{"id": "mathd_algebra_441", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_algebra_441", "source_split": "test", "license": "MIT", "informal": "Assuming $x\\ne0$, simplify $\\frac{12}{x \\cdot x} \\cdot \\frac{x^4}{14x}\\cdot \\frac{35}{3x}$. Show that it is 10.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_algebra_441 (x : ℝ) (h₀ : x ≠ 0) :\n    12 / (x * x) * (x ^ 4 / (14 * x)) * (35 / (3 * x)) = 10 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_algebra_441 (x : ℝ) (h₀ : x ≠ 0) :\n    12 / (x * x) * (x ^ 4 / (14 * x)) * (35 / (3 * x)) = 10 := by\n", "area": "algebra", "difficulty": "easy", "split": "dev", "seed": 2026}
{"id": "mathd_algebra_209", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_algebra_209", "source_split": "test", "license": "MIT", "informal": "Suppose that $h(x)=f^{-1}(x)$. If $h(2)=10$, $h(10)=1$ and $h(1)=2$, what is $f(f(10))$? Show that it is 1.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_algebra_209 (σ : Equiv ℝ ℝ) (h₀ : σ.2 2 = 10) (h₁ : σ.2 10 = 1) (h₂ : σ.2 1 = 2) :\n    σ.1 (σ.1 10) = 1 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_algebra_209 (σ : Equiv ℝ ℝ) (h₀ : σ.2 2 = 10) (h₁ : σ.2 10 = 1) (h₂ : σ.2 1 = 2) :\n    σ.1 (σ.1 10) = 1 := by\n", "area": "algebra", "difficulty": "easy", "split": "dev", "seed": 2026}
{"id": "mathd_algebra_69", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_algebra_69", "source_split": "valid", "license": "MIT", "informal": "An audience of $450$ people is seated in an auditorium. Each row contains the same number of seats and each seat in the auditorium is occupied. With three fewer seats per row, and five extra rows, the same audience could still be seated, occupying all seats. How many rows does the auditorium have? Show that it is 25.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_algebra_69 (rows seats : ℕ) (h₀ : rows * seats = 450)\n  (h₁ : (rows + 5) * (seats - 3) = 450) : rows = 25 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_algebra_69 (rows seats : ℕ) (h₀ : rows * seats = 450)\n  (h₁ : (rows + 5) * (seats - 3) = 450) : rows = 25 := by\n", "area": "algebra", "difficulty": "easy", "split": "dev", "seed": 2026}
{"id": "mathd_algebra_206", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_algebra_206", "source_split": "valid", "license": "MIT", "informal": "The polynomial $p(x) = x^2+ax+b$ has distinct roots $2a$ and $b$. Find $a+b$. Show that it is -1.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_algebra_206 (a b : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f x = x ^ 2 + a * x + b) (h₁ : 2 * a ≠ b)\n  (h₂ : f (2 * a) = 0) (h₃ : f b = 0) : a + b = -1 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_algebra_206 (a b : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f x = x ^ 2 + a * x + b) (h₁ : 2 * a ≠ b)\n  (h₂ : f (2 * a) = 0) (h₃ : f b = 0) : a + b = -1 := by\n", "area": "algebra", "difficulty": "easy", "split": "test", "seed": 2026}
{"id": "mathd_algebra_455", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_algebra_455", "source_split": "valid", "license": "MIT", "informal": "At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice.  At her fifth practice she made 48 free throws.  How many free throws did she make at the first practice? Show that it is 3.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_algebra_455 (x : ℝ) (h₀ : 2 * (2 * (2 * (2 * x))) = 48) : x = 3 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_algebra_455 (x : ℝ) (h₀ : 2 * (2 * (2 * (2 * x))) = 48) : x = 3 := by\n", "area": "algebra", "difficulty": "easy", "split": "test", "seed": 2026}
{"id": "mathd_algebra_234", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_algebra_234", "source_split": "valid", "license": "MIT", "informal": "What is the sixth term in the geometric sequence $\\frac{27}{125}, \\frac{9}{25}, \\frac{3}{5},\\ldots$?  Express your answer as a common fraction. Show that it is \\frac{25}{9}.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_algebra_234 (d : ℝ) (h₀ : 27 / 125 * d = 9 / 25) : 3 / 5 * d ^ 3 = 25 / 9 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_algebra_234 (d : ℝ) (h₀ : 27 / 125 * d = 9 / 25) : 3 / 5 * d ^ 3 = 25 / 9 := by\n", "area": "algebra", "difficulty": "easy", "split": "test", "seed": 2026}
