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4302c7830e25eb00
import Mathlib theorem amc12a_2019_p21 (z : ℂ) (h₀ : z = (1 + Complex.I) / Real.sqrt 2) : ((∑ k : ℤ in Finset.Icc 1 12, z ^ k ^ 2) * (∑ k : ℤ in Finset.Icc 1 12, 1 / z ^ k ^ 2)) = 36 := by
z : ℂ h₀ : z = (1 + Complex.I) / ↑√2 ⊢ (∑ k ∈ Finset.Icc 1 12, z ^ k ^ 2) * ∑ k ∈ Finset.Icc 1 12, 1 / z ^ k ^ 2 = 36
38a3b362ac4b77c8
import Mathlib theorem amc12a_2015_p10 (x y : ℤ) (h₀ : 0 < y) (h₁ : y < x) (h₂ : x + y + x * y = 80) : x = 26 := by
x y : ℤ h₀ : 0 < y h₁ : y < x h₂ : x + y + x * y = 80 ⊢ x = 26
f897047cc20bfb4d
import Mathlib theorem amc12a_2008_p8 (x y : ℝ) (h₀ : 0 < x ∧ 0 < y) (h₁ : y ^ 3 = 1) (h₂ : 6 * x ^ 2 = 2 * (6 * y ^ 2)) : x ^ 3 = 2 * Real.sqrt 2 := by
x y : ℝ h₀ : 0 < x ∧ 0 < y h₁ : y ^ 3 = 1 h₂ : 6 * x ^ 2 = 2 * (6 * y ^ 2) ⊢ x ^ 3 = 2 * √2
9d8b27f7b8ae39f5
import Mathlib theorem mathd_algebra_182 (y : ℂ) : 7 * (3 * y + 2) = 21 * y + 14 := by
y : ℂ ⊢ 7 * (3 * y + 2) = 21 * y + 14
7ca640a6b5fee9be
import Mathlib theorem aime_1984_p5 (a b : ℝ) (h₀ : Real.logb 8 a + Real.logb 4 (b ^ 2) = 5) (h₁ : Real.logb 8 b + Real.logb 4 (a ^ 2) = 7) : a * b = 512 := by
a b : ℝ h₀ : logb 8 a + logb 4 (b ^ 2) = 5 h₁ : logb 8 b + logb 4 (a ^ 2) = 7 ⊢ a * b = 512
e11b01863ba91f10
import Mathlib theorem mathd_numbertheory_780 (m x : ℤ) (h₀ : 0 ≤ x) (h₁ : 10 ≤ m ∧ m ≤ 99) (h₂ : 6 * x % m = 1) (h₃ : (x - 6 ^ 2) % m = 0) : m = 43 := by
m x : ℤ h₀ : 0 ≤ x h₁ : 10 ≤ m ∧ m ≤ 99 h₂ : 6 * x % m = 1 h₃ : (x - 6 ^ 2) % m = 0 ⊢ m = 43
f2b3cbe0e9377859
import Mathlib theorem mathd_algebra_116 (k x : ℝ) (h₀ : x = (13 - Real.sqrt 131) / 4) (h₁ : 2 * x ^ 2 - 13 * x + k = 0) : k = 19 / 4 := by
k x : ℝ h₀ : x = (13 - √131) / 4 h₁ : 2 * x ^ 2 - 13 * x + k = 0 ⊢ k = 19 / 4
2701d78a75a79cc5
import Mathlib theorem mathd_numbertheory_13 (u v : ℕ) (S : Set ℕ) (h₀ : ∀ n : ℕ, n ∈ S ↔ 0 < n ∧ 14 * n % 100 = 46) (h₁ : IsLeast S u) (h₂ : IsLeast (S \ {u}) v) : (u + v : ℚ) / 2 = 64 := by
u v : ℕ S : Set ℕ h₀ : ∀ (n : ℕ), n ∈ S ↔ 0 < n ∧ 14 * n % 100 = 46 h₁ : IsLeast S u h₂ : IsLeast (S \ {u}) v ⊢ (↑u + ↑v) / 2 = 64
f92de45dcfb1fa02
import Mathlib theorem mathd_numbertheory_169 : Nat.gcd 20! 200000 = 40000 := by
⊢ 20!.gcd 200000 = 40000
1aa15fdcb17604bc
import Mathlib theorem amc12a_2009_p9 (a b c : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f (x + 3) = 3 * x ^ 2 + 7 * x + 4) (h₁ : ∀ x, f x = a * x ^ 2 + b * x + c) : a + b + c = 2 := by
a b c : ℝ f : ℝ → ℝ h₀ : ∀ (x : ℝ), f (x + 3) = 3 * x ^ 2 + 7 * x + 4 h₁ : ∀ (x : ℝ), f x = a * x ^ 2 + b * x + c ⊢ a + b + c = 2
3856741db21e9cc6
import Mathlib theorem amc12a_2019_p9 (a : ℕ → ℚ) (h₀ : a 1 = 1) (h₁ : a 2 = 3 / 7) (h₂ : ∀ n, a (n + 2) = a n * a (n + 1) / (2 * a n - a (n + 1))) : ↑(a 2019).den + (a 2019).num = 8078 := by
a : ℕ → ℚ h₀ : a 1 = 1 h₁ : a 2 = 3 / 7 h₂ : ∀ (n : ℕ), a (n + 2) = a n * a (n + 1) / (2 * a n - a (n + 1)) ⊢ ↑(a 2019).den + (a 2019).num = 8078
7559fc4bae243b02
import Mathlib theorem mathd_algebra_13 (a b : ℝ) (h₀ : ∀ x, x - 3 ≠ 0 ∧ x - 5 ≠ 0 → 4 * x / (x ^ 2 - 8 * x + 15) = a / (x - 3) + b / (x - 5)) : a = -6 ∧ b = 10 := by
a b : ℝ h₀ : ∀ (x : ℝ), x - 3 ≠ 0 ∧ x - 5 ≠ 0 → 4 * x / (x ^ 2 - 8 * x + 15) = a / (x - 3) + b / (x - 5) ⊢ a = -6 ∧ b = 10
4b1f8de70266d4f7
