{"name": "amc12a_2019_p21", "split": "valid", "informal_prefix": "/-- Let $z=\\frac{1+i}{\\sqrt{2}}.$What is $\\left(z^{1^2}+z^{2^2}+z^{3^2}+\\dots+z^{{12}^2}\\right) \\cdot \\left(\\frac{1}{z^{1^2}}+\\frac{1}{z^{2^2}}+\\frac{1}{z^{3^2}}+\\dots+\\frac{1}{z^{{12}^2}}\\right)?$\n\n$\\textbf{(A) } 18 \\qquad \\textbf{(B) } 72-36\\sqrt2 \\qquad \\textbf{(C) } 36 \\qquad \\textbf{(D) } 72 \\qquad \\textbf{(E) } 72+36\\sqrt2$ Show that it is \\textbf{(C) }36.-/\n", "formal_statement": "theorem amc12a_2019_p21 (z : ℂ) (h₀ : z = (1 + Complex.I) / Real.sqrt 2) :\n ((∑ k : ℤ in Finset.Icc 1 12, z ^ k ^ 2) * (∑ k : ℤ in Finset.Icc 1 12, 1 / z ^ k ^ 2)) = 36 := by\n", "goal": "z : ℂ\nh₀ : z = (1 + Complex.I) / ↑√2\n⊢ (∑ k ∈ Finset.Icc 1 12, z ^ k ^ 2) * ∑ k ∈ Finset.Icc 1 12, 1 / z ^ k ^ 2 = 36", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "amc12a_2015_p10", "split": "valid", "informal_prefix": "/-- Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$?\n\n$ \\textbf{(A)}\\ 8 \\qquad\\textbf{(B)}\\ 10 \\qquad\\textbf{(C)}\\ 15 \\qquad\\textbf{(D)}\\ 18 \\qquad\\textbf{(E)}\\ 26$ Show that it is \\textbf{(E)}\\ 26.-/\n", "formal_statement": "theorem amc12a_2015_p10 (x y : ℤ) (h₀ : 0 < y) (h₁ : y < x) (h₂ : x + y + x * y = 80) : x = 26 := by\n", "goal": "x y : ℤ\nh₀ : 0 < y\nh₁ : y < x\nh₂ : x + y + x * y = 80\n⊢ x = 26", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "amc12a_2008_p8", "split": "valid", "informal_prefix": "/-- What is the [[volume]] of a [[cube]] whose [[surface area]] is twice that of a cube with volume 1? \n\n$\\mathrm{(A)}\\ \\sqrt{2}\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 2\\sqrt{2}\\qquad\\mathrm{(D)}\\ 4\\qquad\\mathrm{(E)}\\ 8$ Show that it is \\mathrm{(C)}.-/\n", "formal_statement": "theorem amc12a_2008_p8 (x y : ℝ) (h₀ : 0 < x ∧ 0 < y) (h₁ : y ^ 3 = 1)\n (h₂ : 6 * x ^ 2 = 2 * (6 * y ^ 2)) : x ^ 3 = 2 * Real.sqrt 2 := by\n", "goal": "x y : ℝ\nh₀ : 0 < x ∧ 0 < y\nh₁ : y ^ 3 = 1\nh₂ : 6 * x ^ 2 = 2 * (6 * y ^ 2)\n⊢ x ^ 3 = 2 * √2", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "mathd_algebra_182", "split": "valid", "informal_prefix": "/-- Expand the following expression: $7(3y+2)$ Show that it is 21y+14.-/\n", "formal_statement": "theorem mathd_algebra_182 (y : ℂ) : 7 * (3 * y + 2) = 21 * y + 14 := by\n", "goal": "y : ℂ\n⊢ 7 * (3 * y + 2) = 21 * y + 14", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "aime_1984_p5", "split": "valid", "informal_prefix": "/-- Determine the value of $ab$ if $\\log_8a+\\log_4b^2=5$ and $\\log_8b+\\log_4a^2=7$. Show that it is 512.-/\n", "formal_statement": "theorem aime_1984_p5 (a b : ℝ) (h₀ : Real.logb 8 a + Real.logb 4 (b ^ 2) = 5)\n (h₁ : Real.logb 8 b + Real.logb 4 (a ^ 2) = 7) : a * b = 512 := by\n", "goal": "a b : ℝ\nh₀ : logb 8 a + logb 4 (b ^ 2) = 5\nh₁ : logb 8 b + logb 4 (a ^ 2) = 7\n⊢ a * b = 512", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "mathd_numbertheory_780", "split": "valid", "informal_prefix": "/-- Suppose $m$ is a two-digit positive integer such that $6^{-1}\\pmod m$ exists and $6^{-1}\\equiv 6^2\\pmod m$. What is $m$? Show that it is 43.-/\n", "formal_statement": "theorem mathd_numbertheory_780 (m x : ℤ) (h₀ : 0 ≤ x) (h₁ : 10 ≤ m ∧ m ≤ 99) (h₂ : 6 * x % m = 1)\n (h₃ : (x - 6 ^ 2) % m = 0) : m = 43 := by\n", "goal": "m x : ℤ\nh₀ : 0 ≤ x\nh₁ : 10 ≤ m ∧ m ≤ 99\nh₂ : 6 * x % m = 1\nh₃ : (x - 6 ^ 2) % m = 0\n⊢ m = 43", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "mathd_algebra_116", "split": "valid", "informal_prefix": "/-- For what real value of $k$ is $\\frac{13-\\sqrt{131}}{4}$ a root of $2x^2-13x+k$? Show that it is $\\frac{19}{4}$.