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putnam_1963_b2 | abbrev putnam_1963_b2_solution : Prop := sorry
-- True
/--
Let $S$ be the set of all numbers of the form $2^m3^n$, where $m$ and $n$ are integers, and let $P$ be the set of all positive real numbers. Is $S$ dense in $P$?
-/
theorem putnam_1963_b2
(S : Set β)
(hS : S = {2 ^ m * 3 ^ n | (m : β€) (n : β€)})
: closure S β Set.Ioi (0 : β) β putnam_1963_b2_solution :=
sorry | b039d7145c470c58 |
putnam_1986_b3 | theorem putnam_1986_b3
(n p : β)
(nppos : n > 0 β§ p > 0)
(pprime : Nat.Prime p)
(cong : Polynomial β€ β Polynomial β€ β β€ β Prop)
(hcong : β f g m, cong f g m β β i : β, m β£ (f - g).coeff i)
(f g h r s : Polynomial β€)
(hcoprime : cong (r * f + s * g) 1 p)
(hprod : cong (f * g) h p)
: (β F G : Polynomial β€, cong F f p β§ cong G g p β§ cong (F * G) h (p ^ n)) :=
sorry | 8580651ad4bfabbb |
putnam_1996_b3 | abbrev putnam_1996_b3_solution : β β β := sorry
-- (fun n : β => (2 * n ^ 3 + 3 * n ^ 2 - 11 * n + 18) / 6)
/--
Given that $\{x_1,x_2,\ldots,x_n\}=\{1,2,\ldots,n\}$, find, with proof, the largest possible value, as a function of $n$ (with $n \geq 2$), of $x_1x_2+x_2x_3+\cdots+x_{n-1}x_n+x_nx_1$.
-/
theorem putnam_1996_b3
(n : β) (hn : n β₯ 2) :
IsGreatest
{k | β x : β β β€,
(x '' (Finset.range n) = Set.Icc (1 : β€) n) β§
β i : Fin n, x i * x ((i + 1) % n) = k}
(putnam_1996_b3_solution n) :=
sorry | c5077c43db23b9da |
putnam_1974_b6 | abbrev putnam_1974_b6_solution : (β Γ β Γ β) := sorry
-- ((2^1000 - 1)/3, (2^1000 - 1)/3, 1 + (2^1000 - 1)/3)
/--
For a set with $1000$ elements, how many subsets are there whose candinality is respectively $\equiv 0 \bmod 3, \equiv 1 \bmod 3, \equiv 2 \bmod 3$?
-/
theorem putnam_1974_b6
(n : β€)
(hn : n = 1000)
(count0 count1 count2 : β)
(hcount0 : count0 = {S | S β Finset.Icc 1 n β§ S.card β‘ 0 [MOD 3]}.ncard)
(hcount1 : count1 = {S | S β Finset.Icc 1 n β§ S.card β‘ 1 [MOD 3]}.ncard)
(hcount2 : count2 = {S | S β Finset.Icc 1 n β§ S.card β‘ 2 [MOD 3]}.ncard)
: (count0, count1, count2) = putnam_1974_b6_solution :=
sorry | 3d65f9291c5b3200 |
putnam_1964_b6 | theorem putnam_1964_b6
(D : Set (EuclideanSpace β (Fin 2)))
(hD : D = {v : EuclideanSpace β (Fin 2) | dist 0 v β€ 1})
(cong : Set (EuclideanSpace β (Fin 2)) β Set (EuclideanSpace β (Fin 2)) β Prop)
(hcong : β A B, cong A B β β f : (EuclideanSpace β (Fin 2)) β (EuclideanSpace β (Fin 2)), B = f '' A β§ β v w : EuclideanSpace β (Fin 2), dist v w = dist (f v) (f w))
: (Β¬β A B : Set (Fin 2 β β), cong A B β§ A β© B = β
β§ A βͺ B = D) :=
sorry | 631798de1c67bc48 |
putnam_2013_b3 | abbrev putnam_2013_b3_solution : Prop := sorry
-- True
/--
Let $\mathcal{P}$ be a nonempty collection of subsets of $\{1,\dots, n\}$ such that: \begin{enumerate} \item[(i)] if $S, S' \in \mathcal{P}$, then $S \cup S' \in \mathcal{P}$ and $S \cap S' \in \mathcal{P}$, and \item[(ii)] if $S \in \mathcal{P}$ and $S \neq \emptyset$, then there is a subset $T \subset S$ such that $T \in \mathcal{P}$ and $T$ contains exactly one fewer element than $S$. \end{enumerate} Suppose that $f: \mathcal{P} \to \mathbb{R}$ is a function such that $f(\emptyset) = 0$ and \[f(S \cup S') = f(S) + f(S') - f(S \cap S') \mbox{ for all $S,S' \in \mathcal{P}$.} \] Must there exist real numbers $f_1,\dots,f_n$ such that\[f(S) = \sum_{i \in S} f_i\] \n for every $S \in \mathcal{P}$?
-/
theorem putnam_2013_b3
: (β (n : β) (P : Finset (Finset (Fin n))) (f : Finset (Fin n) β β),
P β β₯ β (β S β P, β S' β P, S βͺ S' β P β§ S β© S' β P) β
(β S β P, S β β₯ β β T β P, T β S β§ Finset.card T + 1 = Finset.card S) β
f β₯ = 0 β (β S β P, β S' β P, f (S βͺ S') = f S + f S' - f (S β© S')) β
β r : Fin n β β, β S β P, f S = β i in S, r i)
β putnam_2013_b3_solution :=
sorry | 185a012e14172598 |
putnam_2003_b3 | theorem putnam_2003_b3 (n : β) :
n ! = β i in Finset.Icc 1 n, ((List.range βn / iββ).map succ).foldl Nat.lcm 1 :=
sorry | baf0aacbeaf4dad1 |
putnam_1970_a2 | theorem putnam_1970_a2
(A B C D E F G : β)
(hle : B^2 - 4*A*C < 0)
: β Ξ΄ > 0, Β¬β x y : β, x^2 + y^2 β Set.Ioo 0 (Ξ΄^2) β§
A*x^2 + B*x*y + C*y^2 + D*x^3 + E*x^2*y + F*x*y^2 + G*y^3 = 0 :=
sorry | e758caba328bad48 |
putnam_2008_b4 | theorem putnam_2008_b4
(p : β)
(hp : Nat.Prime p)
(h : Polynomial β€)
(hh : β i j : Finset.range (p ^ 2), h.eval i β‘ h.eval j [ZMOD p ^ 2] β i = j)
: (β i j : Finset.range (p ^ 3), h.eval i β‘ h.eval j [ZMOD p ^ 3] β i = j) :=
sorry | 589761e4f74cda80 |
putnam_2018_b4 | theorem putnam_2018_b4
(a : β)
(x : β β β)
(hx0 : x 0 = 1)
(hx12 : x 1 = a β§ x 2 = a)
(hxn : β n, x (n + 3) = 2 * (x (n + 2)) * (x (n + 1)) - x n) :
(β n, x n = 0) β (β c, c > 0 β§ Function.Periodic x c) :=
sorry | 1a04f661e9c83461 |
putnam_1968_b5 | abbrev putnam_1968_b5_solution : β β β := sorry
-- fun p => p^2 + p
/--
Let $p$ be a prime number. Find the number of distinct $2 \times 2$ matrices $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ such that $a, b, c, d \in \{0, 1, ..., p - 1\}$, $a + d \equiv 1 \pmod p$, and $ad - bc \equiv 0 \pmod p$.
-/
theorem putnam_1968_b5
(p : β)
(hp : Prime p)
: {M : Matrix (Fin 2) (Fin 2) (ZMod p) | M 0 0 + M 1 1 = 1 β§ M 0 0 * M 1 1 - M 0 1 * M 1 0 = 0}.ncard = putnam_1968_b5_solution p :=
sorry | 892df511eaf6c029 |
putnam_1985_a3 | abbrev putnam_1985_a3_solution : β β β := sorry
-- fun d β¦ exp d - 1
/--
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 $\lim_{n \to \infty} a_n(n)$.
-/
theorem putnam_1985_a3
(d : β)
(a : β β β β β)
(ha0 : β m : β, a m 0 = d / 2 ^ m)
(ha : β m : β, β j : β, a m (j + 1) = (a m j) ^ 2 + 2 * a m j)
: Tendsto (fun n β¦ a n n) atTop (π (putnam_1985_a3_solution d)) :=
sorry | 96110cd8c90a15d9 |
putnam_1978_b5 | abbrev putnam_1978_b5_solution : Polynomial β := sorry
-- 4 * X ^ 4 - 4 * X ^ 2 + 1
/--
Find the real polynomial $p(x)$ of degree $4$ with largest possible coefficient of $x^4$ such that $p([-1, 1]) \subseteq [0, 1]$.
