1)) / ((g x) ^ n)) : (βˆ€ n > 0, I (n + 1) > I n) ∧ Tendsto I atTop atTop := sorry
3af9bcbaa227bb22
putnam_2007_a3
abbrev putnam_2007_a3_solution : β„• β†’ β„š := sorry -- fun k ↦ (k)! * (k + 1)! / ((3 * k + 1) * (2 * k)!) /-- Let $k$ be a positive integer. Suppose that the integers $1, 2, 3, \dots, 3k+1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $3$? Your answer should be in closed form, but may include factorials. -/ theorem putnam_2007_a3 (k : β„•) (kpos : k > 0) (perms : Set (Fin (3 * k + 1) β†’ β„€)) (goodperms : Set (Fin (3 * k + 1) β†’ β„€)) (hgoodperms : goodperms = {f ∈ perms | Β¬βˆƒ j : Fin (3 * k + 1), 3 ∣ βˆ‘ i : Fin (3 * k + 1), ite (i ≀ j) (f i) 0}) (hperms : perms = {f : Fin (3 * k + 1) β†’ β„€ | βˆ€ y ∈ Icc 1 (3 * k + 1), βˆƒ! x : Fin (3 * k + 1), f x = y}) : (goodperms.ncard = perms.ncard * (putnam_2007_a3_solution k)) := sorry
401603179a5464db
putnam_1974_b2
theorem putnam_1974_b2 (y : ℝ β†’ ℝ) (ycontdiff : ContDiff ℝ 1 y) (limy : Tendsto (fun x : ℝ => (deriv y x)^2 + (y x)^3) atTop (𝓝 0)) : Tendsto y atTop (𝓝 0) ∧ Tendsto (deriv y) atTop (𝓝 0) := sorry
84557fb2db81cc21
putnam_1999_a4
abbrev putnam_1999_a4_solution : ℝ := sorry -- 9/32 /-- Sum the series \[\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{m^2 n}{3^m(n3^m+m3^n)}.\] -/ theorem putnam_1999_a4 : Tendsto (fun i => βˆ‘ m in Finset.range i, βˆ‘' n : β„•, (((m + 1)^2*(n+1))/(3^(m + 1) * ((n+1)*3^(m + 1) + (m + 1)*3^(n+1))) : ℝ)) atTop (𝓝 putnam_1999_a4_solution) := sorry
cf3dc16f15c5604d
putnam_1964_b2
theorem putnam_1964_b2 (S : Type*) [Fintype S] [Nonempty S] (P : Finset (Set S)) (hPP : βˆ€ T ∈ P, βˆ€ U ∈ P, T ∩ U β‰  βˆ…) (hPS : Β¬βˆƒ T : Set S, T βˆ‰ P ∧ (βˆ€ U ∈ P, T ∩ U β‰  βˆ…)) : (P.card = 2 ^ (Fintype.card S - 1)) := sorry
93a69211f817ce71
putnam_2021_a3
abbrev putnam_2021_a3_solution : Set β„• := sorry -- {3 * m ^ 2 | m > 0} /-- Determine all positive integers $N$ for which the sphere $x^2+y^2+z^2=N$ has an inscribed regular tetrahedron whose vertices have integer coordinates. -/ theorem putnam_2021_a3 (N : β„•) (Nsphere : Set (EuclideanSpace ℝ (Fin 3))) (hNsphere : Nsphere = {p | (p 0) ^ 2 + (p 1) ^ 2 + (p 2) ^ 2 = (N : ℝ)}) (intcoords : (EuclideanSpace ℝ (Fin 3)) β†’ Prop) (intcoords_def : βˆ€ p, intcoords p ↔ βˆ€ i : Fin 3, p i = round (p i)) : (0 < N ∧ βˆƒ A B C D : EuclideanSpace ℝ (Fin 3), A ∈ Nsphere ∧ B ∈ Nsphere ∧ C ∈ Nsphere ∧ D ∈ Nsphere ∧ intcoords A ∧ intcoords B ∧ intcoords C ∧ intcoords D ∧ (βˆƒ s > 0, dist A B = s ∧ dist A C = s ∧ dist A D = s ∧ dist B C = s ∧ dist B D = s ∧ dist C D = s)) ↔ N ∈ putnam_2021_a3_solution := sorry
8dc16a1abd9a5115
putnam_1981_b3
theorem putnam_1981_b3 (P : β„• β†’ Prop) (hP : βˆ€ n, P n ↔ βˆ€ p : β„•, (Nat.Prime p ∧ p ∣ n^2 + 3) β†’ βˆƒ k : β„•, (p : β„€) ∣ (k : β„€)^2 + 3 ∧ k^2 < n) : βˆ€ n : β„•, βˆƒ m : β„•, (m : β„€) > n ∧ P m := sorry
69455b04ba7935f9
putnam_2014_b3
theorem putnam_2014_b3 (m n : β„•) (A : Matrix (Fin m) (Fin n) β„š) (mnpos : 0 < m ∧ 0 < n) (Aprime : {p : β„• | p.Prime ∧ βˆƒ (i : Fin m) (j : Fin n), |A i j| = p}.encard β‰₯ m + n) : A.rank β‰₯ 2 := sorry
09ad2f6350fe0c39
putnam_1963_b6
theorem putnam_1963_b6 (d : β„•) (S : Set (Fin d β†’ ℝ) β†’ Set (Fin d β†’ ℝ)) (hS : S = fun A : Set (Fin d β†’ ℝ) => ⋃ p ∈ A, ⋃ q ∈ A, segment ℝ p q) (A : β„• β†’ Set (Fin d β†’ ℝ)) (ddim : 1 ≀ d ∧ d ≀ 3) (hA0 : Nonempty (A 0)) (hAn : βˆ€ n β‰₯ 1, A n = S (A (n - 1))) : βˆ€ n β‰₯ 2, A n = A (n + 1) := sorry
0efae5679cacf7cf
putnam_2007_b5
theorem putnam_2007_b5 (k : β„•) (kpos : k > 0) : (βˆƒ P : Finset.range k β†’ Polynomial ℝ, βˆ€ n : β„€, ⌊(n : ℝ) / kβŒ‹ ^ k = βˆ‘ i : Finset.range k, (P i).eval (n : ℝ) * ⌊(n : ℝ) / kβŒ‹ ^ (i : β„•)) := sorry
