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putnam_2007_a1 | abbrev putnam_2007_a1_solution : Set β := sorry
-- {2 / 3, 3 / 2, (13 + β601) / 12, (13 - β601) / 12}
/--
Find all values of $\alpha$ for which the curves $y = \alpha*x^2 + \alpha*x + 1/24$ and $x = \alpha*y^2 + \alpha*y + 1/24$ are tangent to each other.
-/
theorem putnam_2007_a1
(P : (β β β) β Prop)
(P_def : β f, P f β β x y, f x = y β§ f y = x β§ deriv f x * deriv f y = 1)
(Ξ± : β) :
P (fun t β¦ Ξ± * t ^ 2 + Ξ± * t + 1 / 24) β Ξ± β putnam_2007_a1_solution :=
sorry | dc281f82829d463e |
putnam_2017_a1 | abbrev putnam_2017_a1_solution : Set β€ := sorry
-- {x : β€ | x > 0 β§ (x = 1 β¨ 5 β£ x)}
/--
Let $S$ be the smallest set of positive integers such that (a) $2$ is in $S$, (b) $n$ is in $S$ whenever $n^2$ is in $S$, and (c) $(n+5)^2$ is in $S$ whenever $n$ is in $S$. Which positive integers are not in $S$?.
-/
theorem putnam_2017_a1
(IsQualifying : Set β€ β Prop)
(IsQualifying_def : β S, IsQualifying S β
(β n β S, 0 < n) β§
2 β S β§
(β n > 0, n ^ 2 β S β n β S) β§
(β n β S, (n + 5) ^ 2 β S))
(S : Set β€)
(hS : IsLeast IsQualifying S) :
SαΆ β© {n | 0 < n} = putnam_2017_a1_solution :=
sorry | 71bf451d20ae8bbc |
putnam_1970_a4 | theorem putnam_1970_a4
(x : β β β)
(hxlim : Tendsto (fun n => x (n+2) - x n) atTop (π 0))
: Tendsto (fun n => (x (n+1) - x (n))/(n+1)) atTop (π 0) :=
sorry | 2f3009b9e8fe511c |
putnam_2018_b2 | theorem putnam_2018_b2
(n : β)
(hn : n > 0)
(f : β β β β β)
(hf : β z : β, f n z = β i in Finset.range n, (n - i) * z^i)
: β z : β, βzβ β€ 1 β f n z β 0 :=
sorry | 855548fadff25ed1 |
putnam_2008_b2 | abbrev putnam_2008_b2_solution : β := sorry
-- -1
/--
Let $F_0(x)=\ln x$. For $n \geq 0$ and $x>0$, let $F_{n+1}(x)=\int_0^x F_n(t)\,dt$. Evaluate $\lim_{n \to \infty} \frac{n!F_n(1)}{\ln n}$.
-/
theorem putnam_2008_b2
(F : β β β β β)
(hF0 : β x : β, F 0 x = Real.log x)
(hFn : β n : β, β x > 0, F (n + 1) x = β« t in Set.Ioo 0 x, F n t)
: Tendsto (fun n : β => ((n)! * F n 1) / Real.log n) atTop (π putnam_2008_b2_solution) :=
sorry | 4b0014f08d8ea741 |
putnam_1992_a1 | theorem putnam_1992_a1
(f : β€ β β€) :
(f = fun n β¦ 1 - n) β
(β n : β€, f (f n) = n) β§ (β n : β€, f (f (n + 2) + 2) = n) β§ (f 0 = 1) :=
sorry | 198846b5aa1a01eb |
putnam_1978_b3 | theorem putnam_1978_b3
(P : β+ β Polynomial β)
(hP1 : P 1 = 1 + X)
(hP2 : P 2 = 1 + 2 * X)
(hPodd : β n, P (2 * n + 1) = P (2 * n) + C ((n : β) + 1) * X * P (2 * n - 1))
(hPeven : β n, P (2 * n + 2) = P (2 * n + 1) + C ((n : β) + 1) * X * P (2 * n))
(a : β+ β β)
(haroot : β n, (P n).eval (a n) = 0)
(haub : β n, β x, (P n).eval x = 0 β x β€ a n)
: (StrictMono a β§ Tendsto a atTop (π 0)) :=
sorry | 62e406761e8d5e3d |
putnam_1995_a5 | abbrev putnam_1995_a5_solution : Prop := sorry
-- True
/--
Let $x_{1},x_{2},\dots,x_{n}$ be differentiable (real-valued) functions of a single variable $f$ which satisfy \begin{align*} \frac{dx_{1}}{dt} &= a_{11}x_{1} + a_{12}x_{2} + \cdots + a_{1n}x_{n} \ \frac{dx_{2}}{dt} &= a_{21}x_{1} + a_{22}x_{2} + \cdots + a_{2n}x_{n} \ \vdots && \vdots \ \frac{dx_{n}}{dt} &= a_{n1}x_{1} + a_{n2}x_{2} + \cdots + a_{nn}x_{n} \end{align*} for some constants $a_{ij}>0$. Suppose that for all $i$, $x_{i}(t) \to 0$ as $t \to \infty$. Are the functions $x_{1},x_{2},\dots,x_{n}$ necessarily linearly dependent?
-/
theorem putnam_1995_a5 :
putnam_1995_a5_solution β
(β (n : β) (x : Fin n β (β β β)) (a : Fin n β Fin n β β),
(0 < n) β
(β i, Differentiable β (x i)) β
(β i j, a i j > 0) β
(β t i, (deriv (x i)) t = β j : Fin n, (a i j) * ((x j) t)) β
(β i, Tendsto (x i) atTop (π 0)) β
Β¬(β b : Fin n β β, (β t : β, β i : Fin n, (b i) * ((x i) t) = 0) β
(β i, b i = 0))) :=
sorry | f7b0a2cf562e84c9 |
putnam_1985_a5 | abbrev putnam_1985_a5_solution : Set β := sorry
-- {3, 4, 7, 8}
/--
Let $I_m = \int_0^{2\pi} \cos(x)\cos(2x)\cdots \cos(mx)\,dx$. For which integers $m$, $1 \leq m \leq 10$ is $I_m \neq 0$?
-/
theorem putnam_1985_a5
(I : β β β)
(hI : I = fun (m : β) β¦ β« x in (0)..(2 * Real.pi), β k in Finset.Icc 1 m, cos (k * x)) :
{m β Finset.Icc 1 10 | I m β 0} = putnam_1985_a5_solution :=
sorry | ef7a41ef54fc02ed |
putnam_2000_a5 | theorem putnam_2000_a5
(r : β)
(z : EuclideanSpace β (Fin 2))
(p : Fin 3 β (EuclideanSpace β (Fin 2)))
(rpos : r > 0)
(pdiff : β n m, (n β m) β (p n β p m))
(pint : β n i, p n i = round (p n i))
(pcirc : β n, p n β Metric.sphere z r) :
β n m, (n β m) β§ (dist (p n) (p m) β₯ r ^ ((1 : β) / 3)) :=
sorry | ef7fb266130f62be |
putnam_2010_a5 | theorem putnam_2010_a5
(G : Type*) [Group G]
(i : G βͺ (Fin 3 β β))
(h : β a b, (i a) Γβ (i b) = i (a * b) β¨ (i a) Γβ (i b) = 0)
(a b : G) :
(i a) Γβ (i b) = 0 :=
sorry | 4d71dcea75a93105 |
putnam_2022_b1 | theorem putnam_2022_b1
(P : Polynomial β€)
(b : β β β)
(Pconst : P.coeff 0 = 0)
(Podd : Odd (P.coeff 1))
(hB : β x : β, HasSum (fun i => b i * x ^ i) (Real.exp (aeval x P))) :
β k : β, b k β 0 :=
sorry | eaa4336414551d9a |
putnam_2013_a3 | theorem putnam_2013_a3
(n : β)
(a : Set.Icc 0 n β β)
(x : β)
(hx : 0 < x β§ x < 1)
(hsum : (β i : Set.Icc 0 n, a i / (1 - x ^ (i.1 + 1))) = 0)
: β y : β, 0 < y β§ y < 1 β§ (β i : Set.Icc 0 n, a i * y ^ i.1) = 0 :=
sorry | c0cb05f1ff5f150f |
putnam_2003_a3 | abbrev putnam_2003_a3_solution : β := sorry
-- 2 * Real.sqrt 2 - 1
/--
Find the minimum value of $|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x|$ for real numbers $x$.
-/
theorem putnam_2003_a3
(f : β β β)
(hf : β x : β, f x = |Real.sin x + Real.cos x + Real.tan x + 1 / Real.tan x + 1 / Real.cos x + 1 / Real.sin x|) :
IsLeast (Set.range f) putnam_2003_a3_solution :=
sorry | 866efe559bcc093d |
putnam_1974_a6 | abbrev putnam_1974_a6_solution : β := sorry
-- 25
/--
Given $n$, let $k(n)$ be the minimal degree of any monic integral polynomial $f$ such that the value of $f(x)$ is divisible by $n$ for every integer $x$. Find the value of $k(1000000)$.