-/
theorem putnam_1978_b5
(S : Set (Polynomial β))
(hS : S = {p : Polynomial β | p.degree = 4 β§ β x β Icc (-1) 1, p.eval x β Icc 0 1})
: (putnam_1978_b5_solution β S β§ (β p β S, p.coeff 4 β€ putnam_1978_b5_solution.coeff 4)) :=
sorry | 701748b818a933bd |
putnam_1995_a3 | theorem putnam_1995_a3
(relation : (Fin 9 β β€) β (Fin 9 β β€) β Prop)
(digits_to_num : (Fin 9 β β€) β β€)
(hdigits_to_num : digits_to_num = fun dig => β i : Fin 9, (dig i) * 10^i.1)
(hrelation : β d e : (Fin 9 β β€), relation d e β (β i : Fin 9, d i < 10 β§ d i β₯ 0 β§ e i < 10 β§ e i β₯ 0) β§ (β i : Fin 9, 7 β£ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j))))
: β d e f : (Fin 9 β β€), ((relation d e) β§ (relation e f)) β (β i : Fin 9, 7 β£ d i - f i) :=
sorry | 0c477dbe4635975a |
putnam_2010_a3 | theorem putnam_2010_a3
(h : β Γ β β β)
(a b M : β)
(H : ContDiff β 1 h)
(H' : β x, h x = a * (fderiv β h (1, 0) x) + (fderiv β h (0, 1) x))
(H'' : β x, |h x| β€ M) : h = 0 :=
sorry | f0acd96a1d540fa5 |
putnam_1977_a6 | abbrev putnam_1977_a6_solution : Prop := sorry
-- True
/--
Let $X$ be the square $[0, 1] \times [0, 1]$, and let $f : X \to \mathbb{R}$ be continuous. If $\int_Y f(x, y) \, dx \, dy = 0$ for all squares $Y$ such that
\begin{itemize}
\item[(1)] $Y \subseteq X$,
\item[(2)] $Y$ has sides parallel to those of $X$,
\item[(3)] at least one of $Y$'s sides is contained in the boundary of $X$,
\end{itemize}
is it true that $f(x, y) = 0$ for all $x, y$?
-/
theorem putnam_1977_a6
(X : Set (β Γ β))
(hX : X = Set.prod (Icc 0 1) (Icc 0 1))
(room : (β Γ β) β β)
(hroom : room = fun (a,b) β¦ min (min a (1 - a)) (min b (1 - b)))
: ((β f : (β Γ β) β β, Continuous f β (β P β X, β« x in (P.1 - room P)..(P.1 + room P), β« y in (P.2 - room P)..(P.2 + room P), f (x, y) = 0) β (β P β X, f P = 0)) β putnam_1977_a6_solution) :=
sorry | f9e08fb9aaaf0cff |
putnam_1967_a6 | abbrev putnam_1967_a6_solution : β := sorry
-- 8
/--
Given real numbers $\{a_i\}$ and $\{b_i\}$, ($i=1,2,3,4$), such that $a_1b_2-a_2b_1 \neq 0$. Consider the set of all solutions $(x_1,x_2,x_3,x_4)$ of the simultaneous equations $a_1x_1+a_2x_2+a_3x_3+a_4x_4=0$ and $b_1x_1+b_2x_2+b_3x_3+b_4x_4=0$, for which no $x_i$ ($i=1,2,3,4$) is zero. Each such solution generates a $4$-tuple of plus and minus signs $(\text{signum }x_1,\text{signum }x_2,\text{signum }x_3,\text{signum }x_4)$. Determine, with a proof, the maximum number of distinct $4$-tuples possible.
-/
theorem putnam_1967_a6
(abneq0 : (Fin 4 β β) β (Fin 4 β β) β Prop)
(habneq0 : abneq0 = (fun a b : Fin 4 β β => a 0 * b 1 - a 1 * b 0 β 0))
(numtuples : (Fin 4 β β) β (Fin 4 β β) β β)
(hnumtuples : β a b : Fin 4 β β, numtuples a b = {s : Fin 4 β β | β x : Fin 4 β β, (β i : Fin 4, x i β 0) β§ (β i : Fin 4, a i * x i) = 0 β§ (β i : Fin 4, b i * x i) = 0 β§ (β i : Fin 4, s i = Real.sign (x i))}.encard)
: (β a b : Fin 4 β β, abneq0 a b β§ numtuples a b = putnam_1967_a6_solution) β§ (β a b : Fin 4 β β, abneq0 a b β numtuples a b β€ putnam_1967_a6_solution) :=
sorry | a7ceea55dcd5d16a |
putnam_2003_a5 | theorem putnam_2003_a5
(n : β) (npos : n > 0)
(dyckpath : (m : β) β Set ((Fin (2 * m)) β β€))
(hdyckpath : dyckpath = fun m β¦ {p |
range p β {-1, 1} β§ β k, p k = 0 β§ β j, β k, ite (k β€ j) (p k) 0 β₯ 0})
(noevenreturn : (m : β) β Set ((Fin (2 * m)) β β€))
(hnoevenreturn : noevenreturn = fun m β¦ {p |
Β¬β i j, i < j β§ p i = 1 β§ (β k β Ioc i j, p i = -1) β§
Even (j.1 - i.1) β§ β k, ite (k β€ j) (p k) 0 = 0})
: β f : ((Fin (2 * n)) β β€) β (Fin (2 * (n - 1)) β β€),
β y β dyckpath (n - 1), β! x, x β dyckpath n β© noevenreturn n β§ f x = y :=
sorry | 66e908b70c22ba70 |
putnam_2013_a5 | theorem putnam_2013_a5
(m : β)
(area2 : (Fin 2 β β) β (Fin 2 β β) β (Fin 2 β β) β β)
(area3 : (Fin 3 β β) β (Fin 3 β β) β (Fin 3 β β) β β)
(areadef2 : (Fin m β Fin m β Fin m β β) β Prop)
(areadef3 : (Fin m β Fin m β Fin m β β) β Prop)
(mge3 : m β₯ 3)
(harea2 : β a b c, area2 a b c = (volume (convexHull β {a, b, c})).toReal)
(harea3 : β a b c, area3 a b c = (ΞΌH[2] (convexHull β {a, b, c})).toReal)
(hareadef2 : β a, areadef2 a β β A : Fin m β (Fin 2 β β), (β i : Fin m, β j : Fin m, β k : Fin m, if (i < j β§ j < k) then (a i j k * area2 (A i) (A j) (A k)) else 0) β₯ 0)
(hareadef3 : β a, areadef3 a β β A : Fin m β (Fin 3 β β), (β i : Fin m, β j : Fin m, β k : Fin m, if (i < j β§ j < k) then (a i j k * area3 (A i) (A j) (A k)) else 0) β₯ 0)
: β a, areadef2 a β areadef3 a :=
sorry | 09dadb84ed35d39f |
putnam_1989_b6 | theorem putnam_1989_b6
(n : β) [NeZero n]
(I : (Fin n β β) β Fin (n + 2) β β)
(I_def : β x i, I x i = if i = 0 then 0 else if i = - 1 then 1 else x (i : β).pred)
(X : Set (Fin n β β))
(X_def : β x, x β X β 0 < x 0 β§ x (-1) < 1 β§ β i, i + 1 < n β x i < x (i + 1))
(S : (β β β) β (Fin (n + 2) β β) β β)
(S_def : β f x, S f x = β i in Finset.Iic n, (x (i + 1) - x i) * f (i + 1)) :
β P : Polynomial β,
P.degree = n β§
(β t β Icc 0 1, P.eval t β Icc 0 1) β§
(β f : β β β, f 1 = 0 β ContinuousOn f (Icc 0 1) β
β« x, S f (I x) ββ[|X] = β« t in (0)..1, f t * P.eval t) :=
sorry | 3b47611a00a667ae |
putnam_1999_b6 | theorem putnam_1999_b6
(S : Finset β€)
(hSgt : β s, s β S β s > 1)
(hSgcd : β n, β s β S, Int.gcd s n = 1 β¨ Int.gcd s n = s) :
βα΅ (s β S) (t β S), Prime (Int.gcd s t) :=
sorry | c82e786c67b07261 |
putnam_1996_a5 | theorem putnam_1996_a5
(p : β)
(hpprime : Prime p)
(hpge3 : p > 3)
(k : β)
(hk : k = Nat.floor (2*p/(3 : β)))
: p^2 β£ β i in Finset.Icc 1 k, Nat.choose p i :=
sorry | 5850cdd3b7ec6fef |
putnam_1986_a5 | theorem putnam_1986_a5
(n : β) (hn : 1 β€ n)
(f : Fin n β ((Fin n β β) β β))
(hf : β i, ContDiff β 2 (f i))
(C : Fin n β Fin n β β)
(hf' : β i j : Fin n, β x : Fin n β β, fderiv β (f i) x (Pi.single j 1) - fderiv β (f j) x (Pi.single i 1) = C i j)
: β g : (Fin n β β) β β, β i : Fin n, IsLinearMap β (Ξ» x β¦ f i x + fderiv β g x (Pi.single i 1)) :=
sorry | 51160fabb4ba5bfe |
putnam_1981_a1 | abbrev putnam_1981_a1_solution : β := sorry
-- 1/8
/--
Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$.
-/
theorem putnam_1981_a1
(P : β β β β Prop)
(hP : β n k, P n k β 5^k β£ β m in Finset.Icc 1 n, (m^m : β€))
(E : β β β)
(hE : β n β Ici 1, P n (E n) β§ β k : β, P n k β k β€ E n) :
Tendsto (fun n : β => ((E n) : β)/n^2) atTop (π putnam_1981_a1_solution) :=
sorry | 9813035730f27b9b |
putnam_2004_a1 | abbrev putnam_2004_a1_solution : Prop := sorry
-- True
/--
Basketball star Shanille O'Keal's team statistician keeps track of the number, $S(N)$, of successful free throws she has made in her first $N$ attempts of the season. Early in the season, $S(N)$ was less than $80\%$ of $N$, but by the end of the season, $S(N)$ was more than $80\%$ of $N$. Was there necessarily a moment in between when $S(N)$ was exactly $80\%$ of $N$?