f8c04b94868a178e
putnam_2018_a6
theorem putnam_2018_a6 (A B C D : EuclideanSpace ℝ (Fin 2)) (PPprops : (EuclideanSpace ℝ (Fin 2)) β†’ (EuclideanSpace ℝ (Fin 2)) β†’ Prop) (hPPprops : βˆ€ P1 P2, PPprops P1 P2 ↔ P1 β‰  P2 ∧ (βˆƒ q : β„š, (dist P1 P2) ^ 2 = q)) (ABCDnoline : Β¬Collinear ℝ {A, B, C} ∧ Β¬Collinear ℝ {A, B, D} ∧ Β¬Collinear ℝ {A, C, D} ∧ Β¬Collinear ℝ {B, C, D}) (ABCDsqrrat : PPprops A B ∧ PPprops A C ∧ PPprops A D ∧ PPprops B C ∧ PPprops B D ∧ PPprops C D) : βˆƒ q : β„š, (MeasureTheory.volume (convexHull ℝ {A, B, C}) / MeasureTheory.volume (convexHull ℝ {A, B, D})).toReal = q := sorry
d37e57c80fc12cb7
putnam_2008_a6
theorem putnam_2008_a6 : βˆƒ c : ℝ, c > 0 ∧ βˆ€ G [Group G] (fing : Fintype G), fing.card > 1 β†’ βˆƒ s : List G, s.length ≀ c * Real.log (fing.card : ℝ) ∧ βˆ€ g : G, βˆƒ t : List G, t.Sublist s ∧ t.prod = g := sorry
3dd246f25e079383
putnam_1992_b5
abbrev putnam_1992_b5_solution : Prop := sorry -- False /-- Let $D_n$ denote the value of the $(n-1) \times (n-1)$ determinant \[ \left[ \begin{array}{cccccc} 3 & 1 & 1 & 1 & \cdots & 1 \\ 1 & 4 & 1 & 1 & \cdots & 1 \\ 1 & 1 & 5 & 1 & \cdots & 1 \\ 1 & 1 & 1 & 6 & \cdots & 1 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & 1 & \cdots & n+1 \end{array} \right]. \] Is the set $\left\{ \frac{D_n}{n!} \right\}_{n \geq 2}$ bounded? -/ theorem putnam_1992_b5 (D : β„• β†’ β„š) (hD : βˆ€ n, D n = Matrix.det (fun i j : Fin (n - 1) ↦ ite (i = j) ((i : β„•) + 3 : β„š) 1)) : putnam_1992_b5_solution ↔ (Bornology.IsBounded {x | βˆƒ n β‰₯ 2, D n / factorial n = x}) := sorry
f3eb7c554c502eb1
putnam_1982_b5
theorem putnam_1982_b5 (T : Set ℝ) (hT : T = Ioi (Real.exp (Real.exp 1))) (S : ℝ β†’ β„• β†’ ℝ) (hS : βˆ€ x ∈ T, S x 0 = (Real.exp 1) ∧ βˆ€ n : β„•, S x (n + 1) = Real.logb (S x n) x) (g : ℝ β†’ ℝ) : βˆ€ x ∈ T, (βˆƒ L : ℝ, Tendsto (S x) atTop (𝓝 L)) ∧ (βˆ€ x ∈ T, Tendsto (S x) atTop (𝓝 (g x))) β†’ ContinuousOn g T := sorry
f54a48576fb94620
putnam_1995_b1
theorem putnam_1995_b1 (Pi : Finpartition (Finset.range 9) β†’ (Finset.range 9) β†’ β„•) (Pi_def : βˆ€ partition k, Pi partition k = (Exists.choose (Finpartition.exists_mem partition k.2)).card) : βˆ€ Pt1 Pt2, βˆƒ x y : Finset.range 9, x β‰  y ∧ Pi Pt1 x = Pi Pt1 y ∧ Pi Pt2 x = Pi Pt2 y := sorry
97bc2d5e556b2325
putnam_1985_b1
abbrev putnam_1985_b1_solution : Fin 5 β†’ β„€ := sorry -- fun i ↦ i - 2 /-- Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial \[ p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5) \] has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved. -/ theorem putnam_1985_b1 (p : (Fin 5 β†’ β„€) β†’ (Polynomial ℝ)) (hp : p = fun m ↦ ∏ i : Fin 5, ((X : Polynomial ℝ) - m i)) (numnzcoeff : Polynomial ℝ β†’ β„•) (hnumnzcoeff : numnzcoeff = fun p ↦ {j ∈ Finset.range (p.natDegree + 1) | coeff p j β‰  0}.card) : (Injective putnam_1985_b1_solution ∧ βˆ€ m : Fin 5 β†’ β„€, Injective m β†’ numnzcoeff (p putnam_1985_b1_solution) ≀ numnzcoeff (p m)) := sorry
1cd2a135e3317ce2
putnam_2000_b1
theorem putnam_2000_b1 (N : β„•) (a b c : Fin N β†’ β„€) (Nge1 : N β‰₯ 1) (hodd : βˆ€ j : Fin N, Odd (a j) ∨ Odd (b j) ∨ Odd (c j)) : (βˆƒ r s t : β„€, {j : Fin N | Odd (r * a j + s * b j + t * c j)}.ncard β‰₯ (4 * N : ℝ) / 7) := sorry
d682645997fe5e07
putnam_2010_b1
abbrev putnam_2010_b1_solution : Prop := sorry -- False /-- Is there an infinite sequence of real numbers $a_1, a_2, a_3, \dots$ such that \[ a_1^m + a_2^m + a_3^m + \cdots = m \] for every positive integer $m$? -/ theorem putnam_2010_b1 : (βˆƒ a : β„• β†’ ℝ, βˆ€ m : β„•, m > 0 β†’ βˆ‘' i : β„•, (a i)^m = m) ↔ putnam_2010_b1_solution := sorry