-/
theorem putnam_1974_a6
(hdivnallx : Polynomial β€ β Prop)
(hdivnallx_def : hdivnallx = fun f => Monic f β§ (β x : β€, (10^6 : β€) β£ f.eval x))
: sInf {d : β | β f : Polynomial β€, hdivnallx f β§ d = f.natDegree} = putnam_1974_a6_solution :=
sorry | c7007e4f9de05873 |
putnam_1964_a6 | theorem putnam_1964_a6
(S : Finset β)
(pairs : Set (β Γ β))
(hpairs : pairs = {(a, b) | (a β S) β§ (b β S) β§ (a < b)})
(distance : β Γ β β β)
(hdistance : distance = fun (a, b) β¦ b - a)
(hrepdist : β p β pairs, (β m β pairs, distance m > distance p) β β q β pairs, q β p β§ distance p = distance q)
: (β p q : pairs, q β p β β r : β, distance p / distance q = r) :=
sorry | f217b6a03ce6bac2 |
putnam_1986_a3 | abbrev putnam_1986_a3_solution : β := sorry
-- Real.pi / 2
/--
Evaluate $\sum_{n=0}^\infty \mathrm{Arccot}(n^2+n+1)$, where $\mathrm{Arccot}\,t$ for $t \geq 0$ denotes the number $\theta$ in the interval $0 < \theta \leq \pi/2$ with $\cot \theta = t$.
-/
theorem putnam_1986_a3
(cot : β β β)
(fcot : cot = fun ΞΈ β¦ cos ΞΈ / sin ΞΈ)
(arccot : β β β)
(harccot : β t : β, t β₯ 0 β arccot t β Set.Ioc 0 (Real.pi / 2) β§ cot (arccot t) = t)
: (β' n : β, arccot (n ^ 2 + n + 1) = putnam_1986_a3_solution) :=
sorry | e63058cdff236457 |
putnam_1996_a3 | abbrev putnam_1996_a3_solution : Prop := sorry
-- False
/--
Suppose that each of 20 students has made a choice of anywhere from 0 to 6 courses from a total of 6 courses offered. Prove or disprove: there are 5 students and 2 courses such that all 5 have chosen both courses or all 5 have chosen neither course.
-/
theorem putnam_1996_a3 :
(β choices : Fin 20 β Set (Fin 6),
β (students : Finset (Fin 20)) (courses : Finset (Fin 6)),
students.card = 5 β§
courses.card = 2 β§
(βcourses β β s β students, choices s β¨ βcourses β β s β students, (choices s)αΆ))
β putnam_1996_a3_solution :=
sorry | 2f88722f3195292b |
putnam_1973_a2 | abbrev putnam_1973_a2_solution : Prop := sorry
-- True
/--
Consider an infinite series whose $n$th term is given by $\pm \frac{1}{n}$, where the actual values of the $\pm$ signs repeat in blocks of $8$ (so the $\frac{1}{9}$ term has the same sign as the $\frac{1}{1}$ term, and so on). Call such a sequence balanced if each block contains four $+$ and four $-$ signs. Prove that being balanced is a sufficient condition for the sequence to converge. Is being balanced also necessary for the sequence to converge?
-/
theorem putnam_1973_a2
(L : List β)
(hL : L.length = 8 β§ β i : Fin L.length, L[i] = 1 β¨ L[i] = -1)
(pluses : β)
(hpluses : pluses = {i : Fin L.length | L[i] = 1}.ncard)
(S : β β β)
(hS : S = fun n : β β¦ β i in Finset.Icc 1 n, L[i % 8]/i)
: (pluses = 4 β β l : β, Tendsto S atTop (π l)) β§ (putnam_1973_a2_solution β ((β l : β, Tendsto S atTop (π l)) β pluses = 4)) :=
sorry | c51693e6eb479ee6 |
putnam_1963_a2 | theorem putnam_1963_a2
(f : β β β)
(hfpos : β n, f n > 0)
(hfinc : StrictMonoOn f (Set.Ici 1))
(hf2 : f 2 = 2)
(hfmn : β m n, m > 0 β n > 0 β IsRelPrime m n β f (m * n) = f m * f n)
: β n > 0, f n = n :=
sorry | 8ce4a2f57fd2f72d |
putnam_1987_a2 | abbrev putnam_1987_a2_solution : β := sorry
-- 1984
/--
The sequence of digits $123456789101112131415161718192021 \dots$ is obtained by writing the positive integers in order. If the $10^n$-th digit in this sequence occurs in the part of the sequence in which the $m$-digit numbers are placed, define $f(n)$ to be $m$. For example, $f(2)=2$ because the $100$th digit enters the sequence in the placement of the two-digit integer $55$. Find, with proof, $f(1987)$.
-/
theorem putnam_1987_a2
(seqind seqsize f : β β β)
(hseqind : seqind 1 = 1 β§ β i β₯ 2, seqind i = seqind (i - 1) + (Nat.digits 10 (i - 1)).length)
(hseqsize : β i β₯ 1, β j : Fin ((Nat.digits 10 i).length), seqsize (seqind i + j) = (Nat.digits 10 i).length)
(hf : β n : β, f n = seqsize (10 ^ n)) :
f 1987 = putnam_1987_a2_solution :=
sorry | 54933b2da1297c4a |
putnam_1988_b1 | theorem putnam_1988_b1
: β a β₯ 2, β b β₯ 2, β x y z : β€, x > 0 β§ y > 0 β§ z > 0 β§ a * b = x * y + x * z + y * z + 1 :=
sorry | 87eff43764113c48 |
putnam_2020_b6 | theorem putnam_2020_b6
(n : β)
(npos : n > 0)
: β k in Finset.Icc 1 n, ((-1) ^ Int.floor (k * (Real.sqrt 2 - 1)) : β) β₯ 0 :=
sorry | c4f422b1777e8f8e |
putnam_1998_b1 | abbrev putnam_1998_b1_solution : β := sorry
-- 6
/--
Find the minimum value of \[\frac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\] for $x>0$.
-/
theorem putnam_1998_b1
: sInf {((x + 1/x)^6 - (x^6 + 1/x^6) - 2)/((x + 1/x)^3 + (x^3 + 1/x^3)) | x > (0 : β)} = putnam_1998_b1_solution :=
sorry | f176c62769ce4d83 |
putnam_2002_a2 | theorem putnam_2002_a2
(unit_sphere : Set (EuclideanSpace β (Fin 3)))
(hsphere : unit_sphere = sphere 0 1)
(hemi : EuclideanSpace β (Fin 3) β Set (EuclideanSpace β (Fin 3)))
(hhemi : hemi = fun V β¦ {P : EuclideanSpace β (Fin 3) | βͺP, Vβ«_β β₯ 0})
: (β (S : Set (EuclideanSpace β (Fin 3))), S β unit_sphere β§ S.encard = 5 β β V : EuclideanSpace β (Fin 3), V β 0 β§ (S β© hemi V).encard β₯ 4) :=
sorry | 152f16dbdcdf9d5b |
putnam_2012_a2 | theorem putnam_2012_a2
(S : Type*) [CommSemigroup S]
(a b c : S)
(hS : β x y : S, β z : S, x * z = y)
(habc : a * c = b * c)
: a = b :=
sorry | 9c82af6e9122e750 |
putnam_1962_a3 | theorem putnam_1962_a3
(A B C A' B' C' P Q R : EuclideanSpace β (Fin 2))
(k : β)
(hk : k > 0)
(hABC : Β¬Collinear β {A, B, C})
(hA' : A' β segment β B C β§ dist C A' / dist A' B = k)
(hB' : B' β segment β C A β§ dist A B' / dist B' C = k)
(hC' : C' β segment β A B β§ dist B C' / dist C' A = k)
(hP : P β segment β B B' β§ P β segment β C C')
(hQ : Q β segment β C C' β§ Q β segment β A A')
(hR : R β segment β A A' β§ R β segment β B B')
: (volume (convexHull β {P, Q, R})).toReal / (volume (convexHull β {A, B, C})).toReal = (k - 1)^2 / (k^2 + k + 1) :=
sorry | 323992d1505ba6eb |
putnam_1972_a3 | abbrev putnam_1972_a3_solution : Set (β β β) := sorry
-- {f | β A B : β, β x β Set.Icc 0 1, f x = A * x + B}
/--
We call a function $f$ from $[0,1]$ to the reals to be supercontinuous on $[0,1]$ if the Cesaro-limit exists for the sequence $f(x_1), f(x_2), f(x_3), \dots$ whenever it does for the sequence $x_1, x_2, x_3 \dots$. Find all supercontinuous functions on $[0,1]$.
