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|---|---|---|---|---|---|---|---|---|
Let $A$ be the set of all integers $n$ such that $1 \leq n \leq 2021$ and $\gcd(n,2021)=1$. For every nonnegative integer $j$, let $S(j)=\sum_{n \in A}n^j$. Determine all values of $j$ such that $S(j)$ is a multiple of $2021$. | the set of all j such that neither 42 nor 46 divides j | Show that the values of $j$ in question are those not divisible by either $42$ or $46$. | open Filter Topology | [] | @Eq (Set Nat) answer
(@setOf Nat fun (j_1 : Nat) =>
And (Not (@Dvd.dvd Nat Nat.instDvd (@OfNat.ofNat Nat (nat_lit 42) (instOfNatNat (nat_lit 42))) j_1))
(Not (@Dvd.dvd Nat Nat.instDvd (@OfNat.ofNat Nat (nat_lit 46) (instOfNatNat (nat_lit 46))) j_1))) | Set β | [
{
"t": "β",
"v": null,
"name": "j"
},
{
"t": "Finset β",
"v": null,
"name": "A"
},
{
"t": "β β β",
"v": null,
"name": "S"
},
{
"t": "A = {n | 1 β€ n β§ n β€ 2021 β§ Nat.gcd n 2021 = 1}",
"v": null,
"name": "hA"
},
{
"t": "β j' : β, S j' = β n in A, n ^ j'",
"v": null,
"name": "hS"
},
{
"t": "(2021 β£ S j) β j β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2021_a5",
"tags": [
"number_theory"
]
} |
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? | True | Show that it does follow that $P(2)$ is a composite integer. | open Filter Topology | [] | @Eq Prop answer True | Prop | [
{
"t": "(β P, (β n : β, P.coeff n = 0 β¨ P.coeff n = 1) β\n (β Q R : Polynomial β€, Q.degree > 0 β§ R.degree > 0 β§ P = Q * R) β\n (P.eval 2 β 0 β§ P.eval 2 β 1 β§ Β¬Prime (P.eval 2)))\n β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2021_a6",
"tags": [
"number_theory",
"algebra"
]
} |
Determine the maximum value of the sum $S = \sum_{n=1}^\infty \frac{n}{2^n}(a_1a_2 \dots a_n)^{1/n}$ over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying $\sum_{k=1}^\infty a_k=1$. | 2 / 3 | Show that the answer is $2/3$. | open Filter Topology | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))) | β | [
{
"t": "IsGreatest\n {S | β a : β+ β β, (β' k, a k = 1) β§ (β k, 0 β€ a k) β§\n S = β' n : β+, n / 2 ^ (n : β) * (β k in Finset.Icc 1 n, a k) ^ (1 / n : β)}\n answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2021_b2",
"tags": [
"analysis"
]
} |
Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\mathbb{R}^2$, and define $\rho(x,y)=yh_x-xh_y$. Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $\mathcal{S}$ of radius $r$ whose center is a distance $d$ away from the origin such that the integral of $\rho$ over the interior of $\mathcal{S}$ is zero. | True | Show that the given statement is true. | open Filter Topology Metric | [] | @Eq Prop answer True | Prop | [
{
"t": "((EuclideanSpace β (Fin 2)) β β) β (EuclideanSpace β (Fin 2)) β β",
"v": null,
"name": "rho"
},
{
"t": "β (h : (EuclideanSpace β (Fin 2)) β β) (p : EuclideanSpace β (Fin 2)),\n rho h p = (p 1) *\n deriv (fun x' : β => h ((WithLp.equiv 2 (EuclideanSpace β (Fin 2))).symm ![x', p 1])) (p 0)\n - (p 0) * deriv (fun y' : β => h ((WithLp.equiv 2 (EuclideanSpace β (Fin 2))).symm ![p 0, y'])) (p 1)",
"v": null,
"name": "rho_def"
},
{
"t": "(β h : (EuclideanSpace β (Fin 2)) β β, ContDiff β 2 h β\n (β d > 0, β r > 0, d > r β (β c : EuclideanSpace β (Fin 2), dist c 0 = d β§ (β« p in (ball c r), rho h p) = 0))) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2021_b3",
"tags": [
"analysis"
]
} |
Determine all ordered pairs of real numbers $(a,b)$ such that the line $y = ax+b$ intersects the curve $y = \ln(1+x^2)$ in exactly one point. | {(a, b) | (a = 0 β§ b = 0) β¨ 1 β€ |a| β¨ (0 < |a| β§ |a| < 1 β§ letI rm := (1 - β(1 - a ^ 2)) / a; letI rp := (1 + β(1 - a ^ 2)) / a; (b < Real.log (1 + rm ^ 2) - a * rm β¨ b > Real.log (1 + rp ^ 2) - a * rp))} | Show that the solution is the set of ordered pairs $(a,b)$ which satisfy at least one of (1) $a = b = 0$, (2) $|a| \geq 1$, and (3) $0 < |a| < 1$ and $b < \log(1 + r_{-}^2) - ar_{-}$ or $b > \log(1 + r_{+}^2) - ar_{+}$ where $r_{\pm} = \frac{1 \pm \sqrt{1 - a^2}}{a}$. | open Polynomial | [] | @Eq (Set (Prod Real Real)) answer
(@setOf (Prod Real Real) fun (x : Prod Real Real) =>
_example.match_1 (fun (x_1 : Prod Real Real) => Prop) x fun (a b : Real) =>