6f698b814fd07be3
putnam_2022_a5
abbrev putnam_2022_a5_solution : β„• := sorry -- 290 /-- Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays? -/ theorem putnam_2022_a5 (IsValidMove : Set (Fin 2022) β†’ Set (Fin 2022) β†’ Prop) (IsValidMove_def : βˆ€ x y, IsValidMove x y ↔ (x = y ∧ βˆ€ i < 2021, i βˆ‰ x β†’ i + 1 ∈ x) ∨ βˆƒ i < 2021, i βˆ‰ x ∧ i + 1 βˆ‰ x ∧ y = x βˆͺ {i, i + 1}) (IsValidGame : List (Set (Fin 2022)) β†’ Prop) (IsValidGame_def : βˆ€ g, IsValidGame g ↔ (βˆƒ gt, g = βˆ… :: gt) ∧ g.Chain' IsValidMove) (ConformsToStrategy : List (Set (Fin 2022)) β†’ (Set (Fin 2022) β†’ Set (Fin 2022)) β†’ Prop) (ConformsToStrategy_def : βˆ€ g s, ConformsToStrategy g s ↔ βˆ€ (i) (h : i + 1 < g.length), Even i β†’ g[i + 1] = s g[i]) : IsGreatest {n | βˆƒ s, (βˆ€ x, IsValidMove x (s x)) ∧ βˆ€ g, IsValidGame g β†’ ConformsToStrategy g s β†’ βˆƒ gh x, g = gh ++ [x] ∧ n ≀ xᢜ.ncard} putnam_2022_a5_solution := sorry
801c2afeea8551e0
putnam_1967_b4
theorem putnam_1967_b4 (n : β„•) (lockers : β„• β†’ Set.Icc 1 n β†’ Bool) (npos : n β‰₯ 1) (hlockers0 : βˆ€ i : Set.Icc 1 n, lockers 0 i = false) (hlockersk : βˆ€ k ∈ Set.Icc 1 n, βˆ€ i : Set.Icc 1 n, lockers k i = if k ∣ i then !(lockers (k - 1) i) else (lockers (k - 1) i)) : βˆ€ i : Set.Icc 1 n, lockers n i ↔ (βˆƒ j : β„€, j ^ 2 = i) := sorry
387764c8bc3ec09d
putnam_1983_b4
theorem putnam_1983_b4 (f : β„• β†’ β„€) (a : β„• β†’ β„•) (hf : f = fun (n : β„•) ↦ n + Int.floor (√n)) (ha0 : a 0 > 0) (han : βˆ€ n : β„•, a (n + 1) = f (a n)) : (βˆƒ i : β„•, βˆƒ s : β„€, a i = s ^ 2) := sorry
7425198cbd334f35
putnam_1993_b4
theorem putnam_1993_b4 (K : ℝ Γ— ℝ β†’ ℝ) (f g : ℝ β†’ ℝ) (Kpos : βˆ€ x y : Set.Icc (0 : ℝ) 1, K (x, y) > 0) (Kcont : ContinuousOn K {(x, y) : ℝ Γ— ℝ | x ∈ Set.Icc 0 1 ∧ y ∈ Set.Icc 0 1}) (fgpos : βˆ€ 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 : βˆ€ x : Set.Icc (0 : ℝ) 1, (∫ y in Set.Ioo 0 1, f y * K (x, y)) = g x ∧ (∫ y in Set.Ioo 0 1, g y * K (x, y)) = f x) : βˆ€ x : Set.Icc (0 : ℝ) 1, f x = g x := sorry
99f34e09d17013ef
putnam_1971_b1
theorem putnam_1971_b1 (S : Type*) [Mul S] (hself : βˆ€ x : S, x * x = x) (h2 : βˆ€ x y z : S, (x * y) * z = (y * z) * x) : (βˆ€ x y z : S, (x * y) * z = x * (y * z)) ∧ (βˆ€ x y : S, x * y = y * x) := sorry
922a95598124c41a
putnam_2016_b4
abbrev putnam_2016_b4_solution : β„• β†’ ℝ := sorry -- (fun n : β„• => (2 * n)! / (4 ^ n * (n)!)) /-- Let $A$ be a $2n \times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be $0$ or $1$, each with probability $1/2$. Find the expected value of $\det(A-A^t)$ (as a function of $n$), where $A^t$ is the transpose of $A$. -/ theorem putnam_2016_b4 (n : β„•) (npos : n β‰₯ 1) (mats01 : Set (Matrix (Fin (2 * n)) (Fin (2 * n)) ℝ)) (hmats01 : mats01 = {A | βˆ€ i j : Fin (2 * n), A i j = 0 ∨ A i j = 1}) : (βˆ‘' A : mats01, (A.1 - (Matrix.transpose A)).det) / mats01.ncard = putnam_2016_b4_solution n := sorry
e6e5a8688794d903
putnam_2006_b4
abbrev putnam_2006_b4_solution : β„• β†’ β„• := sorry -- fun k ↦ 2 ^ k /-- Let $Z$ denote the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^n$.) Given a vector subspace $V$ of $\mathbb{R}^n$, let $Z(V)$ denote the number of members of $Z$ that lie in $V$. Let $k$ be given, $0 \leq k \leq n$. Find the maximum, over all vector subspaces $V \subseteq \mathbb{R}^n$ of dimension $k$, of the number of points in $V \cap Z$. -/ theorem putnam_2006_b4 (n k : β„•) (npos : n > 0) (hk : k ≀ n) (Z : Set (Fin n β†’ ℝ)) (hZ : Z = {P : Fin n β†’ ℝ | βˆ€ j : Fin n, P j = 0 ∨ P j = 1}) : IsGreatest {y | βˆƒ V : Subspace ℝ (Fin n β†’ ℝ), Module.rank ℝ V = k ∧ (Z ∩ V).ncard = y} (putnam_2006_b4_solution k) := sorry