import Mathlib theorem induction_sum2kp1npqsqm1 (n : ℕ) : ∑ k in Finset.range n, (2 * k + 3) = (n + 1) ^ 2 - 1 := by
n : ℕ ⊢ ∑ k ∈ Finset.range n, (2 * k + 3) = (n + 1) ^ 2 - 1
593155e948849d68
import Mathlib theorem aime_1991_p6 (r : ℝ) (h₀ : (∑ k in Finset.Icc (19 : ℕ) 91, Int.floor (r + k / 100)) = 546) : Int.floor (100 * r) = 743 := by
r : ℝ h₀ : ∑ k ∈ Finset.Icc 19 91, ⌊r + ↑k / 100⌋ = 546 ⊢ ⌊100 * r⌋ = 743
c78cf03da3a6e721
import Mathlib theorem mathd_numbertheory_149 : (∑ k in Finset.filter (fun x => x % 8 = 5 ∧ x % 6 = 3) (Finset.range 50), k) = 66 := by
⊢ ∑ k ∈ Finset.filter (fun x => x % 8 = 5 ∧ x % 6 = 3) (Finset.range 50), k = 66
7e3c8086c3ca0fac
import Mathlib theorem imo_1984_p2 (a b : ℤ) (h₀ : 0 < a ∧ 0 < b) (h₁ : ¬7 ∣ a) (h₂ : ¬7 ∣ b) (h₃ : ¬7 ∣ a + b) (h₄ : 7 ^ 7 ∣ (a + b) ^ 7 - a ^ 7 - b ^ 7) : 19 ≤ a + b := by
a b : ℤ h₀ : 0 < a ∧ 0 < b h₁ : ¬7 ∣ a h₂ : ¬7 ∣ b h₃ : ¬7 ∣ a + b h₄ : 7 ^ 7 ∣ (a + b) ^ 7 - a ^ 7 - b ^ 7 ⊢ 19 ≤ a + b
ca716533245de685
import Mathlib theorem amc12a_2008_p4 : (∏ k in Finset.Icc (1 : ℕ) 501, ((4 : ℝ) * k + 4) / (4 * k)) = 502 := by
⊢ ∏ k ∈ Finset.Icc 1 501, (4 * ↑k + 4) / (4 * ↑k) = 502
dd1b0df921c83516
import Mathlib theorem imo_2006_p3 (a b c : ℝ) : a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2) ≤ 9 * Real.sqrt 2 / 32 * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2 := by
a b c : ℝ ⊢ a * b * (a ^ 2 - b ^ 2) + b * c * (b ^ 2 - c ^ 2) + c * a * (c ^ 2 - a ^ 2) ≤ 9 * √2 / 32 * (a ^ 2 + b ^ 2 + c ^ 2) ^ 2
996b38a1fc8ee61a
import Mathlib theorem mathd_algebra_462 : ((1 : ℚ) / 2 + 1 / 3) * (1 / 2 - 1 / 3) = 5 / 36 := by
⊢ (1 / 2 + 1 / 3) * (1 / 2 - 1 / 3) = 5 / 36
aaf43df039ab2ec3
import Mathlib theorem imo_1964_p1_2 (n : ℕ) : ¬7 ∣ 2 ^ n + 1 := by
n : ℕ ⊢ ¬7 ∣ 2 ^ n + 1
95ccd12e5096d350
import Mathlib theorem mathd_numbertheory_221 (S : Finset ℕ) (h₀ : ∀ x : ℕ, x ∈ S ↔ 0 < x ∧ x < 1000 ∧ x.divisors.card = 3) : S.card = 11 := by
S : Finset ℕ h₀ : ∀ (x : ℕ), x ∈ S ↔ 0 < x ∧ x < 1000 ∧ x.divisors.card = 3 ⊢ S.card = 11
095e7c1c8e3af948
import Mathlib theorem mathd_numbertheory_64 : IsLeast { x : ℕ | 30 * x ≡ 42 [MOD 47] } 39 := by
⊢ IsLeast {x | 30 * x ≡ 42 [MOD 47]} 39
e296ed472df217f2
import Mathlib theorem imo_1987_p4 (f : ℕ → ℕ) : ∃ n, f (f n) ≠ n + 1987 := by
f : ℕ → ℕ ⊢ ∃ n, f (f n) ≠ n + 1987
ab530dce9367f974
import Mathlib theorem mathd_numbertheory_33 (n : ℕ) (h₀ : n < 398) (h₁ : n * 7 % 398 = 1) : n = 57 := by
n : ℕ h₀ : n < 398 h₁ : n * 7 % 398 = 1 ⊢ n = 57
95192a71852300c8
import Mathlib theorem amc12_2001_p9 (f : ℝ → ℝ) (h₀ : ∀ x > 0, ∀ y > 0, f (x * y) = f x / y) (h₁ : f 500 = 3) : f 600 = 5 / 2 := by
f : ℝ → ℝ h₀ : ∀ x > 0, ∀ y > 0, f (x * y) = f x / y h₁ : f 500 = 3 ⊢ f 600 = 5 / 2
3e2594781b6b2c0b
import Mathlib theorem imo_1965_p1 (x : ℝ) (h₀ : 0 ≤ x) (h₁ : x ≤ 2 * Real.pi) (h₂ : 2 * Real.cos x ≤ abs (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x)))) (h₃ : abs (Real.sqrt (1 + Real.sin (2 * x)) - Real.sqrt (1 - Real.sin (2 * x))) ≤ Real.sqrt 2) : Real.pi / 4 ≤ x ∧ x ≤ 7 * Real.pi / 4 :...
x : ℝ h₀ : 0 ≤ x h₁ : x ≤ 2 * π h₂ : 2 * x.cos ≤ |√(1 + (2 * x).sin) - √(1 - (2 * x).sin)| h₃ : |√(1 + (2 * x).sin) - √(1 - (2 * x).sin)| ≤ √2 ⊢ π / 4 ≤ x ∧ x ≤ 7 * π / 4
6233383f12eabd6c
import Mathlib theorem mathd_numbertheory_48 (b : ℕ) (h₀ : 0 < b) (h₁ : 3 * b ^ 2 + 2 * b + 1 = 57) : b = 4 := by