-/
theorem putnam_2004_a1
(S : (β β Fin 2) β β β β)
(hS : β attempts, β N β₯ 1, S attempts N = (β i : Fin N, (attempts i).1) / N) :
(β attempts a b,
(1 β€ a β§ a < b β§ S attempts a < 0.8 β§ S attempts b > 0.8) β
(β c : β, a < c β§ c < b β§ S attempts c = 0.8))
β putnam_2004_a1_solution :=
sorry | a37d15ee1e52be11 |
putnam_2014_a1 | theorem putnam_2014_a1
(f : β β β)
(hf : β x : β, f x = (1 - x + x ^ 2) * Real.exp x)
(hfdiff : ContDiff β β€ f)
(c : β β β)
(hc : β k : β, c k = taylorCoeffWithin f k Set.univ 0)
: β k : β, c k β 0 β β q : β, c k = q β§ (q.num = 1 β¨ Prime q.num.natAbs) :=
sorry | 3a79b3513e88e5a0 |
putnam_1963_a4 | theorem putnam_1963_a4
(T : (β β β) β (β β β))
(T_def : β a n, T a n = n * ((1 + a (n + 1)) / a n - 1))
(P : (β β β) β β β Prop)
(P_def : β a C, P a C β C β€ limsup (T a) atTop β¨ Β¬ BddAbove (range (T a))) :
(β a, (β n, 0 < a n) β P a 1) β§ (β C > 1, β a, (β n, 0 < a n) β§ Β¬ P a C) :=
sorry | b72f9f8c37d3ffd9 |
putnam_1973_a4 | abbrev putnam_1973_a4_solution : β := sorry
-- 3
/--
How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line?
-/
theorem putnam_1973_a4
(f : β β β)
(hf : f = fun x => 2^x - 1 - x^2)
: putnam_1973_a4_solution = {x : β | f x = 0}.ncard :=
sorry | 2f69a243a37a2dcb |
putnam_1987_a4 | abbrev putnam_1987_a4_solution : β := sorry
-- (5 / 3) * sqrt 30
/--
Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that
\[
P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$},
\]
and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.
-/
theorem putnam_1987_a4
(P : MvPolynomial (Fin 3) β)
(hPreal : β i : Fin 3 ββ β, (coeff i P).im = 0)
(F : β β β β β)
(vars : β β β β β β (Fin 3 β β))
(hvars : vars = fun a b c β¦ fun i β¦ ite (i = 0) a (ite (i = 1) b c))
(h : β x y z u : β, eval (vars (u * x) (u * y) (u * z)) P = u ^ 2 * F (y - x) (z - x))
(hPval : eval (vars 1 0 0) P = 4 β§ eval (vars 0 1 0) P = 5 β§ eval (vars 0 0 1) P = 6)
(A B C : β)
(hPABC : eval (vars A B C) P = 0)
(habs : βB - Aβ = 10)
: (βC - Aβ = putnam_1987_a4_solution) :=
sorry | 2fc94706a7ec30e2 |
putnam_1997_a4 | theorem putnam_1997_a4
(G : Type*)
[Group G]
(Ο : G β G)
(hΟ : β g1 g2 g3 h1 h2 h3 : G, (g1 * g2 * g3 = 1 β§ h1 * h2 * h3 = 1) β Ο g1 * Ο g2 * Ο g3 = Ο h1 * Ο h2 * Ο h3)
: β a : G, let Ο := fun g => a * Ο g; β x y : G, Ο (x * y) = Ο x * Ο y :=
sorry | 33f486c7439b4386 |
putnam_1975_a1 | abbrev putnam_1975_a1_solution : ((β€ Γ β€) β β€) Γ ((β€ Γ β€) β β€) := sorry
-- (fun (a, b) => a + b + 1, fun (a, b) => a - b)
/--
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.
-/
theorem putnam_1975_a1
(nab nxy : (β€ Γ β€ Γ β€) β Prop)
(hnab : nab = fun (n, a, b) => n = (a^2 + (a : β))/2 + (b^2 + (b : β))/2)
(hnxy : nxy = fun (n, x, y) => 4*n + 1 = x^2 + y^2)
: (β n a b : β€, nab (n, a, b) β nxy (n, putnam_1975_a1_solution.1 (a, b), putnam_1975_a1_solution.2 (a, b))) β§
β n : β€, (β x y : β€, nxy (n, x, y)) β β a b : β€, nab (n, a, b) :=
sorry | edb35adf7bfd4c56 |
putnam_1965_a1 | abbrev putnam_1965_a1_solution : β := sorry
-- Real.pi / 15
/--
Let $\triangle ABC$ satisfy $\angle CAB < \angle BCA < \frac{\pi}{2} < \angle ABC$. If the bisector of the external angle at $A$ meets line $BC$ at $P$, the bisector of the external angle at $B$ meets line $CA$ at $Q$, and $AP = BQ = AB$, find $\angle CAB$.
-/
theorem putnam_1965_a1
(A B C X Y : EuclideanSpace β (Fin 2))
(hABC : Β¬Collinear β {A, B, C})
(hangles : β C A B < β B C A β§ β B C A < Ο/2 β§ Ο/2 < β A B C)
(hX : Collinear β {X, B, C} β§ β X A B = (Ο - β C A B)/2 β§ dist A X = dist A B)
(hY : Collinear β {Y, C, A} β§ β Y B C = (Ο - β A B C)/2 β§ dist B Y = dist A B)
: β C A B = putnam_1965_a1_solution :=
sorry | 801a7e53041baba6 |
putnam_2012_a4 | theorem putnam_2012_a4
(IsFiniteAP : Set β€ β Prop)
(IsFiniteAP_def : β s,
IsFiniteAP s β β n : β, β a d : β€, 0 < d β§ s = {a + i * d | i : Fin n})
(q r : β€)
(A B : Fin 2 β β)
(T : Set β€)
(S : Set β€)
(qpos : q > 0)
(ABlt : A 0 < A 1 β§ B 0 < B 1)
(hT : T = {x : β€ | β b m : β€, (b : β) β Set.Icc (B 0) (B 1) β§ x = b + m * q})
(hS : S = {a : β€ | (a : β) β Set.Icc (A 0) (A 1) β§ r * a β T})
(prod_lt : (A 1 - A 0) * (B 1 - B 0) < q) :
IsFiniteAP {x | x β S β§ (x : β) β Set.Icc (A 0) (A 1)} :=
sorry | 2452deb7e5980507 |
putnam_1972_a5 | theorem putnam_1972_a5
(n : β)
(hn : n > 1)
: Β¬((n : β€) β£ 2^n - 1) :=
sorry | a058f7bd349016a4 |
putnam_1962_a5 | abbrev putnam_1962_a5_solution : β β β := sorry
-- fun n : β => n * (n + 1) * 2^(n - 2)
/--
Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \]
-/
theorem putnam_1962_a5
: β n β₯ 2, putnam_1962_a5_solution n = β k in Finset.Icc 1 n, Nat.choose n k * k^2 :=
sorry | 398b11f0294a4b06 |
putnam_1993_a6 | theorem putnam_1993_a6
(seq : β β β€)
(hseq23 : β n, seq n = 2 β¨ seq n = 3)
(hseq2inds : β n, seq n = 2 β (β N : β, n = β i : Fin N, (seq i + 1)))
: β r : β, β n, seq n = 2 β (β m : β€, n + 1 = 1 + Int.floor (r * m)) :=
sorry | 48de00a3fed833a4 |
putnam_1983_a6 | abbrev putnam_1983_a6_solution : β := sorry
-- 2 / 9
/--
Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$.
-/
theorem putnam_1983_a6
(F : β β β)
(hF : F = fun a β¦ (a ^ 4 / exp (a ^ 3)) * β« x in (0)..a, β« y in (0)..(a - x), exp (x ^ 3 + y ^ 3))
: (Tendsto F atTop (π putnam_1983_a6_solution)) :=
sorry | 0fe13e9d9a270a3e |
putnam_2019_b5 | abbrev putnam_2019_b5_solution : β Γ β := sorry
-- β¨2019, 1010β©
/--
Let $F_m$ be the $m$th Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_m = F_{m-1} + F_{m-2}$ for all $m \geq 3$. Let $p(x)$ be the polynomial of degree $1008$ such that $p(2n + 1) = F_{2n+1}$ for $n = 0,1,2,\ldots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$.