Or
(And (@Eq Real a (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))
(@Eq Real b (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))))
(Or
(@LE.le Real Real.instLE (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@abs Real Real.lattice Real.instAddGroup a))
(And
(@LT.lt Real Real.instLT (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))
(@abs Real Real.lattice Real.instAddGroup a))
(And
(@LT.lt Real Real.instLT (@abs Real Real.lattice Real.instAddGroup a)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(Or
(@LT.lt Real Real.instLT b
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(Real.log
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(Real.sqrt
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
a (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))
a)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) a
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(Real.sqrt
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) a
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))
a))))
(@GT.gt Real Real.instLT b
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(Real.log
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(Real.sqrt
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
a (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))
a)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) a
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(Real.sqrt
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) a
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))
a))))))))) | Set (β Γ β) | [
{
"t": "{a : (β Γ β) | β! x : β, a.1 * x + a.2 = Real.log (1 + x^2)} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2022_a1",
"tags": [
"algebra"
]
} |
Let $n$ be an integer with $n \geq 2$. Over all real polynomials $p(x)$ of degree $n$, what is the largest possible number of negative coefficients of $p(x)^2$? | $2n - 2$ | Show that the solution is $2n - 2$. | open Polynomial | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) =>
@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat)
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) n_1)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n β₯ 2",
"v": null,
"name": "hn"
},
{
"t": "Set β[X]",
"v": null,
"name": "S"
},
{
"t": "S = {P | natDegree P = n}",
"v": null,
"name": "hS"
},
{
"t": "β[X] β β",
"v": null,
"name": "negs"
},
{
"t": "β P : β[X], negs P = β i in Finset.range (P.natDegree + 1), if P.coeff i < 0 then 1 else 0",
"v": null,
"name": "hnegs"
},
{
"t": "sSup {negs (P^2) | P β S} = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2022_a2",
"tags": [
"algebra"
]
} |
Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays? | 290 | Show that the solution is 290. | null | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 290) (instOfNatNat (nat_lit 290))) | β | [
{
"t": "Set (Fin 2022) β Set (Fin 2022) β Prop",
"v": null,
"name": "IsValidMove"
},
{
"t": "β x y, IsValidMove x y β\n (x = y β§ β i < 2021, i β x β i + 1 β x) β¨\n β i < 2021, i β x β§ i + 1 β x β§ y = x βͺ {i, i + 1}",
"v": null,
"name": "IsValidMove_def"
},
{
"t": "List (Set (Fin 2022)) β Prop",
"v": null,
"name": "IsValidGame"
},
{
"t": "β g, IsValidGame g β (β gt, g = β
:: gt) β§ g.Chain' IsValidMove",
"v": null,
"name": "IsValidGame_def"
},
{
"t": "List (Set (Fin 2022)) β (Set (Fin 2022) β Set (Fin 2022)) β Prop",
"v": null,
"name": "ConformsToStrategy"
},
{
"t": "β g s, ConformsToStrategy g s β\n β (i) (h : i + 1 < g.length), Even i β g[i + 1] = s g[i]",
"v": null,
"name": "ConformsToStrategy_def"
},
{
"t": "IsGreatest\n {n | β s, (β x, IsValidMove x (s x)) β§ β g,\n IsValidGame g β ConformsToStrategy g s β β gh x, g = gh ++ [x] β§ n β€ xαΆ.ncard}\n answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2022_a5",
"tags": [
"combinatorics"
]
} |