-/
theorem putnam_1972_a3
(climit_exists : (β β β) β Prop)
(supercontinuous : (β β β) β Prop)
(hclimit_exists : β x, climit_exists x β β C : β, Tendsto (fun n => (β i in Finset.range n, (x i))/(n : β)) atTop (π C))
(hsupercontinuous : β f, supercontinuous f β β (x : β β β), (β i : β, x i β Icc 0 1) β climit_exists x β climit_exists (fun i => f (x i))) :
{f | supercontinuous f} = putnam_1972_a3_solution :=
sorry | 338a897dcbc74c58 |
putnam_2015_a6 | theorem putnam_2015_a6
(n : β)
(A B M : Matrix (Fin n) (Fin n) β)
(npos : n > 0)
(hmul : A * M = M * B)
(hpoly : Matrix.charpoly A = Matrix.charpoly B)
: β X : Matrix (Fin n) (Fin n) β, (A - M * X).det = (B - X * M).det :=
sorry | 065610d5c5c22e3f |
putnam_1990_a6 | abbrev putnam_1990_a6_solution : β := sorry
-- 17711
/--
If $X$ is a finite set, let $|X|$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $\{1,2,\dots,n\}$ \emph{admissible} if $s>|T|$ for each $s \in S$, and $t>|S|$ for each $t \in T$. How many admissible ordered pairs of subsets of $\{1,2,\dots,10\}$ are there? Prove your answer.
-/
theorem putnam_1990_a6 :
((Finset.univ : Finset <| Finset (Set.Icc 1 10) Γ Finset (Set.Icc 1 10)).filter
fun β¨S, Tβ© β¦ (β s β S, T.card < s) β§ (β t β T, S.card < t)).card =
putnam_1990_a6_solution :=
sorry | 90045cb1da5a6106 |
putnam_1980_a6 | abbrev putnam_1980_a6_solution : β := sorry
-- 1 / Real.exp 1
/--
Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0 \leq x \leq 1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that $u \leq \int_0^1|f'(x)-f(x)|\,dx$ for all $f$ in $C$.
-/
theorem putnam_1980_a6
(C : Set (β β β))
(hC : C = {f : β β β | ContDiffOn β 1 f (Set.Icc 0 1) β§ f 0 = 0 β§ f 1 = 1}) :
IsGreatest
{(u : β) | β f β C, u β€ (β« x in Set.Ioo 0 1, |deriv f x - f x|)}
putnam_1980_a6_solution :=
sorry | 5b4f35c7ea302eca |
putnam_2009_b3 | abbrev putnam_2009_b3_solution : Set β€ := sorry
-- {n : β€ | β k β₯ 1, n = 2 ^ k - 1}
/--
Call a subset $S$ of $\{1, 2, \dots, n\}$ \emph{mediocre} if it has the following property: Whenever $a$ and $b$ are elements of $S$ whose average is an integer, that average is also an element of $S$. Let $A(n)$ be the number of mediocre subsets of $\{1,2,\dots,n\}$. [For instance, every subset of $\{1,2,3\}$ except $\{1,3\}$ is mediocre, so $A(3) = 7$.] Find all positive integers $n$ such that $A(n+2) - 2A(n+1) + A(n) = 1$.
-/
theorem putnam_2009_b3
(mediocre : β€ β Set β€ β Prop)
(hmediocre : β n S, mediocre n S β (S β Icc 1 n) β§ β a β S, β b β S, 2 β£ a + b β (a + b) / 2 β S)
(A : β€ β β€)
(hA : A = fun n β¦ ({S : Set β€ | mediocre n S}.ncard : β€))
: ({n : β€ | n > 0 β§ A (n + 2) - 2 * A (n + 1) + A n = 1} = putnam_2009_b3_solution) :=
sorry | f1f6e2307105ff38 |
putnam_2019_b3 | theorem putnam_2019_b3
(n : β)
(hn : n > 0)
(Q : Matrix (Fin n) (Fin n) β)
(hQ0 : β i j : Fin n, i β j β dotProduct (Q i) (Q j) = 0 β§ dotProduct (Qα΅ i) (Qα΅ j) = 0)
(hQ1 : β i : Fin n, dotProduct (Q i) (Q i) = 1 β§ dotProduct (Qα΅ i) (Qα΅ i) = 1)
(u : Matrix (Fin n) (Fin 1) β)
(hu : uα΅*u = 1)
(P : Matrix (Fin n) (Fin n) β)
(hP : P = 1 - 2 * (u * uα΅))
: (Q - 1).det β 0 β (P * Q - 1).det = 0 :=
sorry | 003c3d6b7e4ae77e |
putnam_1971_a5 | abbrev putnam_1971_a5_solution : β€ Γ β€ := sorry
-- (11, 8)
/--
After each play of a certain game of solitaire, the player receives either $a$ or $b$ points, where $a$ and $b$ are positive integers with $a > b$; scores accumulate from play to play. If there are $35$ unattainable scores, one of which is $58$, find $a$ and $b$.
-/
theorem putnam_1971_a5
(a b : β€)
(hab : a > 0 β§ b > 0 β§ a > b)
(pab : β€ β β€ β Prop)
(hpab : β x y, pab x y β
{s : β | Β¬β m n : β, m*x + n*y = s}.ncard = 35 β§
Β¬β m n : β, m*x + n*y = 58) :
pab a b β a = putnam_1971_a5_solution.1 β§ b = putnam_1971_a5_solution.2 :=
sorry | 5cecbd61d48c2b4d |
putnam_2024_b4 | abbrev putnam_2024_b4_solution : β := sorry
--(1 - rexp (- 2))/2
/--
Let $n$ be a positive integer. Set $a_{n, 0} = 1$. For $k \geq 0$
choose an integer $m_{n, k}$ uniformly at random from the set
$\{1, 2, \ldots, n\}$, and let
$$a_{n, k+1} = \begin{cases}
a_{n, k} + 1 & \text{if } m_{n, k} > a_{n, k} \\
a_{n, k} & \text{ if } m_{n, k} = a_{n, k} \\
a_{n, k} -1 & \text{if } m_{n, k} < a_{n, k} \end{cases}$$.
Let $E(n)$ be the expected value of $a_{n, n}$. Determine
$\lim_{n \to \infty} E(n)/n$.
-/
theorem putnam_2024_b4
{Ξ© : Type*}
[MeasureSpace Ξ©]
[IsProbabilityMeasure (β : Measure Ξ©)]
(m a : β β β β Ξ© β β€)
(hβ : β n > 0, a n 0 = 1)
(hβ : β n k, 0 < n β pdf.IsUniform (m n k) (Set.Icc 1 n) β (ProbabilityTheory.uniformOn <| Set.Icc 1 n))
(hβ : β n k Ο, 0 < n β
a n (k + 1) Ο =
if m n k Ο > a n k Ο then
a n k Ο + 1
else if m n k Ο = a n k Ο then
a n k Ο
else
a n k Ο - 1)
(hβ : ProbabilityTheory.iIndepFun inferInstance m.uncurry β) :
Filter.Tendsto (fun n => (β« Ο, a n n Ο : β) / n) Filter.atTop (π putnam_2024_b4_solution) :=
sorry | 535809caa3501e5b |
putnam_1966_a1 | theorem putnam_1966_a1
(f : β€ β β€)
(hf : f = fun n : β€ => β m in Finset.Icc 0 n, (if Even m then m / 2 else (m - 1)/2))
: β x y : β€, x > 0 β§ y > 0 β§ x > y β x * y = f (x + y) - f (x - y) :=
sorry | 100412fdc6624f04 |
putnam_2011_a4 | abbrev putnam_2011_a4_solution : Set β := sorry
-- {n : β | Odd n}
/--
For which positive integers $n$ is there an $n \times n$ matrix with integer entries such that every dot product of a row with itself is even, while every dot product of two different rows is odd?
-/
theorem putnam_2011_a4
(nmat : β β Prop)
(hnmat : β n, nmat n β
β A : Matrix (Fin n) (Fin n) β€,
(β r, Even ((A r) β¬α΅₯ (A r))) β§
Pairwise fun r1 r2 β¦ Odd ((A r1) β¬α΅₯ (A r2)))
(n : β) (npos : 0 < n) :
nmat n β n β putnam_2011_a4_solution :=
sorry | c4c7db6f47109dce |
putnam_1969_b2 | abbrev putnam_1969_b2_solution : Prop := sorry
-- False
/--
Show that a finite group can not be the union of two of its proper subgroups. Does the statement remain true if 'two' is replaced by 'three'?
