Formal Problem-Solving
Collection
This collection is part of the official implementation of Beyond Theorem Proving: Formulation, Framework and Benchmark for Formal Problem-Solving. • 5 items • Updated
• 3
Datasets:
informal_problem stringlengths 47 738 | informal_answer stringlengths 1 204 | informal_solution stringlengths 16 303 | header stringlengths 8 63 ⌀ | intros listlengths 0 2 | formal_answer stringlengths 20 24.1k | formal_answer_type stringlengths 1 88 | outros listlengths 1 13 | metainfo dict |
|---|---|---|---|---|---|---|---|---|
Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having 0 as a left-hand endpoint, such that for every positive member $x$ of $I$ the average of $f$ over the closed interval $[0, x]$ is equal to the geometric mean of the numbers $f(0)$ and $f(x)$. | the set of functions $f(x) = \frac{a}{(1 - c x)^2}$ where $a \geq 0$ | Show that \[ f(x) = \frac{a}{(1 - cx)^2} \begin{cases} \text{for } 0 \le x < \frac{1}{c}, & \text{if } c > 0\\ \text{for } 0 \le x < \infty, & \text{if } c \le 0, \end{cases} \] where $a > 0$. | open MeasureTheory Set | [] | @Eq (Set (Real → Real)) answer
(@setOf (Real → Real) fun (f : Real → Real) =>
@Exists Real fun (a : Real) =>
@Exists Real fun (c : Real) =>
And (@GE.ge Real Real.instLE a (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)))
(@Eq (Real → Real) f fun (x : Real) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) a
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) c x))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) | Set (ℝ → ℝ) | [
{
"t": "Set ℝ → (ℝ → ℝ) → Prop",
"v": null,
"name": "P"
},
{
"t": "∀ s f, P s f ↔ 0 ≤ f ∧ ∀ x ∈ s, ⨍ t in Ico 0 x, f t = √(f 0 * f x)",
"v": null,
"name": "P_def"
},
{
"t": "answer = {f : ℝ → ℝ | P (Ioi 0) f ∨ (∃ e > 0, P (Ioo 0 e) f)}",
"v": null,
"name": "h_answer"
}
... | {
"problem_name": "putnam_1962_a2",
"tags": [
"analysis"
]
} |
Evaluate in closed form \[ \sum_{k=1}^n {n \choose k} k^2. \] | $n(n+1)2^{n-2}$ | Show that the expression equals $n(n+1)2^{n-2}$. | null | [] | @Eq (Nat → Nat) answer fun (n_1 : Nat) =>
@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat)
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) n_1
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) | ℕ → ℕ | [
{
"t": "ℕ",
"v": null,
"name": "n"
},
{
"t": "n ≥ 2",
"v": null,
"name": "hn"
},
{
"t": "answer n = ∑ k in Finset.Icc 1 n, Nat.choose n k * k^2",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1962_a5",
"tags": [
"algebra",
"combinatorics"
]
} |
Find an integral formula (i.e., a function $z$ such that $y(x) = \int_{1}^{x} z(t) dt$) for the solution of the differential equation $$\delta (\delta - 1) (\delta - 2) \cdots (\delta - n + 1) y = f(x)$$ with the initial conditions $y(1) = y'(1) = \cdots = y^{(n-1)}(1) = 0$, where $n \in \mathbb{N}$, $f$ is continuous for all $x \ge 1$, and $\delta$ denotes $x\frac{d}{dx}$. | $(x - t)^{n-1} \cdot f(t) / ((n-1)! \cdot t^n)$ | Show that the solution is $$y(x) = \int_{1}^{x} \frac{(x - t)^{n - 1} f(t)}{(n - 1)!t^n} dt$$. | open Nat Set Topology Filter | [] | @Eq ((Real → Real) → Nat → Real → Real → Real) answer fun (f_1 : Real → Real) (n_1 : Nat) (x t : Real) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x t)
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(f_1 t))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@Nat.cast Real Real.instNatCast
(Nat.factorial
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) t n_1)) | (ℝ → ℝ) → ℕ → ℝ → ℝ → ℝ | [
{
"t": "ℕ → (ℝ → ℝ) → (ℝ → ℝ)",
"v": null,
"name": "P"
},
{
"t": "P 0 = id ∧ ∀ i y, P (i + 1) y = P i (fun x ↦ x * deriv y x - i * y x)",
"v": null,
"name": "hP"
},
{
"t": "ℕ",
"v": null,
"name": "n"
},
{
"t": "0 < n",
"v": null,
"name": "hn"
},
{
... | {
"problem_name": "putnam_1963_a3",
"tags": [
"analysis"
]
} |
For what integer $a$ does $x^2-x+a$ divide $x^{13}+x+90$? | 2 | Show that $a=2$. | open Topology Filter Polynomial | [] | @Eq Int answer (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2))) | ℤ | [
{
"t": "ℤ",
"v": null,
"name": "a"
},
{
"t": "Polynomial.X^2 - Polynomial.X + (Polynomial.C a) ∣ (Polynomial.X ^ 13 + Polynomial.X + (Polynomial.C 90))",
"v": null,
"name": "h_div"
},
{
"t": "answer = a",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1963_b1",
"tags": [
"algebra"
]
} |
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 $P$? | True | Show that $S$ is dense in $P$. | open Topology Filter Polynomial | [] | @Eq Prop answer True | Prop | [
{
"t": "Set ℝ",
"v": null,
"name": "S"
},
{
"t": "S = {2 ^ m * 3 ^ n | (m : ℤ) (n : ℤ)}",
"v": null,
"name": "hS"
},
{
"t": "answer = (closure S ⊇ Set.Ioi (0 : ℝ))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1963_b2",
"tags": [
"analysis"
]
} |
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 $y$. | the set of functions of the form $A \sinh(k u)$, $A u$, or $A \sin(k u)$ | Show that the solution is the sets of functions $f(u)=A\sinh ku$, $f(u)=Au$, and $f(u)=A\sin ku$ with $A,k \in \mathbb{R}$. | open Topology Filter Polynomial | [] | @Eq (Set (Real → Real)) answer
(@Union.union (Set (Real → Real)) (@Set.instUnion (Real → Real))
(@Union.union (Set (Real → Real)) (@Set.instUnion (Real → Real))
(@setOf (Real → Real) fun (x : Real → Real) =>
@Exists Real fun (A : Real) =>
@Exists Real fun (k : Real) =>
@Eq (Real → Real)
(fun (u : Real) =>
@HMul.hMul Real Real Real (@instHMul Real Real.instMul) A
(Real.sinh (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) k u)))
x)
(@setOf (Real → Real) fun (x : Real → Real) =>
@Exists Real fun (A : Real) =>
@Eq (Real → Real) (fun (u : Real) => @HMul.hMul Real Real Real (@instHMul Real Real.instMul) A u) x))
(@setOf (Real → Real) fun (x : Real → Real) =>
@Exists Real fun (A : Real) =>
@Exists Real fun (k : Real) =>
@Eq (Real → Real)
(fun (u : Real) =>
@HMul.hMul Real Real Real (@instHMul Real Real.instMul) A
(Real.sin (@HMul.hMul Real Real Real (@instHMul Real Real.instMul) k u)))
x)) | Set (ℝ → ℝ) | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f"
},
{
"t": "f ∈ answer",
"v": null,
"name": "h_answer"
},
{
"t": "ContDiff ℝ 1 f",
"v": null,
"name": "h_cont_diff"
},
{
"t": "Differentiable ℝ (deriv f)",
"v": null,
"name": "h_diff"
},
{
"t": "∀ x y : ℝ, (f x... | {
"problem_name": "putnam_1963_b3",
"tags": [
"analysis"
]
} |
Let $\alpha$ be a real number. Find all continuous real-valued functions $f : [0, 1] \to (0, \infty)$ such that
\begin{align*}
\int_0^1 f(x) dx &= 1, \\
\int_0^1 x f(x) dx &= \alpha, \\
\int_0^1 x^2 f(x) dx &= \alpha^2. \\
\end{align*} | the empty set | Prove that there are no such functions. | open Set | [] | @Eq (Real → Set (Real → Real)) answer fun (x : Real) =>
@EmptyCollection.emptyCollection (Set (Real → Real)) (@Set.instEmptyCollection (Real → Real)) | ℝ → Set (ℝ → ℝ) | [
{
"t": "ℝ",
"v": null,
"name": "α"
},
{
"t": "answer α = {f : ℝ → ℝ | (∀ x ∈ Icc 0 1, f x > 0) ∧ ContinuousOn f (Icc 0 1) ∧ ∫ x in (0)..1, f x = 1 ∧ ∫ x in (0)..1, x * f x = α ∧ ∫ x in (0)..1, x^2 * f x = α^2}",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1964_a2",
"tags": [
"analysis",
"algebra"
]
} |
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 $\angle CAB$. | π / 15 | Show that the solution is $\angle CAB = \frac{\pi}{15}$. | open EuclideanGeometry Real | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi
(@OfNat.ofNat Real (nat_lit 15)
(@instOfNatAtLeastTwo Real (nat_lit 15) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 13) (instOfNatNat (nat_lit 13))))))) | ℝ | [
{
"t": "EuclideanSpace ℝ (Fin 2)",
"v": null,
"name": "A"
},
{
"t": "EuclideanSpace ℝ (Fin 2)",
"v": null,
"name": "B"
},
{
"t": "EuclideanSpace ℝ (Fin 2)",
"v": null,
"name": "C"
},
{
"t": "EuclideanSpace ℝ (Fin 2)",
"v": null,
"name": "X"
},
{
"t... | {
"problem_name": "putnam_1965_a1",
"tags": [
"geometry"
]
} |
How many orderings of the integers from $1$ to $n$ satisfy the condition that, for every integer $i$ except the first, there exists some earlier integer in the ordering which differs from $i$ by $1$? | $2^{n-1}$ | There are $2^{n-1}$ such orderings. | open EuclideanGeometry Topology Filter Complex | [] | @Eq (Nat → Nat) answer fun (n_1 : Nat) =>
@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1 (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) | ℕ → ℕ | [
{
"t": "ℕ",
"v": null,
"name": "n"
},
{
"t": "n > 0",
"v": null,
"name": "npos"
},
{
"t": "{p ∈ permsOfFinset (Finset.Icc 1 n) | ∀ m ∈ Finset.Icc 2 n, ∃ k ∈ Finset.Ico 1 m, p m = p k + 1 ∨ p m = p k - 1}.card = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1965_a5",
"tags": [
"combinatorics"
]
} |
Find $$\lim_{n \to \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos^2\left(\frac{\pi}{2n}(x_1 + x_2 + \cdots + x_n)\right) dx_1 dx_2 \cdots dx_n.$$ | 1 / 2 | Show that the limit is $\frac{1}{2}$. | open EuclideanGeometry Topology Filter Complex | [] | @Eq Real answer
(@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 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))) | ℝ | [
{
"t": "Tendsto (fun n : ℕ ↦ ∫ x in {x : Fin (n+1) → ℝ | ∀ k : Fin (n+1), x k ∈ Set.Icc 0 1}, (Real.cos (Real.pi/(2 * (n+1)) * ∑ k : Fin (n+1), x k))^2) atTop (𝓝 answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1965_b1",
"tags": [
"analysis"
]
} |
Consider polynomial forms $ax^2-bx+c$ with integer coefficients which have two distinct zeros in the open interval $0<x<1$. Exhibit with a proof the least positive integer value of $a$ for which such a polynomial exists. | 5 | Show that the minimum possible value for $a$ is $5$. | open Polynomial | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) | ℕ | [
{
"t": "Set ℤ",
"v": null,
"name": "S"
},
{
"t": "S = {a | ∃ P : Polynomial ℤ, P.degree = 2 ∧ (∃ z1 z2 : Set.Ioo (0 : ℝ) 1, z1 ≠ z2 ∧ aeval (z1 : ℝ) P = 0 ∧ aeval (z2 : ℝ) P = 0) ∧P.coeff 2 = a ∧ a > 0}",
"v": null,
"name": "hS"
},
{
"t": "IsLeast S answer",
"v": null,
"n... | {
"problem_name": "putnam_1967_a3",
"tags": [
"algebra"
]
} |
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. | 8 | Show that the maximum number of distinct $4$-tuples is eight. | open Nat Topology Filter | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) | ℕ | [
{
"t": "Fin 4 → ℝ",
"v": null,
"name": "a"
},
{
"t": "Fin 4 → ℝ",
"v": null,
"name": "b"
},
{
"t": "a 0 * b 1 - a 1 * b 0 ≠ 0",
"v": null,
"name": "abneq0"
},
{
"t": "ℕ",
"v": null,
"name": "numtuples"
},
{
"t": "numtuples = {s : Fin 4 → ℝ | ∃ x : ... | {
"problem_name": "putnam_1967_a6",
"tags": [
"algebra",
"geometry"
]
} |
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 $P \in V$. | 8 | The supremum is $8$. | open Finset Polynomial | [] | @Eq Real answer
(@OfNat.ofNat Real (nat_lit 8)
(@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))) | ℝ | [
{
"t": "Set ℝ[X]",
"v": null,
"name": "V"
},
{
"t": "V = {P : ℝ[X] | P.degree = 2 ∧ ∀ x ∈ Set.Icc 0 1, |P.eval x| ≤ 1}",
"v": null,
"name": "V_def"
},
{
"t": "sSup {|(derivative P).eval 0| | P ∈ V} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1968_a5",
"tags": [
"algebra"
]
} |
Find all polynomials of the form $\sum_{0}^{n} a_{i} x^{n-i}$ with $n \ge 1$ and $a_i = \pm 1$ for all $0 \le i \le n$ whose roots are all real. | {X - 1, -(X - 1), X + 1, -(X + 1), X^2 + X - 1, -(X^2 + X - 1), X^2 - X - 1, -(X^2 - X - 1), X^3 + X^2 - X - 1, -(X^3 + X^2 - X - 1), X^3 - X^2 - X + 1, -(X^3 - X^2 - X + 1)} | The set of such polynomials is $$\{\pm (x - 1), \pm (x + 1), \pm (x^2 + x - 1), \pm (x^2 - x - 1), \pm (x^3 + x^2 - x - 1), \pm (x^3 - x^2 - x + 1)\}.$$ | open Finset Polynomial | [] | @Eq (Set (@Polynomial Complex Complex.instSemiring)) answer
(@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))
(@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing)
(@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring)))))
(@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring) (@Polynomial.one Complex Complex.instSemiring))))
(@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing)
(@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring)
(@Polynomial.one Complex Complex.instSemiring)))))
(@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing))
(@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial.add' Complex Complex.instSemiring))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat (@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@Polynomial.X Complex Complex.instSemiring))
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring)
(@Polynomial.one Complex Complex.instSemiring))))
(@Insert.insert (@Polynomial Complex Complex.instSemiring) (Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing)
(@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing))
(@HAdd.hAdd (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHAdd (@Polynomial Complex Complex.instSemiring)
(@Polynomial.add' Complex Complex.instSemiring))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@Polynomial.X Complex Complex.instSemiring))
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring)
(@Polynomial.one Complex Complex.instSemiring)))))
(@Insert.insert (@Polynomial Complex Complex.instSemiring)
(Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@Polynomial.X Complex Complex.instSemiring))
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring)
(@Polynomial.one Complex Complex.instSemiring))))
(@Insert.insert (@Polynomial Complex Complex.instSemiring)
(Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing)
(@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial.sub Complex Complex.instRing))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@Polynomial.X Complex Complex.instSemiring))
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring)
(@Polynomial.one Complex Complex.instSemiring)))))
(@Insert.insert (@Polynomial Complex Complex.instSemiring)
(Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring) (@Polynomial.sub Complex Complex.instRing))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial.sub Complex Complex.instRing))
(@HAdd.hAdd (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@instHAdd (@Polynomial Complex Complex.instSemiring)
(@Polynomial.add' Complex Complex.instSemiring))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@Polynomial.X Complex Complex.instSemiring))