{"id": "mathd_algebra_160", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_algebra_160", "source_split": "test", "license": "MIT", "informal": "For each plumbing repair job, Mr. Wrench charges $N$ dollars for coming out to the house plus $x$ dollars per hour that he works at the house. He charged $\\$97$ for a one-hour repair job and $\\$265$ for a five-hour repair job. What is his charge for a two-hour repair job? Show that it is \\$ 139.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_algebra_160 (n x : ℝ) (h₀ : n + x = 97) (h₁ : n + 5 * x = 265) : n + 2 * x = 139 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_algebra_160 (n x : ℝ) (h₀ : n + x = 97) (h₁ : n + 5 * x = 265) : n + 2 * x = 139 := by\n", "area": "algebra", "difficulty": "easy", "split": "test", "seed": 2026}
{"id": "mathd_algebra_288", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_algebra_288", "source_split": "test", "license": "MIT", "informal": "A point $(x,y)$ on the coordinate plane with both coordinates negative is a distance of 6 units from the $x$-axis. It is a distance of 15 units from the point $(8,3)$. It is a distance $\\sqrt{n}$ from the origin. What is $n$? Show that it is 52.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_algebra_288 (x y : ℝ) (n : NNReal) (h₀ : x < 0 ∧ y < 0) (h₁ : abs y = 6)\n    (h₂ : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) = 15)\n    (h₃ : Real.sqrt (x ^ 2 + y ^ 2) = Real.sqrt n) : n = 52 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_algebra_288 (x y : ℝ) (n : NNReal) (h₀ : x < 0 ∧ y < 0) (h₁ : abs y = 6)\n    (h₂ : Real.sqrt ((x - 8) ^ 2 + (y - 3) ^ 2) = 15)\n    (h₃ : Real.sqrt (x ^ 2 + y ^ 2) = Real.sqrt n) : n = 52 := by\n", "area": "algebra", "difficulty": "easy", "split": "test", "seed": 2026}
{"id": "mathd_numbertheory_353", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_numbertheory_353", "source_split": "test", "license": "MIT", "informal": "Let $S = 2010 + 2011 + \\cdots + 4018$. Compute the residue of $S$, modulo 2009. Show that it is 0.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_numbertheory_353 (s : ℕ) (h₀ : s = ∑ k ∈ Finset.Icc 2010 4018, k) : s % 2009 = 0 := by\n", "formal_statement_original": "theorem mathd_numbertheory_353 (s : ℕ) (h₀ : s = ∑ k in Finset.Icc 2010 4018, k) : s % 2009 = 0 := by\n", "converted": "big_operator_in_to_mem", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_numbertheory_353 (s : ℕ) (h₀ : s = ∑ k ∈ Finset.Icc 2010 4018, k) : s % 2009 = 0 := by\n", "area": "number_theory", "difficulty": "easy", "split": "dev", "seed": 2026}
{"id": "mathd_numbertheory_232", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_numbertheory_232", "source_split": "valid", "license": "MIT", "informal": "Compute $(3^{-1}+5^{-1})^{-1}\\pmod{31}$. Express your answer as an integer from $0$ to $30$, inclusive. Show that it is 29.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_numbertheory_232 (x y z : ZMod 31) (h₀ : x = 3⁻¹) (h₁ : y = 5⁻¹)\n  (h₂ : z = (x + y)⁻¹) : z = 29 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_numbertheory_232 (x y z : ZMod 31) (h₀ : x = 3⁻¹) (h₁ : y = 5⁻¹)\n  (h₂ : z = (x + y)⁻¹) : z = 29 := by\n", "area": "number_theory", "difficulty": "easy", "split": "dev", "seed": 2026}
{"id": "mathd_numbertheory_430", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_numbertheory_430", "source_split": "test", "license": "MIT", "informal": "If $A$, $B$, and $C$ represent three distinct digits from 1 to 9 and they satisfy the following equations, what is the value of the sum $A+B+C$? (In the equation below, $AA$ represents a two-digit number both of whose digits are $A$.) $$A+B=C$$$$AA-B=2\\times C$$$$C\\times B=AA+A$$ 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 mathd_numbertheory_430 (a b c : ℕ) (h₀ : 1 ≤ a ∧ a ≤ 9) (h₁ : 1 ≤ b ∧ b ≤ 9)\n    (h₂ : 1 ≤ c ∧ c ≤ 9) (h₃ : a ≠ b) (h₄ : a ≠ c) (h₅ : b ≠ c) (h₆ : a + b = c)\n    (h₇ : 10 * a + a - b = 2 * c) (h₈ : c * b = 10 * a + a + a) : a + b + c = 8 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_numbertheory_430 (a b c : ℕ) (h₀ : 1 ≤ a ∧ a ≤ 9) (h₁ : 1 ≤ b ∧ b ≤ 9)\n    (h₂ : 1 ≤ c ∧ c ≤ 9) (h₃ : a ≠ b) (h₄ : a ≠ c) (h₅ : b ≠ c) (h₆ : a + b = c)\n    (h₇ : 10 * a + a - b = 2 * c) (h₈ : c * b = 10 * a + a + a) : a + b + c = 8 := by\n", "area": "number_theory", "difficulty": "easy", "split": "test", "seed": 2026}
{"id": "mathd_numbertheory_328", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_numbertheory_328", "source_split": "test", "license": "MIT", "informal": "What is the remainder when 5 to the 999,999th power is divided by 7? Show that it is 6.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_numbertheory_328 : 5 ^ 999999 % 7 = 6 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_numbertheory_328 : 5 ^ 999999 % 7 = 6 := by\n", "area": "number_theory", "difficulty": "easy", "split": "test", "seed": 2026}