-/
theorem putnam_1969_b2
(P : β β Prop)
(P_def : β n, P n β β (G : Type) [Group G] [Finite G],
β H : Fin n β Subgroup G, (β i, H i < β€) β β i, (H i : Set G) < β€) :
P 2 β§ (P 3 β putnam_1969_b2_solution) :=
sorry | c184cff6254ac63c |
putnam_1994_a4 | theorem putnam_1994_a4
(A B : Matrix (Fin 2) (Fin 2) β€)
(ABinv : Nonempty (Invertible A) β§
Nonempty (Invertible (A + B)) β§
Nonempty (Invertible (A + 2 * B)) β§
Nonempty (Invertible (A + 3 * B)) β§
Nonempty (Invertible (A + 4 * B)))
: Nonempty (Invertible (A + 5 * B)) :=
sorry | e4b40ea6b74370fd |
putnam_1979_b2 | abbrev putnam_1979_b2_solution : β Γ β β β := sorry
-- fun (a, b) => (Real.exp (-1))*(b^b/a^a)^(1/(b-a))
/--
If $0 < a < b$, find $$\lim_{t \to 0} \left( \int_{0}^{1}(bx + a(1-x))^t dx \right)^{\frac{1}{t}}$$ in terms of $a$ and $b$.
-/
theorem putnam_1979_b2
: β a b : β, 0 < a β§ a < b β Tendsto (fun t : β => (β« x in Icc 0 1, (b*x + a*(1 - x))^t)^(1/t)) (π[β ] 0) (π (putnam_1979_b2_solution (a, b))) :=
sorry | a92b461714bf5f5a |
putnam_1972_b5 | theorem putnam_1972_b5
(A B C D : EuclideanSpace β (Fin 3))
(hnonplanar : Β¬Coplanar β {A, B, C, D})
(hangles : β A B C = β C D A β§ β B C D = β D A B)
: dist A B = dist C D β§ dist B C = dist D A :=
sorry | 99f84dcd457b70ed |
putnam_1962_b5 | theorem putnam_1962_b5
(n : β€)
(ng1 : n > 1)
: (3 * (n : β) + 1) / (2 * n + 2) < β i : Finset.Icc 1 n, ((i : β) / n) ^ (n : β) β§ β i : Finset.Icc 1 n, ((i : β) / n) ^ (n : β) < 2 :=
sorry | 3b8b2ad457bdf985 |
putnam_2012_b4 | abbrev putnam_2012_b4_solution : Prop := sorry
-- True
/--
Suppose that $a_0 = 1$ and that $a_{n+1} = a_n + e^{-a_n}$ for $n=0,1,2,\dots$. Does $a_n - \log n$
have a finite limit as $n \to \infty$? (Here $\log n = \log_e n = \ln n$.)
-/
theorem putnam_2012_b4
(a : β β β)
(ha0 : a 0 = 1)
(han : β n : β, a (n + 1) = a n + exp (-a n))
: ((β L : β, Tendsto (fun n β¦ a n - Real.log n) atTop (π L)) β putnam_2012_b4_solution) :=
sorry | ce253d9604e7e3c5 |
putnam_1975_b1 | abbrev putnam_1975_b1_solution : β€ := sorry
-- 7
/--
Let $H$ be a subgroup of the additive group of ordered pairs of integers under componentwise addition. If $H$ is generated by the elements $(3, 8)$, $(4, -1)$, and $(5, 4)$, then $H$ is also generated by two elements $(1, b)$ and $(0, a)$ for some integer $b$ and positive integer $a$. Find $a$.
-/
theorem putnam_1975_b1
(H : Set (β€ Γ β€))
(hH : H = {(x, y) : (β€ Γ β€) | β u v w : β€, (x, y) = (u*3 + v*4 + w*5, u*8 + v*(-1) + w*4)})
: (β b : β€, H = {(x, y) : (β€ Γ β€) | β u v : β€, (x, y) = (u, u*b + v*putnam_1975_b1_solution)}) β§ putnam_1975_b1_solution > 0 :=
sorry | df643ba70c65f458 |
putnam_1965_b1 | abbrev putnam_1965_b1_solution : β := sorry
-- 1 / 2
/--
Find $$\lim_{n \to \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos^2\left(\frac{\pi}{2n}(x_1 + x_2 + \cdots + x_n)\right) dx_1 dx_2 \cdots dx_n.$$
-/
theorem putnam_1965_b1
: Tendsto (fun n : β β¦ β« x in {x : Fin (n+1) β β | β k : Fin (n+1), x k β Set.Icc 0 1}, (Real.cos (Real.pi/(2 * (n+1)) * β k : Fin (n+1), x k))^2) atTop (π putnam_1965_b1_solution) :=
sorry | 424ce1c7af19c406 |
putnam_1987_b4 | abbrev putnam_1987_b4_solution : Prop Γ β Γ Prop Γ β := sorry
-- (True, -1, True, 0)
/--
Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \cos y_n - y_n \sin y_n$ and $y_{n+1}= x_n \sin y_n + y_n \cos y_n$ for $n=1,2,3,\dots$. For each of $\lim_{n\to \infty} x_n$ and $\lim_{n \to \infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist.
-/
theorem putnam_1987_b4
(x y : β β β)
(hxy1 : (x 1, y 1) = (0.8, 0.6))
(hx : β n β₯ 1, x (n + 1) = (x n) * cos (y n) - (y n) * sin (y n))
(hy : β n β₯ 1, y (n + 1) = (x n) * sin (y n) + (y n) * cos (y n)) :
let (existsx, limx, existsy, limy) := putnam_1987_b4_solution
((β c : β, Tendsto x atTop (π c)) β existsx) β§
(existsx β Tendsto x atTop (π limx)) β§
((β c : β, Tendsto y atTop (π c)) β existsy) β§
(existsy β Tendsto y atTop (π limy)) :=
sorry | afc66948594fb538 |
putnam_1997_b4 | theorem putnam_1997_b4
(a : β β β β β€)
(ha : β m n, a m n = coeff ((1 + X + X ^ 2) ^ m) n)
(k : β) :
(β i in Finset.Iic β2 * (k : β) / 3ββ, (-1) ^ i * a (k - i) i) β Icc 0 1 :=
sorry | 719e573780ea2a71 |
putnam_1979_a4 | abbrev putnam_1979_a4_solution : Prop := sorry
-- True
/--
Let $A$ be a set of $2n$ points in the plane, $n$ colored red and $n$ colored blue, such that no three points in $A$ are collinear. Must there exist $n$ closed straight line segments, each connecting one red and one blue point in $A$, such that no two of the $n$ line segments intersect?
-/
theorem putnam_1979_a4
(A : Finset (Fin 2 β β) Γ Finset (Fin 2 β β) β Prop)
(hA : A = fun (R, B) => R.card = B.card β§ R β© B = β
β§
β u : Finset (Fin 2 β β), u β R βͺ B β u.card = 3 β Β¬Collinear β (u : Set (Fin 2 β β)))
(w : (Fin 2 β β) Γ (Fin 2 β β) β β β (Fin 2 β β))
(hw : w = fun (P, Q) => fun x : β => fun i : Fin 2 => x * P i + (1 - x) * Q i) :
putnam_1979_a4_solution β
(β R B, A (R, B) β
β v : Finset ((Fin 2 β β) Γ (Fin 2 β β)),
(β L β v, β M β v, L β M β β x β Icc 0 1, β y β Icc 0 1,
Real.sqrt ((w (L.1, L.2) x 0 - w (M.1, M.2) y 0)^2 + (w (L.1, L.2) x 1 - w (M.1, M.2) y 1)^2) β 0) β§
v.card = R.card β§ β L β v, L.1 β R β§ L.2 β B) :=
sorry | af9e3eced5a7be57 |
putnam_1994_b2 | abbrev putnam_1994_b2_solution : Set β := sorry
-- {c : β | c < 243 / 8}
/--
For which real numbers $c$ is there a straight line that intersects the curve $x^4+9x^3+cx^2+9x+4$ in four distinct points?
-/
theorem putnam_1994_b2
(c : β) :
(β m b : β,
{x : β | m * x + b = x ^ 4 + 9 * x ^ 3 + c * x ^ 2 + 9 * x + 4}.encard = 4)
β c β putnam_1994_b2_solution :=
sorry | 2d51744f0dcd3b0f |
putnam_1969_a4 | theorem putnam_1969_a4
: Tendsto (fun n => β i in Finset.Icc (1 : β€) n, (-1)^(i+1)*(i : β)^(-i)) atTop (π (β« x in Ioo (0 : β) 1, x^x)) :=
sorry | 1b44abb7a3d546e7 |
putnam_1984_b2 | abbrev putnam_1984_b2_solution : β := sorry
-- 8
/--
Find the minimum value of $(u-v)^2+(\sqrt{2-u^2}-\frac{9}{v})^2$ for $0<u<\sqrt{2}$ and $v>0$.