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring)
(@Polynomial.one Complex Complex.instSemiring))))
(@Insert.insert (@Polynomial Complex Complex.instSemiring)
(Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@Neg.neg (@Polynomial Complex Complex.instSemiring) (@Polynomial.neg' Complex Complex.instRing)
(@HSub.hSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial.sub Complex Complex.instRing))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial.sub Complex Complex.instRing))
(@HAdd.hAdd (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@instHAdd (@Polynomial Complex Complex.instSemiring)
(@Polynomial.add' Complex Complex.instSemiring))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@Polynomial.X Complex Complex.instSemiring))
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring)
(@Polynomial.one Complex Complex.instSemiring)))))
(@Insert.insert (@Polynomial Complex Complex.instSemiring)
(Set (@Polynomial Complex Complex.instSemiring))
(@Set.instInsert (@Polynomial Complex Complex.instSemiring))
(@HAdd.hAdd (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@instHAdd (@Polynomial Complex Complex.instSemiring)
(@Polynomial.add' Complex Complex.instSemiring))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial.sub Complex Complex.instRing))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial.sub Complex Complex.instRing))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@Polynomial.X Complex Complex.instSemiring))
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring)
(@Polynomial.one Complex Complex.instSemiring))))
(@Singleton.singleton (@Polynomial Complex Complex.instSemiring)
(Set (@Polynomial Complex Complex.instSemiring))
(@Set.instSingletonSet (@Polynomial Complex Complex.instSemiring))
(@Neg.neg (@Polynomial Complex Complex.instSemiring)
(@Polynomial.neg' Complex Complex.instRing)
(@HAdd.hAdd (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@instHAdd (@Polynomial Complex Complex.instSemiring)
(@Polynomial.add' Complex Complex.instSemiring))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial.sub Complex Complex.instRing))
(@HSub.hSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial Complex Complex.instSemiring) (@Polynomial Complex Complex.instSemiring)
(@instHSub (@Polynomial Complex Complex.instSemiring)
(@Polynomial.sub Complex Complex.instRing))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
(@HPow.hPow (@Polynomial Complex Complex.instSemiring) Nat
(@Polynomial Complex Complex.instSemiring)
(@instHPow (@Polynomial Complex Complex.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Complex Complex.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Complex Complex.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Complex Complex.instSemiring)
(@Polynomial.semiring Complex Complex.instSemiring)))))
(@Polynomial.X Complex Complex.instSemiring)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@Polynomial.X Complex Complex.instSemiring))
(@OfNat.ofNat (@Polynomial Complex Complex.instSemiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Complex Complex.instSemiring)
(@Polynomial.one Complex Complex.instSemiring))))))))))))))))) | Set ℂ[X] | [
{
"t": "{P : ℂ[X] | P.natDegree ≥ 1 ∧ (∀ k ∈ Set.Icc 0 P.natDegree, P.coeff k = 1 ∨ P.coeff k = -1) ∧\n ∀ z : ℂ, P.eval z = 0 → ∃ r : ℝ, r = z} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1968_a6",
"tags": [
"algebra"
]
} |
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 $ad - bc \equiv 0 \pmod p$. | $p^2 + p$ | There are $p^2 + p$ such matrices. | open Finset Polynomial Topology Filter Metric | [] | @Eq (Nat → Nat) answer fun (p_1 : Nat) =>
@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat)
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) p_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
p_1 | ℕ → ℕ | [
{
"t": "ℕ",
"v": null,
"name": "p"
},
{
"t": "Nat.Prime p",
"v": null,
"name": "hp"
},
{
"t": "{M : Matrix (Fin 2) (Fin 2) (ZMod p) | M 0 0 + M 1 1 = 1 ∧ M 0 0 * M 1 1 - M 0 1 * M 1 0 = 0}.ncard = answer p",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1968_b5",
"tags": [
"linear_algebra",
"number_theory",
"combinatorics"
]
} |
What are the possible ranges (across all real inputs $x$ and $y$) of a polynomial $f(x, y)$ with real coefficients? | {{x} | x : ℝ} ∪ {Set.Ici x | x : ℝ} ∪ {Set.Iic x | x : ℝ} ∪ {Set.Iio x | x : ℝ} ∪ {Set.Ioi x | x : ℝ} ∪ {Set.univ} | Show that the possibles ranges are a single point, any half-open or half-closed semi-infinite interval, or all real numbers. | open Matrix Filter Topology Set Nat | [] | @Eq (Set (Set Real)) answer
(@Union.union (Set (Set Real)) (@Set.instUnion (Set Real))
(@Union.union (Set (Set Real)) (@Set.instUnion (Set Real))
(@Union.union (Set (Set Real)) (@Set.instUnion (Set Real))
(@Union.union (Set (Set Real)) (@Set.instUnion (Set Real))
(@Union.union (Set (Set Real)) (@Set.instUnion (Set Real))
(@setOf (Set Real) fun (x : Set Real) =>
@Exists Real fun (x_1 : Real) =>
@Eq (Set Real) (@Singleton.singleton Real (Set Real) (@Set.instSingletonSet Real) x_1) x)
(@setOf (Set Real) fun (x : Set Real) =>
@Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Ici Real Real.instPreorder x_1) x))
(@setOf (Set Real) fun (x : Set Real) =>
@Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Iic Real Real.instPreorder x_1) x))
(@setOf (Set Real) fun (x : Set Real) =>
@Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Iio Real Real.instPreorder x_1) x))
(@setOf (Set Real) fun (x : Set Real) =>
@Exists Real fun (x_1 : Real) => @Eq (Set Real) (@Set.Ioi Real Real.instPreorder x_1) x))
(@Singleton.singleton (Set Real) (Set (Set Real)) (@Set.instSingletonSet (Set Real)) (@Set.univ Real))) | Set (Set ℝ) | [
{
"t": "answer = {{z : ℝ | ∃ x : Fin 2 → ℝ, MvPolynomial.eval x f = z} | f : MvPolynomial (Fin 2) ℝ}",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1969_a1",
"tags": [
"algebra",
"set_theory"
]
} |
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'? | False | Show that the statement is no longer true if 'two' is replaced by 'three'. | open Matrix Filter Topology Set Nat | [] | @Eq Prop answer False | Prop | [
{
"t": "ℕ → Prop",
"v": null,
"name": "P"
},
{
"t": "∀ n, P n ↔ ∀ (G : Type) [Group G] [Finite G],\n ∀ H : Fin n → Subgroup G, (∀ i, H i < ⊤) → ⋃ i, (H i : Set G) < ⊤",
"v": null,
"name": "P_def"
},
{
"t": "answer ↔ (P 3)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1969_b2",
"tags": [
"abstract_algebra"
]
} |
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. | (3, 1444) | The maximum attainable length is $3$; the smallest such square is $38^2 = 1444$. | open Metric Set EuclideanGeometry | [] | @Eq (Prod Nat Nat) answer
(@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
(@OfNat.ofNat Nat (nat_lit 1444) (instOfNatNat (nat_lit 1444)))) | ℕ × ℕ | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "L"
},
{
"t": "∀ n : ℕ, L n ≤ (Nat.digits 10 n).length ∧\n(∀ k : ℕ, k < L n → (Nat.digits 10 n)[k]! = (Nat.digits 10 n)[0]!) ∧\n(L n ≠ (Nat.digits 10 n).length → (Nat.digits 10 n)[L n]! ≠ (Nat.digits 10 n)[0]!)",
"v": null,
"name": "hL"
},
{
"t"... | {
"problem_name": "putnam_1970_a3",
"tags": [
"number_theory"
]
} |
Evaluate the infinite product $\lim_{n \to \infty} \frac{1}{n^4} \prod_{i = 1}^{2n} (n^2 + i^2)^{1/n}$. | $e^{2 \ln 5 - 4 + 2 \arctan 2}$ | Show that the solution is $e^{2 \log(5) - 4 + 2 arctan(2)}$. | open Metric Set EuclideanGeometry Filter Topology | [] | @Eq Real answer
(Real.exp
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@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))))))
(Real.log
(@OfNat.ofNat Real (nat_lit 5)
(@instOfNatAtLeastTwo Real (nat_lit 5) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))
(@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))))))
(Real.arctan
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))))) | ℝ | [
{
"t": "Tendsto (fun n => 1/(n^4) * ∏ i in Finset.Icc (1 : ℤ) (2*n), ((n^2 + i^2) : ℝ)^((1 : ℝ)/n)) atTop (𝓝 answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1970_b1",
"tags": [
"analysis"
]
} |
Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$. | {Polynomial.X} | Show that the only such polynomial is the identity function. | open Set | [] | @Eq (Set (@Polynomial Real Real.semiring)) answer
(@Singleton.singleton (@Polynomial Real Real.semiring) (Set (@Polynomial Real Real.semiring))
(@Set.instSingletonSet (@Polynomial Real Real.semiring)) (@Polynomial.X Real Real.semiring)) | Set (Polynomial ℝ) | [
{
"t": "Polynomial ℝ",
"v": null,
"name": "P"
},
{
"t": "∀ P : Polynomial ℝ, P ∈ answer ↔ (P.eval 0 = 0 ∧ (∀ x : ℝ, P.eval (x^2 + 1) = (P.eval x)^2 + 1))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1971_a2",
"tags": [
"algebra"
]
} |
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 $b$. | (11, 8) | Show that the solution is $a = 11$ and $b = 8$. | open Set MvPolynomial | [] | @Eq (Prod Int Int) answer
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 11) (@instOfNat (nat_lit 11)))
(@OfNat.ofNat Int (nat_lit 8) (@instOfNat (nat_lit 8)))) | ℤ × ℤ | [
{
"t": "ℤ",
"v": null,
"name": "a"
},
{
"t": "ℤ",
"v": null,
"name": "b"
},
{
"t": "a > 0 ∧ b > 0 ∧ a > b",
"v": null,
"name": "hab"
},
{
"t": "ℤ → ℤ → Prop",
"v": null,
"name": "pab"
},
{
"t": "∀ x y, pab x y ↔\n {s : ℕ | ¬∃ m n : ℕ, m*x + n*y ... | {
"problem_name": "putnam_1971_a5",
"tags": [
"number_theory"
]
} |
Find all functions $F : \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$ that satisfy $F(x) + F\left(\frac{x - 1}{x}\right) = 1 + x$ for all $x \in \mathbb{R} \setminus \{0, 1\}$. | $\left\{x \mapsto \frac{x^3 - x^2 - 1}{2x(x - 1)}\right\}$ | The only such function is $F(x) = \frac{x^3 - x^2 - 1}{2x(x - 1)}$. | open Set MvPolynomial | [] | @Eq (Set (Real → Real)) answer
(@Singleton.singleton (Real → Real) (Set (Real → Real)) (@Set.instSingletonSet (Real → Real)) fun (x : Real) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) x
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
(@HPow.hPow Real Nat Real (@instHPow Real Nat (@Monoid.toNatPow Real Real.instMonoid)) x
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@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))))))
x)
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))) | Set (ℝ → ℝ) | [
{
"t": "Set ℝ",
"v": null,
"name": "S"
},
{
"t": "S = univ \\ {0, 1}",
"v": null,
"name": "hS"
},
{
"t": "(ℝ → ℝ) → Prop",
"v": null,
"name": "P"
},
{
"t": "P = fun (F : ℝ → ℝ) => ∀ x ∈ S, F x + F ((x - 1)/x) = 1 + x",
"v": null,
"name": "hP"
},
{
... | {
"problem_name": "putnam_1971_b2",
"tags": [
"algebra"
]
} |
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 $[0,1]$. | the set of all linear functions on [0,1] | Show that the solution is the set of affine functions. | open EuclideanGeometry Filter Topology Set | [] | @Eq (Set (Real → Real)) answer
(@setOf (Real → Real) fun (f : Real → Real) =>
@Exists Real fun (A : Real) =>
@Exists Real fun (B : Real) =>
∀ (x : Real),
@Membership.mem Real (Set Real) (@Set.instMembership Real)
(@Set.Icc Real Real.instPreorder (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
x →
@Eq Real (f x)
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul) A x) B)) | Set (ℝ → ℝ) | [
{
"t": "(ℕ → ℝ) → Prop",
"v": null,
"name": "climit_exists"
},
{
"t": "(ℝ → ℝ) → Prop",
"v": null,
"name": "supercontinuous"
},
{
"t": "∀ x, climit_exists x ↔ ∃ C : ℝ, Tendsto (fun n => (∑ i in Finset.range n, (x i))/(n : ℝ)) atTop (𝓝 C)",
"v": null,
"name": "hclimit_exi... | {
"problem_name": "putnam_1972_a3",
"tags": [
"analysis"
]
} |
Let $x : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function whose second derivative is nonstrictly decreasing. If $x(t) - x(0) = s$, $x'(0) = 0$, and $x'(t) = v$ for some $t > 0$, find the maximum possible value of $t$ in terms of $s$ and $v$. | $2s / v$ | Show that the maximum possible time is $t = \frac{2s}{v}$. | open EuclideanGeometry Filter Topology Set MeasureTheory Metric | [] | @Eq (Real → Real → Real) answer fun (s_1 v_1 : Real) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@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))))))
s_1)
v_1 | ℝ → ℝ → ℝ | [
{
"t": "ℝ",
"v": null,
"name": "s"
},
{
"t": "ℝ",
"v": null,
"name": "v"
},
{
"t": "s > 0",
"v": null,
"name": "hs"
},
{
"t": "v > 0",
"v": null,
"name": "hv"
},
{
"t": "ℝ → (ℝ → ℝ) → Prop",
"v": null,
"name": "valid"
},
{
"t": "∀ t... | {
"problem_name": "putnam_1972_b2",
"tags": [
"analysis"
]
} |
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? | True | Show that the condition is necessary. | open Nat Set MeasureTheory Topology Filter | [] | @Eq Prop answer True | Prop | [
{
"t": "List ℝ",
"v": null,
"name": "L"
},
{
"t": "L.length = 8 ∧ ∀ i : Fin L.length, L[i] = 1 ∨ L[i] = -1",
"v": null,
"name": "hL"
},
{
"t": "ℕ",
"v": null,
"name": "pluses"
},
{
"t": "pluses = {i : Fin L.length | L[i] = 1}.ncard",
"v": null,
"name": "hp... | {
"problem_name": "putnam_1973_a2",
"tags": [
"analysis"
]
} |
How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line? | 3 | Show that the solution is 3. | open Nat Set MeasureTheory Topology Filter | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))) | ℕ | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f"
},
{
"t": "f = fun x => 2^x - 1 - x^2",
"v": null,
"name": "hf"
},
{
"t": "answer = {x : ℝ | f x = 0}.ncard",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1973_a4",
"tags": [
"analysis"
]
} |
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. | the identity function $f(x) = x$ | Show that one such example where equality holds is the identity function. | open Nat Set MeasureTheory Topology Filter | [] | @Eq (Real → Real) answer fun (x : Real) => x | ℝ → ℝ | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f"
},
{
"t": "(ℝ → ℝ) → Prop",
"v": null,
"name": "hprop"
},
{
"t": "hprop = fun g => ContDiff ℝ 1 g ∧ (∀ x : ℝ, 0 < deriv g x ∧ deriv g x ≤ 1) ∧ g 0 = 0",
"v": null,
"name": "hprop_def"
},
{
"t": "hprop f",
"v": null,
"... | {
"problem_name": "putnam_1973_b4",
"tags": [
"analysis"
]
} |