{"id": "mathd_numbertheory_64", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_numbertheory_64", "source_split": "valid", "license": "MIT", "informal": "What is the smallest positive integer that satisfies the congruence $30x \\equiv 42 \\pmod{47}$? Show that it is 39.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_numbertheory_64 : IsLeast { x : ℕ | 30 * x ≡ 42 [MOD 47] } 39 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_numbertheory_64 : IsLeast { x : ℕ | 30 * x ≡ 42 [MOD 47] } 39 := by\n", "area": "number_theory", "difficulty": "easy", "split": "test", "seed": 2026}
{"id": "mathd_numbertheory_321", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_numbertheory_321", "source_split": "test", "license": "MIT", "informal": "Notice that  \\[35\\cdot40=1400.\\]Find some integer $n$ with $0\\leq n<1399$ such that $n$ is the multiplicative inverse to 160 modulo 1399. Show that it is 1058.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_numbertheory_321 (n : ZMod 1399) (h₁ : n = 160⁻¹) : n = 1058 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_numbertheory_321 (n : ZMod 1399) (h₁ : n = 160⁻¹) : n = 1058 := by\n", "area": "number_theory", "difficulty": "easy", "split": "test", "seed": 2026}
{"id": "mathd_numbertheory_345", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_numbertheory_345", "source_split": "test", "license": "MIT", "informal": "What is the remainder when $2000+2001+2002+2003+2004+2005+2006$ is divided by $7$? Show that it is 0.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_numbertheory_345 : (2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006) % 7 = 0 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_numbertheory_345 : (2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006) % 7 = 0 := by\n", "area": "number_theory", "difficulty": "easy", "split": "test", "seed": 2026}
{"id": "amc12a_2015_p10", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "amc12a_2015_p10", "source_split": "valid", "license": "MIT", "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", "area": "algebra", "difficulty": "medium", "split": "dev", "seed": 2026}
{"id": "aime_1991_p1", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "aime_1991_p1", "source_split": "valid", "license": "MIT", "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", "area": "algebra", "difficulty": "medium", "split": "dev", "seed": 2026}
{"id": "amc12_2001_p9", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "amc12_2001_p9", "source_split": "valid", "license": "MIT", "informal": "Let $f$ be a function satisfying $f(xy) = \\frac{f(x)}y$ for all positive real numbers $x$ and $y$. If $f(500) =3$, what is the value of $f(600)$?\n\n$(\\mathrm{A})\\ 1 \\qquad (\\mathrm{B})\\ 2 \\qquad (\\mathrm{C})\\ \\frac52 \\qquad (\\mathrm{D})\\ 3 \\qquad (\\mathrm{E})\\ \\frac{18}5$ Show that it is \\textbf{C } \\frac52.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem amc12_2001_p9 (f : ℝ → ℝ) (h₀ : ∀ x > 0, ∀ y > 0, f (x * y) = f x / y) (h₁ : f 500 = 3) :\n    f 600 = 5 / 2 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem amc12_2001_p9 (f : ℝ → ℝ) (h₀ : ∀ x > 0, ∀ y > 0, f (x * y) = f x / y) (h₁ : f 500 = 3) :\n    f 600 = 5 / 2 := by\n", "area": "algebra", "difficulty": "medium", "split": "test", "seed": 2026}
{"id": "amc12_2000_p20", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "amc12_2000_p20", "source_split": "test", "license": "MIT", "informal": "If $x,y,$ and $z$ are positive numbers satisfying\n\n$x + \\frac{1}{y} = 4,\\qquad y + \\frac{1}{z} = 1, \\qquad \\text{and} \\qquad z + \\frac{1}{x} = \\frac{7}{3}$\n\nThen what is the value of $xyz$ ?\n\n$\\text {(A)}\\ \\frac{2}{3} \\qquad \\text {(B)}\\ 1 \\qquad \\text {(C)}\\ \\frac{4}{3} \\qquad \\text {(D)}\\ 2 \\qquad \\text {(E)}\\ \\frac{7}{3}$ Show that it is xyz = 1 \\rightarrow B.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem amc12_2000_p20 (x y z : ℝ) (h₀ : 0 < x ∧ 0 < y ∧ 0 < z) (h₁ : x + 1 / y = 4)\n    (h₂ : y + 1 / z = 1) (h₃ : z + 1 / x = 7 / 3) : x * y * z = 1 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem amc12_2000_p20 (x y z : ℝ) (h₀ : 0 < x ∧ 0 < y ∧ 0 < z) (h₁ : x + 1 / y = 4)\n    (h₂ : y + 1 / z = 1) (h₃ : z + 1 / x = 7 / 3) : x * y * z = 1 := by\n", "area": "algebra", "difficulty": "medium", "split": "test", "seed": 2026}
