problem_name stringlengths 14 14 | formal_statement stringlengths 75 1.32k | informal_statement stringlengths 47 898 | informal_solution stringlengths 5 303 | tags listlengths 1 3 | split stringclasses 1
value |
|---|---|---|---|---|---|
putnam_1971_a1 | theorem putnam_1971_a1
(S : Set (ℤ × ℤ × ℤ))
(hS : S.ncard = 9)
(L : (ℤ × ℤ × ℤ) × (ℤ × ℤ × ℤ) → Set (ℝ × ℝ × ℝ))
(hL : L = fun ((a, b, c), (d, e, f)) =>
{(t*a + (1-t)*d, t*b + (1-t)*e, t*c + (1-t)*f) | t ∈ Ioo (0 : ℝ) 1})
: ∃ x y z : ℤ, ∃ P Q : ℤ × ℤ × ℤ, P ∈ S ∧ Q ∈ S ∧ P ≠ Q ∧ ((x : ℝ), (y : ℝ), (z : ℝ)) ∈ L (P, Q) ... | Let $S$ be a set of $9$ lattice points (points with integer coordinates) in $3$-dimensional Euclidean space. Prove that there exists a lattice point along the interior of some line segment that joins two distinct points in $S$. | None. | [
"geometry",
"combinatorics"
] | test |
putnam_1971_a2 | abbrev putnam_1971_a2_solution : Set (Polynomial ℝ) := sorry
theorem putnam_1971_a2
(P : Polynomial ℝ) :
(P.eval 0 = 0 ∧ (∀ x : ℝ, P.eval (x^2 + 1) = (P.eval x)^2 + 1)) ↔ P ∈ putnam_1971_a2_solution :=
sorry | Determine all polynomials $P(x)$ such that $P(x^2 + 1) = (P(x))^2 + 1$ and $P(0) = 0$. | Show that the only such polynomial is the identity function. | [
"algebra"
] | test |
putnam_1971_a3 | theorem putnam_1971_a3
(a b c : ℝ × ℝ)
(R : ℝ)
(habclattice : a.1 = round a.1 ∧ a.2 = round a.2 ∧ b.1 = round b.1 ∧ b.2 = round b.2 ∧ c.1 = round c.1 ∧ c.2 = round c.2)
(habcneq : a ≠ b ∧ a ≠ c ∧ b ≠ c)
(hR : R > 0)
(hcircle : ∃ C : ℝ × ℝ,
√((a.1 - C.1)^2 + (a.2 - C.2)^2) = R ∧
√((b.... | The three vertices of a triangle of sides $a,b,c$ are lattice points and lie on a circle of radius $R$. Show that $abc \geq 2R$. | None. | [
"geometry"
] | test |
putnam_1971_a4 | theorem putnam_1971_a4
(ε : ℝ)
(hε : 0 < ε ∧ ε < 1)
(P : ℕ → ℝ → MvPolynomial (Fin 2) ℝ)
(hP : P = fun (n : ℕ) (δ : ℝ) => (MvPolynomial.X 0 + MvPolynomial.X 1)^n * ((MvPolynomial.X 0)^2 - (MvPolynomial.C (2 - δ))*(MvPolynomial.X 0)*(MvPolynomial.X 1) + (MvPolynomial.X 1)^2))
: ∃ N : ℕ, ∀ n ≥ N, ∀ i : Fin 2 →₀ ℕ, MvPoly... | Show that for $\epsilon \in (0,1)$, the expression $(x + y)^n (x^2 - 2-\epsilon)xy + y^2)$ is a polynomial with positive coefficients for $n$ sufficiently large, where $n$ is an integer. | None. | [
"analysis"
] | test |
putnam_1971_a5 | abbrev putnam_1971_a5_solution : ℤ × ℤ := sorry
theorem putnam_1971_a5
(a b : ℤ)
(hab : a > 0 ∧ b > 0 ∧ a > b)
(pab : ℤ → ℤ → Prop)
(hpab : ∀ x y, pab x y ↔
{s : ℕ | ¬∃ m n : ℕ, m*x + n*y = s}.ncard = 35 ∧
¬∃ m n : ℕ, m*x + n*y = 58) :
pab a b ↔ a = putnam_1971_a5_solution.1 ∧ b = putnam... | 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 $b$. | Show that the solution is $a = 11$ and $b = 8$. | [
"number_theory"
] | test |
putnam_1971_a6 | theorem putnam_1971_a6
(c : ℝ)
(hc : ∀ n : ℤ, n > 0 → ∃ m : ℤ, (n : ℝ)^c = m)
: ∃ m : ℤ, m ≥ 0 ∧ c = m :=
sorry | Let $c$ be a real number such that $n^c$ is an integer for every positive integer $n$. Show that $c$ is a non-negative integer. | None. | [
"number_theory"
] | test |
putnam_1971_b1 | theorem putnam_1971_b1
(S : Type*) [Mul S]
(hself : ∀ x : S, x * x = x)
(h2 : ∀ x y z : S, (x * y) * z = (y * z) * x)
: (∀ x y z : S, (x * y) * z = x * (y * z)) ∧ (∀ x y : S, x * y = y * x) :=
sorry | Let $S$ be a set and let $\cdot$ be a binary operation on $S$ satisfying the two following laws: (1) for all $x$ in $S$, $x = x \cdot x$, (2) for all $x,y,z$ in $S$, $(x \cdot y) \cdot z) = (y \cdot z) \cdot x$. Show that $\cdot$ is associative and commutative. | None. | [
"abstract_algebra"
] | test |
putnam_1971_b2 | abbrev putnam_1971_b2_solution : Set (ℝ → ℝ) := sorry
theorem putnam_1971_b2
(S : Set ℝ)
(hS : S = univ \ {0, 1})
(P : (ℝ → ℝ) → Prop)
(hP : P = fun (F : ℝ → ℝ) => ∀ x ∈ S, F x + F ((x - 1)/x) = 1 + x)
: (∀ F ∈ putnam_1971_b2_solution, P F) ∧ ∀ f : ℝ → ℝ, P f → ∃ F ∈ putnam_1971_b2_solution, (∀ x ∈ S, f x = F x) :=
sor... | 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 $x \in \mathbb{R} \setminus \{0, 1\}$. | The only such function is $F(x) = \frac{x^3 - x^2 - 1}{2x(x - 1)}$. | [
"algebra"
] | test |
putnam_1971_b3 | theorem putnam_1971_b3
(T : ℝ)
(hT : T > 0)
: MeasureTheory.volume {t : ℝ | t ≥ T ∧ Nat.floor t = 2 * (Nat.floor (t - T))} = 1 :=
sorry | If two cars travel around a track at constant speeds of one lap per hour, starting from the same point but at different times, prove that the total amount of time for which the first car has completed exactly twice as many laps as the second is exactly $1$ hour. | None. | [
"algebra"
] | test |
putnam_1971_b6 | theorem putnam_1971_b6
(δ : ℤ → ℤ)
(hδ : δ = fun n => sSup {t | Odd t ∧ t ∣ n})
: ∀ x : ℤ, x > 0 → |∑ i ∈ Finset.Icc 1 x, (δ i)/(i : ℚ) - 2*x/3| < 1 :=
sorry | Let $\delta(x) be the greatest odd divisor of the positive integer $x$. Show that $|\sum_{n = 1}^x \delta(n)/n - 2x/3| < 1$ for all positive integers $x$. | None. | [
"number_theory"
] | test |
putnam_1972_a1 | theorem putnam_1972_a1
(n : ℕ) (hn : n > 0)
(fourAP : ℤ → ℤ → ℤ → ℤ → Prop)
(hfourAP : ∀ n1 n2 n3 n4, fourAP n1 n2 n3 n4 ↔ n4-n3 = n3-n2 ∧ n3-n2 = n2-n1) :
¬ ∃ r : ℕ, r > 0 ∧ r + 3 ≤ n ∧ fourAP (n.choose r) (n.choose (r+1)) (n.choose (r+2)) (n.choose (r+3)) :=
sorry | Show that there are no four consecutive binomial coefficients ${n \choose r}, {n \choose (r+1)}, {n \choose (r+2)}, {n \choose (r+3)}$ where $n,r$ are positive integers and $r+3 \leq n$, which are in arithmetic progression. | None. | [
"algebra"
] | test |
putnam_1972_a2 | theorem putnam_1972_a2
: (∀ (S : Type*) (_ : Mul S), (∀ x y : S, x * (x * y) = y ∧ ((y * x) * x) = y) → (∀ x y : S, x * y = y * x)) ∧ ∃ (S : Type*) (_ : Mul S), (∀ x y : S, x * (x * y) = y ∧ ((y * x) * x) = y) ∧ ¬(∀ x y z : S, x * (y * z) = (x * y) * z) :=
sorry | Let $S$ be a set and $\cdot$ be a binary operation on $S$ satisfying: (1) for all $x,y$ in $S$, $x \cdot (x \cdot y) = y$ (2) for all $x,y$ in $S$, $(y \cdot x) \cdot x = y$. Show that $\cdot$ is commutative but not necessarily associative. | None. | [
"abstract_algebra"
] | test |