Let $n$ be a positive integer. Determine, in terms of $n$, the largest integer $m$ with the following property: There exist real numbers $x_1,\dots,x_{2n}$ with $-1<x_1<x_2<\cdots<x_{2n}<1$ such that the sum of the lengths of the $n$ intervals $[x_1^{2k-1},x_2^{2k-1}],[x_3^{2k-1},x_4^{2k-1}],\dots,[x_{2n-1}^{2k-1},x_{2n}^{2k-1}]$ is equal to $1$ for all integers $k$ with $1 \leq k \leq m$. | (fun n : β => n) | Show that the largest such $m$ is $n$. | open Set | [] | @Eq (Nat β Nat) answer fun (n_1 : Nat) => n_1 | β β β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "0 < n",
"v": null,
"name": "hn"
},
{
"t": "IsGreatest\n {m : β | β x : β β β,\n StrictMono x β§ -1 < x 1 β§ x (2 * n) < 1 β§\n β k β Icc 1 m, β i in Icc 1 n, ((x (2 * i) : β) ^ (2 * k - 1) - (x (2 * i - 1)) ^ (2 * k - 1)) = 1}\n (answer n)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2022_a6",
"tags": [
"algebra"
]
} |
Let $\times$ represent the cross product in $\mathbb{R}^3$. For what positive integers $n$ does there exist a set $S \subset \mathbb{R}^3$ with exactly $n$ elements such that $S=\{v \times w:v,w \in S\}$? | {1, 7} | Show that the possible values of $n$ are $1$ and $7$. | open Polynomial | [] | @Eq (Set Nat) answer
(@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
(@Singleton.singleton Nat (Set Nat) (@Set.instSingletonSet Nat)
(@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7))))) | Set β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "Finset (Fin 3 β β) β Prop",
"v": null,
"name": "P"
},
{
"t": "β S : Finset (Fin 3 β β), P S β (S = {u : Fin 3 β β | β v w : S, u = crossProduct v w})",
"v": null,
"name": "P_def"
},
{
"t": "(0 < n β§ β S : Finset (Fin 3 β β), S.card = n β§ P S) β n β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2022_b2",
"tags": [
"algebra"
]
} |
Assign to each positive real number a color, either red or blue. Let $D$ be the set of all distances $d>0$ such that there are two points of the same color at distance $d$ apart. Recolor the positive reals so that the numbers in $D$ are red and the numbers not in $D$ are blue. If we iterate this recoloring process, will we always end up with all the numbers red after a finite number of steps? | True | Show that the answer is yes. | open Polynomial | [] | @Eq Prop answer True | Prop | [
{
"t": "(β β Prop) β (β β Prop)",
"v": null,
"name": "recolor"
},
{
"t": "β isred d, recolor isred d β β p q : β, p > 0 β§ q > 0 β§ p < q β§ isred p = isred q β§ q - p = d",
"v": null,
"name": "hrecolor"
},
{
"t": "(β isred, (β k, β p > 0, (recolor^[k] isred) p)) β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2022_b3",
"tags": [
"algebra"
]
} |
Find all integers $n$ with $n \geq 4$ for which there exists a sequence of distinct real numbers $x_1,\dots,x_n$ such that each of the sets $\{x_1,x_2,x_3\},\{x_2,x_3,x_4\},\dots,\{x_{n-2},x_{n-1},x_n\},\{x_{n-1},x_n,x_1\}$, and $\{x_n,x_1,x_2\}$ forms a $3$-term arithmetic progression when arranged in increasing order. | {n : β | 3 β£ n β§ n β₯ 9} | Show that the values of $n$ in question are the multiples of $3$ starting with $9$. | open Polynomial | [] | @Eq (Set Nat) answer
(@setOf Nat fun (n_1 : Nat) =>
And (@Dvd.dvd Nat Nat.instDvd (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) n_1)
(@GE.ge Nat instLENat n_1 (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9))))) | Set β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n β₯ 4",
"v": null,
"name": "hn"
},
{
"t": "β β β β β β Prop",
"v": null,
"name": "ap3"
},
{
"t": "(β β β) β Prop",
"v": null,
"name": "xprog"
},
{
"t": "β x0 x1 x2, ap3 x0 x1 x2 β β o0 o1 o2 : β, (o0 < o1 β§ o1 < o2 β§ ({o0, o1, o2} : Set β) = {x0, x1, x2}) β (o1 - o0 = o2 - o1)",
"v": null,
"name": "hap3"
},
{
"t": "β x, xprog x β\n ((β i j : Fin n, i.1 β j.1 β x i.1 β x j.1) β§ (β i : Fin n, ap3 (x i.1) (x ((i.1 + 1) % n)) (x ((i.1 + 2) % n))))",
"v": null,
"name": "hxprog"
},
{
"t": "(β x, xprog x) β n β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2022_b4",
"tags": [
"algebra"
]
} |
For $0 \leq p \leq 1/2$, let $X_1,X_2,\dots$ be independent random variables such that
$X_i=\begin{cases}
1 & \text{with probability $p$,} \\
-1 & \text{with probability $p$,} \\
0 & \text{with probability $1-2p$,}
\end{cases}$