-/
theorem putnam_1984_b2
(f : β β β β β)
(hf : β u v : β, f u v = (u - v) ^ 2 + (Real.sqrt (2 - u ^ 2) - 9 / v) ^ 2) :
IsLeast {y | βα΅ (u : Set.Ioo 0 β2) (v > 0), f u v = y} putnam_1984_b2_solution :=
sorry | 7c0ab9c8f0dd975a |
putnam_2001_b2 | abbrev putnam_2001_b2_solution : Set (β Γ β) := sorry
-- {((3 ^ ((1 : β) / 5) + 1) / 2, (3 ^ ((1 : β) / 5) - 1) / 2)}
/--
Find all pairs of real numbers $(x,y)$ satisfying the system of equations
\begin{align*}
\frac{1}{x}+\frac{1}{2y}&=(x^2+3y^2)(3x^2+y^2) \\
\frac{1}{x}-\frac{1}{2y}&=2(y^4-x^4).
\end{align*}
-/
theorem putnam_2001_b2
(x y : β)
(hx : x β 0)
(hy : y β 0)
(eq1 eq2 : Prop)
(heq1 : eq1 β (1 / x + 1 / (2 * y) = (x ^ 2 + 3 * y ^ 2) * (3 * x ^ 2 + y ^ 2)))
(heq2 : eq2 β (1 / x - 1 / (2 * y) = 2 * (y ^ 4 - x ^ 4))) :
eq1 β§ eq2 β (x, y) β putnam_2001_b2_solution :=
sorry | e94ea7031451eb3a |
putnam_2011_b2 | abbrev putnam_2011_b2_solution : Set β := sorry
-- {2, 5}
/--
Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0$. Which primes appear in seven or more elements of $S$?
-/
theorem putnam_2011_b2
(S : Set (Fin 3 β β))
(t : β)
(hS : S = {s : Fin 3 β β | (s 0).Prime β§ (s 1).Prime β§ (s 2).Prime β§ β x : β, (s 0) * x ^ 2 + (s 1) * x + (s 2) = 0}) :
(t.Prime β§ ({s β S | β i : Fin 3, s i = t}.encard β₯ 7))
β t β putnam_2011_b2_solution :=
sorry | 236b9e65124ae824 |
putnam_2023_a6 | abbrev putnam_2023_a6_solution : Set β := sorry
-- {n : β | 0 < n}
/--
Alice and Bob play a game in which they take turns choosing integers from $1$ to $n$. Before any integers are chosen, Bob selects a goal of 'odd' or 'even'. On the first turn, Alice chooses one of the $n$ integers. On the second turn, Bob chooses one of the remaining integers. They continue alternately choosing one of the integers that has not yet been chosen, until the $n$th turn, which is forced and ends the game. Bob wins if the parity of $\{k : \mbox{the number $k$ was chosen on the $k$th turn}\}$ matches his goal. For which values of $n$ does Bob have a winning strategy?
-/
theorem putnam_2023_a6
(IsValidGame : List β β Prop)
(IsValidGame_def : β g, IsValidGame g β g.Nodup β§ (β i β g, i β Icc 1 g.length))
(parityOf : List β β ZMod 2)
(parityOf_def : β g, parityOf g = ((range g.length).filter fun i β¦ g[i]! = i + 1).card)
(ConformsToStrategy : List β β (List β β β) β Prop)
(ConformsToStrategy_def : β g s, ConformsToStrategy g s β
β (i) (h : i < g.length), Odd i β g[i] = s (g.take i))
(IsWinningFor : β β (List β β β) β Prop)
(IsWinningFor_def : β n s, IsWinningFor n s β
β p, β g, g.length = n β IsValidGame g β ConformsToStrategy g s β parityOf g = p) :
{n : β | 0 < n β§ β s, IsWinningFor n s} = putnam_2023_a6_solution :=
sorry | cd230851416cd5dd |
putnam_2006_b6 | abbrev putnam_2006_b6_solution : β β β := sorry
-- fun k => ((k+1)/k)^k
/--
Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \[ a_{n+1} = a_n + \frac{1}{\sqrt[k]{a_n}} \] for $n > 0$. Evaluate \[\lim_{n \to \infty} \frac{a_n^{k+1}}{n^k}.\]
-/
theorem putnam_2006_b6
(k : β)
(hk : k > 1)
(a : β β β)
(ha0 : a 0 > 0)
(ha : β n : β, a (n + 1) = a n + 1/((a n)^((1 : β)/k)))
: Tendsto (fun n => (a n)^(k+1)/(n ^ k)) atTop (π (putnam_2006_b6_solution k)) :=
sorry | 5463d574e46b3040 |
putnam_2016_b6 | abbrev putnam_2016_b6_solution : β := sorry
-- 1
/--
Evaluate $\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k} \sum_{n=0}^\infty \frac{1}{k2^n+1}$.
-/
theorem putnam_2016_b6 :
β' k : β, ((-1 : β) ^ ((k + 1 : β€) - 1) / (k + 1 : β)) * β' n : β, (1 : β) / ((k + 1) * (2 ^ n) + 1) = putnam_2016_b6_solution :=
sorry | 5d7fc8fd14d11d50 |
putnam_2024_a2 | abbrev putnam_2024_a2_solution : Set β[X] := sorry
-- {s β’ X + C a | (s : β€Λ£) (a : β)}
/--
For which real polynomials $p$ is there a real polynomial $q$ such that
$p(p(x)) - x = (p(x) - x)^2q(x)$ for all real $x$?
-/
theorem putnam_2024_a2 :
{ p : Polynomial β | β (q : Polynomial β),
β x, p.eval (p.eval x) - x = (p.eval x - x) ^ 2 * q.eval x } = putnam_2024_a2_solution :=
sorry | 237c86d43a7d5525 |
putnam_1971_b3 | theorem putnam_1971_b3
(T : β)
(hT : T > 0)
: MeasureTheory.volume {t : β | t β₯ T β§ Nat.floor t = 2 * (Nat.floor (t - T))} = 1 :=
sorry | 6e3061a1afaee02d |
putnam_2019_a5 | abbrev putnam_2019_a5_solution : β β β := sorry
-- (fun p : β => (p - 1) / 2)
/--
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by $q(x)=\sum_{k=1}^{p-1} a_kx^k$, where $a_k=k^{(p-1)/2}\mod{p}$. Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.
-/
theorem putnam_2019_a5
(p : β)
(q : Polynomial (ZMod p))
(a : β β ZMod p)
(npoly : β β Polynomial (ZMod p))
(podd : Odd p)
(pprime : p.Prime)
(hq : β k : β, q.coeff k = a k)
(ha0 : a 0 = 0 β§ β k > p - 1, a k = 0)
(haother : β k : Set.Icc 1 (p - 1), a k = ((k : β) ^ ((p - 1) / 2)) % p)
(hnpoly : β n x, (npoly n).eval x = (x - 1) ^ n) :
IsGreatest {n | npoly n β£ q} (putnam_2019_a5_solution p) :=
sorry | 0ce3fd2e5f08ac75 |
putnam_2009_a5 | abbrev putnam_2009_a5_solution : Prop := sorry
-- False
/--
Is there a finite abelian group $G$ such that the product of the orders of all its elements is 2^{2009}?
-/
theorem putnam_2009_a5
: (β (G : Type*) (_ : CommGroup G) (_ : Fintype G), β g : G, orderOf g = 2^2009) β putnam_2009_a5_solution :=
sorry | 2db7be6af3abc680 |
putnam_1993_b6 | theorem putnam_1993_b6
(S : Fin 3 β β)
(f : Fin 3 β Fin 3 β (Fin 3 β β) β (Fin 3 β β))
(Spos : β i : Fin 3, S i > 0)
(hf : β i j k : Fin 3, (i β j β§ i β k β§ j β k) β β S' : Fin 3 β β, if S' i β€ S' j then ((f i j S') i = 2 * S' i β§ (f i j S') j = S' j - S' i β§ (f i j S') k = S' k) else (f i j S' = S'))
: β (Ss : β β (Fin 3 β β)) (N : β), Ss 0 = S β§ (β i : Fin 3, Ss N i = 0) β§ (β n : Fin N, β i j : Fin 3, i β j β§ f i j (Ss n) = Ss ((n : β) + 1)) :=
sorry | 866f00cb4441d656 |
putnam_1983_b6 | theorem putnam_1983_b6
(n : β)
(npos : n > 0)
(Ξ± : β)
(hΞ± : Ξ± ^ (2 ^ n + 1) - 1 = 0 β§ Ξ± β 1)
: (β p q : Polynomial β€, (aeval Ξ± p) ^ 2 + (aeval Ξ± q) ^ 2 = -1) :=
sorry | 1b4edde165f51740 |
putnam_1977_b6 | theorem putnam_1977_b6
{G : Type*}
[Group G]
(H : Subgroup G)
(h : β)
(h_def : h = Nat.card H)
(a : G)
(ha : β x : H, (x*a)^3 = 1)
(P : Set G)
(hP : P = {g : G | β xs : List H, (xs.length β₯ 1) β§ g = (List.map (fun h : H => h*a) xs).prod})
: (Finite P) β§ (P.ncard β€ 3*h^2) :=
sorry | 50e1fb025d0d1767 |
putnam_1967_b6 | theorem putnam_1967_b6
(f : β β β β β)
(fdiff : (β y : β, Differentiable β (fun x : β => f x y)) β§ (β x : β, Differentiable β (fun y : β => f x y)))
(fbound : β x y : β, (x ^ 2 + y ^ 2 β€ 1) β |f x y| β€ 1)
: β x0 y0 : β, (x0 ^ 2 + y0 ^ 2 < 1) β§ ((deriv (fun x : β => f x y0) x0) ^ 2 + (deriv (fun y : β => f x0 y) y0) ^ 2 β€ 16) :=
sorry | e3aefdc1bfcdb1cc |
putnam_2010_b3 | abbrev putnam_2010_b3_solution : Set β := sorry
-- Ici 1005
/--
There are $2010$ boxes labeled $B_1, B_2, \dots, B_{2010}$, and $2010n$ balls have been distributed among them, for some positive integer $n$. You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving \emph{exactly} $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?