b : ℕ h₀ : 0 < b h₁ : 3 * b ^ 2 + 2 * b + 1 = 57 ⊢ b = 4
65b56dd5b158b479
import Mathlib theorem numbertheory_sqmod4in01d (a : ℤ) : a ^ 2 % 4 = 0 ∨ a ^ 2 % 4 = 1 := by
a : ℤ ⊢ a ^ 2 % 4 = 0 ∨ a ^ 2 % 4 = 1
ad9eb02ddb681cc1
import Mathlib theorem mathd_numbertheory_466 : (∑ k in Finset.range 11, k) % 9 = 1 := by
⊢ (∑ k ∈ Finset.range 11, k) % 9 = 1
11b5722d8a500b1d
import Mathlib theorem mathd_algebra_48 (q e : ℂ) (h₀ : q = 9 - 4 * Complex.I) (h₁ : e = -3 - 4 * Complex.I) : q - e = 12 := by
q e : ℂ h₀ : q = 9 - 4 * Complex.I h₁ : e = -3 - 4 * Complex.I ⊢ q - e = 12
59c1ada943500f1a
import Mathlib theorem amc12_2000_p15 (f : ℂ → ℂ) (h₀ : ∀ x, f (x / 3) = x ^ 2 + x + 1) (h₁ : Fintype (f ⁻¹' {7})) : (∑ y in (f ⁻¹' {7}).toFinset, y / 3) = -1 / 9 := by
f : ℂ → ℂ h₀ : ∀ (x : ℂ), f (x / 3) = x ^ 2 + x + 1 h₁ : Fintype ↑(f ⁻¹' {7}) ⊢ ∑ y ∈ (f ⁻¹' {7}).toFinset, y / 3 = -1 / 9
a096805797f5dfd1
import Mathlib theorem mathd_numbertheory_132 : 2004 % 12 = 0 := by
⊢ 2004 % 12 = 0
7a42500799290a6b
import Mathlib theorem amc12a_2009_p5 (x : ℝ) (h₀ : x ^ 3 - (x + 1) * (x - 1) * x = 5) : x ^ 3 = 125 := by
x : ℝ h₀ : x ^ 3 - (x + 1) * (x - 1) * x = 5 ⊢ x ^ 3 = 125
4fb65db037d80b12
import Mathlib theorem mathd_numbertheory_188 : Nat.gcd 180 168 = 12 := by
⊢ Nat.gcd 180 168 = 12
1eb8300c0a409f42
import Mathlib theorem mathd_algebra_224 (S : Finset ℕ) (h₀ : ∀ n : ℕ, n ∈ S ↔ Real.sqrt n < 7 / 2 ∧ 2 < Real.sqrt n) : S.card = 8 := by
S : Finset ℕ h₀ : ∀ (n : ℕ), n ∈ S ↔ √↑n < 7 / 2 ∧ 2 < √↑n ⊢ S.card = 8
e7cb9b8380d9c339
import Mathlib theorem induction_divisibility_3divnto3m2n (n : ℕ) : 3 ∣ n ^ 3 + 2 * n := by
n : ℕ ⊢ 3 ∣ n ^ 3 + 2 * n
0324044868d977ad
import Mathlib theorem induction_sum_1oktkp1 (n : ℕ) : (∑ k in Finset.range n, (1 : ℝ) / ((k + 1) * (k + 2))) = n / (n + 1) := by
n : ℕ ⊢ ∑ k ∈ Finset.range n, 1 / ((↑k + 1) * (↑k + 2)) = ↑n / (↑n + 1)
2d4c2fcd5b1dc617
import Mathlib theorem mathd_numbertheory_32 (S : Finset ℕ) (h₀ : ∀ n : ℕ, n ∈ S ↔ n ∣ 36) : (∑ k in S, k) = 91 := by
S : Finset ℕ h₀ : ∀ (n : ℕ), n ∈ S ↔ n ∣ 36 ⊢ ∑ k ∈ S, k = 91
33cf4cd6541974a0
import Mathlib theorem mathd_algebra_422 (x : ℝ) (σ : Equiv ℝ ℝ) (h₀ : ∀ x, σ.1 x = 5 * x - 12) (h₁ : σ.1 (x + 1) = σ.2 x) : x = 47 / 24 := by
x : ℝ σ : ℝ ≃ ℝ h₀ : ∀ (x : ℝ), σ.toFun x = 5 * x - 12 h₁ : σ.toFun (x + 1) = σ.invFun x ⊢ x = 47 / 24
aac6247adecc7c3c
import Mathlib theorem amc12b_2002_p11 (a b : ℕ) (h₀ : Nat.Prime a) (h₁ : Nat.Prime b) (h₂ : Nat.Prime (a + b)) (h₃ : Nat.Prime (a - b)) : Nat.Prime (a + b + (a - b + (a + b))) := by
a b : ℕ h₀ : a.Prime h₁ : b.Prime h₂ : (a + b).Prime h₃ : (a - b).Prime ⊢ (a + b + (a - b + (a + b))).Prime
ec83b4154c4c24fd
import Mathlib theorem mathd_algebra_73 (p q r x : ℂ) (h₀ : (x - p) * (x - q) = (r - p) * (r - q)) (h₁ : x ≠ r) : x = p + q - r := by
p q r x : ℂ h₀ : (x - p) * (x - q) = (r - p) * (r - q) h₁ : x ≠ r ⊢ x = p + q - r
59640eba40e7b6a3
import Mathlib theorem mathd_numbertheory_109 (v : ℕ → ℕ) (h₀ : ∀ n, v n = 2 * n - 1) : (∑ k in Finset.Icc 1 100, v k) % 7 = 4 := by
v : ℕ → ℕ h₀ : ∀ (n : ℕ), v n = 2 * n - 1 ⊢ (∑ k ∈ Finset.Icc 1 100, v k) % 7 = 4
85ee06a42648499e
import Mathlib theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3 (x y z : ℤ) (h₀ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z) : x + y + z + 6 ∣ x ^ 3 + y ^ 3 + z ^ 3 := by