{"id": "amc12a_2008_p25", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "amc12a_2008_p25", "source_split": "test", "license": "MIT", "informal": "A sequence $(a_1,b_1)$, $(a_2,b_2)$, $(a_3,b_3)$, $\\ldots$ of points in the coordinate plane satisfies\n\n$(a_{n + 1}, b_{n + 1}) = (\\sqrt {3}a_n - b_n, \\sqrt {3}b_n + a_n)$  for $n = 1,2,3,\\ldots$.\n\nSuppose that $(a_{100},b_{100}) = (2,4)$.  What is $a_1 + b_1$?\n\n$\\mathrm{(A)}\\ -\\frac{1}{2^{97}}\\qquad\\mathrm{(B)}\\ -\\frac{1}{2^{99}}\\qquad\\mathrm{(C)}\\ 0\\qquad\\mathrm{(D)}\\ \\frac{1}{2^{98}}\\qquad\\mathrm{(E)}\\ \\frac{1}{2^{96}}$ Show that it is \\mathrm{(D)}.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem amc12a_2008_p25 (a b : ℕ → ℝ) (h₀ : ∀ n, a (n + 1) = Real.sqrt 3 * a n - b n)\n    (h₁ : ∀ n, b (n + 1) = Real.sqrt 3 * b n + a n) (h₂ : a 100 = 2) (h₃ : b 100 = 4) :\n    a 1 + b 1 = 1 / 2 ^ 98 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem amc12a_2008_p25 (a b : ℕ → ℝ) (h₀ : ∀ n, a (n + 1) = Real.sqrt 3 * a n - b n)\n    (h₁ : ∀ n, b (n + 1) = Real.sqrt 3 * b n + a n) (h₂ : a 100 = 2) (h₃ : b 100 = 4) :\n    a 1 + b 1 = 1 / 2 ^ 98 := by\n", "area": "algebra", "difficulty": "medium", "split": "test", "seed": 2026}
{"id": "algebra_absapbon1pabsapbleqsumabsaon1pabsa", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "algebra_absapbon1pabsapbleqsumabsaon1pabsa", "source_split": "test", "license": "MIT", "informal": "Show that for any real numbers $a$ and $b$, $\\frac{|a+b|}{1+|a+b|}\\leq \\frac{|a|}{1+|a|}+\\frac{|b|}{1+|b|}$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem algebra_absapbon1pabsapbleqsumabsaon1pabsa (a b : ℝ) :\n    abs (a + b) / (1 + abs (a + b)) ≤ abs a / (1 + abs a) + abs b / (1 + abs b) := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem algebra_absapbon1pabsapbleqsumabsaon1pabsa (a b : ℝ) :\n    abs (a + b) / (1 + abs (a + b)) ≤ abs a / (1 + abs a) + abs b / (1 + abs b) := by\n", "area": "algebra", "difficulty": "medium", "split": "test", "seed": 2026}
{"id": "amc12b_2003_p6", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "amc12b_2003_p6", "source_split": "valid", "license": "MIT", "informal": "The second and fourth terms of a geometric sequence are $2$ and $6$. Which of the following is a possible first term?\n\n$\\textbf{(A) } -\\sqrt{3}  \\qquad\\textbf{(B) } -\\frac{2\\sqrt{3}}{3} \\qquad\\textbf{(C) } -\\frac{\\sqrt{3}}{3} \\qquad\\textbf{(D) } \\sqrt{3} \\qquad\\textbf{(E) } 3$ Show that it is \\textbf{(B)}\\ -\\frac{2\\sqrt{3}}{3}.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem amc12b_2003_p6 (a r : ℝ) (u : ℕ → ℝ) (h₀ : ∀ k, u k = a * r ^ k) (h₁ : u 1 = 2)\n  (h₂ : u 3 = 6) : u 0 = 2 / Real.sqrt 3 ∨ u 0 = -(2 / Real.sqrt 3) := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem amc12b_2003_p6 (a r : ℝ) (u : ℕ → ℝ) (h₀ : ∀ k, u k = a * r ^ k) (h₁ : u 1 = 2)\n  (h₂ : u 3 = 6) : u 0 = 2 / Real.sqrt 3 ∨ u 0 = -(2 / Real.sqrt 3) := by\n", "area": "algebra", "difficulty": "medium", "split": "test", "seed": 2026}
{"id": "amc12b_2004_p3", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "amc12b_2004_p3", "source_split": "valid", "license": "MIT", "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", "area": "number_theory", "difficulty": "medium", "split": "dev", "seed": 2026}
{"id": "induction_divisibility_3div2tooddnp1", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "induction_divisibility_3div2tooddnp1", "source_split": "valid", "license": "MIT", "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", "area": "number_theory", "difficulty": "medium", "split": "dev", "seed": 2026}
{"id": "amc12b_2002_p3", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "amc12b_2002_p3", "source_split": "valid", "license": "MIT", "informal": "For how many positive integers $n$ is $n^2 - 3n + 2$ a [[prime]] number?\n\n$\\mathrm{(A)}\\ \\text{none}\n\\qquad\\mathrm{(B)}\\ \\text{one}\n\\qquad\\mathrm{(C)}\\ \\text{two}\n\\qquad\\mathrm{(D)}\\ \\text{more\\ than\\ two,\\ but\\ finitely\\ many}\n\\qquad\\mathrm{(E)}\\ \\text{infinitely\\ many}$ Show that it is \\mathrm{(B)}\\ \\text{one}.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem amc12b_2002_p3 (S : Finset ℕ)\n  (h₀ : ∀ n : ℕ, n ∈ S ↔ 0 < n ∧ Nat.Prime (n ^ 2 + 2 - 3 * n)) :\n  S.card = 1 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem amc12b_2002_p3 (S : Finset ℕ)\n  (h₀ : ∀ n : ℕ, n ∈ S ↔ 0 < n ∧ Nat.Prime (n ^ 2 + 2 - 3 * n)) :\n  S.card = 1 := by\n", "area": "number_theory", "difficulty": "medium", "split": "test", "seed": 2026}
{"id": "amc12a_2008_p15", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "amc12a_2008_p15", "source_split": "valid", "license": "MIT", "informal": "Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?\n\n$\\mathrm{(A)}\\ 0\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 8$ Show that it is D.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem amc12a_2008_p15 (k : ℕ) (h₀ : k = 2008 ^ 2 + 2 ^ 2008) : (k ^ 2 + 2 ^ k) % 10 = 6 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem amc12a_2008_p15 (k : ℕ) (h₀ : k = 2008 ^ 2 + 2 ^ 2008) : (k ^ 2 + 2 ^ k) % 10 = 6 := by\n", "area": "number_theory", "difficulty": "medium", "split": "test", "seed": 2026}