-/
theorem putnam_2019_b5
(F : β β β€)
(P : Polynomial β)
(hF : β x, x β₯ 1 β F (x + 2) = F (x + 1) + F x)
(F12 : F 1 = 1 β§ F 2 = 1)
(Pdeg: Polynomial.degree P = 1008)
(hp: β n : β, (n β€ 1008) β P.eval (2 * n + 1 : β) = F (2 * n + 1))
: β j k : β, (P.eval 2019 = F j - F k) β β¨j, kβ© = putnam_2019_b5_solution :=
sorry | e235992cbe2b77d5 |
putnam_2009_b5 | theorem putnam_2009_b5
(f : β β β)
(hfdiff : DifferentiableOn β f (Ioi 1))
(hf : β x > 1, deriv f x = (x ^ 2 - (f x) ^ 2) / ((x ^ 2) * ((f x) ^ 2 + 1)))
: (Tendsto f atTop atTop) :=
sorry | 51f0ccf451eb184f |
putnam_1971_a3 | theorem putnam_1971_a3
(a b c : β Γ β)
(R : β)
(habclattice : a.1 = round a.1 β§ a.2 = round a.2 β§ b.1 = round b.1 β§ b.2 = round b.2 β§ c.1 = round c.1 β§ c.2 = round c.2)
(habcneq : a β b β§ a β c β§ b β c)
(hR : R > 0)
(hcircle : β C : β Γ β,
β((a.1 - C.1)^2 + (a.2 - C.2)^2) = R β§
β((b.1 - C.1)^2 + (b.2 - C.2)^2) = R β§
β((c.1 - C.1)^2 + (c.2 - C.2)^2) = R) :
(β((a.1 - b.1)^2 + (a.2 - b.2)^2)) * (β((a.1 - c.1)^2 + (a.2 - c.2)^2)) * (β((b.1 - c.1)^2 + (b.2 - c.2)^2)) β₯ 2 * R :=
sorry | bf796c9fcc314eb9 |
putnam_2016_a6 | abbrev putnam_2016_a6_solution : β := sorry
-- 5 / 6
/--
Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree $3$ that has a root in the interval $[0,1]$,
\[
\int_0^1 \left| P(x) \right|\,dx \leq C \max_{x \in [0,1]} \left| P(x) \right|.
\]
-/
theorem putnam_2016_a6
(p : β β Prop)
(hp : β c, p c β
β P : Polynomial β, P.degree = 3 β
(β x β Icc 0 1, P.eval x = 0) β
β« x in (0)..1, |P.eval x| β€ c * (sSup {y | β x β Icc 0 1, y = |P.eval x|})) :
IsLeast p putnam_2016_a6_solution :=
sorry | 64c13f15bee305cd |
putnam_2023_b6 | abbrev putnam_2023_b6_solution : β β β€ := sorry
-- (fun n : β => (-1) ^ (β(n / 2 : β)ββ + 1) * 2 * β(n / 2 : β)ββ)
/--
Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\dots,n\}$, let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai+bj=n$. Let $S$ be the $n$-by-$n$ matrix whose $(i,j)$ entry is $s(i,j)$. For example, when $n=5$, we have $S = \begin{bmatrix}
6 & 3 & 2 & 2 & 2 \\
3 & 0 & 1 & 0 & 1 \\
2 & 1 & 0 & 0 & 1 \\
2 & 0 & 0 & 0 & 1 \\
2 & 1 & 1 & 1 & 2
\end{bmatrix}$. Compute the determinant of $S$.
-/
theorem putnam_2023_b6
(n : β)
(S : Matrix (Fin n) (Fin n) β€)
(npos : n > 0)
(hS : β i j : Fin n, S i j = β' a : β, β' b : β, if a * (i.1 + 1) + b * (j.1 + 1) = (n : β€) then 1 else 0)
: S.det = putnam_2023_b6_solution n :=
sorry | 03808b9463123954 |
putnam_2011_a2 | abbrev putnam_2011_a2_solution : β := sorry
-- 3/2
/--
Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1 = b_1 = 1$ and $b_n = b_{n-1} a_n - 2$ for$n=2,3,\dots$. Assume that the sequence $(b_j)$ is bounded. Prove tha \[ S = \sum_{n=1}^\infty \frac{1}{a_1...a_n} \] converges, and evaluate $S$.
-/
theorem putnam_2011_a2
(a b : β β β)
(habn : β n : β, a n > 0 β§ b n > 0)
(hab1 : a 0 = 1 β§ b 0 = 1)
(hb : β n β₯ 1, b n = b (n-1) * a n - 2)
(hbnd : β B : β, β n : β, |b n| β€ B)
: Tendsto (fun n => β i : Fin n, 1/(β j : Fin (i + 1), (a j))) atTop (π putnam_2011_a2_solution) :=
sorry | 8ac568b6de75e402 |
putnam_1984_a2 | abbrev putnam_1984_a2_solution : β := sorry
-- 2
/--
Express $\sum_{k=1}^\infty (6^k/(3^{k+1}-2^{k+1})(3^k-2^k))$ as a rational number.
-/
theorem putnam_1984_a2
: β' k : Set.Ici 1, (6 ^ (k : β) / ((3 ^ ((k : β) + 1) - 2 ^ ((k : β) + 1)) * (3 ^ (k : β) - 2 ^ (k : β)))) = putnam_1984_a2_solution :=
sorry | a21c34f3b53d4fb6 |
putnam_1969_b4 | theorem putnam_1969_b4
(Ξ : β β EuclideanSpace β (Fin 2))
--Note: the problem doesn't say what regularity conditions we should impose on `Ξ` - hopefully continuity is enough.
(Ξ_cts : ContinuousOn Ξ (Set.Icc 0 1))
(hΞ : eVariationOn Ξ (Set.Icc 0 1) = 1) :
letI : Module.Oriented β (EuclideanSpace β (Fin 2)) (Fin 2) := β¨Basis.orientation <| Pi.basisFun _ _β©
letI : Fact (Module.finrank β (EuclideanSpace β (Fin 2)) = 2) := β¨finrank_euclideanSpace_finβ©
β (a b c d : EuclideanSpace β (Fin 2)), Ξ '' (Set.Icc 0 1) β
convexHull β {a, b, c, d} β§ dist a b * dist b c = 1/4 β§
--`abcd` is a rectangle
letI Ο_div_2 := Real.Angle.coe <| Ο / 2
(β‘ a b c = Ο_div_2 β§ β‘ b c d = Ο_div_2 β§ β‘ c d a = Ο_div_2 β§ β‘ d a b = Ο_div_2) :=
sorry | bc5d4f246edbd719 |
putnam_1994_a3 | theorem putnam_1994_a3
(T : Set (EuclideanSpace β (Fin 2)))
(hT : T = convexHull β {(WithLp.equiv 2 (EuclideanSpace β (Fin 2))).symm ![0,0], (WithLp.equiv 2 (EuclideanSpace β (Fin 2))).symm ![1,0], (WithLp.equiv 2 (EuclideanSpace β (Fin 2))).symm ![0,1]})
(Tcolors : T β Fin 4)
: β p q : T, Tcolors p = Tcolors q β§ dist p.1 q.1 β₯ 2 - Real.sqrt 2 :=
sorry | 8095f770b73394ae |
putnam_1979_b5 | theorem putnam_1979_b5
(C : Set (β Γ β))
(hC : IsClosed C β§ Convex β C β§ (0,0) β C)
(hCnint : β a b : β€, (a β 0 β¨ b β 0) β ((a : β), (b : β)) β C)
(hCeven : volume (C β© {p : β Γ β | p.1 β₯ 0 β§ p.2 β₯ 0}) = volume (C β© {p : β Γ β | p.1 β₯ 0 β§ p.2 β€ 0}) β§ volume (C β© {p : β Γ β | p.1 β₯ 0 β§ p.2 β€ 0}) = volume (C β© {p : β Γ β | p.1 β€ 0 β§ p.2 β₯ 0}) β§ volume (C β© {p : β Γ β | p.1 β€ 0 β§ p.2 β₯ 0}) = volume (C β© {p : β Γ β | p.1 β€ 0 β§ p.2 β€ 0}))
: volume C β€ 4 :=
sorry | 97f8eddf5979aa2a |
putnam_1984_a3 | abbrev putnam_1984_a3_solution : MvPolynomial (Fin 3) β := sorry
-- (MvPolynomial.X 2) ^ 2 * ((MvPolynomial.X 0) ^ 2 - (MvPolynomial.X 1) ^ 2)
/--
Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$, and let $M_n$ denote the $2n \times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by
\[
m_{ij}=\begin{cases}
x & \text{if }i=j, \\
a & \text{if }i \neq j\text{ and }i+j\text{ is even}, \\
b & \text{if }i \neq j\text{ and }i+j\text{ is odd}.
\end{cases}
\]
Thus, for example, $M_2=\begin{pmatrix} x & b & a & b \\ b & x & b & a \\ a & b & x & b \\ b & a & b & x \end{pmatrix}$. Express $\lim_{x \to a} \det M_n/(x-a)^{2n-2}$ as a polynomial in $a$, $b$, and $n$, where $\det M_n$ denotes the determinant of $M_n$.
-/
theorem putnam_1984_a3
(n : β)
(a b : β)
(Mn : β β Matrix (Fin (2 * n)) (Fin (2 * n)) β)
(polyabn : Fin 3 β β)
(npos : n > 0)
(aneb : a β b)
(hMn : Mn = fun x : β => fun i j : Fin (2 * n) => if i = j then x else if Even (i.1 + j.1) then a else b)
(hpolyabn : polyabn 0 = a β§ polyabn 1 = b β§ polyabn 2 = n)
: Tendsto (fun x : β => (Mn x).det / (x - a) ^ (2 * n - 2)) (π[β ] a) (π (MvPolynomial.eval polyabn putnam_1984_a3_solution)) :=
sorry | 769a67dce5329b85 |
putnam_1969_b5 | theorem putnam_1969_b5
(a : β β β)
(ha : StrictMono a β§ (β x : β, a x > 0))
(hinvasum : β C : β, Tendsto (fun n => β i : Fin n, 1/(a i)) atTop (π C))
(k : β β β)
(hk : k = fun x => {n | a n β€ x}.ncard)
: Tendsto (fun t => (k t)/t) atTop (π 0) :=
sorry | f1e82ccf9dbe473a |
putnam_1966_a6 | theorem putnam_1966_a6
(a : β β (β β β))
(ha : β n β₯ 1, a n n = n β§ β m β₯ 1, m < n β a n m = m * Real.sqrt (1 + a n (m + 1)))
: Tendsto (fun n => a n 1) atTop (π 3) :=
sorry | c29d95fe5816eb66 |
putnam_1976_a6 | theorem putnam_1976_a6
(f : β β β)
(hfdiff : ContDiff β 2 f)
(hfbd : β x : β, |f x| β€ 1)
(hf0 : (f 0)^2 + (deriv f 0)^2 = 4)
: β y : β, (f y) + (iteratedDeriv 2 f y) = 0 :=
sorry | 8bd245da851bf6ee |
putnam_2001_a3 | abbrev putnam_2001_a3_solution : Set β€ := sorry
-- {m : β€ | β k : β€, k^2 = m β¨ 2*k^2 = m}
/--
For each integer $m$, consider the polynomial
\[P_m(x)=x^4-(2m+4)x^2+(m-2)^2.\] For what values of $m$ is $P_m(x)$
the product of two non-constant polynomials with integer coefficients?