putnam_1972_a3 | abbrev putnam_1972_a3_solution : Set (ℝ → ℝ) := sorry
theorem putnam_1972_a3
(climit_exists : (ℕ → ℝ) → Prop)
(supercontinuous : (ℝ → ℝ) → Prop)
(hclimit_exists : ∀ x, climit_exists x ↔ ∃ C : ℝ, Tendsto (fun n => (∑ i ∈ Finset.range n, (x i))/(n : ℝ)) atTop (𝓝 C))
(hsupercontinuous : ∀ f, supercontinuo... | 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 $[0,1]$. | Show that the solution is the set of affine functions. | [
"analysis"
] | test |
putnam_1972_a5 | theorem putnam_1972_a5
(n : ℕ)
(hn : n > 1)
: ¬((n : ℤ) ∣ 2^n - 1) :=
sorry | Show that if $n$ is an integer greater than $1$, then $n$ does not divide $2^n - 1$. | None. | [
"number_theory"
] | test |
putnam_1972_a6 | theorem putnam_1972_a6
(f : ℝ → ℝ)
(n : ℤ)
(hn : n ≥ 0)
(hfintegrable: IntegrableOn f (Icc 0 1))
(hfint : ∀ i ∈ Icc 0 (n-1), ∫ x in Icc 0 1, x^i*(f x) = 0)
(hfintlast : ∫ x in Icc 0 1, x^n*(f x) = 1)
: ∃ S, S ⊆ Icc (0 : ℝ) 1 ∧ MeasurableSet S ∧ volume S > 0 ∧ ∀ x ∈ S, |f x| ≥ 2^n * (n + 1) :=
sorry | Let $f$ be an integrable function in $0 \leq x \leq 1$ and suppose for all $0 \leq i \leq n-1, \int_0^1 x^i f(x) dx = 0$. Further suppose that $\int_0^1 x^n f(x) dx = 1$. Show that $|f(x)| \geq 2^n(n+1)$ on a set of positive measure. | None. | [
"analysis"
] | test |
putnam_1972_b1 | theorem putnam_1972_b1
(S : ℝ → ℝ)
(hS : S = fun (x : ℝ) => ∑' n : ℕ, x^n * (x - 1)^(2*n) / (Nat.factorial n))
(p : ℕ → ℝ)
(hp : ∃ a : ℝ, a > 0 ∧ ∀ x ∈ ball 0 a, ∑' n : ℕ, (p n)*x^n = S x)
: ¬∃ n : ℕ, p n = 0 ∧ p (n + 1) = 0 ∧ p (n + 2) = 0 :=
sorry | Prove that no three consecutive coefficients of the power series of $$\sum_{n = 0}^{\infty} \frac{x^n(x - 1)^{2n}}{n!}$$ all equal $0$. | None. | [
"analysis"
] | test |
putnam_1972_b2 | noncomputable abbrev putnam_1972_b2_solution : ℝ → ℝ → ℝ := sorry
theorem putnam_1972_b2
(s v : ℝ)
(hs : s > 0)
(hv : v > 0)
(valid : ℝ → (ℝ → ℝ) → Prop)
(hvalid : ∀ t x, valid t x ↔
DifferentiableOn ℝ x (Set.Icc 0 t) ∧ DifferentiableOn ℝ (deriv x) (Set.Icc 0 t) ∧
AntitoneOn (deriv (deri... | 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 $v$. | Show that the maximum possible time is $t = \frac{2s}{v}$. | [
"analysis"
] | test |
putnam_1972_b3 | theorem putnam_1972_b3
(G : Type*) [Group G]
(A B : G)
(hab : A * B * A = B * A^2 * B ∧ A^3 = 1 ∧ (∃ n : ℤ, n > 0 ∧ B^(2*n - 1) = 1))
: B = 1 :=
sorry | Let $A$ and $B$ be two elements in a group such that $ABA = BA^2B$, $A^3 = 1$, and $B^{2n-1} = 1$ for some positive integer $n$. Prove that $B = 1$. | None. | [
"abstract_algebra"
] | test |
putnam_1972_b4 | theorem putnam_1972_b4
(n : ℕ)
(hn : n > 1)
(vars : ℤ → ℤ → ℤ → (Fin 3 → ℤ))
(hvars : vars = fun a b c ↦ fun i ↦ ite (i = 0) a (ite (i = 1) b c))
: ∃ P : MvPolynomial (Fin 3) ℤ, ∀ x : ℤ, x = MvPolynomial.eval (vars (x^n) (x^(n+1)) (x + x^(n+2))) P :=
sorry | Let $n \geq 2$ be an integer. Show that there exists a polynomial $P(x,y,z)$ with integral coefficients such that $x \equiv P(x^n, x^{n+1}, x + x^{n+2})$. | None. | [
"algebra"
] | test |
putnam_1972_b5 | theorem putnam_1972_b5
(A B C D : EuclideanSpace ℝ (Fin 3))
(hnonplanar : ¬Coplanar ℝ {A, B, C, D})
(hangles : ∠ A B C = ∠ C D A ∧ ∠ B C D = ∠ D A B)
: dist A B = dist C D ∧ dist B C = dist D A :=
sorry | Let $ABCD$ be a skew (non-planar) quadrilateral. Prove that if $\angle ABC = \angle CDA$ and $\angle BCD = \angle DAB$, then $AB = CD$ and $AD = BC$. | None. | [
"geometry"
] | test |
putnam_1972_b6 | theorem putnam_1972_b6
(k : ℕ)
(hk : k ≥ 1)
(n : Fin k → ℤ)
(hn : ∀ i : Fin k, n i > 0)
(hn' : ∀ i j : Fin k, i < j → n i < n j)
(zpoly : ℂ → ℂ)
(hzpoly : zpoly = fun z => 1 + ∑ i : Fin k, z^(n i))
: ∀ z : ℂ, z ∈ ball 0 ((Real.sqrt 5 - 1)/2) → zpoly z ≠ 0 :=
sorry | Let $n_1 < n_2 < \dots < n_k$ be a set of positive integers. Prove that the polynomail $1 + z^{n_1} + z^{n_2} + \dots + z^{n_k}$ has not roots inside the circle $|z| < (\frac{\sqrt{5}-1}{2}$. | None. | [
"analysis"
] | test |
putnam_1973_a1 | theorem putnam_1973_a1
(A B C X Y Z : EuclideanSpace ℝ (Fin 2))
(hnoncol : ¬Collinear ℝ {A, B, C})
(hX : X ∈ segment ℝ B C)
(hY : Y ∈ segment ℝ C A)
(hZ : Z ∈ segment ℝ A B)
: ((dist B X ≤ dist X C ∧ dist C Y ≤ dist Y A ∧ dist A Z ≤ dist Z B) →
volume (convexHull ℝ {X, Y, Z}) ≥ (1/4) * volume (convexHull ℝ {A, B, C})) ... | Let $\triangle ABC$ be any triangle in the Euclidean plane, and let points $X$, $Y$, and $Z$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively. If $BX \le XC$, $CY \le YA$, and $AZ \le ZB$, prove that $[\triangle XYZ] \ge \frac{1}{4} [\triangle ABC]$. Regardless of this constraint on $X$,... | None. | [
"geometry"
] | test |
putnam_1973_a2 | abbrev putnam_1973_a2_solution : Prop := sorry
theorem putnam_1973_a2
(L : List ℝ)
(hL : L.length = 8 ∧ ∀ i : Fin L.length, L[i] = 1 ∨ L[i] = -1)
(pluses : ℕ)
(hpluses : pluses = {i : Fin L.length | L[i] = 1}.ncard)
(S : ℕ → ℝ)
(hS : S = fun n : ℕ ↦ ∑ i ∈ Finset.Icc 1 n, L[i % 8]/i)
: (pluses = 4 → ∃ l : ℝ, Tendsto S a... | 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 ... | Show that the condition is necessary. | [
"analysis"
] | test |
putnam_1973_a3 | theorem putnam_1973_a3
(b : ℤ → ℝ)
(hb : b = fun (n : ℤ) => sInf {k + (n : ℝ)/(k : ℝ) | k > (0 : ℤ)})
: ∀ n : ℤ, n > 0 → floor (b n) = floor (Real.sqrt (4 * n + 1)) :=
sorry | Let $n$ be a fixed positive integer and let $b(n)$ be the minimum value of $k + \frac{n}{k}$ as $k$ is allowed to range through all positive integers. Prove that $b(n)$ and $\sqrt{4n + 1}$ have the same integer part. | None. | [
"number_theory"
] | test |
putnam_1973_a4 | abbrev putnam_1973_a4_solution : ℕ := sorry
theorem putnam_1973_a4
(f : ℝ → ℝ)
(hf : f = fun x => 2^x - 1 - x^2)
: putnam_1973_a4_solution = {x : ℝ | f x = 0}.ncard :=
sorry | How many zeros does the function $f(x) = 2^x - 1 - x^2$ have on the real line? | Show that the solution is 3. | [
"analysis"
] | test |
putnam_1973_a6 | theorem putnam_1973_a6
(nint : ℕ → (Fin 7 → (ℝ × ℝ)) → ℕ)
(h_nint : nint = fun n lines => {p : ℝ × ℝ | ∃! S : Set (Fin 7), S.ncard = n ∧ p ∈ ⋂ i ∈ S, {pts | pts.2 = (lines i).1 * pts.1 + (lines i).2}}.ncard)