dbe94be1a48a57ac
putnam_1976_b5
abbrev putnam_1976_b5_solution : β„• β†’ Polynomial β„€ := sorry -- fun n => C (Nat.factorial n) /-- Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$ -/ theorem putnam_1976_b5 : βˆ€ n : β„•, βˆ‘ k in Finset.range (n + 1), C ((-(1 : β„€))^k * Nat.choose n k) * (X - (C (k : β„€)))^n = putnam_1976_b5_solution n := sorry
a82bb6ac6dc0eb89
putnam_1966_b5
theorem putnam_1966_b5 (S : Finset (EuclideanSpace ℝ (Fin 2))) (hcard : S.card β‰₯ 3) (hS : βˆ€ s βŠ† S, s.card = 3 β†’ Β¬Collinear ℝ s.toSet) : βˆƒ L : ZMod S.card β†’ (EuclideanSpace ℝ (Fin 2)), (βˆ€ p ∈ S, βˆƒ! i : ZMod S.card, p = L i) ∧ βˆ€ i j : ZMod S.card, i β‰  j β†’ (βˆ€ I : EuclideanSpace ℝ (Fin 2), (I ∈ segment ℝ (L i) (L (i + 1)) ∧ I ∈ segment ℝ (L j) (L (j + 1))) β†’ I = L i ∨ I = L (i + 1) ∨ I = L j ∨ I = L (j + 1)) := sorry
ed5d5598b7362eb1
putnam_2023_a4
theorem putnam_2023_a4 (v : Fin 12 β†’ EuclideanSpace ℝ (Fin 3)) (hv : letI Ο† : ℝ := (1 + √5) / 2 letI e : (Fin 3 β†’ ℝ) ≃ EuclideanSpace ℝ (Fin 3) := (WithLp.equiv _ _).symm letI s := √(1 + Ο† ^ 2) βˆƒ g : EuclideanSpace ℝ (Fin 3) ≃ₗᡒ[ℝ] EuclideanSpace ℝ (Fin 3), g ∘ v = s⁻¹ β€’ e ∘ ![![1, Ο†, 0], ![-1, Ο†, 0], ![ 1, -Ο†, 0], ![-1, -Ο†, 0], ![Ο†, 0, 1], ![ Ο†, 0, -1], ![-Ο†, 0, 1], ![-Ο†, 0, -1], ![0, 1, Ο†], ![ 0, -1, Ο†], ![ 0, 1, -Ο†], ![ 0, -1, -Ο†]]) (w : EuclideanSpace ℝ (Fin 3)) (Ξ΅ : ℝ) (hΞ΅ : Ξ΅ > 0) : βˆƒ a : Fin 12 β†’ β„€, β€–βˆ‘ i, a i β€’ v i - wβ€– < Ξ΅ := sorry
2e9a705d5040e3bb
putnam_1969_a6
theorem putnam_1969_a6 (x : β„• β†’ ℝ) (y : β„• β†’ ℝ) (hy1 : βˆ€ n β‰₯ 2, y n = x (n-1) + 2 * (x n)) (hy2 : βˆƒ c : ℝ, Tendsto y atTop (𝓝 c)) : βˆƒ C : ℝ, Tendsto x atTop (𝓝 C) := sorry
8eb86142cbe439a0
putnam_1979_a6
theorem putnam_1979_a6 (n : β„•) (p : β„• β†’ ℝ) (hp : βˆ€ i ∈ Finset.range n, p i ∈ Icc 0 1) : βˆƒ x ∈ Icc 0 1, (βˆ€ i ∈ Finset.range n, x β‰  p i) ∧ βˆ‘ i in Finset.range n, 1/|x - p i| ≀ 8*n*βˆ‘ i in Finset.range n, (1 : ℝ)/(2*i + 1) := sorry
bf182b3f0c78cd2b
putnam_1987_b6
theorem putnam_1987_b6 (p : β„•) (F : Type*) [Field F] [Fintype F] (S : Set F) (hp : Odd p ∧ Nat.Prime p) (Fcard : Fintype.card F = p ^ 2) (Snz : βˆ€ x ∈ S, x β‰  0) (Scard : S.ncard = ((p : β„€) ^ 2 - 1) / 2) (hS : βˆ€ a : F, a β‰  0 β†’ Xor' (a ∈ S) (-a ∈ S)) : (Even ((S ∩ {x | βˆƒ a ∈ S, x = 2 * a}).ncard)) := sorry
0ab58e1c5ffe2a39
putnam_2020_a2
abbrev putnam_2020_a2_solution : β„• β†’ β„• := sorry -- fun k ↦ 4 ^ k /-- Let $k$ be a nonnegative integer. Evaluate \[ \sum_{j=0}^k 2^{k-j} \binom{k+j}{j}. \] -/ theorem putnam_2020_a2 (k : β„•) : (βˆ‘ j in Finset.Icc 0 k, 2 ^ (k - j) * Nat.choose (k + j) j = putnam_2020_a2_solution k) := sorry
1ef22069305e95b4
putnam_1988_a5
theorem putnam_1988_a5 : (βˆƒ f : ℝ β†’ ℝ, (βˆ€ x > 0, f (f x) = 6 * x - f x ∧ f x > 0) ∧ (βˆ€ g : ℝ β†’ ℝ, (βˆ€ x > 0, g (g x) = 6 * x - g x ∧ g x > 0) β†’ (βˆ€ x > 0, f x = g x))) := sorry
3458e2dd6ece982e
putnam_1965_b3
theorem putnam_1965_b3 : {(a, b, c) : β„€ Γ— β„€ Γ— β„€ | a > 0 ∧ a ≀ b ∧ c > 0 ∧ a^2 + b^2 = c^2 ∧ a*b/(2 : β„š) = 2*(a + b + c)}.ncard = 3 := sorry
be013e20faf74bd4
putnam_1998_a5
theorem putnam_1998_a5 (k : β„•) (c : Fin k β†’ (EuclideanSpace ℝ (Fin 2))) (r : Fin k β†’ ℝ) (hr : βˆ€ i, r i > 0) (E : Set (EuclideanSpace ℝ (Fin 2))) (hE : E βŠ† ⋃ i, ball (c i) (r i)) : βˆƒ (n : β„•) (t : Fin n β†’ Fin k), (βˆ€ i j, i β‰  j β†’ (ball (c (t i)) (r (t i)) ∩ ball (c (t j)) (r (t j)) = βˆ…)) ∧ E βŠ† ⋃ i : Fin n, ball (c (t i)) (3 * (r (t i))) := sorry
3df95b52b64d12ec
