uuid stringlengths 16 16 | formal_statement stringlengths 108 665 | goal_state stringlengths 12 476 |
|---|---|---|
f512fdbf83cae765 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_89 (b : ℝ) (h₀ : b ≠ 0) :
(7 * b ^ 3) ^ 2 * (4 * b ^ 2) ^ (-(3 : ℤ)) = (49 / 64) := by | b : ℝ
h₀ : b ≠ 0
⊢ (7 * b ^ 3) ^ 2 * (4 * b ^ 2) ^ (-3) = 49 / 64 |
31f891f3dacd0a32 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem imo_1966_p4 (n : ℕ) (x : ℝ) (h₀ : ∀ t : ℕ, t ≤ n → ∀ k : ℤ, x ≠ k * π / 2 ^ t)
(h₁ : 0 < n) :
∑ i ∈ Finset.Icc 1 n, 1 / Real.sin (2 ^ i * x) = 1 / Real.tan x - 1 / Real.tan (2 ^ n * x) := by | n : ℕ
x : ℝ
h₀ : ∀ t ≤ n, ∀ (k : ℤ), x ≠ ↑k * π / 2 ^ t
h₁ : 0 < n
⊢ ∑ i ∈ Finset.Icc 1 n, 1 / Real.sin (2 ^ i * x) = 1 / Real.tan x - 1 / Real.tan (2 ^ n * x) |
51a23bae9ef64f96 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_67 (f g : ℝ → ℝ) (h₀ : ∀ x, f x = 5 * x + 3) (h₁ : ∀ x, g x = x ^ 2 - 2) :
g (f (-1)) = (2) := by | f g : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = 5 * x + 3
h₁ : ∀ (x : ℝ), g x = x ^ 2 - 2
⊢ g (f (-1)) = 2 |
e16a2e2b466d89e2 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_numbertheory_326 (n : ℕ) (h₀ : (↑n - 1) * ↑n * (↑n + 1) = (720 : ℤ)) : n + 1 = (10) := by | n : ℕ
h₀ : (↑n - 1) * ↑n * (↑n + 1) = 720
⊢ n + 1 = 10 |
590b7c7da52040ad | import Mathlib
open scoped Nat Real Topology Polynomial
theorem induction_divisibility_3div2tooddnp1 (n : ℕ) : 3 ∣ 2 ^ (2 * n + 1) + 1 := by | n : ℕ
⊢ 3 ∣ 2 ^ (2 * n + 1) + 1 |
4608603f02591b40 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_123 (a b : ℕ) (h₀ : 0 < a ∧ 0 < b) (h₁ : a + b = 20) (h₂ : a = 3 * b) :
a - b = (10) := by | a b : ℕ
h₀ : 0 < a ∧ 0 < b
h₁ : a + b = 20
h₂ : a = 3 * b
⊢ a - b = 10 |
40016e2eaedcb104 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4 (x e : ℂ) (h₀ : x + e = 7)
(h₁ : 2 * x + e = 3) : e = 11 ∧ x = -4 := by | x e : ℂ
h₀ : x + e = 7
h₁ : 2 * x + e = 3
⊢ e = 11 ∧ x = -4 |
56ec66f844159732 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem imo_1993_p5 : ∃ f : ℕ → ℕ, f 1 = 2 ∧ ∀ n, f (f n) = f n + n ∧ ∀ n, f n < f (n + 1) := by | ⊢ ∃ f, f 1 = 2 ∧ ∀ (n : ℕ), f (f n) = f n + n ∧ ∀ (n : ℕ), f n < f (n + 1) |
5a9da85bacc85bae | import Mathlib
open scoped Nat Real Topology Polynomial
theorem numbertheory_prmdvsneqnsqmodpeq0 (n : ℤ) (p : ℕ) (h₀ : Nat.Prime p) :
↑p ∣ n ↔ n ^ 2 % p = 0 := by | n : ℤ
p : ℕ
h₀ : Nat.Prime p
⊢ ↑p ∣ n ↔ n ^ 2 % ↑p = 0 |
736c689d50002a6a | import Mathlib