: ¬ ∃ lines : Fin 7 → (ℝ × ℝ), (∀ i j : Fin 7, i ≠ j → lines i ≠ lines j) ∧ nint 3 lines ≥ 6 ∧ nint 2 lines ≥ 4 ... | Prove that it is impossible for seven distinct straight lines to be situated in the Euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines interest. | None. | [
"geometry"
] | test |
putnam_1973_b1 | theorem putnam_1973_b1
(n : ℕ)
(a : Finset.Icc 1 (2 * n + 1) → ℤ)
(h_remove : ∀ S : Finset (Finset.Icc 1 (2 * n + 1)), S.card = 2*n → ∃ T, T ⊆ S ∧ T.card = n ∧ ∑ i ∈ T, a i = ∑ i ∈ (S \ T), a i)
: ∀ i j : Finset.Icc 1 (2 * n + 1), a i = a j :=
sorry | Let $a_1, \dots, a_{2n + 1}$ be a set of integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sums. Prove $a_1 = a_2 = \dots = a_{2n+1}$. | None. | [
"algebra"
] | test |
putnam_1973_b2 | theorem putnam_1973_b2
(z : ℂ)
(hzrat : ∃ q1 q2 : ℚ, z.re = q1 ∧ z.im = q2)
(hznorm : ‖z‖ = 1)
: ∀ n : ℤ, ∃ q : ℚ, ‖z^(2*n) - 1‖ = q :=
sorry | Let $z = x+iy$ be a complex number with $x$ and $y$ rational and with $\| z \| = 1$. Show thaat the number $\| z^{2n} - 1 \|$ is rational for every integer $n$. | None. | [
"number_theory"
] | test |
putnam_1973_b3 | theorem putnam_1973_b3
(p : ℕ)
(pgt1 : p > 1)
(hprime : ∀ x ∈ Set.Ico 0 p, Nat.Prime (x^2 - x + p))
: ∃! triple : ℤ × ℤ × ℤ, let (a,b,c) := triple; b^2 - 4*a*c = 1 - 4*p ∧ 0 < a ∧ a ≤ c ∧ -a ≤ b ∧ b < a :=
sorry | Let $p > 1$ be an integer with the property that $x^2 - x + p$ is prime for all $x$ in the range $0 < x < p$. Show there exists exactly one triple of integers $a,b,c$ satisfying $b^2 - 4ac = 1 - 4p$, $0 < a \leq c$, and $-a \leq b < a$. | None. | [
"number_theory",
"algebra"
] | test |
putnam_1973_b4 | abbrev putnam_1973_b4_solution : ℝ → ℝ := sorry
theorem putnam_1973_b4
(f : ℝ → ℝ)
(hprop : (ℝ → ℝ) → Prop)
(hprop_def : hprop = fun g => ContDiff ℝ 1 g ∧ (∀ x : ℝ, 0 < deriv g x ∧ deriv g x ≤ 1) ∧ g 0 = 0)
(hf : hprop f)
: (∫ x in Icc 0 1, f x)^2 ≥ ∫ x in Icc 0 1, (f x)^3 ∧ (hprop putnam_1973_b4_solution ∧ (∫ x in Icc... | 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. | Show that one such example where equality holds is the identity function. | [
"analysis"
] | test |
putnam_1974_a1 | abbrev putnam_1974_a1_solution : ℕ := sorry
theorem putnam_1974_a1
(conspiratorial : Set ℤ → Prop)
(hconspiratorial : ∀ 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))) :
IsGreatest {k | ∃ S, S ⊆ Icc... | 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? | Show that the answer is 11. | [
"number_theory"
] | test |
putnam_1974_a3 | abbrev putnam_1974_a3_solution : (Set ℕ) × (Set ℕ) := sorry
theorem putnam_1974_a3
(assmption : ∀ p : ℕ, p.Prime ∧ p > 2 → ((∃ m n : ℤ, p = m^2 + n^2) ↔ p ≡ 1 [MOD 4]))
: ∀ p : ℕ, ((p.Prime ∧ p > 2 ∧ (∃ x y : ℤ, p = x^2 + 16*y^2)) ↔ p ∈ putnam_1974_a3_solution.1) ∧ ((p.Prime ∧ p > 2 ∧ (∃ x y : ℤ, p = 4*x^2 + 4*x*y + 5*... | 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) $4x^2 + 4xy + 5y^2$. | 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 $5 \bmod 8$. | [
"number_theory"
] | test |
putnam_1974_a4 | noncomputable abbrev putnam_1974_a4_solution : ℕ → ℚ := sorry
theorem putnam_1974_a4
(n : ℕ)
(hn : 0 < n) :
(1 : ℚ) / (2 ^ (n - 1)) * ∑ k ∈ Finset.Icc 0 ⌊n / 2⌋₊, (n - 2 * k) * (n.choose k) =
putnam_1974_a4_solution n :=
sorry | Evaluate in closed form: $\frac{1}{2^{n-1}} \sum_{k < n/2} (n-2k)*{n \choose k}$. | Show that the solution is $\frac{n}{2^{n-1}} * {(n-1) \choose \left[ (n-1)/2 \right]}$. | [
"algebra"
] | test |
putnam_1974_a6 | abbrev putnam_1974_a6_solution : ℕ := sorry
theorem putnam_1974_a6
(hdivnallx : Polynomial ℤ → Prop)
(hdivnallx_def : hdivnallx = fun f => Monic f ∧ (∀ x : ℤ, (10^6 : ℤ) ∣ f.eval x))
: sInf {d : ℕ | ∃ f : Polynomial ℤ, hdivnallx f ∧ d = f.natDegree} = putnam_1974_a6_solution :=
sorry | 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 $k(1000000)$. | Show that the answer is 25. | [
"algebra"
] | test |
putnam_1974_b1 | abbrev putnam_1974_b1_solution : (Fin 5 → EuclideanSpace ℝ (Fin 2)) → Prop := sorry
theorem putnam_1974_b1
(d : (Fin 5 → EuclideanSpace ℝ (Fin 2)) → ℝ)
(d_def : ∀ p, d p = ∑ ⟨i, j⟩ : Fin 5 × Fin 5, if i < j then dist (p i) (p j) else 0)
(p : Fin 5 → EuclideanSpace ℝ (Fin 2))
(hp : ∀ i, ‖p i‖ = 1) :
... | Prove that the optimal configuration of 5 (not necessarily distinct) points $p_1, \dots, p_5$ on the unit circle which maximizes the sum of the ten distances \[\Sigma_{i < j}, d(p_i, p_j) \] is the one which evenly spaces the points like a regular pentagon. | None. | [
"algebra",
"geometry"
] | test |
putnam_1974_b2 | theorem putnam_1974_b2
(y : ℝ → ℝ)
(ycontdiff : ContDiff ℝ 1 y)
(limy : Tendsto (fun x : ℝ => (deriv y x)^2 + (y x)^3) atTop (𝓝 0))
: Tendsto y atTop (𝓝 0) ∧ Tendsto (deriv y) atTop (𝓝 0) :=
sorry | Let $y(x)$ be a continuously differentiable real-valued function of a real vairable $x$. Show that if $(y')^2 + y^3 \to 0$ as $x \to +\infty$, then $y(x)$ and $y'(x) \to 0$ as $x \to +\infty$. | None. | [
"analysis"
] | test |
putnam_1974_b3 | theorem putnam_1974_b3
(α : ℝ)
(ha : Real.cos (Real.pi * α) = (1 : ℝ)/3)
: Irrational α :=
sorry | Prove that if $\alpha$ is a real number such that $\cos (\pi \alpha) = 1/3$, the $\alpha$ is irrational. | None. | [
"number_theory"
] | test |
putnam_1974_b4 | theorem putnam_1974_b4
(f : ℝ × ℝ → ℝ)
(hfcontx : ∀ y : ℝ, Continuous (fun x => f ⟨x, y⟩))
(hfconty : ∀ x : ℝ, Continuous (fun y => f ⟨x, y⟩))
: ∃ g : ℕ → (ℝ × ℝ → ℝ), (∀ n : ℕ, Continuous (g n)) ∧ (∀ x y : ℝ, Tendsto (fun n => g n ⟨x, y⟩) atTop (𝓝 (f ⟨x, y⟩))) :=
sorry | Let $f : \mathbb{R} \to \mathbb{R}$ be continuous in each variable seperately. Show that there exists a sequence of continuous functions $g_n : \mathbb{R}^2 \to \mathbb{R}$ such that $f(x,y) = \lim_{n \to \infty} g_n(x,y)$ for all $(x,y) \in \mathbb{R}^2$. | None. | [
"analysis"
] | test |
putnam_1974_b5 | theorem putnam_1974_b5
: ∀ n ≥ 0, ∑ i ∈ Finset.Icc (0 : ℕ) n, (n^i : ℝ)/(Nat.factorial i) > (Real.exp n)/2 :=
sorry | Show that $1 + (n/1!) + (n^2/2!) + \dots + (n^n/n!) > e^n/2$ for every integer $n \geq 0$. | None. | [
"analysis"
] | test |
putnam_1974_b6 | abbrev putnam_1974_b6_solution : (ℕ × ℕ × ℕ) := sorry