putnam_1975_b3
abbrev putnam_1975_b3_solution : β„• β†’ ℝ := sorry -- fun k : β„• => 1/(Nat.factorial k) /-- Let $s_k (a_1, a_2, \dots, a_n)$ denote the $k$-th elementary symmetric function; that is, the sum of all $k$-fold products of the $a_i$. For example, $s_1 (a_1, \dots, a_n) = \sum_{i=1}^{n} a_i$, and $s_2 (a_1, a_2, a_3) = a_1a_2 + a_2a_3 + a_1a_3$. Find the supremum $M_k$ (which is never attained) of $$\frac{s_k (a_1, a_2, \dots, a_n)}{(s_1 (a_1, a_2, \dots, a_n))^k}$$ across all $n$-tuples $(a_1, a_2, \dots, a_n)$ of positive real numbers with $n \ge k$. -/ theorem putnam_1975_b3 : βˆ€ k : β„•, k > 0 β†’ (βˆ€ a : Multiset ℝ, (βˆ€ i ∈ a, i > 0) ∧ card a β‰₯ k β†’ (esymm a k)/(esymm a 1)^k ≀ putnam_1975_b3_solution k) ∧ βˆ€ M : ℝ, M < putnam_1975_b3_solution k β†’ (βˆƒ a : Multiset ℝ, (βˆ€ i ∈ a, i > 0) ∧ card a β‰₯ k ∧ (esymm a k)/(esymm a 1)^k > M) := sorry
f9d18916e89daae1
putnam_2002_b6
theorem putnam_2002_b6 (p : β„•) (hp : Nat.Prime p) (M : Matrix (Fin 3) (Fin 3) (MvPolynomial (Fin 3) β„€)) (hM : M = fun (r c : Fin 3) => ((X c)^(p^(r : β„•)) : MvPolynomial (Fin 3) β„€)) (cong : β„• β†’ MvPolynomial (Fin 3) β„€ Γ— MvPolynomial (Fin 3) β„€ β†’ Prop) (hcong : cong = fun p : β„• => fun (f, g) => βˆ€ n : Fin 3 β†’β‚€ β„•, Int.ModEq p (f.coeff n) (g.coeff n)) : βˆƒ S : Finset (MvPolynomial (Fin 3) β„€), cong p ((det M), (∏ s in S, s)) ∧ βˆ€ s ∈ S, (βˆƒ a b c : β„€, s = (C a)*(X 0) + (C b)*(X 1) + (C c)*(X 2)) := sorry
4a84e27500e71e6b
putnam_2012_b6
theorem putnam_2012_b6 (p : β„•) (hpodd : Odd p) (hpprime : Nat.Prime p) (hpmod3 : p ≑ 2 [MOD 3]) (P : Equiv.Perm (Fin p)) (hP : βˆ€ i : Fin p, P i = (i * i * i)) : Equiv.Perm.signAux P = 1 ↔ (p ≑ 3 [MOD 4]) := sorry
b7edc025afdf52f8
putnam_2005_b2
abbrev putnam_2005_b2_solution : Set (β„• Γ— (β„• β†’ β„€)) := sorry -- {(n, k) : β„• Γ— (β„• β†’ β„€) | (n = 1 ∧ k 0 = 1) ∨ (n = 3 ∧ (k '' {0, 1, 2} = {2, 3, 6})) ∨ (n = 4 ∧ (βˆ€ i : Fin 4, k i = 4))} /-- Find all positive integers $n,k_1,\dots,k_n$ such that $k_1+\cdots+k_n=5n-4$ and $\frac{1}{k_1}+\cdots+\frac{1}{k_n}=1$. -/ theorem putnam_2005_b2 : {((n : β„•), (k : β„• β†’ β„€)) | (n > 0) ∧ (βˆ€ i ∈ Finset.range n, k i > 0) ∧ (βˆ‘ i in Finset.range n, k i = 5 * n - 4) ∧ (βˆ‘ i : Finset.range n, (1 : ℝ) / (k i) = 1)} = putnam_2005_b2_solution := sorry
541be56cee7e74bd
putnam_2015_b2
abbrev putnam_2015_b2_solution : Prop := sorry -- True /-- Given a list of the positive integers $1,2,3,4,\dots$, take the first three numbers $1,2,3$ and their sum $6$ and cross all four numbers off the list. Repeat with the three smallest remaining numbers $4,5,7$ and their sum $16$. Continue in this way, crossing off the three smallest remaining numbers and their sum, and consider the sequence of sums produced: $6,16,27,36,\dots$. Prove or disprove that there is some number in the sequence whose base $10$ representation ends with $2015$. -/ theorem putnam_2015_b2 (sets : β„• β†’ Set β„•) (Smin3 : Set β„• β†’ (Fin 3 β†’ β„•)) (sums : β„• β†’ β„•) (hsets0 : sets 0 = Set.Ici 1) (hmin3 : βˆ€ S : Set β„•, S.encard β‰₯ 3 β†’ ((Smin3 S) 0 = sInf S ∧ (Smin3 S) 1 = sInf (S \ {(Smin3 S) 0}) ∧ (Smin3 S) 2 = sInf (S \ {(Smin3 S) 0, (Smin3 S) 1}))) (hsums : βˆ€ n : β„•, sums n = (Smin3 (sets n)) 0 + (Smin3 (sets n)) 1 + (Smin3 (sets n)) 2) (hsetsn : βˆ€ n : β„•, sets (n + 1) = sets n \ {(Smin3 (sets n)) 0, (Smin3 (sets n)) 1, (Smin3 (sets n)) 2, sums n}) : (βˆƒ n : β„•, List.IsPrefix [5, 1, 0, 2] (Nat.digits 10 (sums n))) ↔ putnam_2015_b2_solution := sorry
d864ebc2a87941c3
putnam_1990_b2