for all $i \geq 1$. Given a positive integer $n$ and integers $b,a_1,\dots,a_n$, let $P(b,a_1,\dots,a_n)$ denote the probability that $a_1X_1+\dots+a_nX_n=b$. For which values of $p$ is it the case that $P(0,a_1,\dots,a_n) \geq P(b,a_1,\dots,a_n)$ for all positive integers $n$ and all integers $b,a_1,\dots,a_n$? | Set.Icc 0 (1 / 4) | Show that the answer is $p \leq 1/4$. | open Polynomial | [] | @Eq (Set Real) answer
(@Set.Icc Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))) | Set β | [
{
"t": "β",
"v": null,
"name": "p"
},
{
"t": "(n : β) β Set (Fin n β β€)",
"v": null,
"name": "Xset"
},
{
"t": "(n : β) β (Fin n β β€) β β",
"v": null,
"name": "Xprob"
},
{
"t": "(n : β) β β€ β (Fin n β β€) β β",
"v": null,
"name": "P"
},
{
"t": "β n > 0, Xset n = {X : Fin n β β€ | β i : Fin n, X i = 1 β¨ X i = -1 β¨ X i = 0}",
"v": null,
"name": "hXset"
},
{
"t": "β n > 0, β X : Fin n β β€, Xprob n X = β i : Fin n, if (X i = 1 β¨ X i = -1) then p else (1 - 2 * p)",
"v": null,
"name": "hXprob"
},
{
"t": "β n > 0, β (b : β€) (a : Fin n β β€), P n b a = β' X : {X' β Xset n | (β i : Fin n, a i * X' i) = b}, Xprob n X",
"v": null,
"name": "hP"
},
{
"t": "(0 β€ p β§ p β€ 1 / 2 β§ (β n > 0, β (b : β€) (a : Fin n β β€), P n 0 a β₯ P n b a)) β p β answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2022_b5",
"tags": [
"probability",
"algebra"
]
} |
Find all continuous functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ such that $f(xf(y))+f(yf(x))=1+f(x+y)$ for all $x,y>0$. | the set of functions of the form f(x) = 1 / (1 + c * x) where c β₯ 0 | Show that the only such functions are the functions $f(x)=\frac{1}{1+cx}$ for some $c \geq 0$. | open Polynomial | [] | @Eq
(Set
(@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) β
@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))))
answer
(@setOf
(@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) β
@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))))
fun
(f_1 :
@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))) β
@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))) =>
@Exists Real fun (c : Real) =>
And (@GE.ge Real Real.instLE c (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))
(β
(x :
@Set.Elem Real
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))),
@Eq Real
(@Subtype.val Real
(fun (x_1 : Real) =>
@Membership.mem Real (Set Real) (@Set.instMembership Real)
(@Set.Ioi Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))
x_1)
(f_1 x))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c
(@Subtype.val Real
(fun (x_1 : Real) =>
@Membership.mem Real (Set Real) (@Set.instMembership Real)
(@Set.Ioi Real Real.instPreorder
(@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))
x_1)
x)))))) | Set (Set.Ioi (0 : β) β Set.Ioi (0 : β)) | [
{
"t": "Set.Ioi (0 : β) β Set.Ioi (0 : β)",
"v": null,
"name": "f"
},
{
"t": "Continuous f",
"v": null,
"name": "hf"
},
{
"t": "(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 answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2022_b6",
"tags": [
"analysis"
]
} |
For a positive integer $n$, let $f_n(x) = \cos(x) \cos(2x) \cos(3x) \cdots \cos(nx)$. Find the smallest $n$ such that $|f_n''(0)| > 2023$. | 18 | Show that the solution is $n = 18$. | open Nat | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 18) (instOfNatNat (nat_lit 18))) | β | [
{
"t": "β β β β β",
"v": null,
"name": "f"
},
{
"t": "β n > 0, f n = fun x : β => β i in Finset.Icc 1 n, Real.cos (i * x)",
"v": null,
"name": "hf"
},
{
"t": "IsLeast {n | 0 < n β§ |iteratedDeriv 2 (f n) 0| > 2023} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_a1",
"tags": [
"analysis"
]
} |
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2n$; that is to say, $p(x) = x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_1 x + a_0$ for some real coefficients $a_0, \dots, a_{2n-1}$. Suppose that $p(1/k) = k^2$ for all integers $k$ such that $1 \leq |k| \leq n$. Find all other real numbers $x$ for which $p(1/x) = x^2$. | {1/n!, -1/n!} | Show that the other real numbers satisfying $p(1/x) = x^2$ are $\pm \frac{1}{n!}.$ | open Nat | [] | @Eq (Nat β Set Real) answer fun (n_1 : Nat) =>
@Insert.insert Real (Set Real) (@Set.instInsert Real)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@Nat.cast Real Real.instNatCast (Nat.factorial n_1)))