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? | 11 | Show that the answer is 11. | open Set | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 11) (instOfNatNat (nat_lit 11))) | ℕ | [
{
"t": "Set ℤ → Prop",
"v": null,
"name": "conspiratorial"
},
{
"t": "∀ S, conspiratorial S ↔ ∀ a ∈ S, ∀ b ∈ S, ∀ c ∈ S, (a > 0 ∧ b > 0 ∧ c > 0) ∧ ((a ≠ b ∧ b ≠ c ∧ a ≠ c) → (Int.gcd a b > 1 ∨ Int.gcd b c > 1 ∨ Int.gcd a c > 1))",
"v": null,
"name": "hconspiratorial"
},
{
"t": "I... | {
"problem_name": "putnam_1974_a1",
"tags": [
"number_theory"
]
} |
A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squres if and only if $p \equiv 1 \bmod 4$. Find which primes $p > 2$ can be written in each of the following forms, using (not necessarily positive) integers $x$ and $y$: (a) $x^2 + 16y^2$, (b) $4x^2 + 4xy + 5y^2$. | ({p : ℕ | p.Prime ∧ p ≡ 1 [MOD 8]}, {p : ℕ | p.Prime ∧ p ≡ 5 [MOD 8]}) | Show that that the answer to (a) is the set of primes which are $1 \bmod 8$, and the solution to (b) is the set of primes which are $5 \bmod 8$. | open Set | [] | @Eq (Prod (Set Nat) (Set Nat)) answer
(@Prod.mk (Set Nat) (Set Nat)
(@setOf Nat fun (p_1 : Nat) =>
And (Nat.Prime p_1)
(Nat.ModEq (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) p_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))
(@setOf Nat fun (p_1 : Nat) =>
And (Nat.Prime p_1)
(Nat.ModEq (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))) p_1
(@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))))) | (Set ℕ) × (Set ℕ) | [
{
"t": "ℕ",
"v": null,
"name": "p"
},
{
"t": "∀ p : ℕ, p.Prime ∧ p > 2 → ((∃ m n : ℤ, p = m^2 + n^2) ↔ p ≡ 1 [MOD 4])",
"v": null,
"name": "h_assumption"
},
{
"t": "∀ p : ℕ,\n ((p.Prime ∧ p > 2 ∧ (∃ x y : ℤ, p = x^2 + 16*y^2)) ↔ p ∈ answer.1) ∧\n ((p.Prime ∧ p > 2 ∧ (∃ x y ... | {
"problem_name": "putnam_1974_a3",
"tags": [
"number_theory"
]
} |
Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$. | (fun n ↦ (1 : ℚ) / ((2 : ℚ) ^ ((n :ℕ) - 1)) * (n * (n - 1).choose ⌊n / 2⌋₊)) | Show that the solution is $\frac{n}{2^{n-1}} * {(n-1) \choose \left[ (n-1)/2 \right]}$. | open Set Nat | [] | @Eq (Nat → Rat) answer fun (n_1 : Nat) =>
@HMul.hMul Rat Rat Rat (@instHMul Rat Rat.instMul)
(@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv) (@OfNat.ofNat Rat (nat_lit 1) (@Rat.instOfNat (nat_lit 1)))
(@HPow.hPow Rat Nat Rat (@instHPow Rat Nat Rat.instPowNat)
(@OfNat.ofNat Rat (nat_lit 2) (@Rat.instOfNat (nat_lit 2)))
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))
(@HMul.hMul Rat Rat Rat (@instHMul Rat Rat.instMul) (@Nat.cast Rat Rat.instNatCast n_1)
(@Nat.cast Rat Rat.instNatCast
(Nat.choose
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) n_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@Nat.floor Nat Nat.instOrderedSemiring instFloorSemiringNat
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv) n_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))) | ℕ → ℚ | [
{
"t": "ℕ",
"v": null,
"name": "n"
},
{
"t": "0 < n",
"v": null,
"name": "hn"
},
{
"t": "(1 : ℚ) / (2 ^ (n - 1)) * ∑ k in Finset.Icc 0 ⌊n / 2⌋₊, (n - 2 * k) * (n.choose k) = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1974_a4",
"tags": [
"algebra"
]
} |
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 $k(1000000)$. | 25 | Show that the answer is 25. | open Set Nat Polynomial | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 25) (instOfNatNat (nat_lit 25))) | ℕ | [
{
"t": "Polynomial ℤ → Prop",
"v": null,
"name": "hdivnallx"
},
{
"t": "hdivnallx = fun f => Monic f ∧ (∀ x : ℤ, (10^6 : ℤ) ∣ f.eval x)",
"v": null,
"name": "hdivnallx_def"
},
{
"t": "sInf {d : ℕ | ∃ f : Polynomial ℤ, hdivnallx f ∧ d = f.natDegree} = answer",
"v": null,
"... | {
"problem_name": "putnam_1974_a6",
"tags": [
"algebra"
]
} |
For a set with $1000$ elements, how many subsets are there whose candinality is respectively $\equiv 0 \bmod 3, \equiv 1 \bmod 3, \equiv 2 \bmod 3$? | ((2^1000 - 1)/3, (2^1000 - 1)/3, 1 + (2^1000 - 1)/3) | Show that there answer is that there are $(2^1000-1)/3$ subsets of cardinality $\equiv 0 \bmod 3$ and $\equiv 1 \bmod 3$, and $1 + (2^1000-1)/3$ subsets of cardinality $\equiv 2 \bmod 3$. | open Set Nat Polynomial Filter Topology | [] | @Eq (Prod Nat (Prod Nat Nat)) answer
(@Prod.mk Nat (Prod Nat Nat)
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv)
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat)
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@OfNat.ofNat Nat (nat_lit 1000) (instOfNatNat (nat_lit 1000))))
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
(@Prod.mk Nat Nat
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv)
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat)
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@OfNat.ofNat Nat (nat_lit 1000) (instOfNatNat (nat_lit 1000))))
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv)
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat)
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@OfNat.ofNat Nat (nat_lit 1000) (instOfNatNat (nat_lit 1000))))
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))) | ℕ × ℕ × ℕ | [
{
"t": "ℤ",
"v": null,
"name": "n"
},
{
"t": "n = 1000",
"v": null,
"name": "hn"
},
{
"t": "ℕ",
"v": null,
"name": "count0"
},
{
"t": "ℕ",
"v": null,
"name": "count1"
},
{
"t": "ℕ",
"v": null,
"name": "count2"
},
{
"t": "count0 = {S... | {
"problem_name": "putnam_1974_b6",
"tags": [
"set_theory"
]
} |
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. | (fun (a, b) => a + b + 1, fun (a, b) => a - b) | $x = a + b + 1$ and $y = a - b$ (or vice versa). | open Polynomial | [] | @Eq (Prod (Prod Int Int → Int) (Prod Int Int → Int)) answer
(@Prod.mk (Prod Int Int → Int) (Prod Int Int → Int)
(fun (x : Prod Int Int) =>
_example.match_2 (fun (x_1 : Prod Int Int) => Int) x fun (a b : Int) =>
@HAdd.hAdd Int Int Int (@instHAdd Int Int.instAdd) (@HAdd.hAdd Int Int Int (@instHAdd Int Int.instAdd) a b)
(@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1))))
fun (x : Prod Int Int) =>
_example.match_2 (fun (x_1 : Prod Int Int) => Int) x fun (a b : Int) =>
@HSub.hSub Int Int Int (@instHSub Int Int.instSub) a b) | ((ℤ × ℤ) → ℤ) × ((ℤ × ℤ) → ℤ) | [
{
"t": "(ℤ × ℤ × ℤ) → Prop",
"v": null,
"name": "nab"
},
{
"t": "(ℤ × ℤ × ℤ) → Prop",
"v": null,
"name": "nxy"
},
{
"t": "nab = fun (n, a, b) => n = (a^2 + (a : ℚ))/2 + (b^2 + (b : ℚ))/2",
"v": null,
"name": "hnab"
},
{
"t": "nxy = fun (n, x, y) => 4*n + 1 = x^2 +... | {
"problem_name": "putnam_1975_a1",
"tags": [
"algebra",
"number_theory"
]
} |
For which ordered pairs $(b, c)$ of real numbers do both roots of $z^2 + bz + c$ lie strictly inside the unit disk (i.e., $\{|z| < 1\}$) in the complex plane? | $c < 1 \land c - b > -1 \land c + b > -1$ | The desired region is the strict interior of the triangle with vertices $(0, -1)$, $(2, 1)$, and $(-2, 1)$. | open Polynomial | [
{
"t": "ℝ",
"v": null,
"name": "b"
},
{
"t": "ℝ",
"v": null,
"name": "c"
}
] | @Eq Prop (answer (@Prod.mk Real Real b c))
(And (@LT.lt Real Real.instLT c (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))
(And
(@GT.gt Real Real.instLT (@HSub.hSub Real Real Real (@instHSub Real Real.instSub) c b)
(@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))
(@GT.gt Real Real.instLT (@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) c b)
(@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))))) | (ℝ × ℝ) → Prop | [
{
"t": "(∀ z : ℂ, (Polynomial.X^2 + (Polynomial.C (b : ℂ)) * Polynomial.X + (Polynomial.C (c : ℂ)) : Polynomial ℂ).eval z = 0 → ‖z‖ < 1) ↔ answer (b, c)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1975_a2",
"tags": [
"algebra"
]
} |
If $a$, $b$, and $c$ are real numbers satisfying $0 < a < b < c$, at what points in the set $$\{(x, y, z) \in \mathbb{R}^3 : x^b + y^b + z^b = 1, x \ge 0, y \ge 0, z \ge 0\}$$ does $f(x, y, z) = x^a + y^b + z^c$ attain its maximum and minimum? | (fun (a, b, c) => ((a/b)^(1/(b - a)), (1 - ((a/b)^(1/(b - a)))^b)^(1/b), 0), fun (a, b, c) => (0, (1 - ((b/c)^(1/(c - b)))^b)^(1/b), (b/c)^(1/(c - b)))) | $f$ attains its maximum at $\left(x_0, (1 - x_0^b)^{\frac{1}{b}}, 0\right)$ and its minimum at $\left(0, (1 - z_0^b)^{\frac{1}{b}}, z_0\right)$, where $x_0 = \left(\frac{a}{b}\right)^{\frac{1}{b-a}}$ and $z_0 = \left(\frac{b}{c}\right)^{\frac{1}{c-b}}$. | open Polynomial | [] | @Eq
(Prod (Prod Real (Prod Real Real) → Prod Real (Prod Real Real))
(Prod Real (Prod Real Real) → Prod Real (Prod Real Real)))
answer
(@Prod.mk (Prod Real (Prod Real Real) → Prod Real (Prod Real Real))
(Prod Real (Prod Real Real) → Prod Real (Prod Real Real))
(fun (x : Prod Real (Prod Real Real)) =>
_example.match_1 (fun (x_1 : Prod Real (Prod Real Real)) => Prod Real (Prod Real Real)) x
fun (a_1 b_1 c_1 : Real) =>
@Prod.mk Real (Prod Real Real)
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) a_1 b_1)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub) b_1 a_1)))
(@Prod.mk Real Real
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) a_1
b_1)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub) b_1 a_1)))
b_1))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) b_1))
(@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))))
fun (x : Prod Real (Prod Real Real)) =>
_example.match_1 (fun (x_1 : Prod Real (Prod Real Real)) => Prod Real (Prod Real Real)) x
fun (a_1 b_1 c_1 : Real) =>
@Prod.mk Real (Prod Real Real) (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero))
(@Prod.mk Real Real
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) b_1 c_1)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub) c_1 b_1)))
b_1))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) b_1))
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) b_1 c_1)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub) c_1 b_1))))) | ((ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ)) × ((ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ)) | [
{
"t": "ℝ",
"v": null,
"name": "a"
},
{
"t": "ℝ",
"v": null,
"name": "b"
},
{
"t": "ℝ",
"v": null,
"name": "c"
},
{
"t": "0 < a ∧ a < b ∧ b < c",
"v": null,
"name": "hi"
},
{
"t": "(ℝ × ℝ × ℝ) → Prop",
"v": null,
"name": "P"
},
{
"t... | {
"problem_name": "putnam_1975_a3",
"tags": [
"algebra"
]
} |
Let $n = 2m$, where $m$ is an odd integer greater than 1. Let $\theta = e^{2\pi i/n}$. Expression $(1 - \theta)^{-1}$ explicitly as a polynomial in $\theta$ \[ a_k \theta^k + a_{k-1}\theta^{k-1} + \dots + a_1\theta + a_0\], with integer coefficients $a_i$. | $\sum_{j=0}^{(m-1)/2} \theta^{2j+1}$ | Show that the solution is the polynomial $0 + \theta + \theta^3 + \dots + \theta^{m-2}$, alternating consecutive coefficients between 0 and 1. | open Polynomial Real Complex | [] | @Eq (Nat → @Polynomial Int Int.instSemiring) answer fun (m_1 : Nat) =>
@Finset.sum Nat (@Polynomial Int Int.instSemiring)
(@NonUnitalNonAssocSemiring.toAddCommMonoid (@Polynomial Int Int.instSemiring)
(@NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring (@Polynomial Int Int.instSemiring)
(@NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring (@Polynomial Int Int.instSemiring)
(@NonUnitalCommRing.toNonUnitalNonAssocCommRing (@Polynomial Int Int.instSemiring)
(@CommRing.toNonUnitalCommRing (@Polynomial Int Int.instSemiring)
(@Polynomial.commRing Int Int.instCommRing))))))
(Finset.range
(@HDiv.hDiv Nat Nat Nat (@instHDiv Nat Nat.instDiv)
(@HSub.hSub Nat Nat Nat (@instHSub Nat instSubNat) m_1
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
fun (j : Nat) =>
@HPow.hPow (@Polynomial Int Int.instSemiring) Nat (@Polynomial Int Int.instSemiring)
(@instHPow (@Polynomial Int Int.instSemiring) Nat
(@Monoid.toNatPow (@Polynomial Int Int.instSemiring)
(@MonoidWithZero.toMonoid (@Polynomial Int Int.instSemiring)
(@Semiring.toMonoidWithZero (@Polynomial Int Int.instSemiring)
(@Polynomial.semiring Int Int.instSemiring)))))
(@Polynomial.X Int Int.instSemiring)
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat)
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) j)
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) | ℕ → Polynomial ℤ | [
{
"t": "ℕ",
"v": null,
"name": "m"
},
{
"t": "Odd m ∧ m > 1",
"v": null,
"name": "hm"
},
{
"t": "ℂ",
"v": null,
"name": "θ"
},
{
"t": "θ = cexp (2 * Real.pi * I / (2 * m))",
"v": null,
"name": "hθ"
},
{
"t": "1/(1 - θ) = Polynomial.aeval θ (answer ... | {
"problem_name": "putnam_1975_a4",
"tags": [
"algebra"
]
} |
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 $a$. | 7 | $a$ must equal $7$. | open Polynomial Real Complex | [] | @Eq Int answer (@OfNat.ofNat Int (nat_lit 7) (@instOfNat (nat_lit 7))) | ℤ | [
{
"t": "Set (ℤ × ℤ)",
"v": null,
"name": "H"
},
{
"t": "H = {h : (ℤ × ℤ) | ∃ u v w : ℤ, h = (u*3 + v*4 + w*5, u*8 + v*(-1) + w*4)}",
"v": null,
"name": "hH"
},
{
"t": "(∃ b : ℤ, H = {h : (ℤ × ℤ) | ∃ u v : ℤ, h = (u, u*b + v*answer)}) ∧ answer > 0",
"v": null,
"name": "h_a... | {
"problem_name": "putnam_1975_b1",
"tags": [
"abstract_algebra",
"number_theory"
]
} |
Let $s_k (a_1, a_2, \dots, a_n)$ denote the $k$-th elementary symmetric function; that is, the sum of all $k$-fold products of the $a_i$. For example, $s_1 (a_1, \dots, a_n) = \sum_{i=1}^{n} a_i$, and $s_2 (a_1, a_2, a_3) = a_1a_2 + a_2a_3 + a_1a_3$. Find the supremum $M_k$ (which is never attained) of $$\frac{s_k (a_1, a_2, \dots, a_n)}{(s_1 (a_1, a_2, \dots, a_n))^k}$$ across all $n$-tuples $(a_1, a_2, \dots, a_n)$ of positive real numbers with $n \ge k$. | fun k : ℕ => (1: ℝ)/(Nat.factorial k) | The supremum $M_k$ is $\frac{1}{k!}$. | open Polynomial Real Complex Matrix Filter Topology Multiset | [] | @Eq (Nat → Real) answer fun (k : Nat) =>
@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 k)) | ℕ → ℝ | [
{
"t": "∀ k : ℕ, k > 0 → (∀ a : Multiset ℝ, (∀ i ∈ a, i > 0) ∧ card a ≥ k →\n(esymm a k)/(esymm a 1)^k ≤ answer k) ∧\n∀ M : ℝ, M < answer k → (∃ a : Multiset ℝ, (∀ i ∈ a, i > 0) ∧ card a ≥ k ∧\n(esymm a k)/(esymm a 1)^k > M)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1975_b3",
"tags": [
"analysis",
"algebra"
]
} |
Let $C = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ denote the unit circle. Does there exist $B \subseteq C$ for which $B$ is topologically closed and contains exactly one point from each pair of diametrically opposite points in $C$? | False | Such $B$ does not exist. | open Polynomial Real Complex Matrix Filter Topology Multiset | [] | @Eq Prop answer False | Prop | [