x y z : ℤ h₀ : (x - y) ^ 2 + (y - z) ^ 2 + (z - x) ^ 2 = x * y * z ⊢ x + y + z + 6 ∣ x ^ 3 + y ^ 3 + z ^ 3
959e88f44412a4a8
import Mathlib theorem imo_1962_p4 (S : Set ℝ) (h₀ : S = { x : ℝ | Real.cos x ^ 2 + Real.cos (2 * x) ^ 2 + Real.cos (3 * x) ^ 2 = 1 }) : S = { x : ℝ | ∃ m : ℤ, x = Real.pi / 2 + m * Real.pi ∨ x = Real.pi / 4 + m * Real.pi / 2 ∨ x = Real.pi / 6 + m * Real.pi / 6 ∨ x = 5 * Rea...
S : Set ℝ h₀ : S = {x | x.cos ^ 2 + (2 * x).cos ^ 2 + (3 * x).cos ^ 2 = 1} ⊢ S = {x | ∃ m, x = π / 2 + ↑m * π ∨ x = π / 4 + ↑m * π / 2 ∨ x = π / 6 + ↑m * π / 6 ∨ x = 5 * π / 6 + ↑m * π / 6}
1b575acce73e9c03
import Mathlib theorem mathd_numbertheory_236 : 1999 ^ 2000 % 5 = 1 := by
⊢ 1999 ^ 2000 % 5 = 1
d7ddead4e423d260
import Mathlib theorem mathd_numbertheory_24 : (∑ k in Finset.Icc 1 9, 11 ^ k) % 100 = 59 := by
⊢ (∑ k ∈ Finset.Icc 1 9, 11 ^ k) % 100 = 59
52ac6c67f69db4af
import Mathlib theorem algebra_amgm_prod1toneq1_sum1tongeqn (a : ℕ → NNReal) (n : ℕ) (h₀ : Finset.prod (Finset.range n) a = 1) : Finset.sum (Finset.range n) a ≥ n := by
a : ℕ → NNReal n : ℕ h₀ : (Finset.range n).prod a = 1 ⊢ (Finset.range n).sum a ≥ ↑n
79d24166b4d648ff
import Mathlib theorem mathd_algebra_101 (x : ℝ) (h₀ : x ^ 2 - 5 * x - 4 ≤ 10) : x ≥ -2 ∧ x ≤ 7 := by
x : ℝ h₀ : x ^ 2 - 5 * x - 4 ≤ 10 ⊢ x ≥ -2 ∧ x ≤ 7
7faae039f80262a7
import Mathlib theorem mathd_numbertheory_257 (x : ℕ) (h₀ : 1 ≤ x ∧ x ≤ 100) (h₁ : 77 ∣ (∑ k in Finset.range 101, k) - x) : x = 45 := by
x : ℕ h₀ : 1 ≤ x ∧ x ≤ 100 h₁ : 77 ∣ ∑ k ∈ Finset.range 101, k - x ⊢ x = 45
04c0486b1a87b44e
import Mathlib theorem amc12_2000_p5 (x p : ℝ) (h₀ : x < 2) (h₁ : abs (x - 2) = p) : x - p = 2 - 2 * p := by
x p : ℝ h₀ : x < 2 h₁ : |x - 2| = p ⊢ x - p = 2 - 2 * p
233baeacd990da62
import Mathlib theorem mathd_algebra_547 (x y : ℝ) (h₀ : x = 5) (h₁ : y = 2) : Real.sqrt (x ^ 3 - 2 ^ y) = 11 := by
x y : ℝ h₀ : x = 5 h₁ : y = 2 ⊢ √(x ^ 3 - 2 ^ y) = 11
30f95da79bfd3a93
import Mathlib theorem mathd_numbertheory_200 : 139 % 11 = 7 := by
⊢ 139 % 11 = 7
3612dde9f76b7c3f
import Mathlib theorem mathd_algebra_510 (x y : ℝ) (h₀ : x + y = 13) (h₁ : x * y = 24) : Real.sqrt (x ^ 2 + y ^ 2) = 11 := by
x y : ℝ h₀ : x + y = 13 h₁ : x * y = 24 ⊢ √(x ^ 2 + y ^ 2) = 11
4eae6ed0f748e988
import Mathlib theorem mathd_algebra_140 (a b c : ℝ) (h₀ : 0 < a ∧ 0 < b ∧ 0 < c) (h₁ : ∀ x, 24 * x ^ 2 - 19 * x - 35 = (a * x - 5) * (2 * (b * x) + c)) : a * b - 3 * c = -9 := by
a b c : ℝ h₀ : 0 < a ∧ 0 < b ∧ 0 < c h₁ : ∀ (x : ℝ), 24 * x ^ 2 - 19 * x - 35 = (a * x - 5) * (2 * (b * x) + c) ⊢ a * b - 3 * c = -9
fa89a7da02c5e78f
import Mathlib theorem mathd_algebra_455 (x : ℝ) (h₀ : 2 * (2 * (2 * (2 * x))) = 48) : x = 3 := by
x : ℝ h₀ : 2 * (2 * (2 * (2 * x))) = 48 ⊢ x = 3
2f1d523900798a1d
import Mathlib theorem mathd_numbertheory_45 : Nat.gcd 6432 132 + 11 = 23 := by
⊢ Nat.gcd 6432 132 + 11 = 23
9c8a05537acd69b3
import Mathlib theorem aime_1994_p4 (n : ℕ) (h₀ : 0 < n) (h₀ : (∑ k in Finset.Icc 1 n, Int.floor (Real.logb 2 k)) = 1994) : n = 312 := by
n : ℕ h₀✝ : 0 < n h₀ : ∑ k ∈ Finset.Icc 1 n, ⌊logb 2 ↑k⌋ = 1994 ⊢ n = 312
9b7530a5896da2fd
import Mathlib theorem mathd_numbertheory_739 : 9! % 10 = 0 := by
⊢ 9! % 10 = 0
2fda6a1b55630219
import Mathlib theorem mathd_algebra_245 (x : ℝ) (h₀ : x ≠ 0) : (4 / x)⁻¹ * (3 * x ^ 3 / x) ^ 2 * (1 / (2 * x))⁻¹ ^ 3 = 18 * x ^ 8 := by