open scoped Nat Real Topology Polynomial
theorem imo_1964_p1_1 (n : ℕ) (h₀ : 7 ∣ 2 ^ n - 1) : 3 ∣ n := by | n : ℕ
h₀ : 7 ∣ 2 ^ n - 1
⊢ 3 ∣ n |
0b5547c1b197160f | import Mathlib
open scoped Nat Real Topology Polynomial
theorem imo_1990_p3 : {n : ℕ | 1 < n ∧ n ^ 2 ∣ 2 ^ n + 1} = ({3}) := by | ⊢ {n | 1 < n ∧ n ^ 2 ∣ 2 ^ n + 1} = {3} |
740f0526ae8dcf11 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem induction_ineq_nsqlefactn (n : ℕ) (h₀ : 4 ≤ n) : n ^ 2 ≤ n ! := by | n : ℕ
h₀ : 4 ≤ n
⊢ n ^ 2 ≤ n ! |
8ffd002963fc9c7f | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_numbertheory_30 :
(33818 ^ 2 + 33819 ^ 2 + 33820 ^ 2 + 33821 ^ 2 + 33822 ^ 2) % 17 = (0) := by | ⊢ (33818 ^ 2 + 33819 ^ 2 + 33820 ^ 2 + 33821 ^ 2 + 33822 ^ 2) % 17 = 0 |
b125d0c65bbb456d | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_267 (x : ℝ) (h₀ : x ≠ 1) (h₁ : x ≠ -2)
(h₂ : (x + 1) / (x - 1) = (x - 2) / (x + 2)) : x = (0) := by | x : ℝ
h₀ : x ≠ 1
h₁ : x ≠ -2
h₂ : (x + 1) / (x - 1) = (x - 2) / (x + 2)
⊢ x = 0 |
3bbe97444ad40342 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_numbertheory_961 : 2003 % 11 = (1) := by | ⊢ 2003 % 11 = 1 |
a3cb6f1d88897c79 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem induction_seq_mul2pnp1 (n : ℕ) (u : ℕ → ℕ) (h₀ : u 0 = 0)
(h₁ : ∀ n, u (n + 1) = 2 * u n + (n + 1)) : u n = 2 ^ (n + 1) - (n + 2) := by | n : ℕ
u : ℕ → ℕ
h₀ : u 0 = 0
h₁ : ∀ (n : ℕ), u (n + 1) = 2 * u n + (n + 1)
⊢ u n = 2 ^ (n + 1) - (n + 2) |
2eabaae116a89221 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem amc12a_2002_p12 (f : ℝ → ℝ) (k : ℝ) (h₀ : ∀ x, f x = x ^ 2 - 63 * x + k) (h₁ : ∃ x, f x = 0)
(h₂ : f ⁻¹' {0} ⊆ {x : ℝ | ∃ n : ℕ, ↑n = x ∧ Nat.Prime n}) : k = (122) := by | f : ℝ → ℝ
k : ℝ
h₀ : ∀ (x : ℝ), f x = x ^ 2 - 63 * x + k
h₁ : ∃ x, f x = 0
h₂ : f ⁻¹' {0} ⊆ {x | ∃ n, ↑n = x ∧ Nat.Prime n}
⊢ k = 122 |
cf807cbd2525b333 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem algebra_manipexpr_2erprsqpesqeqnrpnesq (e r : ℂ) :
2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 := by | e r : ℂ
⊢ 2 * (e * r) + (e ^ 2 + r ^ 2) = (-r + -e) ^ 2 |
07e000b1fd9760eb | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_119 (d e : ℝ) (h₀ : 2 * d = 17 * e - 8) (h₁ : 2 * e = d - 9) : e = (2) := by | d e : ℝ
h₀ : 2 * d = 17 * e - 8
h₁ : 2 * e = d - 9
⊢ e = 2 |
d23d19ed0f57f528 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem amc12a_2020_p13 (a b c : ℕ) (n : ℝ)
-- With h₀: 1 < n, ℝ should be fine over nnreal.