{
"t": "ℝ × ℝ → Prop",
"v": null,
"name": "P"
},
{
"t": "P = fun (x, y) => x^2 + y^2 = 1",
"v": null,
"name": "hP"
},
{
"t": "(∃ B ⊆ setOf P, IsClosed B ∧ ∀ x y : ℝ, P (x, y) → Xor' ((x, y) ∈ B) ((-x, -y) ∈ B)) ↔ answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1975_b4",
"tags": [
"analysis"
]
} |
Find all integer solutions $(p, r, q, s)$ of the equation $|p^r - q^s| = 1$, where $p$ and $q$ are prime and $r$ and $s$ are greater than $1$. | {(3, 2, 2, 3), (2, 3, 3, 2)} | The only solutions are $(p, r, q, s) = (3, 2, 2, 3)$ and $(p, r, q, s) = (2, 3, 3, 2)$. | null | [] | @Eq (Set (Prod Nat (Prod Nat (Prod Nat Nat)))) answer
(@Insert.insert (Prod Nat (Prod Nat (Prod Nat Nat))) (Set (Prod Nat (Prod Nat (Prod Nat Nat))))
(@Set.instInsert (Prod Nat (Prod Nat (Prod Nat Nat))))
(@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
(@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))
(@Singleton.singleton (Prod Nat (Prod Nat (Prod Nat Nat))) (Set (Prod Nat (Prod Nat (Prod Nat Nat))))
(@Set.instSingletonSet (Prod Nat (Prod Nat (Prod Nat Nat))))
(@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))
(@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
(@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))))) | Set (ℕ × ℕ × ℕ × ℕ) | [
{
"t": "{a : ℕ × ℕ × ℕ × ℕ | Nat.Prime a.1 ∧ Nat.Prime a.2.2.1 ∧ a.2.1 > 1 ∧ a.2.2.2 > 1 ∧ |(a.1^a.2.1 : ℤ) - a.2.2.1^a.2.2.2| = 1} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1976_a3",
"tags": [
"number_theory"
]
} |
Let $r$ be a real root of $P(x) = x^3 + ax^2 + bx - 1$, where $a$ and $b$ are integers and $P$ is irreducible over the rationals. Suppose that $r + 1$ is a root of $x^3 + cx^2 + dx + 1$, where $c$ and $d$ are also integers. Express another root $s$ of $P$ as a function of $r$ that does not depend on the values of $a$, $b$, $c$, or $d$. | $\left(-\frac{1}{r + 1}, -\frac{r + 1}{r}\right)$ | The possible answers are $s = -\frac{1}{r + 1}$ and $s = -\frac{r + 1}{r}$. | open Polynomial | [] | @Eq (Prod (Real → Real) (Real → Real)) answer
(@Prod.mk (Real → Real) (Real → Real)
(fun (r_1 : 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)))
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) r_1
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))
fun (r_1 : Real) =>
@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@Neg.neg Real Real.instNeg
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) r_1
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))
r_1) | (ℝ → ℝ) × (ℝ → ℝ) | [
{
"t": "ℤ",
"v": null,
"name": "a"
},
{
"t": "ℤ",
"v": null,
"name": "b"
},
{
"t": "ℤ",
"v": null,
"name": "c"
},
{
"t": "ℤ",
"v": null,
"name": "d"
},
{
"t": "ℝ",
"v": null,
"name": "r"
},
{
"t": "Polynomial ℚ",
"v": null,
... | {
"problem_name": "putnam_1976_a4",
"tags": [
"algebra"
]
} |
Find $$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left\lfloor \frac{2n}{k} \right\rfloor - 2\left\lfloor \frac{n}{k} \right\rfloor\right).$$ Your answer should be in the form $\ln(a) - b$, where $a$ and $b$ are positive integers. | ln(4) - 1 | The limit equals $\ln(4) - 1$, so $a = 4$ and $b = 1$. | open Polynomial Filter Topology | [] | @Eq (Prod Nat Nat) answer
(@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))
(@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))) | ℕ × ℕ | [
{
"t": "Tendsto (fun n : ℕ => ((1 : ℝ)/n)*∑ k in Finset.Icc (1 : ℤ) n, (Int.floor ((2*n)/k) - 2*Int.floor (n/k))) atTop\n (𝓝 (Real.log answer.1 - answer.2))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1976_b1",
"tags": [
"analysis"
]
} |
Let $G$ be a group generated by two elements $A$ and $B$; i.e., every element of $G$ can be expressed as a finite word $A^{n_1}B^{n_2} \cdots A^{n_{k-1}}B^{n_k}$, where the $n_i$ can assume any integer values and $A^0 = B^0 = 1$. Further assume that $A^4 = B^7 = ABA^{-1}B = 1$, but $A^2 \ne 1$ and $B \ne 1$. Find the number of elements of $G$ than can be written as $C^2$ for some $C \in G$ and express each such square as a word in $A$ and $B$. | (8, {[(0, 0)], [(2, 0)], [(0, 1)], [(0, 2)], [(0, 3)], [(0, 4)], [(0, 5)], [(0, 6)]}) | There are $8$ such squares: $1$, $A^2$, $B$, $B^2$, $B^3$, $B^4$, $B^5$, and $B^6$. | open Polynomial Filter Topology | [] | @Eq (Prod Nat (Set (List (Prod Int Int)))) answer
(@Prod.mk Nat (Set (List (Prod Int Int))) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))
(@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int)))
(@List.cons (Prod Int Int)
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))
(@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))))
(@List.nil (Prod Int Int)))
(@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int)))
(@List.cons (Prod Int Int)
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))
(@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0))))
(@List.nil (Prod Int Int)))
(@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int)))
(@List.cons (Prod Int Int)
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))
(@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1))))
(@List.nil (Prod Int Int)))
(@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int)))
(@List.cons (Prod Int Int)
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))
(@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2))))
(@List.nil (Prod Int Int)))
(@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int)))
(@List.cons (Prod Int Int)
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))
(@OfNat.ofNat Int (nat_lit 3) (@instOfNat (nat_lit 3))))
(@List.nil (Prod Int Int)))
(@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int))) (@Set.instInsert (List (Prod Int Int)))
(@List.cons (Prod Int Int)
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))
(@OfNat.ofNat Int (nat_lit 4) (@instOfNat (nat_lit 4))))
(@List.nil (Prod Int Int)))
(@Insert.insert (List (Prod Int Int)) (Set (List (Prod Int Int)))
(@Set.instInsert (List (Prod Int Int)))
(@List.cons (Prod Int Int)
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))
(@OfNat.ofNat Int (nat_lit 5) (@instOfNat (nat_lit 5))))
(@List.nil (Prod Int Int)))
(@Singleton.singleton (List (Prod Int Int)) (Set (List (Prod Int Int)))
(@Set.instSingletonSet (List (Prod Int Int)))
(@List.cons (Prod Int Int)
(@Prod.mk Int Int (@OfNat.ofNat Int (nat_lit 0) (@instOfNat (nat_lit 0)))
(@OfNat.ofNat Int (nat_lit 6) (@instOfNat (nat_lit 6))))
(@List.nil (Prod Int Int)))))))))))) | ℕ × Set (List (ℤ × ℤ)) | [
{
"t": "Type*",
"v": null,
"name": "G"
},
{
"t": "Group G",
"v": null,
"name": null
},
{
"t": "G",
"v": null,
"name": "A"
},
{
"t": "G",
"v": null,
"name": "B"
},
{
"t": "List (ℤ × ℤ) → G",
"v": null,
"name": "word"
},
{
"t": "word ... | {
"problem_name": "putnam_1976_b2",
"tags": [
"abstract_algebra"
]
} |
Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$ | fun n => C (Nat.factorial n) | The sum equals $n!$. | open Polynomial Filter Topology ProbabilityTheory MeasureTheory | [] | @Eq (Nat → @Polynomial Int Int.instSemiring) answer fun (n : Nat) =>
@DFunLike.coe
(@RingHom Int (@Polynomial Int Int.instSemiring) (@Semiring.toNonAssocSemiring Int Int.instSemiring)
(@Semiring.toNonAssocSemiring (@Polynomial Int Int.instSemiring) (@Polynomial.semiring Int Int.instSemiring)))
Int (fun (x : Int) => @Polynomial Int Int.instSemiring)
(@RingHom.instFunLike Int (@Polynomial Int Int.instSemiring) (@Semiring.toNonAssocSemiring Int Int.instSemiring)
(@Semiring.toNonAssocSemiring (@Polynomial Int Int.instSemiring) (@Polynomial.semiring Int Int.instSemiring)))
(@Polynomial.C Int Int.instSemiring) (@Nat.cast Int instNatCastInt (Nat.factorial n)) | ℕ → Polynomial ℤ | [
{
"t": "∀ n : ℕ, ∑ k in Finset.range (n + 1), C ((-(1 : ℤ))^k * Nat.choose n k) * (X - (C (k : ℤ)))^n = answer n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1976_b5",
"tags": [
"algebra"
]
} |
Show that if four distinct points of the curve $y = 2x^4 + 7x^3 + 3x - 5$ are collinear, then their average $x$-coordinate is some constant $k$. Find $k$. | $-7/8$ | Prove that $k = -\frac{7}{8}$. | null | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@Neg.neg Real Real.instNeg
(@OfNat.ofNat Real (nat_lit 7)
(@instOfNatAtLeastTwo Real (nat_lit 7) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))))))
(@OfNat.ofNat Real (nat_lit 8)
(@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))))) | ℝ | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "y"
},
{
"t": "y = fun x ↦ 2 * x ^ 4 + 7 * x ^ 3 + 3 * x - 5",
"v": null,
"name": "hy"
},
{
"t": "Finset ℝ",
"v": null,
"name": "S"
},
{
"t": "S.card = 4",
"v": null,
"name": "hS"
},
{
"t": "Collinear ℝ {P : Fin 2... | {
"problem_name": "putnam_1977_a1",
"tags": [
"algebra"
]
} |
Find all real solutions $(a, b, c, d)$ to the equations $a + b + c = d$, $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d}$. | $d = a \land b = -c \lor d = b \land a = -c \lor d = c \land a = -b$ | Prove that the solutions are $d = a$ and $b = -c$, $d = b$ and $a = -c$, or $d = c$ and $a = -b$, with $a, b, c, d$ nonzero. | null | [] | @Eq (Real → Real → Real → Real → Prop) answer fun (a_1 b_1 c_1 d_1 : Real) =>
Or (And (@Eq Real d_1 a_1) (@Eq Real b_1 (@Neg.neg Real Real.instNeg c_1)))
(Or (And (@Eq Real d_1 b_1) (@Eq Real a_1 (@Neg.neg Real Real.instNeg c_1)))
(And (@Eq Real d_1 c_1) (@Eq Real a_1 (@Neg.neg Real Real.instNeg b_1)))) | ℝ → ℝ → ℝ → ℝ → Prop | [
{
"t": "ℝ",
"v": null,
"name": "a"
},
{
"t": "ℝ",
"v": null,
"name": "b"
},
{
"t": "ℝ",
"v": null,
"name": "c"
},
{
"t": "ℝ",
"v": null,
"name": "d"
},
{
"t": "answer a b c d ↔\n a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (a + b + c = d ∧ 1 / a + 1 / b + 1... | {
"problem_name": "putnam_1977_a2",
"tags": [
"algebra"
]
} |
Let $f, g, h$ be functions $\mathbb{R} \to \mathbb{R}$. Find an expression for $h(x)$ in terms of $f$ and $g$ such that $f(x) = \frac{h(x + 1) + h(x - 1)}{2}$ and $g(x) = \frac{h(x + 4) + h(x - 4)}{2}$. | $h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$ | Prove that $h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$ suffices. | null | [] | @Eq ((Real → Real) → (Real → Real) → Real → Real) answer fun (f_1 g_1 : Real → Real) (x : Real) =>
@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd)
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (g_1 x)
(f_1
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))))
(f_1
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub) x
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))))
(f_1
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) x
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))))
(f_1
(@HAdd.hAdd Real Real Real (@instHAdd Real Real.instAdd) x
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))) | (ℝ → ℝ) → (ℝ → ℝ) → (ℝ → ℝ) | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "g"
},
{
"t": "ℝ → ℝ",
"v": null,
"name": "h"
},
{
"t": "∀ x, f x = (h (x + 1) + h (x - 1)) / 2",
"v": null,
"name": "hf"
},
{
"t": "∀ x, g x = (h (x + 4) + h (x - 4)) / 2... | {
"problem_name": "putnam_1977_a3",
"tags": [
"algebra"
]
} |
Find $\sum_{n=0}^{\infty} \frac{x^{2^n}}{1 - x^{2^{n+1}}}$ as a rational function of $x$ for $x \in (0, 1)$. | $\frac{x}{1 - x}$ | Prove that the sum equals $\frac{x}{1 - x}$. | open RingHom Set | [] | @Eq (@RatFunc Real Real.commRing) answer
(@HDiv.hDiv (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing)
(@instHDiv (@RatFunc Real Real.commRing) (@RatFunc.instDiv Real Real.commRing Real.instIsDomain))
(@RatFunc.X Real Real.commRing Real.instIsDomain)
(@HSub.hSub (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing) (@RatFunc Real Real.commRing)
(@instHSub (@RatFunc Real Real.commRing) (@RatFunc.instSub Real Real.commRing))
(@OfNat.ofNat (@RatFunc Real Real.commRing) (nat_lit 1)
(@One.toOfNat1 (@RatFunc Real Real.commRing) (@RatFunc.instOne Real Real.commRing)))
(@RatFunc.X Real Real.commRing Real.instIsDomain))) | RatFunc ℝ | [
{
"t": "ℝ",
"v": null,
"name": "x"
},
{
"t": "x ∈ Ioo 0 1",
"v": null,
"name": "hx"
},
{
"t": "answer.eval (id ℝ) x = ∑' n : ℕ, x ^ 2 ^ n / (1 - x ^ 2 ^ (n + 1))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1977_a4",
"tags": [
"algebra",
"analysis"
]
} |
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 $x, y$? | True | Prove that $f(x,y)$ must be identically zero. | open RingHom Set Nat | [] | @Eq Prop answer True | Prop | [
{
"t": "Set (ℝ × ℝ)",
"v": null,
"name": "X"
},
{
"t": "X = Set.prod (Icc 0 1) (Icc 0 1)",
"v": null,
"name": "hX"
},
{
"t": "(ℝ × ℝ) → ℝ",
"v": null,
"name": "room"
},
{
"t": "room = fun (a,b) ↦ min (min a (1 - a)) (min b (1 - b))",
"v": null,
"name": "hr... | {
"problem_name": "putnam_1977_a6",
"tags": [
"analysis"
]
} |
Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$. | 2/3 | Prove that the product equals $\frac{2}{3}$. | open RingHom Set Nat 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))))))) | ℝ | [
{
"t": "Tendsto (fun N ↦ ∏ n in Finset.Icc (2 : ℤ) N, ((n : ℝ) ^ 3 - 1) / (n ^ 3 + 1)) atTop (𝓝 answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1977_b1",
"tags": [
"algebra",
"analysis"
]
} |
An ordered triple $(a, b, c)$ of positive irrational numbers with $a + b + c = 1$ is considered $\textit{balanced}$ if all three elements are less than $\frac{1}{2}$. If a triple is not balanced, we can perform a ``balancing act'' $B$ defined by $B(a, b, c) = (f(a), f(b), f(c))$, where $f(x) = 2x - 1$ if $x > 1/2$ and $f(x) = 2x$ otherwise. Will finitely many iterations of this balancing act always eventually produce a balanced triple? | False | Not necessarily. | open RingHom Set Nat Filter Topology | [] | @Eq Prop answer False | Prop | [
{
"t": "ℝ × ℝ × ℝ → Prop",
"v": null,
"name": "P"
},
{
"t": "ℝ × ℝ × ℝ → Prop",
"v": null,
"name": "balanced"
},
{
"t": "ℝ × ℝ × ℝ → ℝ × ℝ × ℝ",
"v": null,
"name": "B"
},
{
"t": "P = fun (a, b, c) => Irrational a ∧ Irrational b ∧ Irrational c ∧ a > 0 ∧ b > 0 ∧ c >... | {
"problem_name": "putnam_1977_b3",
"tags": [
"algebra"
]
} |
Let $p(x) = 2(x^6 + 1) + 4(x^5 + x) + 3(x^4 + x^2) + 5x^3$. For $k$ with $0 < k < 5$, let
\[
I_k = \int_0^{\infty} \frac{x^k}{p(x)} \, dx.