x : ℝ h₀ : x ≠ 0 ⊢ (4 / x)⁻¹ * (3 * x ^ 3 / x) ^ 2 * (1 / (2 * x))⁻¹ ^ 3 = 18 * x ^ 8
a84cfa87cb3bded8
import Mathlib theorem algebra_apb4leq8ta4pb4 (a b : ℝ) (h₀ : 0 < a ∧ 0 < b) : (a + b) ^ 4 ≤ 8 * (a ^ 4 + b ^ 4) := by
a b : ℝ h₀ : 0 < a ∧ 0 < b ⊢ (a + b) ^ 4 ≤ 8 * (a ^ 4 + b ^ 4)
556c3f5f3e0dd50b
import Mathlib theorem mathd_algebra_28 (c : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f x = 2 * x ^ 2 + 5 * x + c) (h₁ : ∃ x, f x ≤ 0) : c ≤ 25 / 8 := by
c : ℝ f : ℝ → ℝ h₀ : ∀ (x : ℝ), f x = 2 * x ^ 2 + 5 * x + c h₁ : ∃ x, f x ≤ 0 ⊢ c ≤ 25 / 8
c1abccbe022c0cba
import Mathlib theorem mathd_numbertheory_543 : (∑ k in Nat.divisors (30 ^ 4), 1) - 2 = 123 := by
⊢ ∑ k ∈ (30 ^ 4).divisors, 1 - 2 = 123
bc349c831538aca6
import Mathlib theorem mathd_algebra_480 (f : ℝ → ℝ) (h₀ : ∀ x < 0, f x = -x ^ 2 - 1) (h₁ : ∀ x, 0 ≤ x ∧ x < 4 → f x = 2) (h₂ : ∀ x ≥ 4, f x = Real.sqrt x) : f Real.pi = 2 := by
f : ℝ → ℝ h₀ : ∀ x < 0, f x = -x ^ 2 - 1 h₁ : ∀ (x : ℝ), 0 ≤ x ∧ x < 4 → f x = 2 h₂ : ∀ x ≥ 4, f x = √x ⊢ f π = 2
97442ef8a2ddc6eb
import Mathlib theorem mathd_algebra_69 (rows seats : ℕ) (h₀ : rows * seats = 450) (h₁ : (rows + 5) * (seats - 3) = 450) : rows = 25 := by
rows seats : ℕ h₀ : rows * seats = 450 h₁ : (rows + 5) * (seats - 3) = 450 ⊢ rows = 25
4558ece0f712e4ac
import Mathlib theorem mathd_algebra_433 (f : ℝ → ℝ) (h₀ : ∀ x, f x = 3 * Real.sqrt (2 * x - 7) - 8) : f 8 = 1 := by
f : ℝ → ℝ h₀ : ∀ (x : ℝ), f x = 3 * √(2 * x - 7) - 8 ⊢ f 8 = 1
c5c5b79d53ccddc6
import Mathlib theorem mathd_algebra_126 (x y : ℝ) (h₀ : 2 * 3 = x - 9) (h₁ : 2 * -5 = y + 1) : x = 15 ∧ y = -11 := by
x y : ℝ h₀ : 2 * 3 = x - 9 h₁ : 2 * -5 = y + 1 ⊢ x = 15 ∧ y = -11
a7743ff9a3443664
import Mathlib theorem aimeII_2020_p6 (t : ℕ → ℚ) (h₀ : t 1 = 20) (h₁ : t 2 = 21) (h₂ : ∀ n ≥ 3, t n = (5 * t (n - 1) + 1) / (25 * t (n - 2))) : ↑(t 2020).den + (t 2020).num = 626 := by
t : ℕ → ℚ h₀ : t 1 = 20 h₁ : t 2 = 21 h₂ : ∀ n ≥ 3, t n = (5 * t (n - 1) + 1) / (25 * t (n - 2)) ⊢ ↑(t 2020).den + (t 2020).num = 626
fb6543226eecc643
import Mathlib theorem amc12a_2008_p2 (x : ℝ) (h₀ : x * (1 / 2 + 2 / 3) = 1) : x = 6 / 7 := by
x : ℝ h₀ : x * (1 / 2 + 2 / 3) = 1 ⊢ x = 6 / 7
a0aa41ee445d7147
import Mathlib theorem mathd_algebra_35 (p q : ℝ → ℝ) (h₀ : ∀ x, p x = 2 - x ^ 2) (h₁ : ∀ x : ℝ, x ≠ 0 → q x = 6 / x) : p (q 2) = -7 := by
p q : ℝ → ℝ h₀ : ∀ (x : ℝ), p x = 2 - x ^ 2 h₁ : ∀ (x : ℝ), x ≠ 0 → q x = 6 / x ⊢ p (q 2) = -7
62ee38aba94d61f1
import Mathlib theorem algebra_amgm_faxinrrp2msqrt2geq2mxm1div2x : ∀ x > 0, 2 - Real.sqrt 2 ≥ 2 - x - 1 / (2 * x) := by
⊢ ∀ x > 0, 2 - √2 ≥ 2 - x - 1 / (2 * x)
b5a3aa23e805351a
import Mathlib theorem mathd_numbertheory_335 (n : ℕ) (h₀ : n % 7 = 5) : 5 * n % 7 = 4 := by
n : ℕ h₀ : n % 7 = 5 ⊢ 5 * n % 7 = 4
c0b60f68d67e0b49
import Mathlib theorem mathd_numbertheory_35 (S : Finset ℕ) (h₀ : ∀ n : ℕ, n ∣ Nat.sqrt 196) : (∑ k in S, k) = 24 := by
S : Finset ℕ h₀ : ∀ (n : ℕ), n ∣ Nat.sqrt 196 ⊢ ∑ k ∈ S, k = 24
d6fd94d3a965b052
import Mathlib theorem amc12a_2021_p7 (x y : ℝ) : 1 ≤ (x * y - 1) ^ 2 + (x + y) ^ 2 := by
x y : ℝ ⊢ 1 ≤ (x * y - 1) ^ 2 + (x + y) ^ 2
41e8b430cc86774c
import Mathlib theorem mathd_algebra_327 (a : ℝ) (h₀ : 1 / 5 * abs (9 + 2 * a) < 1) : -7 < a ∧ a < -2 := by
a : ℝ h₀ : 1 / 5 * |9 + 2 * a| < 1 ⊢ -7 < a ∧ a < -2