-/
theorem putnam_2001_a3
(P : β€ β Polynomial β€)
(hP : P = fun m : β€ => (Polynomial.X)^4 - (Polynomial.C (2*m + 4))*(Polynomial.X)^2 + Polynomial.C ((m - 2)^2))
: {m : β€ | β a b, P m = a * b β§
(β n β Ici 1, a.coeff n β 0) β§ (β n β Ici 1, b.coeff n β 0)} = putnam_2001_a3_solution :=
sorry | d7bc04f953f7f94a |
putnam_2011_a3 | abbrev putnam_2011_a3_solution : β Γ β := sorry
-- (-1, 2 / Real.pi)
/--
Find a real number $c$ and a positive number $L$ for which $\lim_{r \to \infty} \frac{r^c \int_0^{\pi/2} x^r\sin x\,dx}{\int_0^{\pi/2} x^r\cos x\,dx}=L$.
-/
theorem putnam_2011_a3
: putnam_2011_a3_solution.2 > 0 β§ Tendsto (fun r : β => (r ^ putnam_2011_a3_solution.1 * β« x in Set.Ioo 0 (Real.pi / 2), x ^ r * Real.sin x) / (β« x in Set.Ioo 0 (Real.pi / 2), x ^ r * Real.cos x)) atTop (π putnam_2011_a3_solution.2) :=
sorry | 5ecd38afbff0e836 |
putnam_2024_b3 | theorem putnam_2024_b3
(r : β+ β β)
(r_def : StrictMono r β§
(β x β Set.Ioo 0 (r 1), Real.tan x β x) β§ β n,
0 < r n β§
Real.tan (r n) = r n β§
β x, x β Set.Ioo (r n) (r (n + 1)) β Real.tan x β x)
(n : β+) :
r (n + 1) - r n - Real.pi β Set.Ioo (0 : β) (1 / ((n ^ 2 + n) * Real.pi)) :=
sorry | 880b0097942f29d8 |
putnam_1971_a2 | abbrev putnam_1971_a2_solution : Set (Polynomial β) := sorry
-- {Polynomial.X}
/--
Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$.
-/
theorem putnam_1971_a2
(P : Polynomial β) :
(P.eval 0 = 0 β§ (β x : β, P.eval (x^2 + 1) = (P.eval x)^2 + 1)) β P β putnam_1971_a2_solution :=
sorry | ef7e5e3af5a64a4c |
putnam_2019_b4 | abbrev putnam_2019_b4_solution : β := sorry
-- 2 * Real.log 2 - 1 / 2
/--
Let $\mathcal{F}$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x \geq 1,y \geq 1$ and that satisfy the following two equations (where subscripts denote partial derivatives):
\begin{gather*}
xf_x+yf_y=xy\ln(xy), \\
x^2f_{xx}+y^2f_{yy}=xy.
\end{gather*}
For each $f \in \mathcal{F}$, let $m(f)=\min_{s \geq 1} (f(s+1,s+1)-f(s+1,s)-f(s,s+1)+f(s,s))$. Determine $m(f)$, and show that it is independent of the choice of $f$.
-/
theorem putnam_2019_b4
(f : (Fin 2 β β) β β)
(vec : β β β β (Fin 2 β β))
(fdiff : ContDiff β 2 f)
(hvec : β x y : β, (vec x y) 0 = x β§ (vec x y 1) = y)
(feq1 : β x β₯ 1, β y β₯ 1, x * deriv (fun x' : β => f (vec x' y)) x + y * deriv (fun y' : β => f (vec x y')) y = x * y * Real.log (x * y))
(feq2 : β x β₯ 1, β y β₯ 1, x ^ 2 * iteratedDeriv 2 (fun x' : β => f (vec x' y)) x + y ^ 2 * iteratedDeriv 2 (fun y' : β => f (vec x y')) y = x * y)
: sInf {f (vec (s + 1) (s + 1)) - f (vec (s + 1) s) - f (vec s (s + 1)) + f (vec s s) | s β₯ 1} = putnam_2019_b4_solution :=
sorry | ea66af1f92c8b60d |
putnam_2009_b4 | abbrev putnam_2009_b4_solution : β := sorry
-- 2020050
/--
Say that a polynomial with real coefficients in two variables, $x,y$, is \emph{balanced} if the average value of the polynomial on each circle centered at the origin is $0$. The balanced polynomials of degree at most $2009$ form a vector space $V$ over $\mathbb{R}$. Find the dimension of $V$.
-/
theorem putnam_2009_b4
(IsBalanced : MvPolynomial (Fin 2) β β Prop)
(IsBalanced_def : β P, IsBalanced P β β r > 0,
(β« t in (0 : β)..(2 * Ο), eval ![r * cos t, r * sin t] P) / (2 * Ο * r) = 0)
(V : Submodule β (MvPolynomial (Fin 2) β))
(V_def : β P, P β V β IsBalanced P β§ P.totalDegree β€ 2009) :
Module.rank β V = putnam_2009_b4_solution :=
sorry | 04f8f74bb4326dc7 |
putnam_1990_a1 | abbrev putnam_1990_a1_solution : (β β β€) Γ (β β β€) := sorry
-- (fun n : β => (n)!, fun n : β => 2 ^ n)
/--
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.
-/
theorem putnam_1990_a1
(T : β β β€)
(hT012 : T 0 = 2 β§ T 1 = 3 β§ T 2 = 6)
(hTn : β n, T (n + 3) = (n + 7) * T (n + 2) - 4 * (n + 3) * T (n + 1) + (4 * n + 4) * T n) :
T = putnam_1990_a1_solution.1 + putnam_1990_a1_solution.2 :=
sorry | d2d1dd1ab9d17c87 |
putnam_1962_a4 | theorem putnam_1962_a4
(f : β β β)
(a b : β)
(hdiff : Differentiable β f β§ (Differentiable β (deriv f)))
(hfabs : β x β Set.Icc a b, |f x| β€ 1)
(hfppabs : β x β Set.Icc a b, |(iteratedDeriv 2 f) x| β€ 1)
(hlen2 : b - a β₯ 2)
: β x β Set.Icc a b, |(iteratedDeriv 1 f) x| β€ 2 :=
sorry | 07e44426c898f47e |
putnam_2015_a1 | theorem putnam_2015_a1
(hyperbola : Set (Fin 2 β β))
(hhyperbola : hyperbola = {p | p 1 = 1 / p 0 β§ p 0 > 0})
(A B P : Fin 2 β β)
(PPline : (Fin 2 β β) β (Fin 2 β β) β (β β β))
(hAB : A β hyperbola β§ B β hyperbola β§ A 0 < B 0)
(hP : P β hyperbola β§ A 0 < P 0 β§ P 0 < B 0 β§ (β P', (P' β hyperbola β§ A 0 < P' 0 β§ P' 0 < B 0) β MeasureTheory.volume (convexHull β {A, P', B}) β€ MeasureTheory.volume (convexHull β {A, P, B})))
(hPPline : β P1 P2, P1 0 β P2 0 β PPline P1 P2 = (fun x : β => ((P2 1 - P1 1) / (P2 0 - P1 0)) * (x - P1 0) + P1 1)) :
β« x in Set.Ioo (A 0) (P 0), (PPline A P) x - 1 / x = β« x in Set.Ioo (P 0) (B 0), (PPline P B) x - 1 / x :=
sorry | 2b4e8b85684cc118 |
putnam_2005_a1 | theorem putnam_2005_a1
: β n : β€, n > 0 β (β k : β, β a : Fin k β Fin 2 β β, n = β i : Fin k, 2^(a i 0)*3^(a i 1) β§
(β i j : Fin k, i β j β Β¬(2^(a i 0)*3^(a i 1) β£ 2^(a j 0)*3^(a j 1)))) :=
sorry | 98d9d51fd1a5616b |
putnam_1998_b6 | theorem putnam_1998_b6
: β a b c : β€, β n : β€, n > 0 β§ Β¬(β k : β€, k = Real.sqrt (n^3 + a * n^2 + b * n + c)) :=
sorry | 2f4a4fab61d51fc8 |
putnam_2020_b1 | abbrev putnam_2020_b1_solution : β := sorry
-- 1990
/--
For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13) = 1+1+0+1=3)$. Let
\[
S = \sum_{k=1}^{2020} (-1)^{d(k)} k^3.
\]
Determine $S$ modulo 2020.