-/
theorem putnam_2010_b3
(n : β) (hn : n > 0)
(trans : (β β Fin 2010 β β) β β β Prop)
(htrans : β P T, trans P T β β t : β, t < T β β i j,
i β j β§
P t i β₯ i.1 + 1 β§ P (t + 1) i = P t i - (i.1 + 1) β§ P (t + 1) j = P t j + (i.1 + 1) β§
β k : Fin 2010, k β i β k β j β P (t + 1) k = P t k) :
(β B, β i, B i = 2010 * n β βα΅ (P) (T), P 0 = B β§ trans P T β§ β i, P T i = n)
β n β putnam_2010_b3_solution :=
sorry | 147c81166d626394 |
putnam_2000_b3 | theorem putnam_2000_b3
(N : β) (hN : N > 0)
(a : Fin (N + 1) β β)
(f : β β β)
(mult : (β β β) β β β β)
(M : β β β)
(haN : a N β 0)
(hf : β t, f t = β j : Icc 1 N, a j * Real.sin (2 * Real.pi * j * t))
(hmult : β g : β β β, β t : β, (β c : β, iteratedDeriv c g t β 0) β (iteratedDeriv (mult g t) g t β 0 β§ β k < (mult g t), iteratedDeriv k g t = 0))
(hM : β k, M k = β' t : Ico (0 : β) 1, mult (iteratedDeriv k f) t) :
((β i j : β, i β€ j β M i β€ M j) β§ Tendsto M atTop (π (2 * N))) :=
sorry | b577e1f8143ebb95 |
putnam_1985_b3 | theorem putnam_1985_b3
(a : β β β β β)
(apos : β m n : β, a m n > 0)
(ha : β k : β, k > 0 β {(m, n) : β Γ β | m > 0 β§ n > 0 β§ a m n = k}.encard = 8)
: (β m n, m > 0 β§ n > 0 β§ a m n > m * n) :=
sorry | c808f91c13e59d70 |
putnam_1968_a5 | abbrev putnam_1968_a5_solution : β := sorry
-- 8
/--
Let $V$ be the set of all quadratic polynomials with real coefficients such that $|P(x)| \le 1$ for all $x \in [0, 1]$. Find the supremum of $|P'(0)|$ across all $P \in V$.
-/
theorem putnam_1968_a5
(V : Set β[X])
(V_def : V = {P : β[X] | P.degree = 2 β§ β x β Set.Icc 0 1, |P.eval x| β€ 1})
: sSup {|(derivative P).eval 0| | P β V} = putnam_1968_a5_solution :=
sorry | 790e33efe70b699a |
putnam_1995_b3 | abbrev putnam_1995_b3_solution : β β β€ := sorry
-- fun n => if n = 1 then 45 else if n = 2 then 10 * 45^2 else 0
/--
To each positive integer with $n^{2}$ decimal digits, we associate the determinant of the matrix obtained by writing the digits in order across the rows. For example, for $n=2$, to the integer 8617 we associate $\det \left( \begin{array}{cc} 8 & 6 \ 1 & 7 \end{array} \right) = 50$. Find, as a function of $n$, the sum of all the determinants associated with $n^{2}$-digit integers. (Leading digits are assumed to be nonzero; for example, for $n=2$, there are 9000 determinants.)
-/
theorem putnam_1995_b3
(n : β)
(hn : n > 0)
(digits_set : Set (β β β))
(hdigits_set : digits_set = {f | f 0 β 0 β§ (β i : Fin (n ^ 2), f i β€ 9) β§ (β i β₯ n ^ 2, f i = 0)})
(digits_to_matrix : (β β β) β Matrix (Fin n) (Fin n) β€)
(hdigits_to_matrix : digits_to_matrix = fun f => (fun i j => f (i.1 * n + j.1)))
: β' f : digits_set, (digits_to_matrix f).det = putnam_1995_b3_solution n :=
sorry | ba99130ed6d60ad3 |
putnam_1978_a5 | theorem putnam_1978_a5
(n : β)
(npos : n > 0)
(a : Fin n β β)
(ha : β i : Fin n, a i β Ioo 0 Real.pi)
(ΞΌ : β)
(hΞΌ : ΞΌ = β i : Fin n, a i / n)
: (β i : Fin n, sin (a i) / (a i) β€ (sin ΞΌ / ΞΌ) ^ n) :=
sorry | 25ef58852b4f0683 |
putnam_2008_a4 | abbrev putnam_2008_a4_solution : Prop := sorry
-- False
/--
Define $f : \mathbb{R} \to \mathbb{R} by $f(x) = x$ if $x \leq e$ and $f(x) = x * f(\ln(x))$ if $x > e$. Does $\sum_{n=1}^{\infty} 1/(f(n))$ converge?
-/
theorem putnam_2008_a4
(f : β β β)
(hf : f = fun x => if x β€ Real.exp 1 then x else x * (f (Real.log x)))
: (β r : β, Tendsto (fun N : β => β n in Finset.range N, 1/(f (n + 1))) atTop (π r)) β putnam_2008_a4_solution :=
sorry | 83f1a3fde282cef1 |
putnam_2018_a4 | theorem putnam_2018_a4
(m n : β)
(a : β β β€)
(G : Type*) [Group G]
(g h : G)
(mnpos : m > 0 β§ n > 0)
(mngcd : Nat.gcd m n = 1)
(ha : β k : Set.Icc 1 n, a k = Int.floor (m * k / (n : β)) - Int.floor (m * ((k : β€) - 1) / (n : β)))
(ghprod : ((List.Ico 1 (n + 1)).map (fun k : β => g * h ^ (a k))).prod = 1)
: g * h = h * g :=
sorry | 93caf834393ad979 |
putnam_1970_b2 | theorem putnam_1970_b2
(T : β)
(H : Polynomial β)
(hT : T > 0)
(hH : H.degree β€ 3)
: (H.eval (-T / Real.sqrt 3) + H.eval (T / Real.sqrt 3))/2 = β¨ t in Set.Icc (-T) T, H.eval t :=
sorry | f6f51125aa60956c |
putnam_1973_b4 | abbrev putnam_1973_b4_solution : β β β := sorry
-- (fun x => x)
/--
Suppose $f$ is a function on $[0,1]$ with continuous derivative satisfying $0 < f'(x) \leq 1$ and $f 0 = 0$. Prove that $\left[\int_0^1 f(x) dx\right]]^2 \geq \int_0^1 (f(x))^3 dx$, and find an example where equality holds.
-/
theorem putnam_1973_b4
(f : β β β)
(hprop : (β β β) β Prop)
(hprop_def : hprop = fun g => ContDiff β 1 g β§ (β x : β, 0 < deriv g x β§ deriv g x β€ 1) β§ g 0 = 0)
(hf : hprop f)
: (β« x in Icc 0 1, f x)^2 β₯ β« x in Icc 0 1, (f x)^3 β§ (hprop putnam_1973_b4_solution β§ (β« x in Icc 0 1, putnam_1973_b4_solution x)^2 = β« x in Icc 0 1, (putnam_1973_b4_solution x)^3) :=
sorry | dfba47b9632ab757 |
putnam_2004_b1 | theorem putnam_2004_b1
(n : β)
(P : Polynomial β€)
(r : β)
(Pdeg : P.degree = n)
(Preq0 : Polynomial.aeval r P = 0)
: β i β Finset.range n, β m : β€, m = β j in Finset.range (i + 1), (P.coeff (n - j) * r ^ (i + 1 - j)) :=
sorry | 3a756147ae0b27ae |
putnam_2014_b1 | abbrev putnam_2014_b1_solution : Set β := sorry
-- {n : β | n > 0 β§ Β¬β a β digits 10 n, a = 0}
/--
A \emph{base $10$ over-expansion} of a positive integer $N$ is an expression of the form
\[
N = d_k 10^k + d_{k-1} 10^{k-1} + \cdots + d_0 10^0
\]
with $d_k \neq 0$ and $d_i \in \{0,1,2,\dots,10\}$ for all $i$. For instance, the integer $N = 10$ has two base $10$ over-expansions: $10 = 10 \cdot 10^0$ and the usual base $10$ expansion $10 = 1 \cdot 10^1 + 0 \cdot 10^0$. Which positive integers have a unique base $10$ over-expansion?