theorem putnam_1974_b6
(n : ℤ)
(hn : n = 1000)
(count0 count1 count2 : ℕ)
(hcount0 : count0 = {S | S ⊆ Finset.Icc 1 n ∧ S.card ≡ 0 [MOD 3]}.ncard)
(hcount1 : count1 = {S | S ⊆ Finset.Icc 1 n ∧ S.card ≡ 1 [MOD 3]}.ncard)
(hcount2 : count2 = {S | S ⊆ Finset.Icc 1 n ∧ ... | For a set with $1000$ elements, how many subsets are there whose candinality is respectively $\equiv 0 \bmod 3, \equiv 1 \bmod 3, \equiv 2 \bmod 3$? | 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 $\equiv 2 \bmod 3$. | [
"set_theory"
] | test |
putnam_1975_a1 | abbrev putnam_1975_a1_solution : ((ℤ × ℤ) → ℤ) × ((ℤ × ℤ) → ℤ) := sorry
theorem putnam_1975_a1
(nab nxy : (ℤ × ℤ × ℤ) → Prop)
(hnab : nab = fun (n, a, b) => n = (a^2 + (a : ℚ))/2 + (b^2 + (b : ℚ))/2)
(hnxy : nxy = fun (n, x, y) => 4*n + 1 = x^2 + y^2)
: (∀ n a b : ℤ, nab (n, a, b) → nxy (n, putnam_1975_a1_solution.1 (a... | 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... | $x = a + b + 1$ and $y = a - b$ (or vice versa). | [
"algebra",
"number_theory"
] | test |
putnam_1975_a2 | abbrev putnam_1975_a2_solution : (ℝ × ℝ) → Prop := sorry
theorem putnam_1975_a2
: ∀ b c : ℝ, (∀ z : ℂ, (X^2 + (C (b : ℂ))*X + (C (c : ℂ)) : Polynomial ℂ).eval z = 0 → ‖z‖ < 1) ↔ putnam_1975_a2_solution (b, c) :=
sorry | 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? | The desired region is the strict interior of the triangle with vertices $(0, -1)$, $(2, 1)$, and $(-2, 1)$. | [
"algebra"
] | test |
putnam_1975_a3 | noncomputable abbrev putnam_1975_a3_solution : ((ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ)) × ((ℝ × ℝ × ℝ) → (ℝ × ℝ × ℝ)) := sorry
theorem putnam_1975_a3
(a b c : ℝ)
(hi : 0 < a ∧ a < b ∧ b < c)
(P : (ℝ × ℝ × ℝ) → Prop)
(f : (ℝ × ℝ × ℝ) → ℝ)
(hP : P = fun (x, y, z) => x ≥ 0 ∧ y ≥ 0 ∧ z ≥ 0 ∧ x^b + y^b + z^b = 1)
(hf : f = fun (x, y, z)... | 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? | $f$ 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 $z_0 = \left(\frac{b}{c}\right)^{\frac{1}{c-b}}$. | [
"algebra"
] | test |
putnam_1975_a4 | noncomputable abbrev putnam_1975_a4_solution : ℕ → Polynomial ℤ := sorry
theorem putnam_1975_a4
(m : ℕ)
(hm : Odd m ∧ m > 1)
(θ : ℂ)
(hθ : θ = cexp (2 * Real.pi * I / (2 * m)))
: 1/(1 - θ) = Polynomial.aeval θ (putnam_1975_a4_solution m) :=
sorry | 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 $a_i$. | Show that the solution is the polynomial $0 + \theta + \theta^3 + \dots + \theta^{m-2}$, alternating consecutive coefficients between 0 and 1. | [
"algebra"
] | test |
putnam_1975_a5 | theorem putnam_1975_a5
(I : Interval ℝ)
(f y1 y2 : ℝ → ℝ)
(hfcont : Continuous f)
(hycdiff : ContDiff ℝ 2 y1 ∧ ContDiff ℝ 2 y2)
(hy1sol : ∀ x : ℝ, iteratedDeriv 2 y1 x = (f x) * (y1 x))
(hy2sol : ∀ x : ℝ, iteratedDeriv 2 y2 x = (f x) * (y2 x))
(hylinindep : ∀ c1 c2 : ℝ, (∀ x : ℝ, c1 * y1 x + c2 * y2 x = 0) → (c1 = 0 ∧ ... | On some interval $I$ of the real line, let $y_1(x), y_2(x)$ be linearly independent solutions of the differential equation \[y'' = f(x)y\], where $f(x)$ is a continuous real-valued function. Suppose that $y_1(x) > 0$ and $y_2(x) > 0$ on $I$. Show that there exists a positive constant $c$ such that, on $I$, the function... | None. | [
"analysis"
] | test |
putnam_1975_b1 | abbrev putnam_1975_b1_solution : ℤ := sorry
theorem putnam_1975_b1
(H : Set (ℤ × ℤ))
(hH : H = {(x, y) : (ℤ × ℤ) | ∃ u v w : ℤ, (x, y) = (u*3 + v*4 + w*5, u*8 + v*(-1) + w*4)})
: (∃ b : ℤ, H = {(x, y) : (ℤ × ℤ) | ∃ u v : ℤ, (x, y) = (u, u*b + v*putnam_1975_b1_solution)}) ∧ putnam_1975_b1_solution > 0 :=
sorry | 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 $a$. | $a$ must equal $7$. | [
"abstract_algebra",
"number_theory"
] | test |
putnam_1975_b2 | theorem putnam_1975_b2
(slab : (Fin 3 → ℝ) → ℝ → ℝ → Set (Fin 3 → ℝ))
(hslab : slab = fun normal offset thickness => {x : Fin 3 → ℝ | offset < normal ⬝ᵥ x ∧ normal ⬝ᵥ x < offset + thickness})
(normals : ℕ → (Fin 3 → ℝ))
(offsets : ℕ → ℝ)
(thicknesses : ℕ → ℝ)
(hnormalsunit : ∀ i : ℕ, ‖normals i‖ = 1)
(hthicknessespos :... | In three-dimensional Euclidean space, define a \emph{slab} to be the open set of points lying between two parallel planes. The distance between the planes is called the \emph{thickness} of the slab. Given an infinite sequence $S_1, S_2, \dots$ of slabs of thicknesses $d_1, d_2, \dots,$ respectively, such that $\Sigma_{... | None. | [
"analysis",
"geometry"
] | test |
putnam_1975_b3 | noncomputable abbrev putnam_1975_b3_solution : ℕ → ℝ := sorry
theorem putnam_1975_b3
: ∀ k : ℕ, k > 0 → (∀ a : Multiset ℝ, (∀ i ∈ a, i > 0) ∧ card a ≥ k →
(esymm a k)/(esymm a 1)^k ≤ putnam_1975_b3_solution k) ∧
∀ M : ℝ, M < putnam_1975_b3_solution k → (∃ a : Multiset ℝ, (∀ i ∈ a, i > 0) ∧ card a ≥ k ∧
(esymm a k)/(esy... | 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 $$\frac{s_k (a_1... | The supremum $M_k$ is $\frac{1}{k!}$. | [
"analysis",
"algebra"
] | test |
putnam_1975_b4 | abbrev putnam_1975_b4_solution : Prop := sorry
theorem putnam_1975_b4
(P : ℝ × ℝ → Prop)
(hP : P = fun (x, y) => x^2 + y^2 = 1)
: (∃ B ⊆ setOf P, IsClosed B ∧ ∀ x y : ℝ, P (x, y) → Xor' ((x, y) ∈ B) ((-x, -y) ∈ B)) ↔ putnam_1975_b4_solution :=
sorry | Let $C = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ denote the unit circle. Does there exist $B \subseteq C$ for which $B$ is topologically closed and contains exactly one point from each pair of diametrically opposite points in $C$? | Such $B$ does not exist. | [
"analysis"
] | test |
putnam_1975_b5 | theorem putnam_1975_b5
(e : ℝ)
(he : e = Real.exp 1)
(f : ℕ → ℝ → ℝ)
(h0 : ∀ x : ℝ, f 0 x = e^x)
(hf : ∀ n : ℕ, ∀ x : ℝ, f (n + 1) x = x * (deriv (f n) x))
: ∑' n : ℕ, (f n 1)/(Nat.factorial n) = e^e :=
sorry | Let $f_0(x) = e^x$ and $f_{n+1}(x) = xf'_n(x)$ for all $n \ge 0$. Prove that $$\sum_{n=0}^{\infty} \frac{f_n(1)}{n!} = e^e.$$ | None. | [
"analysis",
"algebra"
] | test |
putnam_1975_b6 | theorem putnam_1975_b6
(s : ℕ → ℝ)
(hs : s = fun (n : ℕ) => ∑ i ∈ Finset.Icc 1 n, 1/(i : ℝ))
: (∀ n : ℕ, n > 1 → n * (n+1 : ℝ)^(1/(n : ℝ)) < n + s n) ∧ (∀ n : ℕ, n > 2 → ((n : ℝ) - 1)*((n : ℝ)^(-1/(n-1 : ℝ))) < n - s n) :=