theorem putnam_1990_b2 (x z : ℝ) (P : β„• β†’ ℝ) (xlt1 : |x| < 1) (zgt1 : |z| > 1) (hP : βˆ€ j β‰₯ 1, P j = (∏ i : Fin j, (1 - z * x ^ (i : β„•))) / (∏ i : Set.Icc 1 j, (z - x ^ (i : β„•)))) : 1 + (βˆ‘' j : Set.Ici 1, (1 + x ^ (j : β„•)) * P j) = 0 := sorry
56940edcec029257
putnam_1990_b3
theorem putnam_1990_b3 (S : Set (Matrix (Fin 2) (Fin 2) β„•)) (hS : βˆ€ A ∈ S, βˆ€ i j : Fin 2, (βˆƒ x : β„€, A i j = x ^ 2) ∧ A i j ≀ 200) : (S.encard > 50387) β†’ (βˆƒ A ∈ S, βˆƒ B ∈ S, A β‰  B ∧ A * B = B * A) := sorry
6107e065fee0cda4
putnam_1980_b3
abbrev putnam_1980_b3_solution : Set ℝ := sorry -- {a : ℝ | a β‰₯ 3} /-- 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.) -/ theorem putnam_1980_b3 (a : ℝ) (u : β„• β†’ ℝ) (hu : u 0 = a ∧ (βˆ€ n : β„•, u (n + 1) = 2 * u n - n ^ 2)) : (βˆ€ n : β„•, u n > 0) ↔ a ∈ putnam_1980_b3_solution := sorry
e87144efaee76557
putnam_1962_b6
theorem putnam_1962_b6 (n : β„•) (a b : β„• β†’ ℝ) (xs : Set ℝ) (f : ℝ β†’ ℝ) (hf : f = fun x : ℝ => βˆ‘ k in Finset.Icc 0 n, ((a k) * Real.sin (k * x) + (b k) * Real.cos (k * x))) (hf1 : βˆ€ x ∈ Set.Icc 0 (2 * Ο€), |f x| ≀ 1) (hxs : xs.ncard = 2 * n ∧ xs βŠ† Set.Ico 0 (2 * Ο€)) (hfxs : βˆ€ x ∈ xs, |f x| = 1) : (Β¬βˆƒ c : ℝ, f = fun x : ℝ => c) β†’ βˆƒ a : ℝ, f = fun x : ℝ => Real.cos (n * x + a) := sorry
60ddd286fd96ba25
putnam_1972_b6
theorem putnam_1972_b6 (k : β„•) (hk : k β‰₯ 1) (n : Fin k β†’ β„€) (hn : βˆ€ i : Fin k, n i > 0) (hn' : βˆ€ i j : Fin k, i < j β†’ n i < n j) (zpoly : β„‚ β†’ β„‚) (hzpoly : zpoly = fun z => 1 + βˆ‘ i : Fin k, z^(n i)) : βˆ€ z : β„‚, z ∈ ball 0 ((Real.sqrt 5 - 1)/2) β†’ zpoly z β‰  0 := sorry
943d2293ffaabc84
putnam_2005_b3
abbrev putnam_2005_b3_solution : Set (ℝ β†’ ℝ) := sorry -- {f : ℝ β†’ ℝ | βˆƒα΅‰ (c > 0) (d > (0 : ℝ)), (d = 1 β†’ c = 1) ∧ (Ioi 0).EqOn f (fun x ↦ c * x ^ d)} /-- Find all differentiable functions $f:(0,\infty) \to (0,\infty)$ for which there is a positive real number $a$ such that $f'(\frac{a}{x})=\frac{x}{f(x)}$ for all $x>0$. -/ theorem putnam_2005_b3 (f : ℝ β†’ ℝ) (hf : βˆ€ x > 0, 0 < f x) (hf' : DifferentiableOn ℝ f (Ioi 0)) : (βˆƒ a > 0, βˆ€ x > 0, deriv f (a / x) = x / f x) ↔ f ∈ putnam_2005_b3_solution := sorry
f82a45083cc0a5ec
putnam_2015_b3
abbrev putnam_2015_b3_solution : Set (Matrix (Fin 2) (Fin 2) ℝ) := sorry -- {A : Matrix (Fin 2) (Fin 2) ℝ | (βˆƒ Ξ± : ℝ, βˆ€ i j : Fin 2, A i j = Ξ± * 1) ∨ (βˆƒ Ξ² : ℝ, A 0 0 = Ξ² * -3 ∧ A 0 1 = Ξ² * -1 ∧ A 1 0 = Ξ² * 1 ∧ A 1 1 = Ξ² * 3)} /-- 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 $S$. -/ theorem putnam_2015_b3 (M : Matrix (Fin 2) (Fin 2) ℝ) (S : Set (Matrix (Fin 2) (Fin 2) ℝ)) (hS : S = {M' | (M' 0 1 - M' 0 0 = M' 1 0 - M' 0 1) ∧ (M' 1 0 - M' 0 1 = M' 1 1 - M' 1 0)}) : (M ∈ S ∧ (βˆƒ k > 1, M ^ k ∈ S)) ↔ M ∈ putnam_2015_b3_solution := sorry
3784e91d14e96e27
putnam_2020_a3
abbrev putnam_2020_a3_solution : Prop := sorry -- False /-- Let $a_0 = \pi/2$, and let $a_n = \sin(a_{n-1})$ for $n \geq 1$. Determine whether \[ \sum_{n=1}^\infty a_n^2 \] converges. -/ theorem putnam_2020_a3 (a : β„• β†’ ℝ) (ha0 : a 0 = Real.pi / 2) (ha : βˆ€ n, a (n+1) = Real.sin (a n)) : (βˆƒ L, Tendsto (fun m : β„• => βˆ‘ n in Finset.Icc 1 m, (a n)^2) atTop (𝓝 L)) ↔ putnam_2020_a3_solution := sorry
0ce6b920dbf938a6
putnam_1988_a4
abbrev putnam_1988_a4_solution : Prop Γ— Prop := sorry -- (True, False) /-- \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} -/ theorem putnam_1988_a4 (p : β„• β†’ Prop) (hp : βˆ€ n, p n ↔ βˆ€ color : (EuclideanSpace ℝ (Fin 2)) β†’ Fin n, βˆƒ p q : EuclideanSpace ℝ (Fin 2), color p = color q ∧ dist p q = 1) : (let (a, b) := putnam_1988_a4_solution; (p 3 ↔ a) ∧ (p 9 ↔ b)) := sorry