\]
For which $k$ is $I_k$ smallest? | 2 | Show that $I_k$ is smallest for $k = 2$. | open Set Polynomial | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))) | ℕ | [
{
"t": "Polynomial ℝ",
"v": null,
"name": "p"
},
{
"t": "p = 2 * (Polynomial.X ^ 6 + 1) + 4 * (Polynomial.X ^ 5 + Polynomial.X) + 3 * (Polynomial.X ^ 4 + Polynomial.X ^ 2) + 5 * Polynomial.X ^ 3",
"v": null,
"name": "hp"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "I"
},
{
... | {
"problem_name": "putnam_1978_a3",
"tags": [
"analysis",
"algebra"
]
} |
Find
\[
\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{i^2j + 2ij + ij^2}.
\] | 7 / 4 | Prove that the sum evaluates to $\frac{7}{4}$. | open Set Real | [] | @Eq Rat answer
(@HDiv.hDiv Rat Rat Rat (@instHDiv Rat Rat.instDiv) (@OfNat.ofNat Rat (nat_lit 7) (@Rat.instOfNat (nat_lit 7)))
(@OfNat.ofNat Rat (nat_lit 4) (@Rat.instOfNat (nat_lit 4)))) | ℚ | [
{
"t": "(∑' i : ℕ+, ∑' j : ℕ+, (1 : ℚ) / (i ^ 2 * j + 2 * i * j + i * j ^ 2)) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1978_b2",
"tags": [
"algebra",
"analysis"
]
} |
Find the real polynomial $p(x)$ of degree $4$ with largest possible coefficient of $x^4$ such that $p([-1, 1]) \subseteq [0, 1]$. | $4x^4 - 4x^2 + 1$ | Prove that $p(x) = 4x^4 - 4x^2 + 1$. | open Set Real Filter Topology Polynomial | [] | @Eq (@Polynomial Real Real.semiring) answer
(@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring))
(@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing))
(@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring))
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 4)
(@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 4)
(@Polynomial.instNatCast Real Real.semiring)
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring)
(@instHPow (@Polynomial Real Real.semiring) Nat
(@Monoid.toNatPow (@Polynomial Real Real.semiring)
(@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring)
(@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring)
(@Polynomial.semiring Real Real.semiring)))))
(@Polynomial.X Real Real.semiring) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))))
(@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring))
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 4)
(@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 4)
(@Polynomial.instNatCast Real Real.semiring)
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring)
(@instHPow (@Polynomial Real Real.semiring) Nat
(@Monoid.toNatPow (@Polynomial Real Real.semiring)
(@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring)
(@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring)
(@Polynomial.semiring Real Real.semiring)))))
(@Polynomial.X Real Real.semiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring)))) | Polynomial ℝ | [
{
"t": "Set (Polynomial ℝ)",
"v": null,
"name": "S"
},
{
"t": "S = {p : Polynomial ℝ | p.degree = 4 ∧ ∀ x ∈ Icc (-1 : ℝ) 1, p.eval x ∈ Icc 0 1}",
"v": null,
"name": "hS"
},
{
"t": "answer ∈ S ∧ (∀ p ∈ S, p.coeff 4 ≤ answer.coeff 4)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1978_b5",
"tags": [
"algebra"
]
} |
For which positive integers $n$ and $a_1, a_2, \dots, a_n$ with $\sum_{i = 1}^{n} a_i = 1979$ does $\prod_{i = 1}^{n} a_i$ attain the greatest value? | Multiset.replicate 659 3 + {2} | $n$ equals $660$; all but one of the $a_i$ equal $3$ and the remaining $a_i$ equals $2$. | null | [] | @Eq (Multiset Nat) answer
(@HAdd.hAdd (Multiset Nat) (Multiset Nat) (Multiset Nat) (@instHAdd (Multiset Nat) (@Multiset.instAdd Nat))
(@Multiset.replicate Nat (@OfNat.ofNat Nat (nat_lit 659) (instOfNatNat (nat_lit 659)))
(@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
(@Singleton.singleton Nat (Multiset Nat) (@Multiset.instSingleton Nat)
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) | Multiset ℕ | [
{
"t": "Multiset ℕ → Prop",
"v": null,
"name": "P"
},
{
"t": "∀ a, P a ↔ Multiset.card a > 0 ∧ (∀ i ∈ a, i > 0) ∧ a.sum = 1979",
"v": null,
"name": "hP"
},
{
"t": "P answer ∧ ∀ a : Multiset ℕ, P a → answer.prod ≥ a.prod",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1979_a1",
"tags": [
"algebra"
]
} |
For which real numbers $k$ does there exist a continuous function $f : \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) = kx^9$ for all real $x$? | $k \geq 0$ | Such a function exists if and only if $k \ge 0$. | null | [] | @Eq (Real → Prop) answer fun (k_1 : Real) =>
@GE.ge Real Real.instLE k_1 (@OfNat.ofNat Real (nat_lit 0) (@Zero.toOfNat0 Real Real.instZero)) | ℝ → Prop | [
{
"t": "ℝ",
"v": null,
"name": "k"
},
{
"t": "answer k ↔ ∃ f : ℝ → ℝ, Continuous f ∧ ∀ x : ℝ, f (f x) = k * x^9",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1979_a2",
"tags": [
"analysis",
"algebra"
]
} |
Let $x_1, x_2, x_3, \dots$ be a sequence of nonzero real numbers such that $$x_n = \frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}}$$ for all $n \ge 3$. For which real values of $x_1$ and $x_2$ does $x_n$ attain integer values for infinitely many $n$? | fun (a, b) => ∃ m : ℤ, a = m ∧ b = m | We must have $x_1 = x_2 = m$ for some integer $m$. | null | [] | @Eq (Prod Real Real → Prop) answer fun (x_1 : Prod Real Real) =>
_example.match_1 (fun (x_2 : Prod Real Real) => Prop) x_1 fun (a b : Real) =>
@Exists Int fun (m : Int) =>
And (@Eq Real a (@Int.cast Real Real.instIntCast m)) (@Eq Real b (@Int.cast Real Real.instIntCast m)) | (ℝ × ℝ) → Prop | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "x"
},
{
"t": "∀ n : ℕ, x n ≠ 0 ∧ (n ≥ 3 → x n = (x (n - 2))*(x (n - 1))/(2*(x (n - 2)) - (x (n - 1))))",
"v": null,
"name": "hx"
},
{
"t": "(∀ m : ℕ, ∃ n : ℕ, n > m ∧ ∃ a : ℤ, a = x n) ↔ answer (x 1, x 2)",
"v": null,
"name": "h_answer"... | {
"problem_name": "putnam_1979_a3",
"tags": [
"algebra"
]
} |
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? | True | Such line segments must exist. | open Set | [] | @Eq Prop answer True | Prop | [
{
"t": "Finset (Fin 2 → ℝ) × Finset (Fin 2 → ℝ) → Prop",
"v": null,
"name": "A"
},
{
"t": "A = fun (R, B) => R.card = B.card ∧ R ∩ B = ∅ ∧\n ∀ u : Finset (Fin 2 → ℝ), u ⊆ R ∪ B → u.card = 3 → ¬Collinear ℝ (u : Set (Fin 2 → ℝ))",
"v": null,
"name": "hA"
},
{
"t": "(Fin 2 → ℝ) ×... | {
"problem_name": "putnam_1979_a4",
"tags": [
"geometry",
"combinatorics"
]
} |
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 $b$. | fun (a, b) => (Real.exp (-1))*(b^b/a^a)^(1/(b-a)) | The limit equals $$e^{-1}\left(\frac{b^b}{a^a}\right)^{\frac{1}{b-a}}.$$ | open Set Topology Filter | [] | @Eq (Prod Real Real → Real) answer fun (x : Prod Real Real) =>
_example.match_1 (fun (x_1 : Prod Real Real) => Real) x fun (a b : Real) =>
@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(Real.exp (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))))
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) b b)
(@HPow.hPow Real Real Real (@instHPow Real Real Real.instPow) a a))
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub) b a))) | ℝ × ℝ → ℝ | [
{
"t": "∀ a b : ℝ, 0 < a ∧ a < b → Tendsto (fun t : ℝ => (∫ x in Icc 0 1, (b*x + a*(1 - x))^t)^(1/t)) (𝓝[≠] 0) (𝓝 (answer (a, b)))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1979_b2",
"tags": [
"analysis"
]
} |
Let $F$ be a finite field with $n$ elements, and assume $n$ is odd. Suppose $x^2 + bx + c$ is an irreducible polynomial over $F$. For how many elements $d \in F$ is $x^2 + bx + c + d$ irreducible? | fun n : ℕ ↦ (n - (1 : ℤ)) / 2 | Show that there are $\frac{n - 1}{2}$ such elements $d$. | open Set Topology Filter Polynomial | [] | @Eq (Nat → Int) answer fun (n_1 : Nat) =>
@HDiv.hDiv Int Int Int (@instHDiv Int Int.instDiv)
(@HSub.hSub Int Int Int (@instHSub Int Int.instSub) (@Nat.cast Int instNatCastInt n_1)
(@OfNat.ofNat Int (nat_lit 1) (@instOfNat (nat_lit 1))))
(@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2))) | ℕ → ℤ | [
{
"t": "Type*",
"v": null,
"name": "F"
},
{
"t": "Field F",
"v": null,
"name": null
},
{
"t": "Fintype F",
"v": null,
"name": null
},
{
"t": "ℕ",
"v": null,
"name": "n"
},
{
"t": "n = Fintype.card F",
"v": null,
"name": "hn"
},
{
"t... | {
"problem_name": "putnam_1979_b3",
"tags": [
"abstract_algebra"
]
} |
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that $3^r \cdot 7^s=\text{lcm}[a,b,c]=\text{lcm}[a,b,d]=\text{lcm}[a,c,d]=\text{lcm}[b,c,d]$. The answer should be a function of $r$ and $s$. (Note that $\text{lcm}[x,y,z]$ denotes the least common multiple of $x,y,z$.) | (fun r s : ℕ => (1 + 4 * r + 6 * r ^ 2) * (1 + 4 * s + 6 * s ^ 2)) | Show that the number is $(1+4r+6r^2)(1+4s+6s^2)$. | null | [] | @Eq (Nat → Nat → Nat) answer fun (r_1 s_1 : Nat) =>
@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat)
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat)
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))
r_1))
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) r_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat)
(@HAdd.hAdd Nat Nat Nat (@instHAdd Nat instAddNat) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))
s_1))
(@HMul.hMul Nat Nat Nat (@instHMul Nat instMulNat) (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))
(@HPow.hPow Nat Nat Nat (@instHPow Nat Nat (@Monoid.toNatPow Nat Nat.instMonoid)) s_1
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) | ℕ → ℕ → ℕ | [
{
"t": "ℕ",
"v": null,
"name": "r"
},
{
"t": "ℕ",
"v": null,
"name": "s"
},
{
"t": "ℕ → ℕ → ℕ → ℕ → Prop",
"v": null,
"name": "abcdlcm"
},
{
"t": "r > 0 ∧ s > 0",
"v": null,
"name": "rspos"
},
{
"t": "∀ a b c d : ℕ, abcdlcm a b c d ↔\n (a > 0 ... | {
"problem_name": "putnam_1980_a2",
"tags": [
"number_theory"
]
} |
Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$. | $\pi / 4$ | Show that the integral is $\pi/4$. | null | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))) | ℝ | [
{
"t": "answer = ∫ x in Set.Ioo 0 (Real.pi / 2), 1 / (1 + (Real.tan x) ^ (Real.sqrt 2))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1980_a3",
"tags": [
"analysis"
]
} |
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 $C$. | $1/e$ | Show that $u=1/e$. | null | [] | @Eq Real answer
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))
(Real.exp (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)))) | ℝ | [
{
"t": "Set (ℝ → ℝ)",
"v": null,
"name": "C"
},
{
"t": "C = {f : ℝ → ℝ | ContDiffOn ℝ 1 f (Set.Icc 0 1) ∧ f 0 = 0 ∧ f 1 = 1}",
"v": null,
"name": "hC"
},
{
"t": "IsGreatest {u : ℝ | ∀ f ∈ C, u ≤ (∫ x in Set.Ioo 0 1, |deriv f x - f x|)} answer",
"v": null,
"name": "h_answe... | {
"problem_name": "putnam_1980_a6",
"tags": [
"analysis"
]
} |
For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$? | $\{c : \mathbb{R} \mid c \geq 1/2\}$ | Show that the inequality holds if and only if $c \geq 1/2$. | open Real | [] | @Eq (Set Real) answer
(@setOf Real fun (c_1 : Real) =>
@GE.ge Real Real.instLE c_1
(@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 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))) | Set ℝ | [
{
"t": "ℝ",
"v": null,
"name": "c"
},
{
"t": "∀ x : ℝ, (exp x + exp (-x)) / 2 ≤ exp (c * x ^ 2) ↔ c ∈ answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1980_b1",
"tags": [
"analysis"
]
} |
For which real numbers $a$ does the sequence defined by the initial condition $u_0=a$ and the recursion $u_{n+1}=2u_n-n^2$ have $u_n>0$ for all $n \geq 0$? (Express the answer in the simplest form.) | $\{a : \mathbb{R} \mid a \geq 3\}$ | Show that $u_n>0$ for all $n \geq 0$ if and only if $a \geq 3$. | null | [] | @Eq (Set Real) answer
(@setOf Real fun (a_1 : Real) =>
@GE.ge Real Real.instLE a_1
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1))))))) | Set ℝ | [
{
"t": "ℝ",
"v": null,
"name": "a"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "u"
},
{
"t": "u 0 = a ∧ (∀ n : ℕ, u (n + 1) = 2 * u n - n ^ 2)",
"v": null,
"name": "hu"
},
{
"t": "(∀ n : ℕ, u n > 0) ↔ a ∈ answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1980_b3",
"tags": [
"algebra"
]
} |
A function $f$ is convex on $[0, 1]$ if and only if $$f(su + (1-s)v) \le sf(u) + (1 - s)f(v)$$ for all $s \in [0, 1]$.