sorry | Show that if $s_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + 1/n, then $n(n+1)^{1/n} < n + s_n$ whenever $n > 1$ and $(n-1)n^{-1/(n-1)} < n - s_n$ whenever $n > 2$. | None. | [
"analysis"
] | test |
putnam_1976_a2 | theorem putnam_1976_a2
(P Q : MvPolynomial (Fin 2) ℤ)
(hP : P = X 0 ^ 2 * X 1 + X 0 * X 1 ^ 2)
(hQ : Q = X 0 ^ 2 + X 0 * X 1 + X 1 ^ 2)
(F G : ℕ → MvPolynomial (Fin 2) ℤ)
(hF : ∀ n, F n = (X 0 + X 1) ^ n - X 0 ^ n - X 1 ^ n)
(hG : ∀ n, G n = (X 0 + X 1) ^ n + X 0 ^ n + X 1 ^ n)
(n : ℕ)
(... | Let $P(x, y) = x^2y + xy^2$, $Q(x, y) = x^2 + xy + y^2$, $F_n(x, y) = (x + y)^n - x^n - y^n$, and $G_n(x, y) = (x + y)^n + x^n + y^n$. Prove that for all positive integers $n$, either $F_n$ or $G_n$ can be represented as a polynomial in $P$ and $Q$ with integer coefficients. | None. | [
"algebra"
] | test |
putnam_1976_a3 | abbrev putnam_1976_a3_solution : Set (ℕ × ℕ × ℕ × ℕ) := sorry
theorem putnam_1976_a3
: {(p, r, q, s) : ℕ × ℕ × ℕ × ℕ | Nat.Prime p ∧ Nat.Prime q ∧ r > 1 ∧ s > 1 ∧ |(p^r : ℤ) - q^s| = 1} = putnam_1976_a3_solution :=
sorry | 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$. | The only solutions are $(p, r, q, s) = (3, 2, 2, 3)$ and $(p, r, q, s) = (2, 3, 3, 2)$. | [
"number_theory"
] | test |
putnam_1976_a4 | noncomputable abbrev putnam_1976_a4_solution : (ℝ → ℝ) × (ℝ → ℝ) := sorry
theorem putnam_1976_a4
(a b c d : ℤ)
(r : ℝ)
(P Q: Polynomial ℚ)
(hP : P = X^3 + (C (a : ℚ))*X^2 + (C (b : ℚ))*X - (C 1) ∧ aeval r P = 0 ∧ Irreducible P)
(hQ : Q = X^3 + (C (c : ℚ))*X^2 + (C (d : ℚ))*X + (C 1) ∧ aeval (r + 1) Q = 0)
: ∃ s : ℝ, a... | 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$, ... | The possible answers are $s = -\frac{1}{r + 1}$ and $s = -\frac{r + 1}{r}$. | [
"algebra"
] | test |
putnam_1976_a6 | theorem putnam_1976_a6
(f : ℝ → ℝ)
(hfdiff : ContDiff ℝ 2 f)
(hfbd : ∀ x : ℝ, |f x| ≤ 1)
(hf0 : (f 0)^2 + (deriv f 0)^2 = 4)
: ∃ y : ℝ, (f y) + (iteratedDeriv 2 f y) = 0 :=
sorry | Suppose that $f : \mathbb{R} \to \mathbb{R}$ is a twice continuously differentiable function such that $|f(x)| \le 1$ for all real $x$ and $(f(0))^2 + (f'(0))^2 = 4$. Prove that $f(y) + f''(y) = 0$ for some real number $y$. | None. | [
"analysis",
"algebra"
] | test |
putnam_1976_b1 | abbrev putnam_1976_b1_solution : ℕ × ℕ := sorry
theorem putnam_1976_b1
: Tendsto (fun n : ℕ => ((1 : ℝ)/n)*∑ k ∈ Finset.Icc (1 : ℤ) n, (Int.floor ((2*n)/k) - 2*Int.floor (n/k))) atTop
(𝓝 (Real.log putnam_1976_b1_solution.1 - putnam_1976_b1_solution.2)) :=
sorry | 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).$$ Your answer should be in the form $\ln(a) - b$, where $a$ and $b$ are positive integers. | The limit equals $\ln(4) - 1$, so $a = 4$ and $b = 1$. | [
"analysis"
] | test |
putnam_1976_b2 | abbrev putnam_1976_b2_solution : ℕ × Set (List (ℤ × ℤ)) := sorry
theorem putnam_1976_b2
(G : Type*) [Group G]
(A B : G)
(word : List (ℤ × ℤ) → G)
(hword : word = fun w : List (ℤ × ℤ) => (List.map (fun t : ℤ × ℤ => A^(t.1)*B^(t.2)) w).prod)
(hG : ∀ g : G, ∃ w : List (ℤ × ℤ), g = word w)
(hA : A^4 = 1 ∧ A^2 ≠ 1)
(hB : B^... | Let $G$ be a group generated by two elements $A$ and $B$; i.e., every element of $G$ can be expressed as a finite word $A^{n_1}B^{n_2} \cdots A^{n_{k-1}}B^{n_k}$, where the $n_i$ can assume any integer values and $A^0 = B^0 = 1$. Further assume that $A^4 = B^7 = ABA^{-1}B = 1$, but $A^2 \ne 1$ and $B \ne 1$. Find the n... | There are $8$ such squares: $1$, $A^2$, $B$, $B^2$, $B^3$, $B^4$, $B^5$, and $B^6$. | [
"abstract_algebra"
] | test |
putnam_1976_b3 | theorem putnam_1976_b3
(Ω : Type*)
[MeasurableSpace Ω]
(μ : Measure Ω)
[IsProbabilityMeasure μ]
(n : ℕ)
(hn : n ≥ 1)
(a : ℝ)
(ha : 0 ≤ a ∧ a < 1/4)
(u : ℕ → ℝ)
(hu01 : u 0 = 1 ∧ u 1 = 1 - a)
(huk : ∀ k ≥ 1, u (k + 1) = u k - a * (u (k-1)))
(hupos : ∀ k : ℕ, u k > 0)
(events : Fin n → Set Ω)
(heventsmeas : ∀ i : Fin n, ... | Suppose that we have $n$ events $A_1, \dots, A_n$, each of which has probability at least $1 - a$ of occufring, where $a < 1/4$. Further suppose that $A_i$ and $A_j$ are mutually independent if $|i-j| > 1$, although $A_i$ and $A_{i+1}$ may be dependent. Assume as known that the recurrence $u_{k+1} = u_k - au_{k-1}$, $u... | None. | [
"probability"
] | test |
putnam_1976_b5 | noncomputable abbrev putnam_1976_b5_solution : ℕ → Polynomial ℤ := sorry
theorem putnam_1976_b5
: ∀ n : ℕ, ∑ k ∈ Finset.range (n + 1), C ((-(1 : ℤ))^k * Nat.choose n k) * (X - (C (k : ℤ)))^n = putnam_1976_b5_solution n :=
sorry | Find $$\sum_{k=0}^{n} (-1)^k {n \choose k} (x - k)^n.$$ | The sum equals $n!$. | [
"algebra"
] | test |
putnam_1976_b6 | theorem putnam_1976_b6
(σ : ℕ → ℤ)
(hσ : σ = fun N : ℕ => ∑ d ∈ Nat.divisors N, (d : ℤ))
(quasiperfect : ℕ → Prop)
(quasiperfect_def : ∀ N, quasiperfect N ↔ σ N = 2*N + 1)
: ∀ N : ℕ, quasiperfect N → ∃ m : ℤ, Odd m ∧ m^2 = N :=
sorry | Let $\sigma(N)$ denote the sum of all positive integer divisors of $N$, including $1$ and $N$. Call a positive integer $N$ \textit{quasiperfect} if $\sigma(N) = 2N + 1$. Prove that every quasiperfect number is the square of an odd integer. | None. | [
"number_theory"
] | test |
putnam_1977_a1 | noncomputable abbrev putnam_1977_a1_solution : ℝ := sorry
theorem putnam_1977_a1
(y : ℝ → ℝ)
(hy : y = fun x ↦ 2 * x ^ 4 + 7 * x ^ 3 + 3 * x - 5)
(S : Finset ℝ)
(hS : S.card = 4)
: (Collinear ℝ {P : Fin 2 → ℝ | P 0 ∈ S ∧ P 1 = y (P 0)} → (∑ x ∈ S, x) / 4 = putnam_1977_a1_solution) :=
sorry | 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$. | Prove that $k = -\frac{7}{8}$. | [
"algebra"
] | test |
putnam_1977_a2 | abbrev putnam_1977_a2_solution : ℝ → ℝ → ℝ → ℝ → Prop := sorry
theorem putnam_1977_a2 :
∀ a b c d : ℝ,
a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 →
(putnam_1977_a2_solution a b c d ↔
(a + b + c = d ∧ 1 / a + 1 / b + 1 / c = 1 / d)) :=
sorry | 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}$. | 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. | [
"algebra"
] | test |
putnam_1977_a3 | abbrev putnam_1977_a3_solution : (ℝ → ℝ) → (ℝ → ℝ) → (ℝ → ℝ) := sorry