4b364f7bd5af11f0
putnam_1965_b2
theorem putnam_1965_b2 (n : β„•) (hn : n > 1) (won : Fin n β†’ Fin n β†’ Bool) (hirrefl : βˆ€ i : Fin n, won i i = false) (hantisymm : βˆ€ i j : Fin n, i β‰  j β†’ won i j = Β¬won j i) (w l : Fin n β†’ β„€) (hw : w = fun r : Fin n => βˆ‘ j : Fin n, (if won r j then 1 else 0)) (hl : l = fun r : Fin n => n - 1 - w r) : βˆ‘ r : Fin n, (w r)^2 = βˆ‘ r : Fin n, (l r)^2 := sorry
e84e8f7f87a75127
putnam_1998_a4
abbrev putnam_1998_a4_solution : Set β„• := sorry -- {n | n ≑ 1 [MOD 6]} /-- Let $A_1=0$ and $A_2=1$. For $n>2$, the number $A_n$ is defined by concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from left to right. For example $A_3=A_2 A_1=10$, $A_4=A_3 A_2 = 101$, $A_5=A_4 A_3 = 10110$, and so forth. Determine all $n$ such that $11$ divides $A_n$. -/ theorem putnam_1998_a4 (A : β„• β†’ List β„•) (hA1 : A 1 = [0]) (hA2 : A 2 = [1]) (hA : βˆ€ n > 0, A (n + 2) = A (n + 1) ++ A n) : {n | 1 ≀ n ∧ 11 ∣ Nat.ofDigits 10 (A n).reverse} = putnam_1998_a4_solution := sorry
949d86cd8f357b64
putnam_1975_b2
theorem putnam_1975_b2 (slab : (Fin 3 β†’ ℝ) β†’ ℝ β†’ ℝ β†’ Set (Fin 3 β†’ ℝ)) (hslab : slab = fun normal offset thickness => {x : Fin 3 β†’ ℝ | offset < normal ⬝α΅₯ x ∧ normal ⬝α΅₯ x < offset + thickness}) (normals : β„• β†’ (Fin 3 β†’ ℝ)) (offsets : β„• β†’ ℝ) (thicknesses : β„• β†’ ℝ) (hnormalsunit : βˆ€ i : β„•, β€–normals iβ€– = 1) (hthicknessespos : βˆ€ i : β„•, thicknesses i > 0) (hthicknessesconv : βˆƒ C : ℝ, Tendsto (fun i : β„• => βˆ‘ j in Finset.range i, thicknesses j) atTop (𝓝 C)) : Set.univ β‰  ⋃ i : β„•, slab (normals i) (offsets i) (thicknesses i) := sorry
163367837a1b1d37
putnam_1984_b1
abbrev putnam_1984_b1_solution : Polynomial ℝ Γ— Polynomial ℝ := sorry -- (Polynomial.X + 3, -Polynomial.X - 2) /-- 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 $n \geq 1$. -/ theorem putnam_1984_b1 (f : β„• β†’ β„€) (hf : βˆ€ n > 0, f n = βˆ‘ i : Set.Icc 1 n, ((i)! : β„€)) : let (P, Q) := putnam_1984_b1_solution; βˆ€ n β‰₯ 1, f (n + 2) = P.eval (n : ℝ) * f (n + 1) + Q.eval (n : ℝ) * f n := sorry
1e4584208b10a382
putnam_1994_b1
abbrev putnam_1994_b1_solution : Set β„€ := sorry -- {n : β„€ | (315 ≀ n ∧ n ≀ 325) ∨ (332 ≀ n ∧ n ≀ 350)} /-- Find all positive integers $n$ that are within $250$ of exactly $15$ perfect squares. -/ theorem putnam_1994_b1 (n : β„€) : n ∈ putnam_1994_b1_solution ↔ (0 < n ∧ {m : β„• | |n - m ^ 2| ≀ 250}.encard = 15) := sorry
c66416b6c99cab31
putnam_1966_b4
theorem putnam_1966_b4 (m n : β„•) (S : Finset β„•) (hS : (βˆ€ i ∈ S, i > 0) ∧ S.card = m * n + 1) : βˆƒ T βŠ† S, (T.card = m + 1 ∧ βˆ€ j ∈ T, βˆ€ i ∈ T, i β‰  j β†’ Β¬(j ∣ i)) ∨ (T.card = n + 1 ∧ βˆ€ i ∈ T, βˆ€ j ∈ T, j < i β†’ j ∣ i) := sorry
4f2bef71963f2ff8
putnam_2023_a5
abbrev putnam_2023_a5_solution : Set β„‚ := sorry -- {-(3^1010 - 1)/2, -(3^1010 - 1)/2 + Real.sqrt (9^1010 - 1) * Complex.I/4, -(3^1010 - 1)/2 - Real.sqrt (9^1010 - 1) * Complex.I/4} /-- For a nonnegative integer $k$, let $f(k)$ be the number of ones in the base 3 representation of $k$. Find all complex numbers $z$ such that \[ \sum_{k=0}^{3^{1010}-1} (-2)^{f(k)} (z+k)^{2023} = 0. \] -/ theorem putnam_2023_a5 : {z : β„‚ | βˆ‘ k in Finset.Icc 0 (3^1010 - 1), (-2)^(num_ones (digits 3 k)) * (z + k)^2023 = 0} = putnam_2023_a5_solution := sorry
1f2ca5db45acbbc5
putnam_2011_b1
theorem putnam_2011_b1 (h k : β„€) (hkpos : h > 0 ∧ k > 0) : βˆ€ Ξ΅ > 0, βˆƒ m n : β„€, m > 0 ∧ n > 0 ∧ Ξ΅ < |h * Real.sqrt m - k * Real.sqrt n| ∧ |h * Real.sqrt m - k * Real.sqrt n| < 2 * Ξ΅ := sorry
66815d046dd27714
putnam_2001_b1