Let $S_t$ denote the set of all nonnegative increasing convex continuous functions $f : [0, 1] \rightarrow \mathbb{R}$ such that $$f(1) - 2f\left(\frac{2}{3}\right) + f\left(\frac{1}{3}\right) \ge t\left(f\left(\frac{2}{3}\right) - 2f\left(\frac{1}{3}\right) + f(0)\right).$$
For which real numbers $t \ge 0$ is $S_t$ closed under multiplication? | $t \le 1$ | $S_t$ is closed under multiplication if and only if $1 \ge t$. | open Set | [] | @Eq (Real → Prop) answer fun (t_1 : Real) =>
@LE.le Real Real.instLE t_1 (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) | ℝ → Prop | [
{
"t": "Set ℝ",
"v": null,
"name": "T"
},
{
"t": "T = Icc 0 1",
"v": null,
"name": "hT"
},
{
"t": "ℝ → (ℝ → ℝ) → Prop",
"v": null,
"name": "P"
},
{
"t": "(ℝ → ℝ) → Prop",
"v": null,
"name": "IsConvex"
},
{
"t": "ℝ → Set (ℝ → ℝ)",
"v": null,
... | {
"problem_name": "putnam_1980_b5",
"tags": [
"analysis",
"algebra"
]
} |
Let $E(n)$ be the greatest integer $k$ such that $5^k$ divides $1^1 2^2 3^3 \cdots n^n$. Find $\lim_{n \rightarrow \infty} \frac{E(n)}{n^2}$. | $\frac{1}{8}$ | The limit equals $\frac{1}{8}$. | open Topology Filter Set Polynomial Function | [] | @Eq Real answer
(@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 8)
(@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))))) | ℝ | [
{
"t": "ℕ → ℕ → Prop",
"v": null,
"name": "P"
},
{
"t": "∀ n k, P n k ↔ 5^k ∣ ∏ m in Finset.Icc 1 n, (m^m : ℤ)",
"v": null,
"name": "hP"
},
{
"t": "ℕ → ℕ",
"v": null,
"name": "E"
},
{
"t": "∀ n ∈ Ici 1, P n (E n) ∧ ∀ k : ℕ, P n k → k ≤ E n",
"v": null,
"na... | {
"problem_name": "putnam_1981_a1",
"tags": [
"analysis",
"number_theory"
]
} |
Does the limit $$lim_{t \rightarrow \infty}e^{-t}\int_{0}^{t}\int_{0}^{t}\frac{e^x - e^y}{x - y} dx dy$$exist? | False | The limit does not exist. | open Topology Filter Set Polynomial Function | [] | @Eq Prop answer False | Prop | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "f"
},
{
"t": "f = fun t : ℝ => Real.exp (-t) * ∫ y in (Ico 0 t), ∫ x in (Ico 0 t), (Real.exp x - Real.exp y) / (x - y)",
"v": null,
"name": "hf"
},
{
"t": "(∃ L : ℝ, Tendsto f atTop (𝓝 L)) ↔ answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1981_a3",
"tags": [
"analysis"
]
} |
Let $P(x)$ be a polynomial with real coefficients; let $$Q(x) = (x^2 + 1)P(x)P'(x) + x((P(x))^2 + (P'(x))^2).$$
Given that $P$ has $n$ distinct real roots all greater than $1$, prove or disprove that $Q$ must have at least $2n - 1$ distinct real roots. | True | $Q(x)$ must have at least $2n - 1$ distinct real roots. | open Topology Filter Set Polynomial Function | [] | @Eq Prop answer True | Prop | [
{
"t": "Polynomial ℝ → Polynomial ℝ",
"v": null,
"name": "Q"
},
{
"t": "Q = fun P : Polynomial ℝ => (Polynomial.X^2 + 1) * P * (Polynomial.derivative P) + Polynomial.X * (P^2 + (Polynomial.derivative P)^2)",
"v": null,
"name": "hQ"
},
{
"t": "Polynomial ℝ → ℝ",
"v": null,
... | {
"problem_name": "putnam_1981_a5",
"tags": [
"algebra"
]
} |
Find the value of $$\lim_{n \rightarrow \infty} \frac{1}{n^5}\sum_{h=1}^{n}\sum_{k=1}^{n}(5h^4 - 18h^2k^2 + 5k^4).$$ | -1 | The limit equals $-1$. | open Topology Filter Set Polynomial Function | [] | @Eq Real answer (@Neg.neg Real Real.instNeg (@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne))) | ℝ | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "f"
},
{
"t": "f = fun n : ℕ => ((1 : ℝ)/n^5) * ∑ h in Finset.Icc 1 n, ∑ k in Finset.Icc 1 n, (5*(h : ℝ)^4 - 18*h^2*k^2 + 5*k^4)",
"v": null,
"name": "hf"
},
{
"t": "Tendsto f atTop (𝓝 answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1981_b1",
"tags": [
"analysis"
]
} |
Determine the minimum value attained by $$(r - 1)^2 + (\frac{s}{r} - 1)^2 + (\frac{t}{s} - 1)^2 + (\frac{4}{t} - 1)^2$$ across all choices of real $r$, $s$, and $t$ that satisfy $1 \le r \le s \le t \le 4$. | 12 - 8 * Real.sqrt 2 | The minimum is $12 - 8\sqrt{2}$. | open Topology Filter Set Polynomial Function | [] | @Eq Real answer
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(@OfNat.ofNat Real (nat_lit 12)
(@instOfNatAtLeastTwo Real (nat_lit 12) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10))))))
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@OfNat.ofNat Real (nat_lit 8)
(@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6))))))
(Real.sqrt
(@OfNat.ofNat Real (nat_lit 2)
(@instOfNatAtLeastTwo Real (nat_lit 2) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))))) | ℝ | [
{
"t": "ℝ × ℝ × ℝ → Prop",
"v": null,
"name": "P"
},
{
"t": "P = fun (r, s, t) => 1 ≤ r ∧ r ≤ s ∧ s ≤ t ∧ t ≤ 4",
"v": null,
"name": "hP"
},
{
"t": "ℝ × ℝ × ℝ → ℝ",
"v": null,
"name": "f"
},
{
"t": "f = fun (r, s, t) => (r - 1)^2 + (s/r - 1)^2 + (t/s - 1)^2 + (4/t... | {
"problem_name": "putnam_1981_b2",
"tags": [
"algebra"
]
} |
Let $V$ be a set of $5$ by $7$ matrices, with real entries and with the property that $rA+sB \in V$ whenever $A,B \in V$ and $r$ and $s$ are scalars (i.e., real numbers). \emph{Prove or disprove} the following assertion: If $V$ contains matrices of ranks $0$, $1$, $2$, $4$, and $5$, then it also contains a matrix of rank $3$. [The rank of a nonzero matrix $M$ is the largest $k$ such that the entries of some $k$ rows and some $k$ columns form a $k$ by $k$ matrix with a nonzero determinant.] | False | Show that the assertion is false. | open Topology Filter Set Polynomial Function | [] | @Eq Prop answer False | Prop | [
{
"t": "Set (Matrix (Fin 5) (Fin 7) ℝ)",
"v": null,
"name": "V"
},
{
"t": "∀ A ∈ V, ∀ B ∈ V, ∀ r s : ℝ, r • A + s • B ∈ V",
"v": null,
"name": "hVAB"
},
{
"t": "∃ A ∈ V, A.rank = 0",
"v": null,
"name": "hVrank0"
},
{
"t": "∃ A ∈ V, A.rank = 1",
"v": null,
... | {
"problem_name": "putnam_1981_b4",
"tags": [
"linear_algebra"
]
} |
Let $B(n)$ be the number of ones in the base two expression for the positive integer $n$. For example, $B(6)=B(110_2)=2$ and $B(15)=B(1111_2)=4$. Determine whether or not $\exp \left(\sum_{n=1}^\infty \frac{B(n)}{n(n+1)}\right)$ is a rational number. Here $\exp(x)$ denotes $e^x$. | True | Show that the expression is a rational number. | open Topology Filter Set Polynomial Function | [] | @Eq Prop answer True | Prop | [
{
"t": "List ℕ → ℤ",
"v": null,
"name": "sumbits"
},
{
"t": "ℕ → ℤ",
"v": null,
"name": "B"
},
{
"t": "∀ bits : List ℕ, sumbits bits = ∑ i : Fin bits.length, (bits[i] : ℤ)",
"v": null,
"name": "hsumbits"
},
{
"t": "∀ n > 0, B n = sumbits (Nat.digits 2 n)",
"v"... | {
"problem_name": "putnam_1981_b5",
"tags": [
"analysis",
"algebra"
]
} |
Let $B_n(x) = 1^x + 2^x + \dots + n^x$ and let $f(n) = \frac{B_n(\log_n 2)}{(n \log_2 n)^2}$. Does $f(2) + f(3) + f(4) + \dots$ converge? | True | Prove that the series converges. | open Set Function Filter Topology Polynomial Real | [] | @Eq Prop answer True | Prop | [
{
"t": "ℕ → ℝ → ℝ",
"v": null,
"name": "B"
},
{
"t": "B = fun (n : ℕ) (x : ℝ) ↦ ∑ k in Finset.Icc 1 n, (k : ℝ) ^ x",
"v": null,
"name": "hB"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "f"
},
{
"t": "f = fun n ↦ B n (logb n 2) / (n * logb 2 n) ^ 2",
"v": null,
"... | {
"problem_name": "putnam_1982_a2",
"tags": [
"algebra"
]
} |
Evaluate $\int_0^{\infty} \frac{\tan^{-1}(\pi x) - \tan^{-1} x}{x} \, dx$. | $\frac{\pi}{2} \log \pi$ | Show that the integral evaluates to $\frac{\pi}{2} \ln \pi$. | open Set Function Filter Topology Polynomial Real | [] | @Eq Real answer
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@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)))))))
(Real.log Real.pi)) | ℝ | [
{
"t": "Tendsto (fun t ↦ ∫ x in (0)..t, (arctan (Real.pi * x) - arctan x) / x) atTop (𝓝 answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1982_a3",
"tags": [
"analysis"
]
} |
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 $$\lim_{n \rightarrow \infty} \sum_{k = 1}^{n} x_{b(k)} = 1.$$ | False | The limit need not equal $1$. | open Set Function Filter Topology Polynomial Real | [] | @Eq Prop answer False | Prop | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "b"
},
{
"t": "ℕ → ℝ",
"v": null,
"name": "x"
},
{
"t": "BijOn b (Ici 1) (Ici 1)",
"v": null,
"name": "h_bij"
},
{
"t": "StrictAntiOn (fun n : ℕ => |x n|) (Ici 1)",
"v": null,
"name": "h_strict_anti"
},
{
"t": "Te... | {
"problem_name": "putnam_1982_a6",
"tags": [
"analysis"
]
} |
Let $A(x, y)$ denote the number of points $(m, n)$ with integer coordinates $m$ and $n$ where $m^2 + n^2 \le x^2 + y^2$. Also, let $g = \sum_{k = 0}^{\infty} e^{-k^2}$. Express the value $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} A(x, y)e^{-x^2 - y^2} dx dy$$ as a polynomial in $g$. | C Real.pi * (2*X - 1)^2 | The desired polynomial is $\pi(2g - 1)^2$. | open Set Function Filter Topology Polynomial Real | [] | @Eq (@Polynomial Real Real.semiring) answer
(@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring))
(@DFunLike.coe
(@RingHom Real (@Polynomial Real Real.semiring) (@Semiring.toNonAssocSemiring Real Real.semiring)
(@Semiring.toNonAssocSemiring (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring)))
Real (fun (x : Real) => @Polynomial Real Real.semiring)
(@RingHom.instFunLike Real (@Polynomial Real Real.semiring) (@Semiring.toNonAssocSemiring Real Real.semiring)
(@Semiring.toNonAssocSemiring (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring)))
(@Polynomial.C Real Real.semiring) Real.pi)
(@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring)
(@instHPow (@Polynomial Real Real.semiring) Nat
(@Monoid.toNatPow (@Polynomial Real Real.semiring)
(@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring)
(@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring) (@Polynomial.semiring Real Real.semiring)))))
(@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing))
(@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring))
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 2)
(@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 2)
(@Polynomial.instNatCast Real Real.semiring)
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))))))
(@Polynomial.X Real Real.semiring))
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))) | Polynomial ℝ | [
{
"t": "ℝ × ℝ → ℕ",
"v": null,
"name": "A"
},
{
"t": "ℝ",
"v": null,
"name": "g"
},
{
"t": "ℝ",
"v": null,
"name": "I"
},
{
"t": "A = fun (x, y) => {a : ℤ × ℤ | a.1^2 + a.2^2 ≤ x^2 + y^2}.ncard",
"v": null,
"name": "hA"
},
{
"t": "g = ∑' k : ℕ, Rea... | {
"problem_name": "putnam_1982_b2",
"tags": [
"analysis"
]
} |
Let $p_n$ denote the probability that $c + d$ will be a perfect square if $c$ and $d$ are selected independently and uniformly at random from $\{1, 2, 3, \dots, n\}$. Express $\lim_{n \rightarrow \infty} p_n \sqrt{n}$ in the form $r(\sqrt{s} - t)$ for integers $s$ and $t$ and rational $r$. | 4/3 * (Real.sqrt 2 - 1) | The limit equals $\frac{4}{3}(\sqrt{2} - 1)$. | open Set Function Filter Topology Polynomial Real | [] | @Eq Real answer
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
(@OfNat.ofNat Real (nat_lit 3)
(@instOfNatAtLeastTwo Real (nat_lit 3) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
(@HSub.hSub Real Real Real (@instHSub Real Real.instSub)
(Real.sqrt
(@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 1) (@One.toOfNat1 Real Real.instOne)))) | ℝ | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "p"
},
{
"t": "p = fun n : ℕ => ({c : Finset.Icc 1 n × Finset.Icc 1 n | ∃ m : ℕ, m^2 = c.1 + c.2}.ncard : ℝ) / n^2",
"v": null,
"name": "hp"
},
{
"t": "Tendsto (fun n : ℕ => p n * Real.sqrt n) atTop (𝓝 answer)",
"v": null,
"name": "h_an... | {
"problem_name": "putnam_1982_b3",
"tags": [
"analysis",
"number_theory",
"probability"
]
} |
Let $n_1, n_2, \dots, n_s$ be distinct integers such that, for every integer $k$, $n_1n_2\cdots n_s$ divides $(n_1 + k)(n_2 + k) \cdots (n_s + k)$. Prove or provide a counterexample to the following claims:
\begin{enumerate}
\item
For some $i$, $|n_i| = 1$.
\item
If all $n_i$ are positive, then $\{n_1, n_2, \dots, n_s\} = \{1, 2, \dots, s\}$.