627a62ca87713166
import Mathlib theorem aime_1984_p15 (x y z w : ℝ) (h₀ : x ^ 2 / (2 ^ 2 - 1) + y ^ 2 / (2 ^ 2 - 3 ^ 2) + z ^ 2 / (2 ^ 2 - 5 ^ 2) + w ^ 2 / (2 ^ 2 - 7 ^ 2) = 1) (h₁ : x ^ 2 / (4 ^ 2 - 1) + y ^ 2 / (4 ^ 2 - 3 ^ 2) + z ^ 2 / (4 ^ 2 - 5 ^ 2) + w ^ 2 / (4 ^ 2 - 7 ^ 2) = ...
x y z w : ℝ h₀ : x ^ 2 / (2 ^ 2 - 1) + y ^ 2 / (2 ^ 2 - 3 ^ 2) + z ^ 2 / (2 ^ 2 - 5 ^ 2) + w ^ 2 / (2 ^ 2 - 7 ^ 2) = 1 h₁ : x ^ 2 / (4 ^ 2 - 1) + y ^ 2 / (4 ^ 2 - 3 ^ 2) + z ^ 2 / (4 ^ 2 - 5 ^ 2) + w ^ 2 / (4 ^ 2 - 7 ^ 2) = 1 h₂ : x ^ 2 / (6 ^ 2 - 1) + y ^ 2 / (6 ^ 2 - 3 ^ 2) + z ^ 2 / (6 ^ 2 - 5 ^ 2) + w ^ 2 / (6 ^ 2 ...
207b0fffd3200c67
import Mathlib theorem algebra_amgm_sqrtxymulxmyeqxpy_xpygeq4 (x y : ℝ) (h₀ : 0 < x ∧ 0 < y) (h₁ : y ≤ x) (h₂ : Real.sqrt (x * y) * (x - y) = x + y) : x + y ≥ 4 := by
x y : ℝ h₀ : 0 < x ∧ 0 < y h₁ : y ≤ x h₂ : √(x * y) * (x - y) = x + y ⊢ x + y ≥ 4
13a1250eba46d0f8
import Mathlib theorem amc12a_2002_p21 (u : ℕ → ℕ) (h₀ : u 0 = 4) (h₁ : u 1 = 7) (h₂ : ∀ n ≥ 2, u (n + 2) = (u n + u (n + 1)) % 10) : ∀ n, (∑ k in Finset.range n, u k) > 10000 → 1999 ≤ n := by
u : ℕ → ℕ h₀ : u 0 = 4 h₁ : u 1 = 7 h₂ : ∀ n ≥ 2, u (n + 2) = (u n + u (n + 1)) % 10 ⊢ ∀ (n : ℕ), ∑ k ∈ Finset.range n, u k > 10000 → 1999 ≤ n
184afa75b49c6932
import Mathlib theorem mathd_algebra_192 (q e d : ℂ) (h₀ : q = 11 - 5 * Complex.I) (h₁ : e = 11 + 5 * Complex.I) (h₂ : d = 2 * Complex.I) : q * e * d = 292 * Complex.I := by
q e d : ℂ h₀ : q = 11 - 5 * Complex.I h₁ : e = 11 + 5 * Complex.I h₂ : d = 2 * Complex.I ⊢ q * e * d = 292 * Complex.I
83fd78760001ecc3
import Mathlib theorem amc12b_2002_p6 (a b : ℝ) (h₀ : a ≠ 0 ∧ b ≠ 0) (h₁ : ∀ x, x ^ 2 + a * x + b = (x - a) * (x - b)) : a = 1 ∧ b = -2 := by
a b : ℝ h₀ : a ≠ 0 ∧ b ≠ 0 h₁ : ∀ (x : ℝ), x ^ 2 + a * x + b = (x - a) * (x - b) ⊢ a = 1 ∧ b = -2
89a8631a4c5bbd4b
import Mathlib theorem mathd_numbertheory_102 : 2 ^ 8 % 5 = 1 := by
⊢ 2 ^ 8 % 5 = 1
b2260ca0363af843
import Mathlib theorem amc12a_2010_p22 (x : ℝ) : 49 ≤ ∑ k:ℤ in Finset.Icc 1 119, abs (↑k * x - 1) := by
x : ℝ ⊢ 49 ≤ ∑ k ∈ Finset.Icc 1 119, |↑k * x - 1|
f2ef7aee87fe1002
import Mathlib theorem mathd_numbertheory_81 : 71 % 3 = 2 := by
⊢ 71 % 3 = 2
e2fd9a4c46858eca
import Mathlib theorem mathd_numbertheory_155 : Finset.card (Finset.filter (fun x => x % 19 = 7) (Finset.Icc 100 999)) = 48 := by
⊢ (Finset.filter (fun x => x % 19 = 7) (Finset.Icc 100 999)).card = 48
4f8666e60b5d71a8
import Mathlib theorem imo_1978_p5 (n : ℕ) (a : ℕ → ℕ) (h₀ : Function.Injective a) (h₁ : a 0 = 0) (h₂ : 0 < n) : (∑ k in Finset.Icc 1 n, (1 : ℝ) / k) ≤ ∑ k in Finset.Icc 1 n, (a k : ℝ) / k ^ 2 := by
n : ℕ a : ℕ → ℕ h₀ : Function.Injective a h₁ : a 0 = 0 h₂ : 0 < n ⊢ ∑ k ∈ Finset.Icc 1 n, 1 / ↑k ≤ ∑ k ∈ Finset.Icc 1 n, ↑(a k) / ↑k ^ 2
f829d6e5cd474673
import Mathlib theorem amc12a_2017_p7 (f : ℕ → ℝ) (h₀ : f 1 = 2) (h₁ : ∀ n, 1 < n ∧ Even n → f n = f (n - 1) + 1) (h₂ : ∀ n, 1 < n ∧ Odd n → f n = f (n - 2) + 2) : f 2017 = 2018 := by
f : ℕ → ℝ h₀ : f 1 = 2 h₁ : ∀ (n : ℕ), 1 < n ∧ Even n → f n = f (n - 1) + 1 h₂ : ∀ (n : ℕ), 1 < n ∧ Odd n → f n = f (n - 2) + 2 ⊢ f 2017 = 2018
b1c4f4388d8bc36a
import Mathlib theorem mathd_numbertheory_42 (S : Set ℕ) (u v : ℕ) (h₀ : ∀ a : ℕ, a ∈ S ↔ 0 < a ∧ 27 * a % 40 = 17) (h₁ : IsLeast S u) (h₂ : IsLeast (S \ {u}) v) : u + v = 62 := by