{"id": "numbertheory_notequiv2i2jasqbsqdiv8", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "numbertheory_notequiv2i2jasqbsqdiv8", "source_split": "test", "license": "MIT", "informal": "Let $a$ and $b$ be two integer numbers. Show that the following statement is false: $a$ and $b$ are both even iff $8 \\mid a^2 + b^2$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem numbertheory_notequiv2i2jasqbsqdiv8 :\n    ¬∀ a b : ℤ, (∃ i j, a = 2 * i ∧ b = 2 * j) ↔ ∃ k, a ^ 2 + b ^ 2 = 8 * k := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem numbertheory_notequiv2i2jasqbsqdiv8 :\n    ¬∀ a b : ℤ, (∃ i j, a = 2 * i ∧ b = 2 * j) ↔ ∃ k, a ^ 2 + b ^ 2 = 8 * k := by\n", "area": "number_theory", "difficulty": "medium", "split": "test", "seed": 2026}
{"id": "numbertheory_3pow2pownm1mod2pownp3eq2pownp2", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "numbertheory_3pow2pownm1mod2pownp3eq2pownp2", "source_split": "test", "license": "MIT", "informal": "Show that for positive integer $n$, the remainder of $3^{2^n} - 1 \\equiv 2^{n+2} \\mod 2^{n+3}$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem numbertheory_3pow2pownm1mod2pownp3eq2pownp2 (n : ℕ) (h₀ : 0 < n) :\n    (3 ^ 2 ^ n - 1) % 2 ^ (n + 3) = 2 ^ (n + 2) := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem numbertheory_3pow2pownm1mod2pownp3eq2pownp2 (n : ℕ) (h₀ : 0 < n) :\n    (3 ^ 2 ^ n - 1) % 2 ^ (n + 3) = 2 ^ (n + 2) := by\n", "area": "number_theory", "difficulty": "medium", "split": "test", "seed": 2026}
{"id": "numbertheory_nckeqnm1ckpnm1ckm1", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "numbertheory_nckeqnm1ckpnm1ckm1", "source_split": "valid", "license": "MIT", "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", "area": "combinatorics", "difficulty": "medium", "split": "dev", "seed": 2026}
{"id": "numbertheory_sumkmulnckeqnmul2pownm1", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "numbertheory_sumkmulnckeqnmul2pownm1", "source_split": "valid", "license": "MIT", "informal": "Show that for positive integers $n$ and $k$, if $k \\leq n$, then $\\sum_{k=1}^n (k*C_n^k) = n * 2^{n-1}$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : ℕ) (h₀ : 0 < n) :\n  (∑ k ∈ Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) := by\n", "formal_statement_original": "theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : ℕ) (h₀ : 0 < n) :\n  (∑ k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) := by\n", "converted": "big_operator_in_to_mem", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem numbertheory_sumkmulnckeqnmul2pownm1 (n : ℕ) (h₀ : 0 < n) :\n  (∑ k ∈ Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) := by\n", "area": "combinatorics", "difficulty": "medium", "split": "test", "seed": 2026}
{"id": "imo_1987_p4", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1987_p4", "source_split": "valid", "license": "MIT", "informal": "Prove that there is no function $f $ from the set of non-negative  integers into itself such that $f(f(n)) = n + 1987 $ for every $n $.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_1987_p4 (f : ℕ → ℕ) : ∃ n, f (f n) ≠ n + 1987 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1987_p4 (f : ℕ → ℕ) : ∃ n, f (f n) ≠ n + 1987 := by\n", "area": "algebra", "difficulty": "hard", "split": "dev", "seed": 2026}
{"id": "imo_2006_p3", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_2006_p3", "source_split": "valid", "license": "MIT", "informal": "Show that for any real numbers $a$, $b$, and $c$, we have $(ab(a^2 - b^2)) + (bc(b^2 - c^2)) + (ca(c^2 - a^2)) \\leq \\frac{9\\sqrt{2}}{32}(a^2 + b^2 + c^2)^2$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_2006_p3 (a b c : ℝ) :\n  a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2) ≤\n  9 * Real.sqrt 2 / 32 * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_2006_p3 (a b c : ℝ) :\n  a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2) ≤\n  9 * Real.sqrt 2 / 32 * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2 := by\n", "area": "algebra", "difficulty": "hard", "split": "dev", "seed": 2026}