\end{enumerate} | (True, True) | Both claims are true. | open Set Function Filter Topology Polynomial Real | [] | @Eq (Prod Prop Prop) answer (@Prod.mk Prop Prop True True) | Prop × Prop | [
{
"t": "Finset ℤ → Prop",
"v": null,
"name": "P"
},
{
"t": "∀ n, P n ↔ n.Nonempty ∧ ∀ k, ∏ i in n, i ∣ ∏ i in n, (i + k)",
"v": null,
"name": "P_def"
},
{
"t": "((∀ n, P n → 1 ∈ n ∨ -1 ∈ n) ↔ answer.1) ∧\n ((∀ n, P n → (∀ i ∈ n, 0 < i) → n = Finset.Icc (1 : ℤ) n.card) ↔ answer... | {
"problem_name": "putnam_1982_b4",
"tags": [
"number_theory"
]
} |
How many positive integers $n$ are there such that $n$ is an exact divisor of at least one of the numbers $10^{40},20^{30}$? | 2301 | Show that the desired count is $2301$. | null | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 2301) (instOfNatNat (nat_lit 2301))) | ℕ | [
{
"t": "{n : ℤ | n > 0 ∧ (n ∣ 10 ^ 40 ∨ n ∣ 20 ^ 30)}.encard = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1983_a1",
"tags": [
"number_theory"
]
} |
Prove or disprove that there exists a positive real number $\alpha$ such that $[\alpha_n] - n$ is even for all integers $n > 0$. (Here $[x]$ denotes the greatest integer less than or equal to $x$.) | True | Prove that such an $\alpha$ exists. | open Nat | [] | @Eq Prop answer True | Prop | [
{
"t": "answer ↔ (∃ α : ℝ, α > 0 ∧ ∀ n : ℕ, n > 0 → Even (⌊α ^ n⌋ - n))",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1983_a5",
"tags": [
"analysis"
]
} |
Let $T$ be the triangle with vertices $(0, 0)$, $(a, 0)$, and $(0, a)$. Find $\lim_{a \to \infty} a^4 \exp(-a^3) \int_T \exp(x^3+y^3) \, dx \, dy$. | 2 / 9 | Show that the integral evaluates to $\frac{2}{9}$. | open Nat Filter Topology Real | [] | @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 9)
(@instOfNatAtLeastTwo Real (nat_lit 9) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7))))))) | ℝ | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "F"
},
{
"t": "F = fun a ↦ (a ^ 4 / exp (a ^ 3)) * ∫ x in (0)..a, ∫ y in (0)..(a - x), exp (x ^ 3 + y ^ 3)",
"v": null,
"name": "hF"
},
{
"t": "Tendsto F atTop (𝓝 answer)",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1983_a6",
"tags": [
"analysis"
]
} |
Let $f(n)$ be the number of ways of representing $n$ as a sum of powers of $2$ with no power being used more than $3$ times. For example, $f(7) = 4$ (the representations are $4 + 2 + 1$, $4 + 1 + 1 + 1$, $2 + 2 + 2 + 1$, $2 + 2 + 1 + 1 + 1$). Can we find a real polynomial $p(x)$ such that $f(n) = [p(n)]$, where $[u]$ denotes the greatest integer less than or equal to $u$? | True | Prove that such a polynomial exists. | open Nat Filter Topology Real | [] | @Eq Prop answer True | Prop | [
{
"t": "ℕ+ → ℕ",
"v": null,
"name": "f"
},
{
"t": "f = fun (n : ℕ+) ↦\n Set.ncard {M : Multiset ℕ |\n (∀ m ∈ M, ∃ k : ℕ, m = (2 ^ k : ℤ)) ∧ \n (∀ m ∈ M, M.count m ≤ 3) ∧ \n (M.sum : ℤ) = n}",
"v": null,
"name": "hf"
},
{
"t": "answer ↔ (∃ p ... | {
"problem_name": "putnam_1983_b2",
"tags": [
"algebra"
]
} |
Define $\left\lVert x \right\rVert$ as the distance from $x$ to the nearest integer. Find $\lim_{n \to \infty} \frac{1}{n} \int_{1}^{n} \left\lVert \frac{n}{x} \right\rVert \, dx$. You may assume that $\prod_{n=1}^{\infty} \frac{2n}{(2n-1)} \cdot \frac{2n}{(2n+1)} = \frac{\pi}{2}$. | $\log \left(\frac{4}{\pi}\right)$ | Show that the limit equals $\ln \left( \frac{4}{\pi} \right)$. | open Nat Filter Topology Real | [] | @Eq Real answer
(Real.log
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid))
(@OfNat.ofNat Real (nat_lit 4)
(@instOfNatAtLeastTwo Real (nat_lit 4) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))))
Real.pi)) | ℝ | [
{
"t": "ℝ → ℝ",
"v": null,
"name": "dist_fun"
},
{
"t": "dist_fun = fun (x : ℝ) ↦ min (x - ⌊x⌋) (⌈x⌉ - x)",
"v": null,
"name": "hdist_fun"
},
{
"t": "Tendsto (fun N ↦ ∏ n in Finset.Icc 1 N, (2 * n / (2 * n - 1)) * (2 * n / (2 * n + 1)) : ℕ → ℝ) atTop (𝓝 (Real.pi / 2))",
"v":... | {
"problem_name": "putnam_1983_b5",
"tags": [
"analysis"
]
} |
Express $\sum_{k=1}^\infty (6^k/(3^{k+1}-2^{k+1})(3^k-2^k))$ as a rational number. | 2 | Show that the sum converges to $2$. | null | [] | @Eq Rat answer (@OfNat.ofNat Rat (nat_lit 2) (@Rat.instOfNat (nat_lit 2))) | ℚ | [
{
"t": "∑' k : Set.Ici 1, (6 ^ (k : ℕ) / ((3 ^ ((k : ℕ) + 1) - 2 ^ ((k : ℕ) + 1)) * (3 ^ (k : ℕ) - 2 ^ (k : ℕ)))) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1984_a2",
"tags": [
"analysis"
]
} |
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 $M_n$. | $(X_2)^2 \cdot ((X_0)^2 - (X_1)^2)$ | Show that $\lim_{x \to a} \frac{\det M_n}{(x-a)^{2n-2}}=n^2(a^2-b^2)$. | open Topology Filter | [] | @Eq (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) answer
(@HMul.hMul (@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@instHMul
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@Distrib.toMul
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@NonUnitalNonAssocSemiring.toDistrib
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@NonUnitalCommRing.toNonUnitalNonAssocCommRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real
Real.instCommSemiring)
(@CommRing.toNonUnitalCommRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.commRing))))))))
(@HPow.hPow
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@instHPow
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat
(@Monoid.toNatPow
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@MonoidWithZero.toMonoid
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@Semiring.toMonoidWithZero
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@CommSemiring.toSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real
Real.instCommSemiring)
(@MvPolynomial.commSemiring Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
Real.instCommSemiring))))))
(@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (nat_lit 2)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 2))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@HSub.hSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@instHSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@SubNegMonoid.toSub
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@AddGroup.toSubNegMonoid
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@AddGroupWithOne.toAddGroup
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@Ring.toAddGroupWithOne
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real
Real.instCommSemiring)
(@CommRing.toRing
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real
Real.instCommSemiring)
(@MvPolynomial.instCommRingMvPolynomial Real
(Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.commRing)))))))
(@HPow.hPow
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@instHPow
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat
(@Monoid.toNatPow
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@MonoidWithZero.toMonoid
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@Semiring.toMonoidWithZero
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real
Real.instCommSemiring)
(@CommSemiring.toSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real
Real.instCommSemiring)
(@MvPolynomial.commSemiring Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
Real.instCommSemiring))))))
(@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (nat_lit 0)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 0))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2))))
(@HPow.hPow
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@instHPow
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring) Nat
(@Monoid.toNatPow
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@MonoidWithZero.toMonoid
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real Real.instCommSemiring)
(@Semiring.toMonoidWithZero
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real
Real.instCommSemiring)
(@CommSemiring.toSemiring
(@MvPolynomial (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real
Real.instCommSemiring)
(@MvPolynomial.commSemiring Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))
Real.instCommSemiring))))))
(@MvPolynomial.X Real (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) Real.instCommSemiring
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))) (nat_lit 1)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))) (nat_lit 1))))
(@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))) | MvPolynomial (Fin 3) ℝ | [
{
"t": "ℕ",
"v": null,
"name": "n"
},
{
"t": "ℝ",
"v": null,
"name": "a"
},
{
"t": "ℝ",
"v": null,
"name": "b"
},
{
"t": "ℝ → Matrix (Fin (2 * n)) (Fin (2 * n)) ℝ",
"v": null,
"name": "Mn"
},
{
"t": "Fin 3 → ℝ",
"v": null,
"name": "polyabn"... | {
"problem_name": "putnam_1984_a3",
"tags": [
"linear_algebra",
"analysis"
]
} |
Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z \leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral $\iiint_R x^1y^9z^8w^4\,dx\,dy\,dz$ in the form $a!b!c!d!/n!$, where $a$, $b$, $c$, $d$, and $n$ are positive integers. | (1, 9, 8, 4, 25) | Show that the integral we desire is $1!9!8!4!/25!$. | open Topology Filter Nat | [] | @Eq (Prod Nat (Prod Nat (Prod Nat (Prod Nat Nat)))) answer
(@Prod.mk Nat (Prod Nat (Prod Nat (Prod Nat Nat))) (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))
(@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 9) (instOfNatNat (nat_lit 9)))
(@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8)))
(@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))
(@OfNat.ofNat Nat (nat_lit 25) (instOfNatNat (nat_lit 25))))))) | ℕ × ℕ × ℕ × ℕ × ℕ | [
{
"t": "Set (Fin 3 → ℝ)",
"v": null,
"name": "R"
},
{
"t": "(Fin 3 → ℝ) → ℝ",
"v": null,
"name": "w"
},
{
"t": "R = {p | (∀ i : Fin 3, p i ≥ 0) ∧ p 0 + p 1 + p 2 ≤ 1}",
"v": null,
"name": "hR"
},
{
"t": "∀ p, w p = 1 - p 0 - p 1 - p 2",
"v": null,
"name": ... | {
"problem_name": "putnam_1984_a5",
"tags": [
"analysis"
]
} |
Let $n$ be a positive integer, and let $f(n)$ denote the last nonzero digit in the decimal expansion of $n!$. For instance, $f(5)=2$.
\begin{enumerate}
\item[(a)] Show that if $a_1,a_2,\dots,a_k$ are \emph{distinct} nonnegative integers, then $f(5^{a_1}+5^{a_2}+\dots+5^{a_k})$ depends only on the sum $a_1+a_2+\dots+a_k$.
\item[(b)] Assuming part (a), we can define $g(s)=f(5^{a_1}+5^{a_2}+\dots+5^{a_k})$, where $s=a_1+a_2+\dots+a_k$. Find the least positive integer $p$ for which $g(s)=g(s + p)$, for all $s \geq 1$, or else show that no such $p$ exists.
\end{enumerate} | 4 | Show that the least such $p$ is $p=4$. | open Topology Filter Function Nat | [] | @Eq Nat answer (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))) | ℕ | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "f"
},
{
"t": "∀ n, some (f n) = (Nat.digits 10 (n !)).find? (fun d ↦ d ≠ 0)",
"v": null,
"name": "hf"
},
{
"t": "ℕ → (ℕ → ℕ) → ℕ → Prop",
"v": null,
"name": "IsPeriodicFrom"
},
{
"t": "∀ x f p, IsPeriodicFrom x f p ↔ Periodic (f... | {
"problem_name": "putnam_1984_a6",
"tags": [
"algebra",
"number_theory"
]
} |
Let $n$ be a positive integer, and define $f(n)=1!+2!+\dots+n!$. Find polynomials $P(x)$ and $Q(x)$ such that $f(n+2)=P(n)f(n+1)+Q(n)f(n)$ for all $n \geq 1$. | $(x + 3, -x - 2)$ | Show that we can take $P(x)=x+3$ and $Q(x)=-x-2$. | open Topology Filter Nat | [] | @Eq (Prod (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)) answer
(@Prod.mk (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring))
(@Polynomial.X Real Real.semiring)
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 3)
(@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 3) (@Polynomial.instNatCast Real Real.semiring)
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1) (instOfNatNat (nat_lit 1)))))))
(@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing))
(@Neg.neg (@Polynomial Real Real.semiring) (@Polynomial.neg' Real Real.instRing)
(@Polynomial.X Real Real.semiring))
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 2)
(@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 2) (@Polynomial.instNatCast Real Real.semiring)
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))) | Polynomial ℝ × Polynomial ℝ | [
{
"t": "ℕ → ℤ",
"v": null,
"name": "f"
},
{
"t": "∀ n > 0, f n = ∑ i : Set.Icc 1 n, ((i)! : ℤ)",
"v": null,
"name": "hf"
},
{
"t": "∀ n ≥ 1, f (n + 2) = (answer.1).eval (n : ℝ) * f (n + 1) + (answer.2).eval (n : ℝ) * f n",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1984_b1",
"tags": [
"algebra"
]
} |
Find the minimum value of $(u-v)^2+(\sqrt{2-u^2}-\frac{9}{v})^2$ for $0<u<\sqrt{2}$ and $v>0$. | 8 | Show that the minimum value is $8$. | open Topology Filter Nat | [] | @Eq Real answer
(@OfNat.ofNat Real (nat_lit 8)
(@instOfNatAtLeastTwo Real (nat_lit 8) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 6) (instOfNatNat (nat_lit 6)))))) | ℝ | [
{
"t": "ℝ → ℝ → ℝ",
"v": null,
"name": "f"
},
{
"t": "∀ u v : ℝ, f u v = (u - v) ^ 2 + (Real.sqrt (2 - u ^ 2) - 9 / v) ^ 2",
"v": null,
"name": "hf"
},
{
"t": "IsLeast {y | ∃ᵉ (u : Set.Ioo 0 √2) (v > 0), f u v = y} answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1984_b2",
"tags": [
"geometry",
"analysis"
]
} |
Prove or disprove the following statement: If $F$ is a finite set with two or more elements, then there exists a binary operation $*$ on F such that for all $x,y,z$ in $F$,
\begin{enumerate}
\item[(i)] $x*z=y*z$ implies $x=y$ (right cancellation holds), and
\item[(ii)] $x*(y*z) \neq (x*y)*z$ (\emph{no} case of associativity holds).
\end{enumerate} | True | Show that the statement is true. | open Topology Filter Nat | [] | @Eq Prop answer True | Prop | [
{
"t": "(∀ (F : Type*) (_ : Fintype F), Fintype.card F ≥ 2 → (∃ mul : F → F → F, ∀ x y z : F, (mul x z = mul y z → x = y) ∧ (mul x (mul y z) ≠ mul (mul x y) z))) ↔ answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1984_b3",
"tags": [
"abstract_algebra"
]
} |
For each nonnegative integer $k$, let $d(k)$ denote the number of $1$'s in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)=2$). Let $m$ be a positive integer. Express $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m$ in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials. | (2, (Polynomial.X * (Polynomial.X - 1)) / 2, Polynomial.X) | Show that $\sum_{k=0}^{2^m-1} (-1)^{d(k)}k^m=(-1)^m2^{m(m-1)/2}m!$. | open Topology Filter Nat | [] | @Eq (Prod Int (Prod (@Polynomial Real Real.semiring) (@Polynomial Nat Nat.instSemiring))) answer
(@Prod.mk Int (Prod (@Polynomial Real Real.semiring) (@Polynomial Nat Nat.instSemiring))
(@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2)))
(@Prod.mk (@Polynomial Real Real.semiring) (@Polynomial Nat Nat.instSemiring)
(@HDiv.hDiv (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHDiv (@Polynomial Real Real.semiring) (@Polynomial.instDiv Real Real.field))
(@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring))
(@Polynomial.X Real Real.semiring)
(@HSub.hSub (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHSub (@Polynomial Real Real.semiring) (@Polynomial.sub Real Real.instRing))
(@Polynomial.X Real Real.semiring)
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring)))))
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 2)
(@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 2)
(@Polynomial.instNatCast Real Real.semiring)
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))))
(@Polynomial.X Nat Nat.instSemiring))) | ℤ × Polynomial ℝ × Polynomial ℕ | [
{
"t": "ℕ",
"v": null,
"name": "m"
},
{
"t": "m > 0",
"v": null,
"name": "mpos"
},
{
"t": "ℕ → ℕ",
"v": null,
"name": "d"
},
{
"t": "List ℕ → ℕ",
"v": null,
"name": "sumbits"
},
{
"t": "∀ bits : List ℕ, sumbits bits = ∑ i : Fin bits.length, bits[i]... | {
"problem_name": "putnam_1984_b5",
"tags": [
"algebra",
"analysis"
]
} |
Determine, with proof, the number of ordered triples $(A_1, A_2, A_3)$ of sets which have the property that
\begin{enumerate}
\item[(i)] $A_1 \cup A_2 \cup A_3 = \{1,2,3,4,5,6,7,8,9,10\}$, and
\item[(ii)] $A_1 \cap A_2 \cap A_3 = \emptyset$.