(@Singleton.singleton Real (Set Real) (@Set.instSingletonSet Real)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@Nat.cast Real Real.instNatCast (Nat.factorial n_1)))) | β β Set β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n > 0 β§ Even n",
"v": null,
"name": "hn"
},
{
"t": "Polynomial β",
"v": null,
"name": "p"
},
{
"t": "Polynomial.Monic p β§ p.degree = 2*n",
"v": null,
"name": "hp"
},
{
"t": "Set β",
"v": null,
"name": "S"
},
{
"t": "S = {x : β | β k : β€, x = k β§ 1 β€ |k| β§ |k| β€ n}",
"v": null,
"name": "hS"
},
{
"t": "β k β S, p.eval (1/k) = k^2",
"v": null,
"name": "hpinv"
},
{
"t": "{x : β | p.eval (1/x) = x^2} \\ S = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_a2",
"tags": [
"algebra"
]
} |
Determine the smallest positive real number $r$ such that there exist differentiable functions $f\colon \mathbb{R} \to \mathbb{R}$ and $g\colon \mathbb{R} \to \mathbb{R}$ satisfying
\begin{enumerate}
\item[(a)] $f(0) > 0$,
\item[(b)] $g(0) = 0$,
\item[(c)] $|f'(x)| \leq |g(x)|$ for all $x$,
\item[(d)] $|g'(x)| \leq |f(x)|$ for all $x$, and
\item[(e)] $f(r) = 0$. \end{enumerate} | Ο/2 | Show that the solution is $r = \pi/2$. | open Nat | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | β | [
{
"t": "answer = sInf {r > 0 | β f g : β β β,\n Differentiable β f β§ Differentiable β g β§\n (β x : β, |deriv f x| β€ |g x| β§ |deriv g x| β€ |f x|) β§\n f 0 > 0 β§ g 0 = 0 β§ f r = 0}",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_a3",
"tags": [
"analysis"
]
} |
For a nonnegative integer $k$, let $f(k)$ be the number of ones in the base 3 representation of $k$. Find all complex numbers $z$ such that \[ \sum_{k=0}^{3^{1010}-1} (-2)^{f(k)} (z+k)^{2023} = 0. \] | {-(3^1010 - 1)/2, -(3^1010 - 1)/2 + Real.sqrt (9^1010 - 1) * Complex.I/4, -(3^1010 - 1)/2 - Real.sqrt (9^1010 - 1) * Complex.I/4} | Show that the solution is the set of complex numbers $\{- \frac{3^{1010} - 1}{2}, - \frac{3^{1010} - 1}{2} \pm \frac{\sqrt{9^{1010} - 1}}{4}i \}$ | open Nat | [] | @Eq (Set Complex) answer
(@Insert.insert Complex (Set Complex) (@Set.instInsert Complex)
(@HDiv.hDiv Complex Complex Complex (@instHDiv Complex (@DivInvMonoid.toDiv Complex Complex.instDivInvMonoid))
(@Neg.neg Complex Complex.instNeg
(@HSub.hSub Complex Complex Complex (@instHSub Complex Complex.instSub)
(@HPow.hPow Complex Nat Complex
(@instHPow Complex Nat
(@Monoid.toNatPow Complex
(@MonoidWithZero.toMonoid Complex (@Semiring.toMonoidWithZero Complex Complex.instSemiring))))
(@OfNat.ofNat Complex (nat_lit 3)
(@instOfNatAtLeastTwo Complex (nat_lit 3)
(@AddMonoidWithOne.toNatCast Complex
(@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@OfNat.ofNat Nat (nat_lit 1010) (instOfNatNat (nat_lit 1010))))
(@OfNat.ofNat Complex (nat_lit 1) (@One.toOfNat1 Complex Complex.instOne))))
(@OfNat.ofNat Complex (nat_lit 2)
(@instOfNatAtLeastTwo Complex (nat_lit 2)
(@AddMonoidWithOne.toNatCast Complex (@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@Insert.insert Complex (Set Complex) (@Set.instInsert Complex)
(@HAdd.hAdd Complex Complex Complex (@instHAdd Complex Complex.instAdd)
(@HDiv.hDiv Complex Complex Complex (@instHDiv Complex (@DivInvMonoid.toDiv Complex Complex.instDivInvMonoid))
(@Neg.neg Complex Complex.instNeg
(@HSub.hSub Complex Complex Complex (@instHSub Complex Complex.instSub)
(@HPow.hPow Complex Nat Complex
(@instHPow Complex Nat
(@Monoid.toNatPow Complex
(@MonoidWithZero.toMonoid Complex (@Semiring.toMonoidWithZero Complex Complex.instSemiring))))
(@OfNat.ofNat Complex (nat_lit 3)
(@instOfNatAtLeastTwo Complex (nat_lit 3)
(@AddMonoidWithOne.toNatCast Complex
(@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@OfNat.ofNat Nat (nat_lit 1010) (instOfNatNat (nat_lit 1010))))
(@OfNat.ofNat Complex (nat_lit 1) (@One.toOfNat1 Complex Complex.instOne))))
(@OfNat.ofNat Complex (nat_lit 2)
(@instOfNatAtLeastTwo Complex (nat_lit 2)
(@AddMonoidWithOne.toNatCast Complex
(@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@HDiv.hDiv Complex Complex Complex (@instHDiv Complex (@DivInvMonoid.toDiv Complex Complex.instDivInvMonoid))
(@HMul.hMul Complex Complex Complex (@instHMul Complex Complex.instMul)
(β(Real.sqrt