-/
theorem putnam_2014_b1
(overexpansion : β β List β β Prop)
(hoverexpansion : overexpansion = fun N d β¦ N = β i : Fin d.length, (d.get i) * 10 ^ i.1 β§ d.getLastI β 0 β§ β a β d, a β Finset.range 11)
(S : Set β)
(hS : β N : β, N β S β N > 0 β§ β! d : List β, overexpansion N d)
: (S = putnam_2014_b1_solution) :=
sorry | c47a0b765c8f9bea |
putnam_1991_b1 | abbrev putnam_1991_b1_solution : Set β€ := sorry
-- {A : β€ | β x > 0, A = x ^ 2}
/--
For each integer $n \geq 0$, let $S(n)=n-m^2$, where $m$ is the greatest integer with $m^2 \leq n$. Define a sequence $(a_k)_{k=0}^\infty$ by $a_0=A$ and $a_{k+1}=a_k+S(a_k)$ for $k \geq 0$. For what positive integers $A$ is this sequence eventually constant?
-/
theorem putnam_1991_b1
(m S : β€ β β€)
(A : β€)
(a : β β β€)
(hm : β n, 0 β€ n β (m n) ^ 2 β€ n β§ (β m' : β€, m' ^ 2 β€ n β m' β€ m n))
(hS : β n, 0 β€ n β S n = n - (m n) ^ 2)
(ha : a 0 = A β§ (β k, a (k + 1) = a k + S (a k)))
(hA : A > 0) :
(β (K : β) (c : β), β k β₯ K, a k = c) β A β putnam_1991_b1_solution :=
sorry | f73f437026f54352 |
putnam_1981_b1 | abbrev putnam_1981_b1_solution : β := sorry
-- -1
/--
Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$
-/
theorem putnam_1981_b1
(f : β β β)
(hf : f = fun n : β => ((1 : β)/n^5) * β h in Finset.Icc 1 n, β k in Finset.Icc 1 n, (5*(h : β)^4 - 18*h^2*k^2 + 5*k^4))
: Tendsto f atTop (π putnam_1981_b1_solution) :=
sorry | a71d97e8349bd247 |
putnam_1996_b5 | abbrev putnam_1996_b5_solution : β β β := sorry
-- (fun n : β β¦ 2 ^ β(n + 2) / 2ββ + 2 ^ β(n + 1) / 2ββ - 2)
/--
Given a finite string $S$ of symbols $X$ and $O$, we write $\Delta(S)$ for the number of $X$'s in $S$ minus the number of $O$'s. For example, $\Delta(XOOXOOX)=-1$. We call a string $S$ \emph{balanced} if every substring $T$ of (consecutive symbols of) $S$ has $-2 \leq \Delta(T) \leq 2$. Thus, $XOOXOOX$ is not balanced, since it contains the substring $OOXOO$. Find, with proof, the number of balanced strings of length $n$.
-/
theorem putnam_1996_b5
(n : β)
(Ξ : (Fin n β β€Λ£) β Fin n β Fin n β β€)
(balanced : (Fin n β β€Λ£) β Prop)
(hΞ : β S, β a b, a β€ b β Ξ S a b = β i in Finset.Icc a b, (S i : β€))
(hbalanced : β S, balanced S β β a b, a β€ b β |Ξ S a b| β€ 2) :
{S : Fin n β β€Λ£ | balanced S}.ncard = putnam_1996_b5_solution n :=
sorry | e1df119f03886b84 |
putnam_1986_b5 | abbrev putnam_1986_b5_solution : Prop := sorry
-- False
/--
Let $f(x,y,z) = x^2+y^2+z^2+xyz$. Let $p(x,y,z), q(x,y,z)$, $r(x,y,z)$ be polynomials with real coefficients satisfying
\[
f(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z).
\]
Prove or disprove the assertion that the sequence $p,q,r$ consists of some permutation of $\pm x, \pm y, \pm z$, where the number of minus signs is $0$ or $2$.
-/
theorem putnam_1986_b5
(f : MvPolynomial (Fin 3) β)
(perms : Set (Set (MvPolynomial (Fin 3) β)))
(hf : f = (X 0) ^ 2 + (X 1) ^ 2 + (X 2) ^ 2 + (X 0) * (X 1) * (X 2))
(hperms : perms = {{X 0, X 1, X 2}, {X 0, -X 1, -X 2}, {-X 0, X 1, -X 2}, {-X 0, -X 1, X 2}}) :
putnam_1986_b5_solution β (β pqr : Fin 3 β MvPolynomial (Fin 3) β,
(β xyz : Fin 3 β β, MvPolynomial.eval (fun i β¦ MvPolynomial.eval xyz (pqr i)) f = MvPolynomial.eval xyz f) β
({pqr 0, pqr 1, pqr 2} β perms)) :=
sorry | 738f1ca15d93f5b0 |
putnam_2021_a1 | abbrev putnam_2021_a1_solution : β := sorry
-- 578
/--
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops.
Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop.
What is the smallest number of hops needed for the grasshopper to reach the point $(2021, 2021)$?
-/
theorem putnam_2021_a1
(P : List (β€ Γ β€) β Prop)
(P_def : β l, P l β l.Chain' fun p q β¦ (p.1 - q.1) ^ 2 + (p.2 - q.2) ^ 2 = 25) :
IsLeast
{k | β l, P ((0, 0) :: l) β§ l.getLast! = (2021, 2021) β§ l.length = k}
putnam_2021_a1_solution :=
sorry | ca24544a4a6e37cb |
putnam_1989_a6 | theorem putnam_1989_a6
(F : Type*) [Field F] [Fintype F]
(hF : Fintype.card F = 2)
(Ξ± : PowerSeries F)
(hΞ± : β n : β, let bin := [1] ++ (digits 2 n) ++ [1]; PowerSeries.coeff F n Ξ± = ite (β i j : Fin bin.length, i < j β bin.get i = 1 β bin.get j = 1 β (β k, i < k β k < j β bin.get k = 0) β Even ((j : β) - (i : β) - 1)) 1 0)
: (Ξ± ^ 3 + PowerSeries.X * Ξ± + 1 = 0) :=
sorry | 60102e0097c302ac |
putnam_1999_a6 | theorem putnam_1999_a6
(a : β€ β β)
(ha1 : a 1 = 1)
(ha2 : a 2 = 2)
(ha3 : a 3 = 24)
(hange4 : β n : β, n β₯ 4 β a n = (6 * (a (n - 1))^2 * (a (n - 3)) - 8 * (a (n - 1)) * (a (n - 2))^2)/(a (n - 2) * a (n - 3)))
: β n, n β₯ 1 β (β k : β€, a n = k * n) :=
sorry | 7b7b7483e31ccce5 |
putnam_2003_b5 | theorem putnam_2003_b5
(A B C : EuclideanSpace β (Fin 2))
(hABC : dist 0 A = 1 β§ dist 0 B = 1 β§ dist 0 C = 1 β§ dist A B = dist A C β§ dist A B = dist B C)
: (β f : β β β, β P : EuclideanSpace β (Fin 2), dist 0 P < 1 β β X Y Z : EuclideanSpace β (Fin 2),
dist X Y = dist P A β§ dist Y Z = dist P B β§ dist X Z = dist P C β§
(MeasureTheory.volume (convexHull β {X, Y, Z})).toReal = f (dist 0 P)) :=
sorry | d67612cbd1a50204 |
putnam_2013_b5 | theorem putnam_2013_b5
(n : β) (hn : n β₯ 1)
(k : Set.Icc 1 n)
(fiter : (Set.Icc 1 n β Set.Icc 1 n) β Prop)
(hfiter : β f, fiter f β β x : Set.Icc 1 n, β j : β, f^[j] x β€ k) :
{f | fiter f}.encard = k * n ^ (n - 1) :=
sorry | 6e6bc982302cf810 |
putnam_1964_b1 | theorem putnam_1964_b1
(a b : β β β)
(h : β n, 0 < a n)
(h' : Summable fun n β¦ (1 : β) / a n)
(h'' : β n, b n = {k | a k β€ n}.ncard) :
Tendsto (fun n β¦ (b n : β) / n) atTop (π 0) :=
sorry | fb0bc1a134118b01 |
putnam_1974_b1 | abbrev putnam_1974_b1_solution : (Fin 5 β EuclideanSpace β (Fin 2)) β Prop := sorry
-- fun p β¦ βα΅ (B > 0) (o : Equiv.Perm (Fin 5)), β i, dist (p (o i)) (p (o (i + 1))) = B
/--
Prove that the optimal configuration of 5 (not necessarily distinct) points $p_1, \dots, p_5$ on the unit circle which maximizes the sum of the ten distances \[\Sigma_{i < j}, d(p_i, p_j) \] is the one which evenly spaces the points like a regular pentagon.