-/
theorem putnam_2020_b1
(d : β β β)
(S : β€)
(hd : d = fun n : β => β i : Fin (Nat.digits 2 n).length, (Nat.digits 2 n)[i]!)
(hS : S = β k : Icc 1 2020, ((-1 : β€)^(d k))*(k : β€)^3)
: S % 2020 = putnam_2020_b1_solution :=
sorry | bfe166d863d570e9 |
putnam_1988_b6 | theorem putnam_1988_b6
(trinums : Set β€)
(htrinums : trinums = {t : β€ | β n : β€, t β₯ 0 β§ t = (n * (n + 1)) / 2})
: {(a, b) : β€ Γ β€ | β t > 0, (a * t + b) β trinums β t β trinums}.encard = β€ :=
sorry | 292629636c97a7a0 |
putnam_2012_a5 | abbrev putnam_2012_a5_solution : Set (β Γ β) := sorry
-- {q | let β¨n, _β© := q; n = 1} βͺ {(2,2)}
/--
Let $\FF_p$ denote the field of integers modulo a prime $p$, and let $n$ be a positive integer. Let $v$ be a fixed vector in $\FF_p^n$, let $M$ be an $n \times n$ matrix with entries of $\FF_p$, and define $G: \FF_p^n \to \FF_p^n$ by $G(x) = v + Mx$. Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x) = G(x)$ and $G^{(k+1)}(x) = G(G^{(k)}(x))$. Determine all pairs $p, n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0)$, $k=1,2,\dots,p^n$ are distinct.
-/
theorem putnam_2012_a5
(n p : β)
(hn : n > 0)
(hp : Nat.Prime p)
{F : Type*} [Field F] [Fintype F]
(hK : Fintype.card F = p)
(G : Matrix (Fin n) (Fin n) F β (Fin n β F) β (Fin n β F) β (Fin n β F))
(hG : β M v x, G M v x = v + mulVec M x) :
(n, p) β putnam_2012_a5_solution β
βα΅ (M : Matrix (Fin n) (Fin n) F) (v : (Fin n β F)),
Β¬(β i j : Finset.range (p^n), i β j β§ (G M v)^[i + 1] 0 = (G M v)^[j + 1] 0) :=
sorry | 5b539c72140f54b9 |
putnam_2002_a5 | theorem putnam_2002_a5
(a : β β β)
(ha : a 0 = 1 β§ β n : β, a (2*n + 1) = a n β§ a (2*n + 2) = a n + a (n + 1))
: β q : β, q > 0 β q β {a (n - 1) / a n | n β Ici 1} :=
sorry | b1c38066bbd001eb |
putnam_1987_a5 | abbrev putnam_1987_a5_solution : Prop := sorry
-- False
/--
Let $\vec{G}(x,y)=\left(\frac{-y}{x^2+4y^2},\frac{x}{x^2+4y^2},0\right)$. Prove or disprove that there is a vector-valued function $\vec{F}(x,y,z)=(M(x,y,z),N(x,y,z),P(x,y,z))$ with the following properties:
\begin{enumerate}
\item[(i)] $M$, $N$, $P$ have continuous partial derivatives for all $(x,y,z) \neq (0,0,0)$;
\item[(ii)] $\text{Curl}\,\vec{F}=\vec{0}$ for all $(x,y,z) \neq (0,0,0)$;
\item[(iii)] $\vec{F}(x,y,0)=\vec{G}(x,y)$.
\end{enumerate}
-/
theorem putnam_1987_a5
(curl : ((Fin 3 β β) β (Fin 3 β β)) β ((Fin 3 β β) β (Fin 3 β β)))
(curl_def : β f x, curl f x = ![
fderiv β f x (Pi.single 1 1) 2 - fderiv β f x (Pi.single 2 1) 1,
fderiv β f x (Pi.single 2 1) 0 - fderiv β f x (Pi.single 0 1) 2,
fderiv β f x (Pi.single 0 1) 1 - fderiv β f x (Pi.single 1 1) 0])
(G : (Fin 2 β β) β (Fin 3 β β))
(G_def : β x y, G ![x, y] = ![(-y / (x ^ 2 + 4 * y ^ 2)), (x / (x ^ 2 + 4 * y ^ 2)), 0]) :
(β F : (Fin 3 β β) β (Fin 3 β β),
ContDiffOn β 1 F {v | v β ![0,0,0]} β§
(β x, x β 0 β curl F x = 0) β§
β x y, F ![x, y, 0] = G ![x, y]) β putnam_1987_a5_solution :=
sorry | 42c4ce572b9a6ea1 |
putnam_1997_a5 | abbrev putnam_1997_a5_solution : Prop := sorry
-- True
/--
Let $N_n$ denote the number of ordered $n$-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ such that $1/a_1 + 1/a_2 +\ldots + 1/a_n=1$. Determine whether $N_{10}$ is even or odd.
-/
theorem putnam_1997_a5
(N : (n : β+) β Set (Fin n β β+))
(hN : N = fun (n : β+) => {t : Fin n β β+ | (β i j : Fin n, i < j β t i <= t j) β§ (β i : Fin n, (1 : β)/(t i) = 1) })
: Odd (N 10).ncard β putnam_1997_a5_solution :=
sorry | eaa9f009fdd7db7b |
putnam_1996_a4 | theorem putnam_1996_a4
(A : Type*)
[Finite A]
(S : Set (A Γ A Γ A))
(hSdistinct : β a b c : A, β¨a, b, cβ© β S β a β b β§ b β c β§ a β c)
(hS1 : β a b c : A, β¨a, b, cβ© β S β β¨b, c, aβ© β S)
(hS2 : β a b c : A, a β c β (β¨a, b, cβ© β S β β¨c, b, aβ© β S))
(hS3 : β a b c d : A, (β¨a, b, cβ© β S β§ β¨c, d, aβ© β S) β (β¨b,c,dβ© β S β§ β¨d,a,bβ© β S))
: β g : A β β, Injective g β§ (β a b c : A, g a < g b β§ g b < g c β β¨a,b,cβ© β S) :=
sorry | f2696344c68b6e93 |
putnam_1986_a4 | abbrev putnam_1986_a4_solution : β Γ β Γ β Γ β Γ β Γ β Γ β := sorry
-- (1, 4, 2, 3, -4, 2, 1)
/--
A \emph{transversal} of an $n\times n$ matrix $A$ consists of $n$ entries of $A$, no two in the same row or column. Let $f(n)$ be the number of $n \times n$ matrices $A$ satisfying the following two conditions:
\begin{enumerate}
\item[(a)] Each entry $\alpha_{i,j}$ of $A$ is in the set
$\{-1,0,1\}$.
\item[(b)] The sum of the $n$ entries of a transversal is the same for all transversals of $A$.
\end{enumerate}
An example of such a matrix $A$ is
\[
A = \left( \begin{array}{ccc} -1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 1 & 0
\end{array}
\right).
\]
Determine with proof a formula for $f(n)$ of the form
\[
f(n) = a_1 b_1^n + a_2 b_2^n + a_3 b_3^n + a_4,
\]
where the $a_i$'s and $b_i$'s are rational numbers.
-/
theorem putnam_1986_a4
(f : β β β)
(hf : f = fun n β¦
Set.ncard {A : Matrix (Fin n) (Fin n) β€ |
(β i j : Fin n, A i j β ({-1, 0, 1} : Set β€)) β§
β S : β€, β Ο : Perm (Fin n), β i : Fin n, A i (Ο i) = S}) :
let (a1, b1, a2, b2, a3, b3, a4) := putnam_1986_a4_solution;
(β n > 0, f n = a1 * b1 ^ n + a2 * b2 ^ n + a3 * b3 ^ n + a4) :=
sorry | 80f272a45d7a05bf |
putnam_2003_a4 | theorem putnam_2003_a4
(a b c A B C : β)
(aAne0 : a β 0 β§ A β 0)
(hle : β x : β, |a * x ^ 2 + b * x + c| β€ |A * x ^ 2 + B * x + C|)
: |b ^ 2 - 4 * a * c| β€ |B ^ 2 - 4 * A * C| :=
sorry | 5e1afdc6fc666a8b |
putnam_2013_a4 | theorem putnam_2013_a4
(n k : β)
(circle : Fin n β Fin 2)
(Z N : Fin n Γ Fin (n + 1) β β€)
(ws : Fin k β Fin n Γ Fin (n + 1))
(Zsum Nsum : β€)
(npos : n β₯ 1) (kpos : k β₯ 1)
(hZ : β w, Z w = β l : {x : Fin n | x < w.2}, if (circle (w.1 + l) = 0) then 1 else 0)
(hN : β w, N w = β l : {x : Fin n | x < w.2}, if (circle (w.1 + l) = 1) then 1 else 0)
(Zle1 : β w w', w.2 = w'.2 β |(Z w : β€) - Z w'| β€ 1)
(hZsum : Zsum = ((1 : β) / k) * β j : Fin k, Z (ws j))
(hNsum : Nsum = ((1 : β) / k) * β j : Fin k, N (ws j)) :
β w, Z w = Zsum β§ N w = Nsum :=
sorry | 204a63ef798410c6 |
putnam_1964_a1 | theorem putnam_1964_a1
(A : Finset (EuclideanSpace β (Fin 2)))
(hAcard : A.card = 6)
(dists : Set β)
(hdists : dists = {d : β | β a b : EuclideanSpace β (Fin 2), a β A β§ b β A β§ a β b β§ d = dist a b})
: (sSup dists / sInf dists β₯ Real.sqrt 3) :=
sorry | ffe2cfb1445dd5cb |
putnam_1974_a1 | abbrev putnam_1974_a1_solution : β := sorry
-- 11
/--
Call a set of positive integers 'conspiratorial' if no three of them are pairwise relatively prime. What is the largest number of elements in any conspiratorial subset of the integers 1 through 16?