\end{enumerate}
Express your answer in the form $2^a 3^b 5^c 7^d$, where $a,b,c,d$ are nonnegative integers. | (10, 10, 0, 0) | Prove that the number of such triples is $2^{10}3^{10}$. | open Set | [] | @Eq (Prod Nat (Prod Nat (Prod Nat Nat))) answer
(@Prod.mk Nat (Prod Nat (Prod Nat Nat)) (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10)))
(@Prod.mk Nat (Prod Nat Nat) (@OfNat.ofNat Nat (nat_lit 10) (instOfNatNat (nat_lit 10)))
(@Prod.mk Nat Nat (@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))
(@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0)))))) | ℕ × ℕ × ℕ × ℕ | [
{
"t": "let (a, b, c, d) := answer;\n {(A1, A2, A3) : Set ℤ × Set ℤ × Set ℤ | A1 ∪ A2 ∪ A3 = Icc 1 10 ∧ A1 ∩ A2 ∩ A3 = ∅}.ncard = 2 ^ a * 3 ^ b * 5 ^ c * 7 ^ d",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1985_a1",
"tags": [
"algebra"
]
} |
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 $\lim_{n \to \infty} a_n(n)$. | $e^d - 1$ | Show that the limit equals $e^d - 1$. | open Set Filter Topology Real | [] | @Eq (Real → Real) answer fun (d_1 : Real) =>
@HSub.hSub Real Real Real (@instHSub Real Real.instSub) (Real.exp d_1)
(@OfNat.ofNat Real (nat_lit 1) (@One.toOfNat1 Real Real.instOne)) | ℝ → ℝ | [
{
"t": "ℝ",
"v": null,
"name": "d"
},
{
"t": "ℕ → ℕ → ℝ",
"v": null,
"name": "a"
},
{
"t": "∀ m : ℕ, a m 0 = d / 2 ^ m",
"v": null,
"name": "ha0"
},
{
"t": "∀ m : ℕ, ∀ j : ℕ, a m (j + 1) = (a m j) ^ 2 + 2 * a m j",
"v": null,
"name": "ha"
},
{
"t":... | {
"problem_name": "putnam_1985_a3",
"tags": [
"analysis"
]
} |
Define a sequence $\{a_i\}$ by $a_1=3$ and $a_{i+1}=3^{a_i}$ for $i \geq 1$. Which integers between $00$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_i$? | {87} | Prove that the only number that occurs infinitely often is $87$. | open Set Filter Topology Real | [] | @Eq (Set (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))))) answer
(@Singleton.singleton (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100))))
(Set (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100)))))
(@Set.instSingletonSet (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100)))))
(@OfNat.ofNat (Fin (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100)))) (nat_lit 87)
(@Fin.instOfNat (@OfNat.ofNat Nat (nat_lit 100) (instOfNatNat (nat_lit 100)))
(@Nat.instNeZeroSucc (@OfNat.ofNat Nat (nat_lit 99) (instOfNatNat (nat_lit 99)))) (nat_lit 87)))) | Set (Fin 100) | [
{
"t": "ℕ → ℕ",
"v": null,
"name": "a"
},
{
"t": "a 1 = 3",
"v": null,
"name": "ha1"
},
{
"t": "∀ i ≥ 1, a (i + 1) = 3 ^ a i",
"v": null,
"name": "ha"
},
{
"t": "{k : Fin 100 | ∀ N : ℕ, ∃ i ≥ N, a i % 100 = k} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1985_a4",
"tags": [
"number_theory"
]
} |
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 $I_m \neq 0$? | {3, 4, 7, 8} | Prove that the integers $m$ with $1 \leq m \leq 10$ and $I_m \neq 0$ are $m = 3, 4, 7, 8$. | open Set Filter Topology Real | [] | @Eq (Set Nat) answer
(@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3)))
(@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4)))
(@Insert.insert Nat (Set Nat) (@Set.instInsert Nat) (@OfNat.ofNat Nat (nat_lit 7) (instOfNatNat (nat_lit 7)))
(@Singleton.singleton Nat (Set Nat) (@Set.instSingletonSet Nat)
(@OfNat.ofNat Nat (nat_lit 8) (instOfNatNat (nat_lit 8))))))) | Set ℕ | [
{
"t": "ℕ → ℝ",
"v": null,
"name": "I"
},
{
"t": "I = fun (m : ℕ) ↦ ∫ x in (0)..(2 * Real.pi), ∏ k in Finset.Icc 1 m, cos (k * x)",
"v": null,
"name": "hI"
},
{
"t": "{m ∈ Finset.Icc 1 10 | I m ≠ 0} = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1985_a5",
"tags": [
"analysis"
]
} |
If $p(x)= a_0 + a_1 x + \cdots + a_m x^m$ is a polynomial with real coefficients $a_i$, then set
\[
\Gamma(p(x)) = a_0^2 + a_1^2 + \cdots + a_m^2.
\]
Let $F(x) = 3x^2+7x+2$. Find, with proof, a polynomial $g(x)$ with real coefficients such that
\begin{enumerate}
\item[(i)] $g(0)=1$, and
\item[(ii)] $\Gamma(f(x)^n) = \Gamma(g(x)^n)$
\end{enumerate}
for every integer $n \geq 1$. | 6x^2 + 5x + 1 | Show that $g(x) = 6x^2 + 5x + 1$ satisfies the conditions. | open Set Filter Topology Real Polynomial | [] | @Eq (@Polynomial Real Real.semiring) answer
(@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring))
(@HAdd.hAdd (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHAdd (@Polynomial Real Real.semiring) (@Polynomial.add' Real Real.semiring))
(@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring))
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 6)
(@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 6)
(@Polynomial.instNatCast Real Real.semiring)
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 4) (instOfNatNat (nat_lit 4))))))
(@HPow.hPow (@Polynomial Real Real.semiring) Nat (@Polynomial Real Real.semiring)
(@instHPow (@Polynomial Real Real.semiring) Nat
(@Monoid.toNatPow (@Polynomial Real Real.semiring)
(@MonoidWithZero.toMonoid (@Polynomial Real Real.semiring)
(@Semiring.toMonoidWithZero (@Polynomial Real Real.semiring)
(@Polynomial.semiring Real Real.semiring)))))
(@Polynomial.X Real Real.semiring) (@OfNat.ofNat Nat (nat_lit 2) (instOfNatNat (nat_lit 2)))))
(@HMul.hMul (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring) (@Polynomial Real Real.semiring)
(@instHMul (@Polynomial Real Real.semiring) (@Polynomial.mul' Real Real.semiring))
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 5)
(@instOfNatAtLeastTwo (@Polynomial Real Real.semiring) (nat_lit 5)
(@Polynomial.instNatCast Real Real.semiring)
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3) (instOfNatNat (nat_lit 3))))))
(@Polynomial.X Real Real.semiring)))
(@OfNat.ofNat (@Polynomial Real Real.semiring) (nat_lit 1)
(@One.toOfNat1 (@Polynomial Real Real.semiring) (@Polynomial.one Real Real.semiring)))) | Polynomial ℝ | [
{
"t": "Polynomial ℝ → ℝ",
"v": null,
"name": "Γ"
},
{
"t": "Polynomial ℝ",
"v": null,
"name": "f"
},
{
"t": "Γ = fun p ↦ ∑ k in Finset.range (p.natDegree + 1), coeff p k ^ 2",
"v": null,
"name": "hΓ"
},
{
"t": "f = 3 * Polynomial.X ^ 2 + 7 * Polynomial.X + 2",
... | {
"problem_name": "putnam_1985_a6",
"tags": [
"algebra"
]
} |
Let $k$ be the smallest positive integer for which there exist distinct integers $m_1, m_2, m_3, m_4, m_5$ such that the polynomial
\[
p(x) = (x-m_1)(x-m_2)(x-m_3)(x-m_4)(x-m_5)
\]
has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_1, m_2, m_3, m_4, m_5$ for which this minimum $k$ is achieved. | fun i : Fin 5 ↦ ↑i - (2 : ℤ) | Show that the minimum $k = 3$ is obtained for $\{m_1, m_2, m_3, m_4, m_5\} = \{-2, -1, 0, 1, 2\}$. | open Set Filter Topology Real Polynomial Function | [] | @Eq (Fin (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) → Int) answer
fun (i : Fin (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5)))) =>
@HSub.hSub Int Int Int (@instHSub Int Int.instSub)
(@Nat.cast Int instNatCastInt (@Fin.val (@OfNat.ofNat Nat (nat_lit 5) (instOfNatNat (nat_lit 5))) i))
(@OfNat.ofNat Int (nat_lit 2) (@instOfNat (nat_lit 2))) | Fin 5 → ℤ | [
{
"t": "(Fin 5 → ℤ) → (Polynomial ℝ)",
"v": null,
"name": "p"
},
{
"t": "p = fun m ↦ ∏ i : Fin 5, ((X : Polynomial ℝ) - m i)",
"v": null,
"name": "hp"
},
{
"t": "Polynomial ℝ → ℕ",
"v": null,
"name": "numnzcoeff"
},
{
"t": "numnzcoeff = fun p ↦ {j ∈ Finset.range (... | {
"problem_name": "putnam_1985_b1",
"tags": [
"algebra"
]
} |
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. | 99 if $n = 101$, otherwise 0 | Show that $f_{100}(1) = 101^{99}$. | open Set Filter Topology Real Polynomial Function | [] | @Eq (Nat → Nat) answer fun (n : Nat) =>
@ite Nat (@Eq Nat n (@OfNat.ofNat Nat (nat_lit 101) (instOfNatNat (nat_lit 101))))
(instDecidableEqNat n (@OfNat.ofNat Nat (nat_lit 101) (instOfNatNat (nat_lit 101))))
(@OfNat.ofNat Nat (nat_lit 99) (instOfNatNat (nat_lit 99)))
(@OfNat.ofNat Nat (nat_lit 0) (instOfNatNat (nat_lit 0))) | ℕ → ℕ | [
{
"t": "ℕ -> Polynomial ℕ",
"v": null,
"name": "f"
},
{
"t": "f 0 = 1",
"v": null,
"name": "hf0x"
},
{
"t": "∀ n ≥ 1, (f n).eval 0 = 0",
"v": null,
"name": "hfn0"
},
{
"t": "∀ n : ℕ, derivative (f (n + 1)) = (n + 1) * (Polynomial.comp (f n) (X + 1))",
"v": nul... | {
"problem_name": "putnam_1985_b2",
"tags": [
"algebra"
]
} |
Evaluate $\int_0^\infty t^{-1/2}e^{-1985(t+t^{-1})}\,dt$. You may assume that $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$. | $\sqrt{\pi / 1985} \cdot e^{-3970}$ | Show that the integral evaluates to $\sqrt{\frac{\pi}{1985}}e^{-3970}$. | open Set Filter Topology Real Polynomial Function | [] | @Eq Real answer
(@HMul.hMul Real Real Real (@instHMul Real Real.instMul)
(Real.sqrt
(@HDiv.hDiv Real Real Real (@instHDiv Real (@DivInvMonoid.toDiv Real Real.instDivInvMonoid)) Real.pi
(@OfNat.ofNat Real (nat_lit 1985)
(@instOfNatAtLeastTwo Real (nat_lit 1985) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 1983) (instOfNatNat (nat_lit 1983))))))))
(Real.exp
(@Neg.neg Real Real.instNeg
(@OfNat.ofNat Real (nat_lit 3970)
(@instOfNatAtLeastTwo Real (nat_lit 3970) Real.instNatCast
(@instNatAtLeastTwo (@OfNat.ofNat Nat (nat_lit 3968) (instOfNatNat (nat_lit 3968))))))))) | ℝ | [
{
"t": "∫ x in Set.univ, Real.exp (- x ^ 2) = Real.sqrt Real.pi",
"v": null,
"name": "fact"
},
{
"t": "∫ t in Set.Ioi 0, t ^ (- (1 : ℝ) / 2) * Real.exp (-1985 * (t + t ^ (-(1 : ℝ)))) = answer",
"v": null,
"name": "h_answer"
}
] | {
"problem_name": "putnam_1985_b5",
"tags": [
"analysis"
]
} |
End of preview. Expand
in Data Studio
Dataset Card for PutnamBench-Solving
This benchmark is part of the official implementation of Beyond Theorem Proving: Formulation, Framework and Benchmark for Formal Problem-Solving.
Our research focuses on:
- What is problem-solving?
- Beyond proving known targets, how can process-verified problem-solving be conducted inside existing formal theorem proving (FTP) environments?
Contribution
- A principled formulation of problem-solving as a deterministic Markov decision process;
- FPS (Formal Problem-Solving), utilizing FTP (formal theorem proving) environments to perform process-verified problem-solving;
- D-FPS (Deductive FPS), decoupling solving and answer verification for better human-alignment;
- RPE (Restricted Propositional Equivalence), a symbolic approach to determine the correctness of answers by formal verification;
- Three benchmarks on problem-solving: FormalMath500, MiniF2F-Solving and PutnamBench-Solving.
Benchmark Details
PutnamBench-Solving is a refactored subset of PutnamBench[8], containing 324 data points with:
- 9 about
Abstract Algebra - 138 about
Algebra - 122 about
Analysis - 14 about
Combinatorics - 28 about
Geometry - 25 about
Linear Algebra - 49 about
Number Theory - 8 about
Probability - 4 about
Set Theory
Direct Use
Formal Problem-Solving (FPS): Given a formal problem, generate a formal solution. The formal solution should solve all goals and provide a direct answer.
Deductive Formal Problem-Solving (D-FPS): Given a formal problem, generate a forward solution and, optionally, a backward proof. The forward solution should use deductive reasoning to derive a direct answer and prove its completeness. The backward proof should prove the answer's soundness.
Formal Theorem Proving (FTP): Given a formal problem and its ground-truth answer, generate a formal proof to prove the ground-truth's correctness.
Dataset Structure
Each problem contains the following fields:
informal_problem: The problem in natural language (including LaTeX).informal_answer: The ground-truth answer in natural language (including LaTeX).informal_solution: A step-by-step solution in natural language (including LaTeX).header: Code that should be executed before initializing the formal problem, e.g.,opens. Ifnull,open BigOperators Real Nat Topologyshould be used.intros: Independent variables $V$ and hypotheses $\Phi$. $V={v_i}{i=1}^n$ is the set of variables independent to the queriable $a$. $\Phi = {\phi_i}{i=1}^p$ is the set of propositions that depend on $V$ (whose all free variables are included in $V$), consisting of conditions that can be used to deduce the answer.outros: Conclusions $\Psi = {\psi_i}_{i=1}^q$ is the set of propositions which depend on $V \cup {a}$, consisting of conclusions that should be satisfied.formal_answer: The ground-truth answer in formal language (Lean 4).formal_answer_type: The type of the ground-truth answer in formal language (Lean 4).metainfo: Meta-information of the problem.
References
[1] Moura, Leonardo de, and Sebastian Ullrich. "The Lean 4 theorem prover and programming language." Automated Deduction–CADE 28: 28th International Conference on Automated Deduction, Virtual Event, July 12–15, 2021, Proceedings 28. Springer International Publishing, 2021.
[2] Community, Mathlib . "The Lean mathematical library.", 10.1145/3372885.3373824. 2019.
[3] Limperg, Jannis, and Asta Halkjær From. "Aesop: White-box best-first proof search for Lean." Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs. 2023.
[4] Aniva, Leni, et al. "Pantograph: A Machine-to-Machine Interaction Interface for Advanced Theorem Proving, High Level Reasoning, and Data Extraction in Lean 4." arXiv preprint arXiv:2410.16429 (2024).
[5] Lightman, Hunter, et al. "Let's verify step by step." The Twelfth International Conference on Learning Representations. 2023.
[6] Hendrycks, Dan, et al. "Measuring mathematical problem solving with the math dataset." arXiv preprint arXiv:2103.03874 (2021).
[7] Zheng, Kunhao, Jesse Michael Han, and Stanislas Polu. "Minif2f: a cross-system benchmark for formal olympiad-level mathematics." arXiv preprint arXiv:2109.00110 (2021).
[8] Tsoukalas, George, et al. "Putnambench: Evaluating neural theorem-provers on the putnam mathematical competition." arXiv preprint arXiv:2407.11214 (2024).
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Collection including purewhite42/putnambench_solving
Papers for purewhite42/putnambench_solving
Pantograph: A Machine-to-Machine Interaction Interface for Advanced Theorem Proving, High Level Reasoning, and Data Extraction in Lean 4
Paper
• 2410.16429 • Published
• 5
PutnamBench: Evaluating Neural Theorem-Provers on the Putnam Mathematical Competition
Paper
• 2407.11214 • Published
MiniF2F: a cross-system benchmark for formal Olympiad-level mathematics
Paper
• 2109.00110 • Published
Measuring Mathematical Problem Solving With the MATH Dataset
Paper
• 2103.03874 • Published
• 6