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@OfNat.ofNat Real (nat_lit 9)
(@instOfNatAtLeastTwo Real (nat_lit 9) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7))))))
(@OfNat.ofNat Nat (nat_lit 1010) (instOfNatNat (nat_lit 1010))))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))))
Complex.I)
(@OfNat.ofNat Complex (nat_lit 4)
(@instOfNatAtLeastTwo Complex (nat_lit 4)
(@AddMonoidWithOne.toNatCast Complex
(@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))))
(@Singleton.singleton Complex (Set Complex) (@Set.instSingletonSet Complex)
(@HSub.hSub Complex Complex Complex (@instHSub Complex Complex.instSub)
(@HDiv.hDiv Complex Complex Complex (@instHDiv Complex (@DivInvMonoid.toDiv Complex Complex.instDivInvMonoid))
(@Neg.neg Complex Complex.instNeg
(@HSub.hSub Complex Complex Complex (@instHSub Complex Complex.instSub)
(@HPow.hPow Complex Nat Complex
(@instHPow Complex Nat
(@Monoid.toNatPow Complex
(@MonoidWithZero.toMonoid Complex (@Semiring.toMonoidWithZero Complex Complex.instSemiring))))
(@OfNat.ofNat Complex (nat_lit 3)
(@instOfNatAtLeastTwo Complex (nat_lit 3)
(@AddMonoidWithOne.toNatCast Complex
(@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@OfNat.ofNat Nat (nat_lit 1010) (instOfNatNat (nat_lit 1010))))
(@OfNat.ofNat Complex (nat_lit 1) (@One.toOfNat1 Complex Complex.instOne))))
(@OfNat.ofNat Complex (nat_lit 2)
(@instOfNatAtLeastTwo Complex (nat_lit 2)
(@AddMonoidWithOne.toNatCast Complex
(@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@HDiv.hDiv Complex Complex Complex (@instHDiv Complex (@DivInvMonoid.toDiv Complex Complex.instDivInvMonoid))
(@HMul.hMul Complex Complex Complex (@instHMul Complex Complex.instMul)
(β(Real.sqrt
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@OfNat.ofNat Real (nat_lit 9)
(@instOfNatAtLeastTwo Real (nat_lit 9) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7))))))
(@OfNat.ofNat Nat (nat_lit 1010) (instOfNatNat (nat_lit 1010))))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))))
Complex.I)
(@OfNat.ofNat Complex (nat_lit 4)
(@instOfNatAtLeastTwo Complex (nat_lit 4)
(@AddMonoidWithOne.toNatCast Complex
(@AddGroupWithOne.toAddMonoidWithOne Complex Complex.addGroupWithOne))
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))))))) | Set β | [
{
"t": "{z : β | β k in Finset.Icc 0 (3^1010 - 1),\n (-2)^(((digits 3 k).filter (fun x => x = 1)).length) * (z + k)^2023 = 0} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_a5",
"tags": [
"algebra"
]
} |
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? | the set of all positive integers | Show that Bob has a winning strategy for all $n$. | open Finset | [] | @Eq (Set Nat) answer
(@setOf Nat fun (n : Nat) => @LT.lt Nat instLTNat (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) n) | Set β | [
{
"t": "List β β Prop",
"v": null,
"name": "IsValidGame"
},
{
"t": "β g, IsValidGame g β g.Nodup β§ (β i β g, i β Icc 1 g.length)",
"v": null,
"name": "IsValidGame_def"
},
{
"t": "List β β ZMod 2",
"v": null,
"name": "parityOf"
},
{
"t": "β g, parityOf g = ((range g.length).filter fun i β¦ g[i]! = i + 1).card",
"v": null,
"name": "parityOf_def"
},
{
"t": "List β β (List β β β) β Prop",
"v": null,
"name": "ConformsToStrategy"
},
{
"t": "β g s, ConformsToStrategy g s β\n β (i) (h : i < g.length), Odd i β g[i] = s (g.take i)",
"v": null,
"name": "ConformsToStrategy_def"
},
{
"t": "β β (List β β β) β Prop",
"v": null,
"name": "IsWinningFor"
},
{
"t": "β n s, IsWinningFor n s β\n β p, β g, g.length = n β IsValidGame g β ConformsToStrategy g s β parityOf g = p",
"v": null,
"name": "IsWinningFor_def"
},
{
"t": "{n : β | 0 < n β§ β s, IsWinningFor n s} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_a6",
"tags": [
"combinatorics"
]
} |
Consider an $m$-by-$n$ grid of unit squares, indexed by $(i,j)$ with $1 \leq i \leq m$ and $1 \leq j \leq n$. There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i,j)$ with $1 \leq i \leq m-1$ and $1 \leq j \leq n-1$. If a coin occupies the square $(i,j)$ with $i \leq m-1$ and $j \leq n-1$ and the squares $(i+1,j)$, $(i,j+1)$, and $(i+1,j+1)$ are unoccupied, then a legal move is to slide the coin from $(i,j)$ to $(i+1,j+1)$. How many distinct configurations of coins can be reached starting from the initial configuration by a (possibly empty) sequence of legal moves? | $\binom{m + n - 2}{m - 1}$ | Show that the number of such configurations is $\binom{m+n-2}{m-1}$. | open Nat | [] | @Eq (Nat β Nat β Nat) answer fun (m_1 n_1 : Nat) =>