theorem putnam_1977_a3
(f g h : ℝ → ℝ)
(hf : ∀ x, f x = (h (x + 1) + h (x - 1)) / 2)
(hg : ∀ x, g x = (h (x + 4) + h (x - 4)) / 2) :
h = putnam_1977_a3_solution f g :=
sorry | 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}$. | Prove that $h(x) = g(x) - f(x - 3) + f(x - 1) + f(x + 1) - f(x + 3)$ suffices. | [
"algebra"
] | test |
putnam_1977_a4 | noncomputable abbrev putnam_1977_a4_solution : RatFunc ℝ := sorry
theorem putnam_1977_a4 :
∀ x ∈ Ioo 0 1,
putnam_1977_a4_solution.eval (id ℝ) x = ∑' n : ℕ, x ^ 2 ^ n / (1 - x ^ 2 ^ (n + 1)) :=
sorry | 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)$. | Prove that the sum equals $\frac{x}{1 - x}$. | [
"algebra",
"analysis"
] | test |
putnam_1977_a5 | theorem putnam_1977_a5
(p m n : ℕ)
(hp : Nat.Prime p)
(hmgen : m ≥ n)
: (choose (p * m) (p * n) ≡ choose m n [MOD p]) :=
sorry | Let $p$ be a prime and $m \geq n$ be non-negative integers. Show that $\binom{pm}{pn} = \binom{m}{n} \pmod p$, where $\binom{m}{n}$ is the binomial coefficient. | None. | [
"algebra",
"number_theory"
] | test |
putnam_1977_a6 | abbrev putnam_1977_a6_solution : Prop := sorry
theorem putnam_1977_a6
(X : Set (ℝ × ℝ))
(hX : X = Set.prod (Icc 0 1) (Icc 0 1))
(room : (ℝ × ℝ) → ℝ)
(hroom : room = fun (a,b) ↦ min (min a (1 - a)) (min b (1 - b)))
: ((∀ f : (ℝ × ℝ) → ℝ, Continuous f → (∀ P ∈ X, ∫ x in (P.1 - room P)..(P.1 + room P), ∫ y in (P.2 - room ... | 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... | Prove that $f(x,y)$ must be identically zero. | [
"analysis"
] | test |
putnam_1977_b1 | noncomputable abbrev putnam_1977_b1_solution : ℝ := sorry
theorem putnam_1977_b1
: Tendsto (fun N ↦ ∏ n ∈ Finset.Icc (2 : ℤ) N, ((n : ℝ) ^ 3 - 1) / (n ^ 3 + 1)) atTop (𝓝 putnam_1977_b1_solution) :=
sorry | Find $\prod_{n=2}^{\infty} \frac{(n^3 - 1)}{(n^3 + 1)}$. | Prove that the product equals $\frac{2}{3}$. | [
"algebra",
"analysis"
] | test |
putnam_1977_b3 | abbrev putnam_1977_b3_solution : Prop := sorry
theorem putnam_1977_b3
(P balanced: ℝ × ℝ × ℝ → Prop)
(B : ℝ × ℝ × ℝ → ℝ × ℝ × ℝ)
(hP : P = fun (a, b, c) => Irrational a ∧ Irrational b ∧ Irrational c ∧ a > 0 ∧ b > 0 ∧ c > 0 ∧ a + b + c = 1)
(hbalanced : balanced = fun (a, b, c) => a < 1/2 ∧ b < 1/2 ∧ c < 1/2)
(hB : B = ... | 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 ... | Not necessarily. | [
"algebra"
] | test |
putnam_1977_b5 | theorem putnam_1977_b5
(n : ℕ)
(hn : n > 1)
(a : Fin n → ℝ)
(A : ℝ)
(hA : A + ∑ i : Fin n, (a i)^2 < (1/((n : ℝ) - 1))*(∑ i : Fin n, a i)^2)
: ∀ i j : Fin n, i < j → A < 2*(a i)*(a j) :=
sorry | If $a_1, a_2, \dots, a_n$ are real numbers with $n > 1$ and $A$ satisfies $$A + \sum_{i = 1}^{n} a_i^2 < \frac{1}{n-1}\left(\sum_{i=1}^{n}a_i\right)^2,$$ prove that $A < 2a_{i}a_{j}$ for all $i, j$ with $1 \le i < j \le n$. | None. | [
"algebra"
] | test |
putnam_1977_b6 | theorem putnam_1977_b6
{G : Type*}
[Group G]
(H : Subgroup G)
[Finite H]
(h : ℕ)
(h_def : h = Nat.card H)
(a : G)
(ha : ∀ x : H, (x*a)^3 = 1)
(P : Set G)
(hP : P = {g : G | ∃ xs : List H, (xs.length ≥ 1) ∧ g = (List.map (fun h : H => h*a) xs).prod})
: (Finite P) ∧ (P.ncard ≤ 3*h^2) :=
sorry | Let $G$ be a group and $H$ be a subgroup of $G$ with $h$ elements. Suppose that $G$ contains some element $a$ such that $(xa)^3 = 1$ for all $x \in H$ (here $1$ represents the identity element of $G$). Let $P$ be the subset of $G$ containing all products of the form $x_1 a x_2 a \cdots x_n a$ with $n \ge 1$ and $x_i \i... | None. | [
"abstract_algebra"
] | test |
putnam_1978_a1 | theorem putnam_1978_a1
(S T : Set ℤ)
(hS : S = {k | ∃ j : ℤ, 0 ≤ j ∧ j ≤ 33 ∧ k = 3 * j + 1})
(hT : T ⊆ S ∧ T.ncard = 20)
: (∃ m ∈ T, ∃ n ∈ T, m ≠ n ∧ m + n = 104) :=
sorry | Let $S = \{1, 4, 7, 10, 13, 16, \dots , 100\}$. Let $T$ be a subset of $20$ elements of $S$. Show that we can find two distinct elements of $T$ with sum $104$. | None. | [
"algebra"
] | test |
putnam_1978_a2 | theorem putnam_1978_a2
(n : ℕ)
(npos : n > 0)
(a b : ℝ)
(hab : a ≠ b)
(c : Fin n → ℝ)
(A : Matrix (Fin n) (Fin n) ℝ)
(hA : A = fun i j ↦ ite (i < j) a (ite (i > j) b (c i)))
(p : ℝ → ℝ)
(hp : p = fun x ↦ ∏ i : Fin n, (c i - x))
: (A.det = ((b * p a - a * p b) / (b - a))) :=
sorry | Let $A$ be the real $n \times n$ matrix $(a_{ij})$ where $a_{ij} = a$ for $i < j$, $b \; (\neq a)$ for $i > j$, and $c_i$ for $i = j$. Show that $\det A = \frac{b p(a) - a p(b)}{b - a}$, where $p(x) = \prod_{i=1}^n (c_i - x)$. | None. | [
"linear_algebra"
] | test |
putnam_1978_a3 | abbrev putnam_1978_a3_solution : ℕ := sorry
theorem putnam_1978_a3
(p : Polynomial ℝ)
(hp : p = 2 * (X ^ 6 + 1) + 4 * (X ^ 5 + X) + 3 * (X ^ 4 + X ^ 2) + 5 * X ^ 3)
(I : ℕ → ℝ)
(hI : I = fun k ↦ ∫ x in Ioi 0, x ^ k / p.eval x) :
IsLeast {y | ∃ k ∈ Ioo 0 5, I k = y} (I putnam_1978_a3_solution) :=
s... | 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? | Show that $I_k$ is smallest for $k = 2$. | [
"analysis",
"algebra"
] | test |
putnam_1978_a4 | theorem putnam_1978_a4
(bypass : (S : Type) → [inst : Mul S] → Prop)
(hbypass : bypass = fun S [Mul S] ↦ ∀ a b c d : S, (a * b) * (c * d) = a * d)
: ((∀ (S : Type) (_ : Mul S), bypass S → ∀ a b c : S, a * b = c → (c * c = c ∧ ∀ d : S, a * d = c * d))
∧ (∃ (S : Type) (_ : Mul S) (_ : Fintype S), bypass S ∧ (∀ a : S, a... | A binary operation (represented by multiplication) on $S$ has the property that $(ab)(cd) = ad$ for all $a, b, c, d$. Show that:
\begin{itemize}
\item[(1)] if $ab = c$, then $cc = c$;
\item[(2)] if $ab = c$, then $ad = cd$ for all $d$.
\end{itemize}
Find a set $S$, and such a binary operation, which also satisfies:
\be... | None. | [
"abstract_algebra"
] | test |
putnam_1978_a5 | theorem putnam_1978_a5
(n : ℕ)
(npos : n > 0)
(a : Fin n → ℝ)
(ha : ∀ i : Fin n, a i ∈ Ioo 0 Real.pi)
(μ : ℝ)
(hμ : μ = ∑ i : Fin n, a i / n)
: (∏ i : Fin n, sin (a i) / (a i) ≤ (sin μ / μ) ^ n) :=
sorry | Let $a_1, a_2, \dots , a_n$ be reals in the interval $(0, \pi)$ with arithmetic mean $\mu$. Show that
\[
\prod_{i=1}^n \left( \frac{\sin a_i}{a_i} \right) \leq \left( \frac{\sin \mu}{\mu} \right)^n.