S : Set ℕ u v : ℕ h₀ : ∀ (a : ℕ), a ∈ S ↔ 0 < a ∧ 27 * a % 40 = 17 h₁ : IsLeast S u h₂ : IsLeast (S \ {u}) v ⊢ u + v = 62
795db81db6b47511
import Mathlib theorem mathd_algebra_110 (q e : ℂ) (h₀ : q = 2 - 2 * Complex.I) (h₁ : e = 5 + 5 * Complex.I) : q * e = 20 := by
q e : ℂ h₀ : q = 2 - 2 * Complex.I h₁ : e = 5 + 5 * Complex.I ⊢ q * e = 20
bc55aac595e82c2c
import Mathlib theorem amc12b_2021_p21 (S : Finset ℝ) (h₀ : ∀ x : ℝ, x ∈ S ↔ 0 < x ∧ x ^ (2 : ℝ) ^ Real.sqrt 2 = Real.sqrt 2 ^ (2 : ℝ) ^ x) : (↑2 ≤ ∑ k in S, k) ∧ (∑ k in S, k) < 6 := by
S : Finset ℝ h₀ : ∀ (x : ℝ), x ∈ S ↔ 0 < x ∧ x ^ 2 ^ √2 = √2 ^ 2 ^ x ⊢ 2 ≤ ∑ k ∈ S, k ∧ ∑ k ∈ S, k < 6
b85784e202806953
import Mathlib theorem mathd_algebra_405 (S : Finset ℕ) (h₀ : ∀ x, x ∈ S ↔ 0 < x ∧ x ^ 2 + 4 * x + 4 < 20) : S.card = 2 := by
S : Finset ℕ h₀ : ∀ (x : ℕ), x ∈ S ↔ 0 < x ∧ x ^ 2 + 4 * x + 4 < 20 ⊢ S.card = 2
f2d146ec4391bcfa
import Mathlib theorem numbertheory_sumkmulnckeqnmul2pownm1 (n : ℕ) (h₀ : 0 < n) : (∑ k in Finset.Icc 1 n, k * Nat.choose n k) = n * 2 ^ (n - 1) := by
n : ℕ h₀ : 0 < n ⊢ ∑ k ∈ Finset.Icc 1 n, k * n.choose k = n * 2 ^ (n - 1)
fed4f7a6c17dc0aa
import Mathlib theorem mathd_algebra_393 (σ : Equiv ℝ ℝ) (h₀ : ∀ x, σ.1 x = 4 * x ^ 3 + 1) : σ.2 33 = 2 := by
σ : ℝ ≃ ℝ h₀ : ∀ (x : ℝ), σ.toFun x = 4 * x ^ 3 + 1 ⊢ σ.invFun 33 = 2
ed6b96fa632ab0ec
import Mathlib theorem amc12b_2004_p3 (x y : ℕ) (h₀ : 2 ^ x * 3 ^ y = 1296) : x + y = 8 := by
x y : ℕ h₀ : 2 ^ x * 3 ^ y = 1296 ⊢ x + y = 8
343bffafae02e014
import Mathlib theorem mathd_numbertheory_303 (S : Finset ℕ) (h₀ : ∀ n : ℕ, n ∈ S ↔ 2 ≤ n ∧ 171 ≡ 80 [MOD n] ∧ 468 ≡ 13 [MOD n]) : (∑ k in S, k) = 111 := by
S : Finset ℕ h₀ : ∀ (n : ℕ), n ∈ S ↔ 2 ≤ n ∧ 171 ≡ 80 [MOD n] ∧ 468 ≡ 13 [MOD n] ⊢ ∑ k ∈ S, k = 111
06269bdcf5a5fcc9
import Mathlib theorem mathd_algebra_151 : Int.ceil (Real.sqrt 27) - Int.floor (Real.sqrt 26) = 1 := by
⊢ ⌈√27⌉ - ⌊√26⌋ = 1
573f1c26c96fa806
import Mathlib theorem amc12a_2011_p18 (x y : ℝ) (h₀ : abs (x + y) + abs (x - y) = 2) : x ^ 2 - 6 * x + y ^ 2 ≤ 8 := by
x y : ℝ h₀ : |x + y| + |x - y| = 2 ⊢ x ^ 2 - 6 * x + y ^ 2 ≤ 8
066b42e19278f1b2
import Mathlib theorem mathd_algebra_15 (s : ℕ → ℕ → ℕ) (h₀ : ∀ a b, 0 < a ∧ 0 < b → s a b = a ^ (b : ℕ) + b ^ (a : ℕ)) : s 2 6 = 100 := by
s : ℕ → ℕ → ℕ h₀ : ∀ (a b : ℕ), 0 < a ∧ 0 < b → s a b = a ^ b + b ^ a ⊢ s 2 6 = 100
a3f84731df8ba905
import Mathlib theorem mathd_numbertheory_211 : Finset.card (Finset.filter (fun n => 6 ∣ 4 * ↑n - (2 : ℤ)) (Finset.range 60)) = 20 := by
⊢ (Finset.filter (fun n => 6 ∣ 4 * ↑n - 2) (Finset.range 60)).card = 20
df2e017834b83107
import Mathlib theorem mathd_numbertheory_640 : (91145 + 91146 + 91147 + 91148) % 4 = 2 := by
⊢ (91145 + 91146 + 91147 + 91148) % 4 = 2
5bcaec464e345b29
import Mathlib theorem amc12b_2003_p6 (a r : ℝ) (u : ℕ → ℝ) (h₀ : ∀ k, u k = a * r ^ k) (h₁ : u 1 = 2) (h₂ : u 3 = 6) : u 0 = 2 / Real.sqrt 3 ∨ u 0 = -(2 / Real.sqrt 3) := by
a r : ℝ u : ℕ → ℝ h₀ : ∀ (k : ℕ), u k = a * r ^ k h₁ : u 1 = 2 h₂ : u 3 = 6 ⊢ u 0 = 2 / √3 ∨ u 0 = -(2 / √3)
c5cb845a6c1c9dd6
import Mathlib theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : ℂ) : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by
a : ℂ ⊢ (a - 10) * (a + 11) = a ^ 2 + a - 110
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minif2f-satp

