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dfe1826 | 1 2 3 4 5 6 | {"id": "amc12a_2015_p10", "split": "dev", "area": "algebra", "difficulty": "medium", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl, MIT)", "source_split": "valid", "reserved": "dev_preview (PLAN.md Phase 7): never in test", "informal": "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.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\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", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem amc12a_2015_p10 (x y : ℤ) (h₀ : 0 < y) (h₁ : y < x) (h₂ : x + y + x * y = 80) : x = 26 := by\n"}
{"id": "aime_1991_p1", "split": "dev", "area": "algebra", "difficulty": "medium", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl, MIT)", "source_split": "valid", "reserved": "dev_preview (PLAN.md Phase 7): never in test", "informal": "Find $x^2+y^2_{}$ if $x_{}^{}$ and $y_{}^{}$ are positive integers such that\n$\\begin{align*}\nxy+x+y&=71, \\\\\nx^2y+xy^2&=880.\n\\end{align*}$ Show that it is 146.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem aime_1991_p1 (x y : ℕ) (h₀ : 0 < x ∧ 0 < y) (h₁ : x * y + (x + y) = 71)\n (h₂ : x ^ 2 * y + x * y ^ 2 = 880) : x ^ 2 + y ^ 2 = 146 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem aime_1991_p1 (x y : ℕ) (h₀ : 0 < x ∧ 0 < y) (h₁ : x * y + (x + y) = 71)\n (h₂ : x ^ 2 * y + x * y ^ 2 = 880) : x ^ 2 + y ^ 2 = 146 := by\n"}
{"id": "amc12b_2004_p3", "split": "dev", "area": "number_theory", "difficulty": "medium", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl, MIT)", "source_split": "valid", "reserved": "dev_preview (PLAN.md Phase 7): never in test", "informal": "If $x$ and $y$ are positive integers for which $2^x3^y=1296$, what is the value of $x+y$?\n\n$(\\mathrm {A})\\ 8 \\qquad (\\mathrm {B})\\ 9 \\qquad (\\mathrm {C})\\ 10 \\qquad (\\mathrm {D})\\ 11 \\qquad (\\mathrm {E})\\ 12$ Show that it is 8.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem amc12b_2004_p3 (x y : ℕ) (h₀ : 2 ^ x * 3 ^ y = 1296) : x + y = 8 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem amc12b_2004_p3 (x y : ℕ) (h₀ : 2 ^ x * 3 ^ y = 1296) : x + y = 8 := by\n"}
{"id": "induction_divisibility_3div2tooddnp1", "split": "dev", "area": "number_theory", "difficulty": "medium", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl, MIT)", "source_split": "valid", "reserved": "dev_preview (PLAN.md Phase 7): never in test", "informal": "For a natural number $n$, show that $3 \\mid (2^{2n+1}+1)$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem induction_divisibility_3div2tooddnp1 (n : ℕ) : 3 ∣ 2 ^ (2 * n + 1) + 1 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem induction_divisibility_3div2tooddnp1 (n : ℕ) : 3 ∣ 2 ^ (2 * n + 1) + 1 := by\n"}
{"id": "numbertheory_nckeqnm1ckpnm1ckm1", "split": "dev", "area": "combinatorics", "difficulty": "medium", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl, MIT)", "source_split": "valid", "reserved": "dev_preview (PLAN.md Phase 7): never in test", "informal": "Show that for positive integers $n$ and $k$ with $k \\leq n$, we have \n$\\binom{n}{k} = \\binom{n-1}{k} + \\binom{n-1}{k-1}$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : ℕ) (h₀ : 0 < n ∧ 0 < k) (h₁ : k ≤ n) :\n Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : ℕ) (h₀ : 0 < n ∧ 0 < k) (h₁ : k ≤ n) :\n Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) := by\n"}
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