{"id": "imo_1965_p2", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1965_p2", "source_split": "test", "license": "MIT", "informal": "Consider the system of equations\n$a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = 0$\n$a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = 0$\n$a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = 0$\nwith unknowns $x_1$, $x_2$, $x_3$. The coefficients satisfy the conditions:\n\n(a) $a_{11}$, $a_{22}$, $a_{33}$ are positive numbers;\n\n(b) the remaining coefficients are negative numbers;\n\n(c) in each equation, the sum of the coefficients is positive.\n\nProve that the given system has only the solution $x_1 = x_2 = x_3 = 0$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_1965_p2 (x y z : ℝ) (a : ℕ → ℝ) (h₀ : 0 < a 0 ∧ 0 < a 4 ∧ 0 < a 8)\n    (h₁ : a 1 < 0 ∧ a 2 < 0) (h₂ : a 3 < 0 ∧ a 5 < 0) (h₃ : a 6 < 0 ∧ a 7 < 0)\n    (h₄ : 0 < a 0 + a 1 + a 2) (h₅ : 0 < a 3 + a 4 + a 5) (h₆ : 0 < a 6 + a 7 + a 8)\n    (h₇ : a 0 * x + a 1 * y + a 2 * z = 0) (h₈ : a 3 * x + a 4 * y + a 5 * z = 0)\n    (h₉ : a 6 * x + a 7 * y + a 8 * z = 0) : x = 0 ∧ y = 0 ∧ z = 0 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1965_p2 (x y z : ℝ) (a : ℕ → ℝ) (h₀ : 0 < a 0 ∧ 0 < a 4 ∧ 0 < a 8)\n    (h₁ : a 1 < 0 ∧ a 2 < 0) (h₂ : a 3 < 0 ∧ a 5 < 0) (h₃ : a 6 < 0 ∧ a 7 < 0)\n    (h₄ : 0 < a 0 + a 1 + a 2) (h₅ : 0 < a 3 + a 4 + a 5) (h₆ : 0 < a 6 + a 7 + a 8)\n    (h₇ : a 0 * x + a 1 * y + a 2 * z = 0) (h₈ : a 3 * x + a 4 * y + a 5 * z = 0)\n    (h₉ : a 6 * x + a 7 * y + a 8 * z = 0) : x = 0 ∧ y = 0 ∧ z = 0 := by\n", "area": "algebra", "difficulty": "hard", "split": "test", "seed": 2026}
{"id": "imo_1993_p5", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1993_p5", "source_split": "valid", "license": "MIT", "informal": "Let $\\mathbb{N} = \\{1,2,3, \\ldots\\}$. Determine if there exists a strictly increasing function $f: \\mathbb{N} \\mapsto \\mathbb{N}$ with the following properties:\n\n(i) $f(1) = 2$;\n\n(ii) $f(f(n)) = f(n) + n, (n \\in \\mathbb{N})$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_1993_p5 : ∃ f : ℕ → ℕ, f 1 = 2 ∧ ∀ n, f (f n) = f n + n ∧ ∀ n, f n < f (n + 1) := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1993_p5 : ∃ f : ℕ → ℕ, f 1 = 2 ∧ ∀ n, f (f n) = f n + n ∧ ∀ n, f n < f (n + 1) := by\n", "area": "algebra", "difficulty": "hard", "split": "test", "seed": 2026}
{"id": "imo_1978_p5", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1978_p5", "source_split": "valid", "license": "MIT", "informal": "Let $f$ be an injective function from ${1,2,3,\\ldots}$ in itself. Prove that for any $n$ we have: $\\sum_{k=1}^{n} f(k)k^{-2} \\geq \\sum_{k=1}^{n} k^{-1}.$", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_1978_p5 (n : ℕ) (a : ℕ → ℕ) (h₀ : Function.Injective a) (h₁ : a 0 = 0) (h₂ : 0 < n) :\n  (∑ k ∈ Finset.Icc 1 n, (1 : ℝ) / k) ≤ ∑ k ∈ Finset.Icc 1 n, (a k : ℝ) / k ^ 2 := by\n", "formal_statement_original": "theorem imo_1978_p5 (n : ℕ) (a : ℕ → ℕ) (h₀ : Function.Injective a) (h₁ : a 0 = 0) (h₂ : 0 < n) :\n  (∑ k in Finset.Icc 1 n, (1 : ℝ) / k) ≤ ∑ k in Finset.Icc 1 n, (a k : ℝ) / k ^ 2 := by\n", "converted": "big_operator_in_to_mem", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1978_p5 (n : ℕ) (a : ℕ → ℕ) (h₀ : Function.Injective a) (h₁ : a 0 = 0) (h₂ : 0 < n) :\n  (∑ k ∈ Finset.Icc 1 n, (1 : ℝ) / k) ≤ ∑ k ∈ Finset.Icc 1 n, (a k : ℝ) / k ^ 2 := by\n", "area": "algebra", "difficulty": "hard", "split": "test", "seed": 2026}
{"id": "imo_1974_p3", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1974_p3", "source_split": "test", "license": "MIT", "informal": "Prove that the number $\\sum^n_{k=0}\\binom{2n+1}{2k+1}2^{3k}$ is not divisible by $5$ for any integer $n\\ge0.$", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_1974_p3 (n : ℕ) :\n    ¬5 ∣ ∑ k ∈ Finset.range (n + 1), Nat.choose (2 * n + 1) (2 * k + 1) * 2 ^ (3 * k) := by\n", "formal_statement_original": "theorem imo_1974_p3 (n : ℕ) :\n    ¬5 ∣ ∑ k in Finset.range (n + 1), Nat.choose (2 * n + 1) (2 * k + 1) * 2 ^ (3 * k) := by\n", "converted": "big_operator_in_to_mem", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1974_p3 (n : ℕ) :\n    ¬5 ∣ ∑ k ∈ Finset.range (n + 1), Nat.choose (2 * n + 1) (2 * k + 1) * 2 ^ (3 * k) := by\n", "area": "number_theory", "difficulty": "hard", "split": "dev", "seed": 2026}
{"id": "imo_1987_p6", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1987_p6", "source_split": "valid", "license": "MIT", "informal": "Let $n $ be an integer greater than or equal to 2.  Prove that if $k^2 + k + n $ is prime for all integers $k $ such that $ 0 \\leq k \\leq \\sqrt{n/3} $, then $k^2 + k + n $ is prime for all integers $k $ such that $0 \\leq k \\leq n - 2 $.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_1987_p6 (p : ℕ) (f : ℕ → ℕ) (h₀ : ∀ x, f x = x ^ 2 + x + p)\n  (h₀ : ∀ k : ℕ, k ≤ Nat.floor (Real.sqrt (p / 3)) → Nat.Prime (f k)) :\n   ∀ i ≤ p - 2, Nat.Prime (f i) := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1987_p6 (p : ℕ) (f : ℕ → ℕ) (h₀ : ∀ x, f x = x ^ 2 + x + p)\n  (h₀ : ∀ k : ℕ, k ≤ Nat.floor (Real.sqrt (p / 3)) → Nat.Prime (f k)) :\n   ∀ i ≤ p - 2, Nat.Prime (f i) := by\n", "area": "number_theory", "difficulty": "hard", "split": "test", "seed": 2026}
