{"problem_id": "mathd_numbertheory_127", "source": "minif2f_v2s", "split": "dev", "area": "number_theory", "difficulty": 2, "informal_statement": "Find the remainder when $1 + 2 + 2^2 + 2^3 + \\dots + 2^{100}$ is divided by 7. Show that it is 3.", "formal_statement": "theorem mathd_numbertheory_127 : (∑ k ∈ Finset.range 101, 2 ^ k) % 7 = 3 := by sorry", "header": "open BigOperators Real Nat Topology Rat"}
{"problem_id": "mathd_algebra_160", "source": "minif2f_v2s", "split": "dev", "area": "algebra", "difficulty": 2, "informal_statement": "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.", "formal_statement": "theorem mathd_algebra_160\n  (n x : ℕ)\n  (f : ℕ → ℕ)\n  (h₀ : f = fun t => n + x * t)\n  (h₁: f 1 = 97)\n  (h₂: f 5 = 265) :\n  f 2 = 139 := by sorry", "header": "open BigOperators Real Nat Topology Rat"}
{"problem_id": "mathd_numbertheory_1124", "source": "minif2f_v2s", "split": "dev", "area": "number_theory", "difficulty": 2, "informal_statement": "The four-digit number $\\underline{374n}$ is divisible by 18. Find the units digit $n$. Show that it is 4.", "formal_statement": "theorem mathd_numbertheory_1124\n  (n x : ℕ)\n  (h₀ : Nat.digits 10 x = [n, 4, 7, 3])\n  (h₁ : 18 ∣ x) : n = 4 := by sorry", "header": "open BigOperators Real Nat Topology Rat"}
{"problem_id": "induction_1pxpownlt1pnx", "source": "minif2f_v2s", "split": "dev", "area": "induction", "difficulty": 2, "informal_statement": "Show that for any real number $x$ and any natural number $n$, if $x > -1$, then $(1+nx)\\leq (1+x)^n$.", "formal_statement": "theorem induction_1pxpownlt1pnx (x : ℝ) (n : ℕ) (h₀ : -1 < x) :\n  1 + (n:ℝ)* x ≤ (1+x)^n := by sorry", "header": "open BigOperators Real Nat Topology Rat"}
{"problem_id": "mathd_algebra_440", "source": "minif2f_v2s", "split": "dev", "area": "algebra", "difficulty": 2, "informal_statement": "Jasmine drank 1.5 pints of water on the first 3 miles of her hike. If she continued at this rate, how many pints of water would she drink in the next 10 miles? Show that it is 5.", "formal_statement": "theorem mathd_algebra_440\n  (x : ℝ)\n  (h₀: x * 3 = 3 / (2:ℝ)):\n  x * 10 = 5 := by sorry", "header": "open BigOperators Real Nat Topology Rat"}
{"problem_id": "mathd_numbertheory_495", "source": "minif2f_v2s", "split": "dev", "area": "number_theory", "difficulty": 2, "informal_statement": "Suppose $a$ and $b$ are positive integers such that the units digit of $a$ is $2$, the units digit of $b$ is $4$, and the greatest common divisor of $a$ and $b$ is $6$.\\n\\n\nShow that the least common multiple of $a$ and $b$ cannot be less than 108.", "formal_statement": "theorem mathd_numbertheory_495\n  (S : Set ℕ)\n  (hS : S = {n | ∃ a b:ℕ, 0 < a ∧ 0 < b ∧ a % 10 = 2 ∧ b % 10 = 4 ∧ Nat.gcd a b = 6 ∧ n = Nat.lcm a b}) :\n  IsLeast S 108 := by sorry", "header": "open BigOperators Real Nat Topology Rat"}
{"problem_id": "mathd_numbertheory_342", "source": "minif2f_v2s", "split": "dev", "area": "number_theory", "difficulty": 2, "informal_statement": "Determine the remainder of 54 (mod 6). Show that it is 0.", "formal_statement": "theorem mathd_numbertheory_342 : 54 % 6 = 0 := by sorry", "header": "open BigOperators Real Nat Topology Rat"}
{"problem_id": "mathd_numbertheory_222", "source": "minif2f_v2s", "split": "dev", "area": "number_theory", "difficulty": 2, "informal_statement": "The least common multiple of two numbers is 3720, and their greatest common divisor is 8. Given that one of the numbers is 120, what is the other number? Show that it is 248.", "formal_statement": "theorem mathd_numbertheory_222 (b : ℕ) (h₀ : Nat.lcm 120 b = 3720) (h₁ : Nat.gcd 120 b = 8) :\n  b = 248 := by sorry", "header": "open BigOperators Real Nat Topology Rat"}