-/
theorem putnam_1974_b1
(d : (Fin 5 β EuclideanSpace β (Fin 2)) β β)
(d_def : β p, d p = β β¨i, jβ© : Fin 5 Γ Fin 5, if i < j then dist (p i) (p j) else 0)
(p : Fin 5 β EuclideanSpace β (Fin 2))
(hp : β i, βp iβ = 1) :
d p = sSup {d q | (q) (hq : β i, βq iβ = 1)} β putnam_1974_b1_solution p :=
sorry | c01c5dff94976787 |
putnam_2003_b4 | theorem putnam_2003_b4
(f : β β β)
(a b c d e : β€)
(r1 r2 r3 r4 : β)
(ane0 : a β 0)
(hf1 : β z, f z = a * z ^ 4 + b * z ^ 3 + c * z ^ 2 + d * z + e)
(hf2 : β z, f z = a * (z - r1) * (z - r2) * (z - r3) * (z - r4)) :
(Β¬Irrational (r1 + r2) β§ r1 + r2 β r3 + r4) β Β¬Irrational (r1 * r2) :=
sorry | d5cf5499f48b34b6 |
putnam_2013_b4 | theorem putnam_2013_b4
(ΞΌ Var M : C(Icc (0 : β) 1, β) β β)
(hΞΌ : β f, ΞΌ f = β« x, f x)
(hVar : β f, Var f = β« x, (f x - ΞΌ f) ^ 2)
(hM : β f : C(Icc (0 : β) 1, β), IsGreatest (range <| abs β f) (M f))
(f g : C(Icc (0 : β) 1, β)) :
Var (f * g) β€ 2 * Var f * (M g) ^ 2 + 2 * Var g * (M f) ^ 2 :=
sorry | f5f67d0ea739aa89 |
putnam_1996_b4 | abbrev putnam_1996_b4_solution : Prop := sorry
-- False
/--
For any square matrix $A$, we can define $\sin A$ by the usual power series: $\sin A=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}A^{2n+1}$. Prove or disprove: there exists a $2 \times 2$ matrix $A$ with real entries such that $\sin A=\begin{pmatrix} 1 & 1996 \\ 0 & 1 \end{pmatrix}$.
-/
theorem putnam_1996_b4
(matsin : Matrix (Fin 2) (Fin 2) β β Matrix (Fin 2) (Fin 2) β)
(mat1996 : Matrix (Fin 2) (Fin 2) β)
(hmatsin : β A, matsin A = β' n : β, ((-(1 : β)) ^ n / (2 * n + 1)!) β’ A ^ (2 * n + 1))
(hmat1996 : mat1996 0 0 = 1 β§ mat1996 0 1 = 1996 β§ mat1996 1 0 = 0 β§ mat1996 1 1 = 1)
: (β A, matsin A = mat1996) β putnam_1996_b4_solution :=
sorry | de47a3f260425db4 |
putnam_1986_b4 | abbrev putnam_1986_b4_solution : Prop := sorry
-- True
/--
For a positive real number $r$, let $G(r)$ be the minimum value of $|r - \sqrt{m^2+2n^2}|$ for all integers $m$ and $n$. Prove or disprove the assertion that $\lim_{r\to \infty}G(r)$ exists and equals $0$.
-/
theorem putnam_1986_b4
(G : β β β)
(hGeq : β r : β, β m n : β€, G r = |r - sqrt (m ^ 2 + 2 * n ^ 2)|)
(hGlb : β r : β, β m n : β€, G r β€ |r - sqrt (m ^ 2 + 2 * n ^ 2)|)
: (Tendsto G atTop (π 0) β putnam_1986_b4_solution) :=
sorry | 9e36e83a7dc5e6a3 |
putnam_1963_b5 | theorem putnam_1963_b5
(a : β€ β β)
(haineq : β n β₯ 1, β k : β€, (n β€ k β§ k β€ 2 * n) β (0 β€ a k β§ a k β€ 100 * a n))
(haseries : β S : β, Tendsto (fun N : β => β n : Fin N, a n) atTop (π S))
: Tendsto (fun n : β€ => n * a n) atTop (π 0) :=
sorry | 09cf1b8c3e6696a2 |
putnam_1970_b3 | theorem putnam_1970_b3
(S : Set (β Γ β))
(a b : β)
(hab : a < b)
(hS : β s β S, s.1 β Ioo a b)
(hSclosed : IsClosed S)
: IsClosed {y | β x : β, β¨x,yβ© β S} :=
sorry | 99a97c71dd42d43e |
putnam_2017_b6 | abbrev putnam_2017_b6_solution : β := sorry
-- 2016! / 1953! - 63! * 2016
/--
Find the number of ordered $64$-tuples $(x_0,x_1,\dots,x_{63})$ such that $x_0,x_1,\dots,x_{63}$ are distinct elements of $\{1,2,\dots,2017\}$ and
\[
x_0 + x_1 + 2x_2 + 3x_3 + \cdots + 63 x_{63}
\]
is divisible by 2017.
-/
theorem putnam_2017_b6
(S : Finset (Finset.range 64 β Finset.Icc 1 2017))
(hs : β x, x β S β (Injective x β§ (2017 β£ (β i : Finset.range 64, if i β€ (β¨1, by norm_numβ© : Finset.range 64) then (x i : β€) else i * (x i : β€))))) :
S.card = putnam_2017_b6_solution :=
sorry | 6f5f334b1b225e1e |
putnam_2007_b6 | theorem putnam_2007_b6
(f : β β β)
(hf : f = fun n β¦ {M : Multiset β | M.sum = (n)! β§ β m β M, β k β Icc 1 n, m = (k)!}.ncard)
: (β C : β, β n : β, n β₯ 2 β n ^ (n ^ 2 / 2 - C * n) * Real.exp (-(n ^ 2) / 4) β€ f n β§ f n β€ n ^ (n ^ 2 / 2 + C * n) * Real.exp (-(n ^ 2) / 4)) :=
sorry | 00f67e7f6e141a6d |
putnam_1992_b6 | theorem putnam_1992_b6
(n : β)
(npos : 0 < n)
(M : Set (Matrix (Fin n) (Fin n) β))
(h1 : 1 β M)
(h2 : β A β M, β B β M, Xor' (A * B β M) (-A * B β M))
(h3 : β A β M, β B β M, (A * B = B * A) β¨ (A * B = -B * A))
(h4 : β A β M, A β 1 β β B β M, A * B = -B * A) :
M.encard β€ n ^ 2 :=
sorry | ce8b3f8c757d0267 |
putnam_2008_a5 | theorem putnam_2008_a5
(n : β)
(nge3 : n β₯ 3)
(f g : Polynomial β)
(hfg : β O z : β, z β 0 β§ β k : β, k β Icc 1 n β (f.eval (k : β)) + Complex.I * (g.eval (k : β)) = O + z * Complex.exp (Complex.I * 2 * Real.pi * k / n))
: (f.natDegree β₯ n - 1 β¨ g.natDegree β₯ n - 1) :=
sorry | 3bc162013071576b |
putnam_2018_a5 | theorem putnam_2018_a5
(f : β β β)
(h0 : f 0 = 0)
(h1 : f 1 = 1)
(hpos : β x : β, f x β₯ 0)
(hf : ContDiff β β€ f)
: β n > 0, β x : β, iteratedDeriv n f x < 0 :=
sorry | b331b74977de4c0c |
putnam_1985_b2 | abbrev putnam_1985_b2_solution : β β β := sorry
-- fun n β¦ ite (n = 101) 99 0
/--
Define polynomials $f_n(x)$ for $n \geq 0$ by $f_0(x)=1$, $f_n(0)=0$ for $n \geq 1$, and
\[
\frac{d}{dx} f_{n+1}(x) = (n+1)f_n(x+1)
\]
for $n \geq 0$. Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.
-/
theorem putnam_1985_b2
(f : β -> Polynomial β)
(hf0x : f 0 = 1)
(hfn0 : β n β₯ 1, (f n).eval 0 = 0)
(hfderiv : β n : β, derivative (f (n + 1)) = (n + 1) * (Polynomial.comp (f n) (X + 1)))
: Nat.factorization ((f 100).eval 1) = putnam_1985_b2_solution :=
sorry | 0722e9e37484a412 |
putnam_1968_a4 | theorem putnam_1968_a4
(n : β)
(S : Fin n β (EuclideanSpace β (Fin 3)))
(hS : β i : Fin n, dist 0 (S i) = 1)
: β i : Fin n, β j : Fin n, (if i < j then (dist (S i) (S j))^2 else (0 : β)) β€ n^2 :=
sorry | 97ddc75f447c1012 |
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ABOUT
This is a HF Dataset version of PutnamBench in Lean 4, put here for convenience. It is NOT my dataset. I am very thankful for the Trishul group for their work.
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