Nat.choose
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) (@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) m_1 n_1)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) m_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) | β β β β β | [
{
"t": "β",
"v": null,
"name": "m"
},
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "β β β β Bool",
"v": null,
"name": "initcoins"
},
{
"t": "initcoins = (fun i j : β => (i β€ m - 2 β§ j β€ n - 2 : Bool))",
"v": null,
"name": "hinitcoins"
},
{
"t": "(β β β β Bool) β (β β β β Bool) β Prop",
"v": null,
"name": "IsLegalMove"
},
{
"t": "β coins1 coins2,\n IsLegalMove coins1 coins2 β\n β i j, i < m - 1 β§ j < n - 1 β§\n coins1 i j β§ !coins1 (i + 1) j β§ !coins1 i (j + 1) β§ !coins1 (i + 1) (j + 1) β§\n !coins2 i j β§ !coins2 (i + 1) j β§ !coins2 i (j + 1) β§ coins2 (i + 1) (j + 1) β§\n (β i' j', ((i', j') β (i, j) β§ (i', j') β (i + 1, j) β§ (i', j') β (i, j + 1) β§ (i', j') β (i + 1, j + 1))\n β coins1 i' j' = coins2 i' j')",
"v": null,
"name": "IsLegalMove_def"
},
{
"t": "List (β β β β Bool) β Prop",
"v": null,
"name": "IsLegalSeq"
},
{
"t": "β seq, IsLegalSeq seq β seq.length β₯ 1 β§ seq[0]! = initcoins β§ (β i < seq.length - 1, IsLegalMove seq[i]! seq[i + 1]!)",
"v": null,
"name": "IsLegalSeq_def"
},
{
"t": "m β₯ 1 β§ n β₯ 1",
"v": null,
"name": "mnpos"
},
{
"t": "{config : β β β β Bool | β seq : List (β β β β Bool), IsLegalSeq seq β§ config = seq.getLast!}.encard = answer m n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_b1",
"tags": [
"combinatorics"
]
} |
For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 * n$. What is the minimum value of $k(n)$? | 3 | Show that the minimum value is 3. | open Nat | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) | β | [
{
"t": "sInf {(digits 2 (2023 * n)).sum | n > 0} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_b2",
"tags": [
"number_theory"
]
} |
A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called zigzag if $k = 1$, or if $y_2 - y_1, y_3 - y_2, \ldots, y_k - y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2,\ldots, X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a(X_1, X_2, \ldots, X_n)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1, i_2, \ldots, i_k$ such that $X_{i_1}, X_{i_2}, \ldots, X_{i_k}$ is zigzag. Find the expected value of $a(X_1, X_2, \ldots, X_n)$ for $n \ge 2$. | (2n + 2) / 3 | Show that the expected value is \frac{2n + 2}{3}. | open ProbabilityTheory Set | [] | @Eq (Nat β Real) answer fun (n_1 : Nat) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@Nat.cast Real Real.instNatCast n_1))
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))) | β β β | [
{
"t": "{k : β} β (Fin k β β) β Prop",
"v": null,
"name": "IsZigZag"
},
{
"t": "β (k : β) [NeZero k] (y : Fin k β β),\n IsZigZag y β k = 1 β¨ ((β i, i + 1 < k β y (i + 1) β y i)) β§\n (β i, i + 2 < k β (y (i + 2) < y (i + 1) β y i < y (i + 1)))",
"v": null,
"name": "IsZigZag_def"
},
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "2 β€ n",
"v": null,
"name": "hn"
},
{
"t": "(Fin n β Icc (0 : β) 1) β β",
"v": null,
"name": "a"
},
{
"t": "β x, IsGreatest {k | β i : Fin k βͺo Fin n, IsZigZag ((β) β x β i)} (a x)",
"v": null,
"name": "ha"
},
{
"t": "πΌ[(β) β a] = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_b3",
"tags": [
"probability",
"combinatorics"
]
} |
For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \geq t_0$ by the following properties:
\begin{enumerate}
\item[(a)] $f(t)$ is continuous for $t \geq t_0$, and is twice differentiable for all $t>t_0$ other than $t_1,\dots,t_n$;
\item[(b)] $f(t_0)=1/2$;
\item[(c)] $\lim_{t \to t_k^+} f'(t)=0$ for $0 \leq k \leq n$;
\item[(d)] For $0 \leq k \leq n-1$, we have $f''(t)=k+1$ when $t_k<t<t_{k+1}$, and $f''(t)=n+1$ when $t>t_n$.