\] | None. | [
"analysis"
] | test |
putnam_1978_a6 | theorem putnam_1978_a6
(S : Finset (EuclideanSpace ℝ (Fin 2)))
(n : ℕ)
(hn : n = S.card)
(npos : n > 0)
: ({pair : Set (EuclideanSpace ℝ (Fin 2)) | ∃ P ∈ S, ∃ Q ∈ S, pair = {P, Q} ∧ dist P Q = 1}.ncard < 2 * (n : ℝ) ^ ((3 : ℝ) / 2)) :=
sorry | Given $n$ distinct points in the plane, prove that fewer than $2n^{3/2}$ pairs of these points are a distance of $1$ apart. | None. | [
"geometry",
"combinatorics"
] | test |
putnam_1978_b2 | abbrev putnam_1978_b2_solution : ℚ := sorry
theorem putnam_1978_b2
: (∑' i : ℕ+, ∑' j : ℕ+, (1 : ℚ) / (i ^ 2 * j + 2 * i * j + i * j ^ 2) = putnam_1978_b2_solution) :=
sorry | Find
\[
\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \frac{1}{i^2j + 2ij + ij^2}.
\] | Prove that the sum evaluates to $\frac{7}{4}$. | [
"algebra",
"analysis"
] | test |
putnam_1978_b3 | theorem putnam_1978_b3
(P : ℕ+ → Polynomial ℝ)
(hP1 : P 1 = 1 + X)
(hP2 : P 2 = 1 + 2 * X)
(hPodd : ∀ n, P (2 * n + 1) = P (2 * n) + C ((n : ℝ) + 1) * X * P (2 * n - 1))
(hPeven : ∀ n, P (2 * n + 2) = P (2 * n + 1) + C ((n : ℝ) + 1) * X * P (2 * n))
(a : ℕ+ → ℝ)
(haroot : ∀ n, (P n).eval (a n) = 0)
(haub : ∀ n, ∀ x, (P... | The polynomials $P_n(x)$ are defined by
\begin{align*}
P_1(x) &= 1 + x, \\
P_2(x) &= 1 + 2x, \\
P_{2n+1}(x) &= P_{2n}(x) + (n + 1) x P_{2n-1}(x), \\
P_{2n+2}(x) &= P_{2n+1}(x) + (n + 1) x P_{2n}(x).
\end{align*}
Let $a_n$ be the largest real root of $P_n(x)$. Prove that $a_n$ is strictly monotonically increasing and te... | None. | [
"algebra",
"analysis"
] | test |
putnam_1978_b4 | theorem putnam_1978_b4 :
∀ N : ℝ,
∃ a b c d : ℤ,
a > N ∧ b > N ∧ c > N ∧ d > N ∧
a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = a * b * c + a * b * d + a * c * d + b * c * d :=
sorry | Show that we can find integers $a, b, c, d$ such that $a^2 + b^2 + c^2 + d^2 = abc + abd + acd + bcd$, and the smallest of $a, b, c, d$ is arbitrarily large. | None. | [
"algebra"
] | test |
putnam_1978_b5 | noncomputable abbrev putnam_1978_b5_solution : Polynomial ℝ := sorry
theorem putnam_1978_b5
(S : Set (Polynomial ℝ))
(hS : S = {p : Polynomial ℝ | p.degree = 4 ∧ ∀ x ∈ Icc (-1) 1, p.eval x ∈ Icc 0 1})
: (putnam_1978_b5_solution ∈ S ∧ (∀ p ∈ S, p.coeff 4 ≤ putnam_1978_b5_solution.coeff 4)) :=
sorry | Find the real polynomial $p(x)$ of degree $4$ with largest possible coefficient of $x^4$ such that $p([-1, 1]) \subseteq [0, 1]$. | Prove that $p(x) = 4x^4 - 4x^2 + 1$. | [
"algebra"
] | test |
putnam_1978_b6 | theorem putnam_1978_b6
(a : ℕ → ℕ → ℝ)
(ha : ∀ i j, a i j ∈ Icc 0 1)
(m n : ℕ)
(mnpos : m > 0 ∧ n > 0)
: ((∑ i ∈ Finset.Icc 1 n, ∑ j ∈ Finset.Icc 1 (m * i), a i j / i) ^ 2 ≤ 2 * m * ∑ i ∈ Finset.Icc 1 n, ∑ j ∈ Finset.Icc 1 (m * i), a i j) :=
sorry | Let $a_{ij}$ be real numbers in $[0, 1]$. Show that
\[
\left( \sum_{i=1}^n \sum_{j=1}^{mi} \frac{a_{ij}}{i} \right) ^2 \leq 2m \sum_{i=1}^n \sum_{j=1}^{mi} a_{ij}.
\] | None. | [
"algebra"
] | test |
putnam_1979_a1 | abbrev putnam_1979_a1_solution : Multiset ℕ := sorry
theorem putnam_1979_a1
(P : Multiset ℕ → Prop)
(hP : ∀ a, P a ↔ Multiset.card a > 0 ∧ (∀ i ∈ a, i > 0) ∧ a.sum = 1979) :
P putnam_1979_a1_solution ∧ ∀ a : Multiset ℕ, P a → putnam_1979_a1_solution.prod ≥ a.prod :=
sorry | 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? | $n$ equals $660$; all but one of the $a_i$ equal $3$ and the remaining $a_i$ equals $2$. | [
"algebra"
] | test |
putnam_1979_a2 | abbrev putnam_1979_a2_solution : ℝ → Prop := sorry
theorem putnam_1979_a2
: ∀ k : ℝ, (∃ f : ℝ → ℝ, Continuous f ∧ ∀ x : ℝ, f (f x) = k*x^9) ↔ putnam_1979_a2_solution k :=
sorry | 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 $x$? | Such a function exists if and only if $k \ge 0$. | [
"analysis",
"algebra"
] | test |
putnam_1979_a3 | abbrev putnam_1979_a3_solution : (ℝ × ℝ) → Prop := sorry
theorem putnam_1979_a3
(x : ℕ → ℝ)
(hx : ∀ n : ℕ, x n ≠ 0 ∧ (n ≥ 3 → x n = (x (n - 2))*(x (n - 1))/(2*(x (n - 2)) - (x (n - 1)))))
: (∀ m : ℕ, ∃ n : ℕ, n > m ∧ ∃ a : ℤ, a = x n) ↔ putnam_1979_a3_solution (x 1, x 2) :=
sorry | 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 $n$? | We must have $x_1 = x_2 = m$ for some integer $m$. | [
"algebra"
] | test |
putnam_1979_a4 | abbrev putnam_1979_a4_solution : Prop := sorry
theorem putnam_1979_a4
(A : Finset (Fin 2 → ℝ) × Finset (Fin 2 → ℝ) → Prop)
(hA : A = fun (R, B) => R.card = B.card ∧ R ∩ B = ∅ ∧
∀ u : Finset (Fin 2 → ℝ), u ⊆ R ∪ B → u.card = 3 → ¬Collinear ℝ (u : Set (Fin 2 → ℝ)))
(w : (Fin 2 → ℝ) × (Fin 2 → ℝ) → ℝ → (... | 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? | Such line segments must exist. | [
"geometry",
"combinatorics"
] | test |
putnam_1979_a5 | theorem putnam_1979_a5
(S : ℝ → ℕ → ℤ)
(hS : S = fun x : ℝ => fun n : ℕ => Int.floor (n*x))
(P : ℝ → Prop)
(hP : ∀ x, P x ↔ x^3 - 10*x^2 + 29*x - 25 = 0)
: ∃ α β : ℝ, α ≠ β ∧ P α ∧ P β ∧ ∀ n : ℕ, ∃ m : ℤ, m > n ∧ ∃ c d : ℕ, S α c = m ∧ S β d = m :=
sorry | Let $S(x)$ denote the sequence $\lfloor 0 \rfloor, \lfloor x \rfloor, \lfloor 2x \rfloor, \lfloor 3x \rfloor, \dots$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Prove that there exist distinct real roots $\alpha$ and $\beta$ of $x^3 - 10x^2 + 29x - 25$ such that infinitely many po... | None. | [
"algebra"
] | test |
putnam_1979_a6 | theorem putnam_1979_a6
(n : ℕ)
(p : ℕ → ℝ)
(hp : ∀ i ∈ Finset.range n, p i ∈ Icc 0 1)
: ∃ x ∈ Icc 0 1, (∀ i ∈ Finset.range n, x ≠ p i) ∧ ∑ i ∈ Finset.range n, 1/|x - p i| ≤ 8*n*∑ i ∈ Finset.range n, (1 : ℝ)/(2*i + 1) :=
sorry | For all $i \in \{0, 1, \dots, n - 1\}$, let $p_i \in [0, 1]$. Prove that there exists some $x \in [0, 1]$ such that $$\sum_{i = 0}^{n - 1} \frac{1}{|x - p_i|} \le 8n\left(\sum_{i = 0}^{n-1} \frac{1}{2i + 1}\right).$$ | None. | [
"algebra"
] | test |
putnam_1979_b2 | noncomputable abbrev putnam_1979_b2_solution : ℝ × ℝ → ℝ := sorry
theorem putnam_1979_b2
: ∀ 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 (putnam_1979_b2_solution (a, b))) :=