theorem putnam_2001_b1 (n : β„•) (hn : n > 0) (hn' : Even n) (nums : Fin n β†’ Fin n β†’ β„€) (colors : Fin n β†’ Fin n β†’ Fin 2) (hnums : βˆ€ k l, nums k l = k * n + l + 1) (hcolorsrows : βˆ€ k, (βˆ‘ l, (if (colors k l = 0) then 1 else 0)) = n / 2) (hcolorscols : βˆ€ l, (βˆ‘ k, (if (colors k l = 0) then 1 else 0)) = n / 2) : (βˆ‘ k, βˆ‘ l, (if (colors k l = 0) then (nums k l) else 0)) = (βˆ‘ k, βˆ‘ l, (if (colors k l = 1) then (nums k l) else 0)) := sorry
cd5f365a9654ac4f
putnam_2024_a1
abbrev putnam_2024_a1_solution : Set β„• := sorry --{1} /-- Determine all positive integers $n$ for which there exist positive integers $a$, $b$ and $c$ satisfying $2a^n + 3b^n = 4c^n$. -/ theorem putnam_2024_a1 : {n : β„• | 0 < n ∧ βˆƒ (a b c : β„•), 0 < a ∧ 0 < b ∧ 0 < c ∧ 2*a^n + 3*b^n = 4*c^n} = putnam_2024_a1_solution := sorry
f87967b5a731b7df
putnam_2016_b5
abbrev putnam_2016_b5_solution : Set (Set.Ioi (1 : ℝ) β†’ Set.Ioi (1 : ℝ)) := sorry -- {f : Set.Ioi (1 : ℝ) β†’ Set.Ioi (1 : ℝ) | βˆƒ c : ℝ, c > 0 ∧ βˆ€ x : Set.Ioi (1 : ℝ), (f x : ℝ) = x ^ c} /-- Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y \in (1,\infty)$ and $x^2 \leq y \leq x^3$, then $(f(x))^2 \leq f(y) \leq (f(x))^3$. -/ theorem putnam_2016_b5 (f : Set.Ioi (1 : ℝ) β†’ Set.Ioi (1 : ℝ)) : f ∈ putnam_2016_b5_solution ↔ (βˆ€ x y : Set.Ioi (1 : ℝ), ((x : ℝ) ^ 2 ≀ y ∧ y ≀ (x : ℝ) ^ 3) β†’ ((f x : ℝ) ^ 2 ≀ f y ∧ f y ≀ (f x : ℝ) ^ 3)) := sorry
2f294b010e3f9a5e
putnam_2006_b5
abbrev putnam_2006_b5_solution : ℝ := sorry -- 1 / 16 /-- For each continuous function $f: [0,1] \to \mathbb{R}$, let $I(f) = \int_0^1 x^2 f(x)\,dx$ and $J(x) = \int_0^1 x \left(f(x)\right)^2\,dx$. Find the maximum value of $I(f) - J(f)$ over all such functions $f$. -/ theorem putnam_2006_b5 (I J : (ℝ β†’ ℝ) β†’ ℝ) (hI : I = fun f ↦ ∫ x in (0)..1, x ^ 2 * (f x)) (hJ : J = fun f ↦ ∫ x in (0)..1, x * (f x) ^ 2) : IsGreatest {y | βˆƒ f : ℝ β†’ ℝ, ContinuousOn f (Icc 0 1) ∧ I f - J f = y} putnam_2006_b5_solution := sorry
1cb77bdbebeb4c00
putnam_1983_b5
abbrev putnam_1983_b5_solution : ℝ := sorry -- log (4 / Real.pi) /-- 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 $\prod_{n=1}^{\infty} \frac{2n}{(2n-1)} \cdot \frac{2n}{(2n+1)} = \frac{\pi}{2}$. -/ theorem putnam_1983_b5 (dist_fun : ℝ β†’ ℝ) (hdist_fun : dist_fun = fun (x : ℝ) ↦ min (x - ⌊xβŒ‹) (⌈xβŒ‰ - x)) (fact : Tendsto (fun N ↦ ∏ n in Finset.Icc 1 N, (2 * n / (2 * n - 1)) * (2 * n / (2 * n + 1)) : β„• β†’ ℝ) atTop (𝓝 (Real.pi / 2))) : (Tendsto (fun n ↦ (1 / n) * ∫ x in (1)..n, dist_fun (n / x) : β„• β†’ ℝ) atTop (𝓝 putnam_1983_b5_solution)) := sorry
fbf624b8930b6dcd
putnam_1993_b5:
theorem putnam_1993_b5: Β¬βˆƒ p : Fin 4 β†’ (EuclideanSpace ℝ (Fin 2)), βˆ€ i j, i β‰  j β†’ (βˆƒ n : β„€, dist (p i) (p j) = n ∧ Odd n) := sorry
35326c51997420cb
putnam_2019_a6
theorem putnam_2019_a6 (g : ℝ β†’ ℝ) (r : ℝ) (hcont : ContinuousOn g (Set.Icc 0 1)) (hdiff : ContDiffOn ℝ 1 g (Set.Ioo 0 1) ∧ DifferentiableOn ℝ (deriv g) (Set.Ioo 0 1)) (hr : r > 1) (hlim : Tendsto (fun x => g x / x ^ r) (𝓝[>] 0) (𝓝 0)) : (Tendsto (deriv g) (𝓝[>] 0) (𝓝 0)) ∨ (Tendsto (fun x : ℝ => sSup {x' ^ r * abs (iteratedDeriv 2 g x') | x' ∈ Set.Ioc 0 x}) (𝓝[>] 0) atTop) := sorry
e5e76acaf00ea014
putnam_1967_b5
theorem putnam_1967_b5 (n : β„•) (hn : n > 0) : (1 : β„š)/2 = βˆ‘ i in Finset.range n, (Nat.choose (n + i - 1) i) * (2 : β„š)^(-(n : β„€) - i) := sorry
daa39e24318255c9
putnam_1977_b5
theorem putnam_1977_b5 (n : β„•) (hn : n > 1) (a : Fin n β†’ ℝ) (A : ℝ) (hA : A + βˆ‘ i : Fin n, (a i)^2 < (1/((n : ℝ) - 1))*(βˆ‘ i : Fin n, a i)^2) : βˆ€ i j : Fin n, i < j β†’ A < 2*(a i)*(a j) := sorry
0a8f12b4dcf0b46f