{"id": "imo_1979_p1", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1979_p1", "source_split": "valid", "license": "MIT", "informal": "If $p$ and $q$ are natural numbers so that$ \\frac{p}{q}=1-\\frac{1}{2}+\\frac{1}{3}-\\frac{1}{4}+ \\ldots -\\frac{1}{1318}+\\frac{1}{1319}, $prove that $p$ is divisible with $1979$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_1979_p1 (p q : ℕ) (h₀ : 0 < q)\n  (h₁ : (∑ k ∈ Finset.Icc (1 : ℕ) 1319, (-1) ^ (k + 1) * ((1 : ℝ) / k)) = p / q) : 1979 ∣ p := by\n", "formal_statement_original": "theorem imo_1979_p1 (p q : ℕ) (h₀ : 0 < q)\n  (h₁ : (∑ k in Finset.Icc (1 : ℕ) 1319, (-1) ^ (k + 1) * ((1 : ℝ) / k)) = p / q) : 1979 ∣ p := by\n", "converted": "big_operator_in_to_mem", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1979_p1 (p q : ℕ) (h₀ : 0 < q)\n  (h₁ : (∑ k ∈ Finset.Icc (1 : ℕ) 1319, (-1) ^ (k + 1) * ((1 : ℝ) / k)) = p / q) : 1979 ∣ p := by\n", "area": "number_theory", "difficulty": "hard", "split": "test", "seed": 2026}
{"id": "imo_1984_p6", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1984_p6", "source_split": "test", "license": "MIT", "informal": "Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_1984_p6 (a b c d k m : ℕ) (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)\n    (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d) (h₂ : a < b ∧ b < c ∧ c < d) (h₃ : a * d = b * c)\n    (h₄ : a + d = 2 ^ k) (h₅ : b + c = 2 ^ m) : a = 1 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1984_p6 (a b c d k m : ℕ) (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)\n    (h₁ : Odd a ∧ Odd b ∧ Odd c ∧ Odd d) (h₂ : a < b ∧ b < c ∧ c < d) (h₃ : a * d = b * c)\n    (h₄ : a + d = 2 ^ k) (h₅ : b + c = 2 ^ m) : a = 1 := by\n", "area": "number_theory", "difficulty": "hard", "split": "test", "seed": 2026}
{"id": "imo_1964_p1_2", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1964_p1_2", "source_split": "valid", "license": "MIT", "informal": "Show that for any natural number $n$, $7$ does not divide $2^n + 1$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_1964_p1_2 (n : ℕ) : ¬7 ∣ 2 ^ n + 1 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1964_p1_2 (n : ℕ) : ¬7 ∣ 2 ^ n + 1 := by\n", "area": "number_theory", "difficulty": "hard", "split": "test", "seed": 2026}
{"id": "putnam_1966_b4", "source": "PutnamBench (lean4/src, informal/putnam.json)", "source_commit": "a635813f5edf297d928f9bd5ef8324a26bdde09c", "source_id": "putnam_1966_b4", "source_split": null, "license": "Apache-2.0 (Lean statement); informal text available with permission from the MAA", "informal": "[Informal statement not redistributed: PutnamBench states it is available with permission from the MAA. Problem: Putnam 1966 B4; source: PutnamBench informal/putnam.json, commit a635813f.]", "header": "import Mathlib\n\nopen Topology Filter\n\n", "source_tags": ["number_theory", "combinatorics"], "formal_statement": "theorem putnam_1966_b4\n(m n : ℕ)\n(S : Finset ℕ)\n(hS : (∀ i ∈ S, i > 0) ∧ S.card = m * n + 1)\n: ∃ T ⊆ S, (T.card = m + 1 ∧ ∀ j ∈ T, ∀ i ∈ T, i ≠ j → ¬(j ∣ i)) ∨ (T.card = n + 1 ∧ ∀ i ∈ T, ∀ j ∈ T, j < i → j ∣ i) := by\n", "statement": "import Mathlib\n\nopen Topology Filter\n\ntheorem putnam_1966_b4\n(m n : ℕ)\n(S : Finset ℕ)\n(hS : (∀ i ∈ S, i > 0) ∧ S.card = m * n + 1)\n: ∃ T ⊆ S, (T.card = m + 1 ∧ ∀ j ∈ T, ∀ i ∈ T, i ≠ j → ¬(j ∣ i)) ∨ (T.card = n + 1 ∧ ∀ i ∈ T, ∀ j ∈ T, j < i → j ∣ i) := by\n", "area": "combinatorics", "difficulty": "hard", "split": "dev", "seed": 2026, "informal_redacted": true}
{"id": "putnam_1965_a4", "source": "PutnamBench (lean4/src, informal/putnam.json)", "source_commit": "a635813f5edf297d928f9bd5ef8324a26bdde09c", "source_id": "putnam_1965_a4", "source_split": null, "license": "Apache-2.0 (Lean statement); informal text available with permission from the MAA", "informal": "[Informal statement not redistributed: PutnamBench states it is available with permission from the MAA. Problem: Putnam 1965 A4; source: PutnamBench informal/putnam.json, commit a635813f.]", "header": "import Mathlib\n\nopen EuclideanGeometry Topology Filter Complex\n\n", "source_tags": ["combinatorics"], "formal_statement": "theorem putnam_1965_a4\n{G B : Type*}\n[Fintype G] [Nonempty G]\n[Fintype B] [Nonempty B]\n(dances : G → B → Prop)\n(h : (¬∃ b : B, ∀ g : G, dances g b) ∧ ∀ g : G, ∃ b : B, dances g b)\n: ∃ g h : G, ∃ b c : B, dances g b ∧ dances h c ∧ ¬dances h b ∧ ¬dances g c := by\n", "statement": "import Mathlib\n\nopen EuclideanGeometry Topology Filter Complex\n\ntheorem putnam_1965_a4\n{G B : Type*}\n[Fintype G] [Nonempty G]\n[Fintype B] [Nonempty B]\n(dances : G → B → Prop)\n(h : (¬∃ b : B, ∀ g : G, dances g b) ∧ ∀ g : G, ∃ b : B, dances g b)\n: ∃ g h : G, ∃ b c : B, dances g b ∧ dances h c ∧ ¬dances h b ∧ ¬dances g c := by\n", "area": "combinatorics", "difficulty": "hard", "split": "dev", "seed": 2026, "informal_redacted": true}
