uuid stringlengths 16 16 | formal_statement stringlengths 64 646 | goal_state stringlengths 12 485 |
|---|---|---|
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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