-/
theorem putnam_1974_a1
(conspiratorial : Set β€ β Prop)
(hconspiratorial : β S, conspiratorial S β β a β S, β b β S, β c β S, (a > 0 β§ b > 0 β§ c > 0) β§ ((a β b β§ b β c β§ a β c) β (Int.gcd a b > 1 β¨ Int.gcd b c > 1 β¨ Int.gcd a c > 1))) :
IsGreatest {k | β S, S β Icc 1 16 β§ conspiratorial S β§ S.encard = k} putnam_1974_a1_solution :=
sorry | 60d08a315d484233 |
putnam_2010_a2 | abbrev putnam_2010_a2_solution : Set (β β β) := sorry
-- {f : β β β | β c d : β, β x : β, f x = c*x + d}
/--
Find all differentiable functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[
f'(x) = \frac{f(x+n)-f(x)}{n}
\]
for all real numbers $x$ and all positive integers $n$.
-/
theorem putnam_2010_a2
: {f : β β β | Differentiable β f β§
β x : β, β n : β€, n > 0 β deriv f x = (f (x + n) - f x)/n} = putnam_2010_a2_solution :=
sorry | 4d77a7aef547218a |
putnam_2000_a2 | theorem putnam_2000_a2 :
β n : β,
β N : β€,
β i : Fin 6 β β,
N > n β§
N = (i 0)^2 + (i 1)^2 β§
N + 1 = (i 2)^2 + (i 3)^2 β§
N + 2 = (i 4)^2 + (i 5)^2 :=
sorry | fab592829ad84d5d |
putnam_2022_b6 | abbrev putnam_2022_b6_solution : Set (Set.Ioi (0 : β) β Set.Ioi (0 : β)) := sorry
-- {f : Set.Ioi (0 : β) β Set.Ioi (0 : β) | β c : β, c β₯ 0 β§ β x : Set.Ioi (0 : β), f x = 1 / (1 + c * x)}
/--
Find all continuous functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that $f(xf(y))+f(yf(x))=1+f(x+y)$ for all $x,y>0$.
-/
theorem putnam_2022_b6
(f : Set.Ioi (0 : β) β Set.Ioi (0 : β))
(hf : Continuous f) :
(β fr : β β β, (β x : Set.Ioi (0 : β), fr x = f x) β§ (β x y : Set.Ioi (0 : β), fr (x * fr y) + fr (y * fr x) = 1 + fr (x + y)))
β f β putnam_2022_b6_solution :=
sorry | d2e1a45a5745ed22 |
putnam_1968_b4 | theorem putnam_1968_b4
(f : β β β)
(hf : Continuous f β§ β r : β, Tendsto (fun y => β« x in ball 0 y, f x) atTop (π r))
: β r : β, Tendsto (fun y => β« x in ball 0 y, f (x - 1/x)) atTop (π r) β§ Tendsto (fun y => β« x in ball 0 y, f x) atTop (π r) :=
sorry | a81fc2bfcca1b679 |
putnam_1978_b4 | theorem putnam_1978_b4 :
β N : β,
β a b c d : β€,
a > N β§ b > N β§ c > N β§ d > N β§
a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = a * b * c + a * b * d + a * c * d + b * c * d :=
sorry | 0c6ba2a0798a0c89 |
putnam_1995_a2 | abbrev putnam_1995_a2_solution : Set (β Γ β) := sorry
-- {x | let β¨a,bβ© := x; a = b}
/--
For what pairs $(a,b)$ of positive real numbers does the improper integral \[ \int_{b}^{\infty} \left( \sqrt{\sqrt{x+a}-\sqrt{x}} - \sqrt{\sqrt{x}-\sqrt{x-b}} \right)\,dx \] converge?
-/
theorem putnam_1995_a2
(habconv : (β Γ β) β Prop)
(habconv_def : habconv = fun β¨a,bβ© =>
β limit : β, Tendsto (fun t : β => β« x in (Set.Icc b t), (sqrt (sqrt (x + a) - sqrt x) - sqrt (sqrt x - sqrt (x - b)))) atTop (π limit))
: β ab : β Γ β, ab.1 > 0 β§ ab.2 > 0 β (habconv ab β ab β putnam_1995_a2_solution) :=
sorry | 552b8614132406f8 |
putnam_2008_b5 | abbrev putnam_2008_b5_solution : Set (β β β) := sorry
-- {fun (x : β) => x + n | n : β€} βͺ {fun (x : β) => -x + n | n : β€}
/--
Find all continuously differentiable functions f : \mathbb{R} \to \mathbb{R} such that for every rational number $q$, the number $f(q)$ is rational and has the same denominator as $q$.
-/
theorem putnam_2008_b5
(fqsat : (β β β) β β β Prop)
(hfqsat : β f q, fqsat f q β ContDiff β 1 f β§ (β p : β, p = f q β§ p.den = q.den))
: β f : (β β β), (β q : β, fqsat f q) β f β putnam_2008_b5_solution :=
sorry | 14d6bf155986a7fe |
putnam_2018_b5 | theorem putnam_2018_b5
(f : (Fin 2 β β) β (Fin 2 β β))
(hβ : ContDiff β 1 f)
(hβ : β x i j, 0 < fderiv β f x (Pi.single i 1) j)
(hβ : β x, 0 < fderiv β f x ![1, 0] 0 * fderiv β f x ![0, 1] 1 -
(1 / 4) * (fderiv β f x ![1, 0] 1 + fderiv β f x ![0, 1] 0) ^ 2) :
Injective f :=
sorry | 67968da07222090a |
putnam_1982_a6 | abbrev putnam_1982_a6_solution : Prop := sorry
-- False
/--
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 $$\lim_{n \rightarrow \infty} \sum_{k = 1}^{n} x_{b(k)} = 1.$$
-/
theorem putnam_1982_a6 :
(β b : β β β,
β x : β β β,
BijOn b (Ici 1) (Ici 1) β
StrictAntiOn (fun n : β => |x n|) (Ici 1) β
Tendsto (fun n : β => |b n - (n : β€)| * |x n|) atTop (π 0) β
Tendsto (fun n : β => β k in Finset.Icc 1 n, x k) atTop (π 1) β
Tendsto (fun n : β => β k in Finset.Icc 1 n, x (b k)) atTop (π 1))
β putnam_1982_a6_solution :=
sorry | 2537ff48a39275c4 |
putnam_1970_a3 | abbrev putnam_1970_a3_solution : β Γ β := sorry
-- (3, 1444)
/--
Find the length of the longest possible sequence of equal nonzero digits (in base 10) in which a perfect square can terminate. Also, find the smallest square that attains this length.
-/
theorem putnam_1970_a3
(L : β β β)
(hL : β n : β, L n β€ (Nat.digits 10 n).length β§
(β k : β, k < L n β (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) β§
(L n β (Nat.digits 10 n).length β (Nat.digits 10 n)[L n]! β (Nat.digits 10 n)[0]!))
: (β n : β, (Nat.digits 10 (n^2))[0]! β 0 β§ L (n^2) = putnam_1970_a3_solution.1) β§
(β n : β, (Nat.digits 10 (n^2))[0]! β 0 β L (n^2) β€ putnam_1970_a3_solution.1) β§
(β m : β, m^2 = putnam_1970_a3_solution.2) β§
L (putnam_1970_a3_solution.2) = putnam_1970_a3_solution.1 β§
(Nat.digits 10 putnam_1970_a3_solution.2)[0]! β 0 β§
β n : β, (Nat.digits 10 (n^2))[0]! β 0 β§ L (n^2) = putnam_1970_a3_solution.1 β n^2 β₯ putnam_1970_a3_solution.2 :=
sorry | 0962cb2520b663af |
putnam_1999_a1 | abbrev putnam_1999_a1_solution : Prop := sorry
-- True
/--
Find polynomials $f(x)$,$g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \mbox{if $x<-1$} \\3x+2 & \mbox{if $-1 \leq x \leq 0$} \\-2x+2 & \mbox{if $x>0$.}\end{cases}\]?
-/
theorem putnam_1999_a1
: putnam_1999_a1_solution β β f g h : Polynomial β, β x : β, |f.eval x| - |g.eval x| + h.eval x = if x < -1 then -1 else (if (x β€ 0) then 3 * x + 2 else -2 * x + 2) :=
sorry | dede9d9b05cb2b93 |
putnam_1989_a1 | abbrev putnam_1989_a1_solution : ββ := sorry
-- 1
/--
How many primes among the positive integers, written as usual in base $10$, are alternating $1$'s and $0$'s, beginning and ending with $1$?
-/
theorem putnam_1989_a1
(pdigalt : List β β Prop)
(hpdigalt : β l, pdigalt l β Odd l.length β§ (β i, l.get i = if Even (i : β) then 1 else 0)) :
{p : β | p.Prime β§ pdigalt (Nat.digits 10 p)}.encard = putnam_1989_a1_solution :=
sorry | a43b50a5a8781215 |
putnam_2021_a6 | abbrev putnam_2021_a6_solution : Prop := sorry
-- True
/--
Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as a product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer?