-/\n", "formal_statement": "theorem mathd_algebra_116 (k x : ℝ) (h₀ : x = (13 - Real.sqrt 131) / 4)\n (h₁ : 2 * x ^ 2 - 13 * x + k = 0) : k = 19 / 4 := by\n", "goal": "k x : ℝ\nh₀ : x = (13 - √131) / 4\nh₁ : 2 * x ^ 2 - 13 * x + k = 0\n⊢ k = 19 / 4", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "mathd_numbertheory_13", "split": "valid", "informal_prefix": "/-- What is the average of the two smallest positive integer solutions to the congruence $$14u \\equiv 46 \\pmod{100}~?$$ Show that it is 64.-/\n", "formal_statement": "theorem mathd_numbertheory_13 (u v : ℕ) (S : Set ℕ)\n (h₀ : ∀ n : ℕ, n ∈ S ↔ 0 < n ∧ 14 * n % 100 = 46) (h₁ : IsLeast S u)\n (h₂ : IsLeast (S \\ {u}) v) : (u + v : ℚ) / 2 = 64 := by\n", "goal": "u v : ℕ\nS : Set ℕ\nh₀ : ∀ (n : ℕ), n ∈ S ↔ 0 < n ∧ 14 * n % 100 = 46\nh₁ : IsLeast S u\nh₂ : IsLeast (S \\ {u}) v\n⊢ (↑u + ↑v) / 2 = 64", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "mathd_numbertheory_169", "split": "valid", "informal_prefix": "/-- What is the greatest common factor of $20 !$ and $200,\\!000$? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.) Show that it is 40,\\!000.-/\n", "formal_statement": "theorem mathd_numbertheory_169 : Nat.gcd 20! 200000 = 40000 := by\n", "goal": "⊢ 20!.gcd 200000 = 40000", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "amc12a_2009_p9", "split": "valid", "informal_prefix": "/-- Suppose that $f(x+3)=3x^2 + 7x + 4$ and $f(x)=ax^2 + bx + c$. What is $a+b+c$?\n\n$\\textbf{(A)}\\ -1 \\qquad \\textbf{(B)}\\ 0 \\qquad \\textbf{(C)}\\ 1 \\qquad \\textbf{(D)}\\ 2 \\qquad \\textbf{(E)}\\ 3$ Show that it is 2.-/\n", "formal_statement": "theorem amc12a_2009_p9 (a b c : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f (x + 3) = 3 * x ^ 2 + 7 * x + 4)\n (h₁ : ∀ x, f x = a * x ^ 2 + b * x + c) : a + b + c = 2 := by\n", "goal": "a b c : ℝ\nf : ℝ → ℝ\nh₀ : ∀ (x : ℝ), f (x + 3) = 3 * x ^ 2 + 7 * x + 4\nh₁ : ∀ (x : ℝ), f x = a * x ^ 2 + b * x + c\n⊢ a + b + c = 2", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "amc12a_2019_p9", "split": "valid", "informal_prefix": "/-- A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \\frac{3}{7}$, and\n$a_n=\\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$for all $n \\geq 3$ Then $a_{2019}$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$\n\n$\\textbf{(A) } 2020 \\qquad\\textbf{(B) } 4039 \\qquad\\textbf{(C) } 6057 \\qquad\\textbf{(D) } 6061 \\qquad\\textbf{(E) } 8078$ Show that it is \\textbf{(E) }8078.-/\n", "formal_statement": "theorem amc12a_2019_p9 (a : ℕ → ℚ) (h₀ : a 1 = 1) (h₁ : a 2 = 3 / 7)\n (h₂ : ∀ n, a (n + 2) = a n * a (n + 1) / (2 * a n - a (n + 1))) :\n ↑(a 2019).den + (a 2019).num = 8078 := by\n", "goal": "a : ℕ → ℚ\nh₀ : a 1 = 1\nh₁ : a 2 = 3 / 7\nh₂ : ∀ (n : ℕ), a (n + 2) = a n * a (n + 1) / (2 * a n - a (n + 1))\n⊢ ↑(a 2019).den + (a 2019).num = 8078", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "mathd_algebra_13", "split": "valid", "informal_prefix": "/-- Find $A$ and $B$ such that\n\\[\\frac{4x}{x^2-8x+15} = \\frac{A}{x-3} + \\frac{B}{x-5}\\]for all $x$ besides 3 and 5. Express your answer as an ordered pair in the form $(A, B).$ Show that it is (-6, 10).-/\n", "formal_statement": "theorem mathd_algebra_13 (a b : ℝ)\n (h₀ : ∀ x, x - 3 ≠ 0 ∧ x - 5 ≠ 0 → 4 * x / (x ^ 2 - 8 * x + 15) = a / (x - 3) + b / (x - 5)) :\n a = -6 ∧ b = 10 := by\n", "goal": "a b : ℝ\nh₀ : ∀ (x : ℝ), x - 3 ≠ 0 ∧ x - 5 ≠ 0 → 4 * x / (x ^ 2 - 8 * x + 15) = a / (x - 3) + b / (x - 5)\n⊢ a = -6 ∧ b = 10", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "induction_sum2kp1npqsqm1", "split": "valid", "informal_prefix": "/-- Show that for positive integer n, $\\sum_{k=0}^{n-1} (2k + 3) = (n + 1)^2 - 1$.-/\n", "formal_statement": "theorem induction_sum2kp1npqsqm1 (n : ℕ) :\n ∑ k in Finset.range n, (2 * k + 3) = (n + 1) ^ 2 - 1 := by\n", "goal": "n : ℕ\n⊢ ∑ k ∈ Finset.range n, (2 * k + 3) = (n + 1) ^ 2 - 1", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n"} {"name": "aime_1991_p6", "split": "valid", "informal_prefix": "/-- Suppose $r^{}_{}$ is a [[real number]] for which\n