sorry | 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 $b$. | The limit equals $$e^{-1}\left(\frac{b^b}{a^a}\right)^{\frac{1}{b-a}}.$$ | [
"analysis"
] | test |
putnam_1979_b3 | abbrev putnam_1979_b3_solution : ℕ → ℤ := sorry
theorem putnam_1979_b3
(F : Type*) [Field F] [Fintype F]
(n : ℕ)
(hn : n = Fintype.card F)
(nodd : Odd n)
(b c : F)
(p : Polynomial F)
(hp : p = X ^ 2 + (C b) * X + (C c) ∧ Irreducible p)
: ({d : F | Irreducible (p + (C d))}.ncard = putnam_1979_b3_solution n) :=
sorry | 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? | Show that there are $\frac{n - 1}{2}$ such elements $d$. | [
"abstract_algebra"
] | test |
putnam_1979_b5 | theorem putnam_1979_b5
(C : Set (ℝ × ℝ))
(hC : IsClosed C ∧ Convex ℝ C ∧ (0,0) ∈ C)
(hCnint : ∀ a b : ℤ, (a ≠ 0 ∨ b ≠ 0) → ((a : ℝ), (b : ℝ)) ∉ C)
(hCeven : volume (C ∩ {p : ℝ × ℝ | p.1 ≥ 0 ∧ p.2 ≥ 0}) = volume (C ∩ {p : ℝ × ℝ | p.1 ≥ 0 ∧ p.2 ≤ 0}) ∧ volume (C ∩ {p : ℝ × ℝ | p.1 ≥ 0 ∧ p.2 ≤ 0}) = volume (C ∩ {p : ℝ × ℝ... | In the plane, let $C$ be a closed convex set that contains $(0,0) but no other point with integer coordinations. Suppose that $A(C)$, the area of $C$, is equally distributed among the four quadrants. Prove that $A(C) \leq 4$. | None. | [
"geometry",
"analysis"
] | test |
putnam_1979_b6 | theorem putnam_1979_b6
(n : ℕ)
(z : Fin n → ℂ)
: (|((∑ i : Fin n, (z i) ^ 2) ^ ((1 : ℂ) / 2)).re| ≤ ∑ i : Fin n, |(z i).re|) :=
sorry | Let $z_i$ be complex numbers for $i = 1, 2, \dots, n$. Show that
\[
\left \lvert \mathrm{Re} \, [(z_1^2 + z_2^2 + \dots + z_n^2)^{1/2} ] \right \rvert \leq \lvert \mathrm{Re} \, z_1 \rvert + \lvert \mathrm{Re} \, z_2 \rvert + \dots + \lvert \mathrm{Re} \, z_n \rvert.
\] | None. | [
"analysis"
] | test |
putnam_1980_a2 | abbrev putnam_1980_a2_solution : ℕ → ℕ → ℕ := sorry
theorem putnam_1980_a2
(r s : ℕ)
(abcdlcm : ℕ → ℕ → ℕ → ℕ → Prop)
(rspos : r > 0 ∧ s > 0)
(habcdlcm : ∀ a b c d : ℕ, abcdlcm a b c d ↔
(a > 0 ∧ b > 0 ∧ c > 0 ∧ d > 0 ∧
(3 ^ r * 7 ^ s = Nat.lcm (Nat.lcm a b) c) ∧
(3 ^ r * 7 ^ s = Nat.l... | 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... | Show that the number is $(1+4r+6r^2)(1+4s+6s^2)$. | [
"number_theory"
] | test |
putnam_1980_a3 | noncomputable abbrev putnam_1980_a3_solution : ℝ := sorry
theorem putnam_1980_a3
: ∫ x in Set.Ioo 0 (Real.pi / 2), 1 / (1 + (Real.tan x) ^ (Real.sqrt 2)) = putnam_1980_a3_solution :=
sorry | Evaluate $\int_0^{\pi/2}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$. | Show that the integral is $\pi/4$. | [
"analysis"
] | test |
putnam_1980_a4 | theorem putnam_1980_a4
(abcvals : ℤ → ℤ → ℤ → Prop)
(habcvals : ∀ a b c : ℤ, abcvals a b c ↔ ¬(a = 0 ∧ b = 0 ∧ c = 0) ∧ |a| < 1000000 ∧ |b| < 1000000 ∧ |c| < 1000000) :
(∃ a b c : ℤ,
abcvals a b c ∧
|a + b * Real.sqrt 2 + c * Real.sqrt 3| < 10 ^ (-(11 : ℝ))) ∧
(∀ a b c : ℤ, abcvals a b c →... | \begin{enumerate}
\item[(a)] Prove that there exist integers $a,b,c$, not all zero and each of absolute value less than one million, such that $|a+b\sqrt{2}+c\sqrt{3}|<10^{-11}$.
\item[(b)] Let $a,b,c$ be integers, not all zero and each of absolute value less than one million. Prove that $|a+b\sqrt{2}+c\sqrt{3}|>10^{-2... | None. | [
"algebra"
] | test |
putnam_1980_a5 | theorem putnam_1980_a5
(P : Polynomial ℝ)
(Pnonconst : P.degree > 0) :
Set.Finite {x : ℝ |
0 = (∫ t in (0)..x, P.eval t * Real.sin t) ∧
0 = (∫ t in (0)..x, P.eval t * Real.cos t)} :=
sorry | Let $P(t)$ be a nonconstant polynomial with real coefficients. Prove that the system of simultaneous equations $0=\int_0^xP(t)\sin t\,dt=\int_0^xP(t)\cos t\,dt$ has only finitely many real solutions $x$. | None. | [
"analysis"
] | test |
putnam_1980_a6 | noncomputable abbrev putnam_1980_a6_solution : ℝ := sorry
theorem putnam_1980_a6
(C : Set (ℝ → ℝ))
(hC : C = {f : ℝ → ℝ | ContDiffOn ℝ 1 f (Set.Icc 0 1) ∧ f 0 = 0 ∧ f 1 = 1}) :
IsGreatest
{(u : ℝ) | ∀ f ∈ C, u ≤ (∫ x in Set.Ioo 0 1, |deriv f x - f x|)}
putnam_1980_a6_solution :=
sorry | 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 $C$. | Show that $u=1/e$. | [
"analysis"
] | test |
putnam_1980_b1 | abbrev putnam_1980_b1_solution : Set ℝ := sorry
theorem putnam_1980_b1
(c : ℝ)
: (∀ x : ℝ, (exp x + exp (-x)) / 2 ≤ exp (c * x ^ 2)) ↔ c ∈ putnam_1980_b1_solution :=
sorry | For which real numbers $c$ is $(e^x+e^{-x})/2 \leq e^{cx^2}$ for all real $x$? | Show that the inequality holds if and only if $c \geq 1/2$. | [
"analysis"
] | test |
putnam_1980_b3 | abbrev putnam_1980_b3_solution : Set ℝ := sorry
theorem putnam_1980_b3
(a : ℝ)
(u : ℕ → ℝ)
(hu : u 0 = a ∧ (∀ n : ℕ, u (n + 1) = 2 * u n - n ^ 2))
: (∀ n : ℕ, u n > 0) ↔ a ∈ putnam_1980_b3_solution :=
sorry | 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.) | Show that $u_n>0$ for all $n \geq 0$ if and only if $a \geq 3$. | [
"algebra"
] | test |
putnam_1980_b4 | theorem putnam_1980_b4
{T : Type}
(X : Finset T)
(A : Fin 1066 → Finset T)
(hX : X.card ≥ 10)
(hA : ∀ i : Fin 1066, A i ⊆ X ∧ (A i).card > ((1 : ℚ)/2) * X.card)
: ∃ Y : Finset T, Y ⊆ X ∧ Y.card = 10 ∧ ∀ i : Fin 1066, ∃ y ∈ Y, y ∈ A i :=
sorry | Let $X$ be a finite set with at least $10$ elements; for each $i \in \{0, 1, ..., 1065\}$, let $A_i \subseteq X$ satisfy $|A_i| > \frac{1}{2}|X|$. Prove that there exist $10$ elements $x_1, x_2, \dots, x_{10} \in X$ such that each $A_i$ contains at least one of $x_1, x_2, \dots, x_{10}$. | None. | [
"set_theory",
"combinatorics"
] | test |
putnam_1980_b5 | abbrev putnam_1980_b5_solution : ℝ → Prop := sorry
theorem putnam_1980_b5
(T : Set ℝ)
(hT : T = Icc 0 1)
(P : ℝ → (ℝ → ℝ) → Prop)
(IsConvex : (ℝ → ℝ) → Prop)
(S : ℝ → Set (ℝ → ℝ))
(P_def : ∀ t f, P t f ↔ f 1 - 2*f (2/3) + f (1/3) ≥ t*(f (2/3) - 2*f (1/3) + f 0))
(IsConvex_def : ∀ f, IsConvex... | 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\le... | $S_t$ is closed under multiplication if and only if $1 \ge t$. | [
"analysis",
"algebra"
] | test |
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