{"id": "putnam_1964_b2", "source": "PutnamBench (lean4/src, informal/putnam.json)", "source_commit": "a635813f5edf297d928f9bd5ef8324a26bdde09c", "source_id": "putnam_1964_b2", "source_split": null, "license": "Apache-2.0 (Lean statement); informal text available with permission from the MAA", "informal": "[Informal statement not redistributed: PutnamBench states it is available with permission from the MAA. Problem: Putnam 1964 B2; source: PutnamBench informal/putnam.json, commit a635813f.]", "header": "import Mathlib\n\nopen Set Function Filter Topology\n\n", "source_tags": ["set_theory", "combinatorics"], "formal_statement": "theorem putnam_1964_b2\n(S : Type*) [Fintype S] [Nonempty S]\n(P : Finset (Set S))\n(hPP : ∀ T ∈ P, ∀ U ∈ P, T ∩ U ≠ ∅)\n(hPS : ¬∃ T : Set S, T ∉ P ∧ (∀ U ∈ P, T ∩ U ≠ ∅))\n: (P.card = 2 ^ (Fintype.card S - 1)) := by\n", "statement": "import Mathlib\n\nopen Set Function Filter Topology\n\ntheorem putnam_1964_b2\n(S : Type*) [Fintype S] [Nonempty S]\n(P : Finset (Set S))\n(hPP : ∀ T ∈ P, ∀ U ∈ P, T ∩ U ≠ ∅)\n(hPS : ¬∃ T : Set S, T ∉ P ∧ (∀ U ∈ P, T ∩ U ≠ ∅))\n: (P.card = 2 ^ (Fintype.card S - 1)) := by\n", "area": "combinatorics", "difficulty": "hard", "split": "test", "seed": 2026, "informal_redacted": true}
{"id": "putnam_1995_b1", "source": "PutnamBench (lean4/src, informal/putnam.json)", "source_commit": "a635813f5edf297d928f9bd5ef8324a26bdde09c", "source_id": "putnam_1995_b1", "source_split": null, "license": "Apache-2.0 (Lean statement); informal text available with permission from the MAA", "informal": "[Informal statement not redistributed: PutnamBench states it is available with permission from the MAA. Problem: Putnam 1995 B1; source: PutnamBench informal/putnam.json, commit a635813f.]", "header": "import Mathlib\n\nopen Filter Topology Real Nat\n\n", "source_tags": ["combinatorics"], "formal_statement": "theorem putnam_1995_b1\n  (Pi : Finpartition (Finset.range 9) → (Finset.range 9) → ℕ)\n  (Pi_def : ∀ partition k, Pi partition k = (Exists.choose (Finpartition.exists_mem partition k.2)).card) :\n  ∀ Pt1 Pt2,\n    ∃ x y : Finset.range 9,\n      x ≠ y ∧\n      Pi Pt1 x = Pi Pt1 y ∧\n      Pi Pt2 x = Pi Pt2 y := by\n", "statement": "import Mathlib\n\nopen Filter Topology Real Nat\n\ntheorem putnam_1995_b1\n  (Pi : Finpartition (Finset.range 9) → (Finset.range 9) → ℕ)\n  (Pi_def : ∀ partition k, Pi partition k = (Exists.choose (Finpartition.exists_mem partition k.2)).card) :\n  ∀ Pt1 Pt2,\n    ∃ x y : Finset.range 9,\n      x ≠ y ∧\n      Pi Pt1 x = Pi Pt1 y ∧\n      Pi Pt2 x = Pi Pt2 y := by\n", "area": "combinatorics", "difficulty": "hard", "split": "test", "seed": 2026, "informal_redacted": true}
{"id": "putnam_1965_b2", "source": "PutnamBench (lean4/src, informal/putnam.json)", "source_commit": "a635813f5edf297d928f9bd5ef8324a26bdde09c", "source_id": "putnam_1965_b2", "source_split": null, "license": "Apache-2.0 (Lean statement); informal text available with permission from the MAA", "informal": "[Informal statement not redistributed: PutnamBench states it is available with permission from the MAA. Problem: Putnam 1965 B2; source: PutnamBench informal/putnam.json, commit a635813f.]", "header": "import Mathlib\n\nopen EuclideanGeometry Topology Filter Complex\n\n", "source_tags": ["combinatorics"], "formal_statement": "theorem putnam_1965_b2\n(n : ℕ)\n(hn : n > 1)\n(won : Fin n → Fin n → Bool)\n(hirrefl : ∀ i : Fin n, won i i = false)\n(hantisymm : ∀ i j : Fin n, i ≠ j → won i j = ¬won j i)\n(w l : Fin n → ℤ)\n(hw : w = fun r : Fin n => ∑ j : Fin n, (if won r j then 1 else 0))\n(hl : l = fun r : Fin n => n - 1 - w r)\n: ∑ r : Fin n, (w r)^2 = ∑ r : Fin n, (l r)^2 := by\n", "statement": "import Mathlib\n\nopen EuclideanGeometry Topology Filter Complex\n\ntheorem putnam_1965_b2\n(n : ℕ)\n(hn : n > 1)\n(won : Fin n → Fin n → Bool)\n(hirrefl : ∀ i : Fin n, won i i = false)\n(hantisymm : ∀ i j : Fin n, i ≠ j → won i j = ¬won j i)\n(w l : Fin n → ℤ)\n(hw : w = fun r : Fin n => ∑ j : Fin n, (if won r j then 1 else 0))\n(hl : l = fun r : Fin n => n - 1 - w r)\n: ∑ r : Fin n, (w r)^2 = ∑ r : Fin n, (l r)^2 := by\n", "area": "combinatorics", "difficulty": "hard", "split": "test", "seed": 2026, "informal_redacted": true}