| {"id": "mathd_algebra_28", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_algebra_28", "source_split": "valid", "license": "MIT", "informal": "What is the largest number $c$ such that $2x^2+5x+c=0$ has at least one real solution? Express your answer as a common fraction. Show that it is \\frac{25}{8}.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_algebra_28 (c : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f x = 2 * x ^ 2 + 5 * x + c)\n (h₁ : ∃ x, f x ≤ 0) : c ≤ 25 / 8 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_algebra_28 (c : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f x = 2 * x ^ 2 + 5 * x + c)\n (h₁ : ∃ x, f x ≤ 0) : c ≤ 25 / 8 := by\n", "area": "algebra", "difficulty": "easy", "split": "demo", "seed": 2026, "note": "Demo only: not in the study; no LLM call was made on it before the demo."} |
| {"id": "mathd_numbertheory_45", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "mathd_numbertheory_45", "source_split": "valid", "license": "MIT", "informal": "What is the result when the greatest common factor of 6432 and 132 is increased by 11? Show that it is 23.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem mathd_numbertheory_45 : Nat.gcd 6432 132 + 11 = 23 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem mathd_numbertheory_45 : Nat.gcd 6432 132 + 11 = 23 := by\n", "area": "number_theory", "difficulty": "easy", "split": "demo", "seed": 2026, "note": "Demo only: not in the study; no LLM call was made on it before the demo."} |
| {"id": "aimeII_2001_p3", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "aimeII_2001_p3", "source_split": "valid", "license": "MIT", "informal": "Given that\n\n$$\n\\begin{align*}x_{1}&=211,\\\\\nx_{2}&=375,\\\\\nx_{3}&=420,\\\\\nx_{4}&=523,\\ \\text{and}\\\\\nx_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\\ \\text{when}\\ n\\geq5, \\end{align*}\n$$\n\nfind the value of $x_{531}+x_{753}+x_{975}$. Show that it is 898.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem aimeII_2001_p3 (x : ℕ → ℤ) (h₀ : x 1 = 211) (h₂ : x 2 = 375) (h₃ : x 3 = 420)\n (h₄ : x 4 = 523) (h₆ : ∀ n ≥ 5, x n = x (n - 1) - x (n - 2) + x (n - 3) - x (n - 4)) :\n x 531 + x 753 + x 975 = 898 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem aimeII_2001_p3 (x : ℕ → ℤ) (h₀ : x 1 = 211) (h₂ : x 2 = 375) (h₃ : x 3 = 420)\n (h₄ : x 4 = 523) (h₆ : ∀ n ≥ 5, x n = x (n - 1) - x (n - 2) + x (n - 3) - x (n - 4)) :\n x 531 + x 753 + x 975 = 898 := by\n", "area": "algebra", "difficulty": "medium", "split": "demo", "seed": 2026, "note": "Demo only: not in the study; no LLM call was made on it before the demo."} |
| {"id": "numbertheory_aneqprodakp4_anmsqrtanp1eq2", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "numbertheory_aneqprodakp4_anmsqrtanp1eq2", "source_split": "valid", "license": "MIT", "informal": "Let $a_0 = 1$. For any positive integer $n$, let $a_{n+1} = \\prod_{k = 1}^n a_k + 4$. Show that for any positive integer $n$, $a_n - \\sqrt{a_{n+1}} = 2$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem numbertheory_aneqprodakp4_anmsqrtanp1eq2 (a : ℕ → ℝ) (h₀ : a 0 = 1)\n (h₁ : ∀ n, a (n + 1) = (∏ k ∈ Finset.range (n + 1), a k) + 4) :\n ∀ n ≥ 1, a n - Real.sqrt (a (n + 1)) = 2 := by\n", "formal_statement_original": "theorem numbertheory_aneqprodakp4_anmsqrtanp1eq2 (a : ℕ → ℝ) (h₀ : a 0 = 1)\n (h₁ : ∀ n, a (n + 1) = (∏ k in Finset.range (n + 1), a k) + 4) :\n ∀ n ≥ 1, a n - Real.sqrt (a (n + 1)) = 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 numbertheory_aneqprodakp4_anmsqrtanp1eq2 (a : ℕ → ℝ) (h₀ : a 0 = 1)\n (h₁ : ∀ n, a (n + 1) = (∏ k ∈ Finset.range (n + 1), a k) + 4) :\n ∀ n ≥ 1, a n - Real.sqrt (a (n + 1)) = 2 := by\n", "area": "number_theory", "difficulty": "medium", "split": "demo", "seed": 2026, "note": "Demo only: not in the study; no LLM call was made on it before the demo."} |
| {"id": "imo_1973_p3", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1973_p3", "source_split": "valid", "license": "MIT", "informal": "Let $a$ and $b$ be real numbers for which the equation\n$x^4 + ax^3 + bx^2 + ax + 1 = 0$\nhas at least one real solution. For all such pairs $(a, b)$, find the minimum value of $a^2 + b^2$.", "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n", "formal_statement": "theorem imo_1973_p3 (a b : ℝ) (h₀ : ∃ x, x ^ 4 + a * x ^ 3 + b * x ^ 2 + a * x + 1 = 0) :\n 4 / 5 ≤ a ^ 2 + b ^ 2 := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1973_p3 (a b : ℝ) (h₀ : ∃ x, x ^ 4 + a * x ^ 3 + b * x ^ 2 + a * x + 1 = 0) :\n 4 / 5 ≤ a ^ 2 + b ^ 2 := by\n", "area": "algebra", "difficulty": "hard", "split": "demo", "seed": 2026, "note": "Demo only: not in the study; no LLM call was made on it before the demo."} |
| {"id": "imo_1964_p1_1", "source": "miniF2F (DeepSeek-Prover-V1.5 Lean 4 port, datasets/minif2f.jsonl)", "source_commit": "2c4ba9119eef74d0d611f494261b2c5bae98c69a", "source_id": "imo_1964_p1_1", "source_split": "valid", "license": "MIT", "informal": "Let $n$ be a natural number. Show that if $7$ divides $2^n-1$, then $3$ divides $n$.", "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_1 (n : ℕ) (h₀ : 7 ∣ 2 ^ n - 1) : 3 ∣ n := by\n", "statement": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem imo_1964_p1_1 (n : ℕ) (h₀ : 7 ∣ 2 ^ n - 1) : 3 ∣ n := by\n", "area": "number_theory", "difficulty": "hard", "split": "demo", "seed": 2026, "note": "Demo only: not in the study; no LLM call was made on it before the demo."} |
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