miniF2F benchmark with initial Lean 4 proof goal_state for both test and validation splits. Intended as the held-out evaluation (test) and hyperparameter-tuning (validation) sets for SATP / sketch-and-prove pipelines trained on NuminaMath-LEAN-satp.

Sibling datasets:

Splits 

from datasets import load_dataset
ds = load_dataset("ChristianZ97/minif2f-satp")
ds["test"]        # 244 rows — held-out evaluation
ds["validation"]  # 244 rows — hyperparameter tuning

Goal-state provenance (read this first)

For every row where the upstream DSP-Plus port already ships a goal field, the published goal_state is byte-for-byte identical to that DSP-Plus value. We do not re-elaborate or re-pretty-print rows that the upstream already provides — the published goal is the upstream goal, character-for-character.

Only the rows DSP-Plus left blank are filled by us, using the same lake harness as the NuminaMath-LEAN-satp training set so the fallback is consistent with the rest of the SATP pipeline.

Split Rows DSP+ stored (byte-equal copy) Lake-fresh fallback (DSP+ blank)
test 244 235 9
validation 244 244 0
Total 488 479 9

The 9 fallback rows in the test split are listed in manifest.json under splits.test.goal_source_counts.

Schema

Column Type Notes
uuid string sha256(canonical(formal_statement))[:16]; same hash scheme as NuminaMath-LEAN-satp.uuid
formal_statement string import Mathlib\n\n + theorem decl ending in := by
goal_state string Pretty-printed Lean goal at proof entry: hypothesis lines + ⊢ + target

Canonicalization

canonical(s) strips -- line comments and collapses whitespace runs.

Source

Vendored from DSP-Plus/datasets/minif2f.jsonl (submodule pin eb98320). DSP-Plus's minif2f is a v4.17-aligned port of the DeepMind miniF2F benchmark — the upstream DeepMind repo pins v4.27.0 and will not elaborate under our v4.17 Mathlib without substantial porting work. DSP-Plus's port is what the rest of the SATP pipeline already uses, so we sourced from it directly to guarantee env-aligned elaboration.

Lean environment

Both DSP-Plus stored goals and our fallback fresh goals were produced under the same Lean toolchain (leanprover/lean4:v4.17.0-rc1) and the same Mathlib commit (8b97781329). Stored values come from DSP-Plus's own elaboration harness; fallback values come from lake env lean --json + trace_state instrumentation.

Citation 

@misc{minif2f,
  author = {Zheng, Kunhao and Han, Jesse Michael and Polu, Stanislas},
  title  = {miniF2F: a cross-system benchmark for formal Olympiad-level mathematics},
  year   = {2021},
  url    = {https://github.com/google-deepmind/miniF2F}
}
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