\end{enumerate}
Considering all choices of $n$ and $t_0,t_1,\dots,t_n$ such that $t_k \geq t_{k-1}+1$ for $1 \leq k \leq n$, what is the least possible value of $T$ for which $f(t_0+T)=2023$? | 29 | Show that the minimum value of $T$ is $29$. | open Nat Topology Filter | [] | @Eq Real answer
(@OfNat.ofNat Real (nat_lit 29)
(@instOfNatAtLeastTwo Real (nat_lit 29) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 27) (instOfNatNat (nat_lit 27)))))) | β | [
{
"t": "β β (β β β) β Set β",
"v": null,
"name": "tne"
},
{
"t": "β n ts, tne n ts = {t | t > ts 0 β§ β i : Fin n, t β ts (i.1 + 1)}",
"v": null,
"name": "htne"
},
{
"t": "IsLeast\n {(T : β) | 0 β€ T β§ β (n : β) (ts : β β β) (f : β β β),\n β k : Fin n, ts (k.1 + 1) β₯ ts k.1 + 1 β§\n ContinuousOn f (Set.Ici (ts 0)) β§\n ContDiffOn β 1 f (tne n ts) β§\n DifferentiableOn β (derivWithin f (tne n ts)) (tne n ts) β§\n f (ts 0) = 0.5 β§\n (β k : Fin (n + 1),\n Tendsto (derivWithin f (tne n ts)) (π[>] (ts k.1)) (π 0)) β§\n (β k : Fin n,\n β t β Set.Ioo (ts k.1) (ts (k.1 + 1)),\n iteratedDerivWithin 2 f (tne n ts) t = k.1 + 1) β§\n (β t > ts n,\n iteratedDerivWithin 2 f (tne n ts) t = n + 1) β§\n f (ts 0 + T) = 2023}\n answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_b4",
"tags": [
"analysis"
]
} |
Determine which positive integers $n$ have the following property: For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi:\{1,2,\dots,n\} \to \{1,2,\dots,n\}$ such that $\pi(\pi(k)) \equiv mk \pmod{n}$ for all $k \in \{1,2,\dots,n\}$. | {n : β | n = 1 β¨ n β‘ 2 [MOD 4]} | Show that the desired property holds if and only if $n=1$ or $n \equiv 2 \pmod{4}$. | open Nat Topology Filter | [] | @Eq (Set Nat) answer
(@setOf Nat fun (n_1 : Nat) =>
Or (@Eq Nat n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(Nat.ModEq (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) n_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) | Set β | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "n β answer β\n 0 < n β§\n (β m : β€, IsRelPrime m n β β p : Equiv.Perm (Fin n),\n β k : Fin n, (p (p k)).1 + 1 β‘ m * (k.1 + 1) [ZMOD n])",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_b5",
"tags": [
"number_theory"
]
} |
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 $S$. | $(-1)^{\lceil n/2 \rceil-1}2\lceil\frac{n}{2}\rceil$ | Show that the determinant equals $(-1)^{\lceil n/2 \rceil-1}2\lceil\frac{n}{2}\rceil$. | open Nat Topology Filter | [] | @Eq (Nat β Int) answer fun (n_1 : Nat) =>
@HMul.hMul Int Int Int (@instHMul Int Int.instMul)
(@HMul.hMul Int Int Int (@instHMul Int Int.instMul)
(@HPow.hPow Int Nat Int (@instHPow Int Nat (@Monoid.toNatPow Int Int.instMonoid))
(@Neg.neg Int Int.instNegInt (@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1))))
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat)
(@Nat.ceil Rat Rat.instOrderedSemiring
(@FloorRing.toFloorSemiring Rat Rat.instLinearOrderedRing Rat.instFloorRing)
(@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv) (@Nat.cast Rat Rat.instNatCast n_1)
(@OfNat.ofNat Rat (nat_lit 2) (@Rat.instOfNat (nat_lit 2)))))
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2))))
(@Nat.cast Int instNatCastInt
(@Nat.ceil Rat Rat.instOrderedSemiring
(@FloorRing.toFloorSemiring Rat Rat.instLinearOrderedRing Rat.instFloorRing)
(@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv) (@Nat.cast Rat Rat.instNatCast n_1)
(@OfNat.ofNat Rat (nat_lit 2) (@Rat.instOfNat (nat_lit 2)))))) | β β β€ | [
{
"t": "β",
"v": null,
"name": "n"
},
{
"t": "Matrix (Fin n) (Fin n) β€",
"v": null,
"name": "S"
},
{
"t": "n > 0",
"v": null,
"name": "npos"
},
{
"t": "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",
"v": null,
"name": "hS"
},
{
"t": "S.det = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_2023_b6",
"tags": [
"linear_algebra"
]
} |
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