-/
theorem putnam_2021_a6 :
(β P, (β n : β, P.coeff n = 0 β¨ P.coeff n = 1) β
(β Q R : Polynomial β€, Q.degree > 0 β§ R.degree > 0 β§ P = Q * R) β
(P.eval 2 β 0 β§ P.eval 2 β 1 β§ Β¬Prime (P.eval 2)))
β putnam_2021_a6_solution :=
sorry | 351bd7c586a5a890 |
putnam_2013_b2 | abbrev putnam_2013_b2_solution : β := sorry
-- 3
/--
Let $C = \bigcup_{N=1}^\infty C_N$, where $C_N$ denotes the set of those `cosine polynomials' of the form
\[
f(x) = 1 + \sum_{n=1}^N a_n \cos(2 \pi n x)
\]
for which:
\begin{enumerate}
\item[(i)]
$f(x) \geq 0$ for all real $x$, and
\item[(ii)]
$a_n = 0$ whenever $n$ is a multiple of $3$.
\end{enumerate}
Determine the maximum value of $f(0)$ as $f$ ranges through $C$, and
prove that this maximum is attained.
-/
theorem putnam_2013_b2
(CN : β β Set (β β β))
(hCN : β N : β, CN N =
{f : β β β |
(β x : β, f x β₯ 0) β§
β a : List β, a.length = N + 1 β§ (β n : Fin (N + 1), 3 β£ (n : β) β a[n]! = 0) β§
β x : β, f x = 1 + β n in Finset.Icc 1 N, a[(n : β)]! * Real.cos (2*Real.pi*n*x)}) :
IsGreatest {f 0 | f β β N β Ici 1, CN N} putnam_2013_b2_solution :=
sorry | 58912708f948fe95 |
putnam_2003_b2 | theorem putnam_2003_b2
(n : β)
(hn : n > 0)
(seq : β β β β β)
(hinit : β j β Icc 1 n, seq 0 j = 1 / j)
(havg : βα΅ (k β Icc 1 (n - 1)) (j β Icc 1 (n - k)),
seq k j = (seq (k - 1) j + seq (k - 1) (j + 1)) / 2) :
(seq (n - 1) 1 < 2 / n) :=
sorry | cae37405a698fc81 |
putnam_1986_b2 | abbrev putnam_1986_b2_solution : Finset (β Γ β Γ β) := sorry
-- {(0, 0, 0), (0, -1, 1), (1, 0, -1), (-1, 1, 0)}
/--
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 $T$.
-/
theorem putnam_1986_b2
: ({T : β Γ β Γ β | β x y z : β, T = (x - y, y - z, z - x) β§ x * (x - 1) + 2 * y * z = y * (y - 1) + 2 * z * x β§ y * (y - 1) + 2 * z * x = z * (z - 1) + 2 * x * y} = putnam_1986_b2_solution) :=
sorry | 7ca24872ebc85edd |
putnam_1996_b2 | theorem putnam_1996_b2
(n : β)
(prododd : β)
(npos : n > 0)
(hprododd : prododd = β i in Finset.range (2 * n), if Odd i then i else 1)
: ((2 * n - 1 : β) / Real.exp 1) ^ ((2 * n - 1 : β) / 2) < prododd β§ prododd < ((2 * n + 1 : β) / Real.exp 1) ^ ((2 * n + 1 : β) / 2) :=
sorry | f764c43ba48d4e7c |
putnam_1991_b6 | abbrev putnam_1991_b6_solution : β β β β β := sorry
-- (fun a b : β => |Real.log (a / b)|)
/--
Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $a^xb^{1-x} \leq a\frac{\sinh ux}{\sinh u}+b\frac{\sinh u(1-x)}{\sinh u}$ for all $u$ with $0<|u| \leq c$ and for all $x$, $0<x<1$. (Note: $\sinh u=(e^u-e^{-u})/2$.)
-/
theorem putnam_1991_b6
(a b : β)
(abpos : a > 0 β§ b > 0) :
IsGreatest {c | β u,
(0 < |u| β§ |u| β€ c) β
(β x β Set.Ioo 0 1, a ^ x * b ^ (1 - x) β€ a * (Real.sinh (u * x) / Real.sinh u) + b * (Real.sinh (u * (1 - x)) / Real.sinh u))}
(putnam_1991_b6_solution a b) :=
sorry | 2ec1dcbb9ed8e25b |
putnam_1973_b3 | theorem putnam_1973_b3
(p : β)
(pgt1 : p > 1)
(hprime : β x β Set.Ico 0 p, Nat.Prime (x^2 - x + p))
: β! triple : β€ Γ β€ Γ β€, let (a,b,c) := triple; b^2 - 4*a*c = 1 - 4*p β§ 0 < a β§ a β€ c β§ -a β€ b β§ b < a :=
sorry | b9fffce99b844962 |
putnam_1963_b3 | abbrev putnam_1963_b3_solution : Set (β β β) := sorry
-- {(fun u : β => A * Real.sinh (k * u)) | (A : β) (k : β)} βͺ {(fun u : β => A * u) | A : β} βͺ {(fun u : β => A * Real.sin (k * u)) | (A : β) (k : β)}
/--
Find every twice-differentiable real-valued function $f$ with domain the set of all real numbers and satisfying the functional equation $(f(x))^2-(f(y))^2=f(x+y)f(x-y)$ for all real numbers $x$ and $y$.
-/
theorem putnam_1963_b3
(f : β β β) :
f β putnam_1963_b3_solution β
(ContDiff β 1 f β§ Differentiable β (deriv f) β§
β x y : β, (f x) ^ 2 - (f y) ^ 2 = f (x + y) * f (x - y)) :=
sorry | 9c8d1c20b5b6a28b |
putnam_2014_b6 | theorem putnam_2014_b6
(f : β β β)
(hlip : β K > 0, β x β Icc 0 1, β y β Icc 0 1, |f x - f y| β€ K * |x - y|)
(hrat : β r β Icc (0 : β) 1, β a b : β€, f r = a + b * r)
: (β I : Finset (Interval β), (β Ii β I, β m c : β, β x β coeHom Ii, f x = c + m * x) β§ Icc 0 1 = β Ii β I, coeHom Ii) :=
sorry | de86d91d593a89a4 |
putnam_2004_b6 | theorem putnam_2004_b6
(A B : Set β)
(N : β β β)
(b : β β β)
(Anempty : A.Nonempty)
(Apos : β a β A, a > 0)
(hN : β x : β, N x = Set.encard {a : A | a β€ x})
(hB : B = {b' > 0 | β a β A, β a' β A, b' = a - a'})
(hbB : Set.range b = B β§ β i : β, b i < b (i + 1)) :
(β r : β, β i : β, (b (i + 1) - b i) β₯ r) β Tendsto (fun x => N x / x) atTop (π 0) :=
sorry | 0203e815dc76cd22 |
putnam_1970_b5 | theorem putnam_1970_b5
(ramp : β€ β (β β β))
(ramp_def : ramp = fun (n : β€) => (fun (x : β) => if x β€ -n then (-n : β) else (if -n < x β§ x β€ n then x else (n : β))))
(F : β β β)
: Continuous F β (β n : β, Continuous ((ramp n) β F)) :=
sorry | e31c61de0129ba50 |
putnam_2018_a3 | abbrev putnam_2018_a3_solution : β := sorry
-- 480/49
/--
Determine the greatest possible value of $\sum_{i=1}^{10} \cos(3x_i)$ for real numbers $x_1, x_2, \ldots, x_{10}$ satisfying $\sum_{i=1}^{10} \cos(x_i) = 0$.
-/
theorem putnam_2018_a3 :
IsGreatest
{β i, Real.cos (3 * x i) | (x : Fin 10 β β) (hx : β i, Real.cos (x i) = 0)}
putnam_2018_a3_solution :=
sorry | c0c2eecfcb2c7327 |
putnam_2008_a3 | theorem putnam_2008_a3
(n : β) (npos : n > 0)
(a : Fin n β β)
(apos : β i : Fin n, a i > 0)
(cont : (Fin n β β) β Prop)
(hcont : β s, cont s β β j k : Fin n, j < k β§ Β¬(s j β£ s k))
(init trans : (β β Fin n β β) β Prop)
(hinit : β P, init P β P 0 = a)
(htrans : β P, trans P β β t : β, cont (P t) β
β j k : Fin n, j < k β§ Β¬(P t j β£ P t k) β§ P (t + 1) j = Nat.gcd (P t j) (P t k) β§ P (t + 1) k = Nat.lcm (P t j) (P t k) β§
β i : Fin n, i β j β i β k β P (t + 1) i = P t i) :
(β f, β P, init P β trans P β β t : β, Β¬cont (P t) β§ P t = f) :=
sorry | 90a1b411182cb7ab |
putnam_1995_b4 | abbrev putnam_1995_b4_solution : β€ Γ β€ Γ β€ Γ β€ := sorry
-- β¨3,1,5,2β©
/--
Evaluate \[ \sqrt[8]{2207 - \frac{1}{2207-\frac{1}{2207-\dots}}}. \] Express your answer in the form $\frac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers.
-/
theorem putnam_1995_b4
(contfrac : β)
(hcontfrac : contfrac = 2207 - 1 / contfrac)
(hcontfrac' : 1 < contfrac) :
let β¨a, b, c, dβ© := putnam_1995_b4_solution
contfrac ^ ((1 : β) / 8) = (a + b * sqrt c) / d :=
sorry | 1cea4910949c423e |
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