(h₀ : 1 < n)
(h₁ : 1 < a ∧ 1 < b ∧ 1 < c)
(h₂ : (n * (n * n ^ (1 / c : ℝ)) ^ (1 / b : ℝ)) ^ (1 / a : ℝ) = (n ^ 25) ^ (1 / 36 : ℝ)) :
b = (3) := by | a b c : ℕ
n : ℝ
h₀ : 1 < n
h₁ : 1 < a ∧ 1 < b ∧ 1 < c
h₂ : (n * (n * n ^ (1 / ↑c)) ^ (1 / ↑b)) ^ (1 / ↑a) = (n ^ 25) ^ (1 / 36)
⊢ b = 3 |
be3b9cd1b7d5d99f | import Mathlib
open scoped Nat Real Topology Polynomial
theorem imo_1977_p5 :
{(a, b) | (a : ℕ) (b : ℕ) (q : ℕ) (r : ℕ)
(h₀ : r < a + b)
(h₁ : a^2 + b^2 = (a + b) * q + r)
(h₂ : q^2 + r = 1977)} =
({(37, 50), (7, 50), (50, 37), (50, 7)}) := by | ⊢ {x | ∃ a b q r, ∃ (_ : r < a + b) (_ : a ^ 2 + b ^ 2 = (a + b) * q + r) (_ : q ^ 2 + r = 1977), (a, b) = x} =
{(37, 50), (7, 50), (50, 37), (50, 7)} |
788a52377215c0f0 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem numbertheory_2dvd4expn (n : ℕ) (h₀ : n ≠ 0) : 2 ∣ 4 ^ n := by | n : ℕ
h₀ : n ≠ 0
⊢ 2 ∣ 4 ^ n |
af485fc2208517df | import Mathlib
open scoped Nat Real Topology Polynomial
theorem amc12a_2010_p11 (x b : ℝ) (h₀ : 0 < b) (h₁ : (7 : ℝ) ^ (x + 7) = 8 ^ x)
(h₂ : x = Real.logb b (7 ^ 7)) : b = (8 / 7) := by | x b : ℝ
h₀ : 0 < b
h₁ : 7 ^ (x + 7) = 8 ^ x
h₂ : x = Real.logb b (7 ^ 7)
⊢ b = 8 / 7 |
51b0f1759fdd3360 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem amc12a_2003_p24 :
IsGreatest {y : ℝ | ∃ a b : ℝ, 1 < b ∧ b ≤ a ∧ y = Real.logb a (a / b) + Real.logb b (b / a)}
(0) := by | ⊢ IsGreatest {y | ∃ a b, 1 < b ∧ b ≤ a ∧ y = Real.logb a (a / b) + Real.logb b (b / a)} 0 |
8b85348f36e047a5 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem amc12a_2002_p1 (f : ℂ → ℂ) (h₀ : ∀ x, f x = (2 * x + 3) * (x - 4) + (2 * x + 3) * (x - 6))
(h₁ : Fintype (f ⁻¹' {0})) : ∑ y ∈ (f ⁻¹' {0}).toFinset, y = (7 / 2) := by | f : ℂ → ℂ
h₀ : ∀ (x : ℂ), f x = (2 * x + 3) * (x - 4) + (2 * x + 3) * (x - 6)
h₁ : Fintype ↑(f ⁻¹' {0})
⊢ ∑ y ∈ (f ⁻¹' {0}).toFinset, y = 7 / 2 |
ed054f1595d12ab7 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_206 (a b : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f x = x ^ 2 + a * x + b) (h₁ : 2 * a ≠ b)
(h₂ : f (2 * a) = 0) (h₃ : f b = 0) : a + b = (-1) := by | a b : ℝ
f : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = x ^ 2 + a * x + b
h₁ : 2 * a ≠ b
h₂ : f (2 * a) = 0
h₃ : f b = 0
⊢ a + b = -1 |
123f866f570d8eeb | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_numbertheory_92 (n : ℕ) (h₀ : 5 * n % 17 = 8) : n % 17 = (5) := by | n : ℕ
h₀ : 5 * n % 17 = 8
⊢ n % 17 = 5 |
04dde96a4ac51f76 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_482 (m n : ℕ) (k : ℝ) (f : ℝ → ℝ) (h₀ : Nat.Prime m) (h₁ : Nat.Prime n)
(h₂ : ∀ x, f x = x ^ 2 - 12 * x + k) (h₃ : f m = 0) (h₄ : f n = 0) (h₅ : m ≠ n) : k = (35) := by | m n : ℕ
k : ℝ
f : ℝ → ℝ
h₀ : Nat.Prime m
h₁ : Nat.Prime n
h₂ : ∀ (x : ℝ), f x = x ^ 2 - 12 * x + k
h₃ : f ↑m = 0
h₄ : f ↑n = 0
h₅ : m ≠ n
⊢ k = 35 |
f9973a93d7e27317 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem amc12b_2002_p3 (S : Finset ℕ)
-- note: we use (n^2 + 2 - 3 * n) over (n^2 - 3 * n + 2) because nat subtraction truncates the latter at 1 and 2
(h₀ : ∀ n : ℕ, n ∈ S ↔ 0 < n ∧ Nat.Prime (n ^ 2 + 2 - 3 * n)) :
S.card = (1) := by | S : Finset ℕ
h₀ : ∀ (n : ℕ), n ∈ S ↔ 0 < n ∧ Nat.Prime (n ^ 2 + 2 - 3 * n)
⊢ S.card = 1 |
a3a1026af7f637be | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_numbertheory_668 (l r : ZMod 7) (h₀ : l = (2 + 3)⁻¹) (h₁ : r = 2⁻¹ + 3⁻¹) :
l - r = (1) := by | l r : ZMod 7
h₀ : l = (2 + 3)⁻¹
h₁ : r = 2⁻¹ + 3⁻¹
⊢ l - r = 1 |
93607103a1d0dd6b | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_251 (x : ℝ) (h₀ : x ≠ 0) (h₁ : 3 + 1 / x = 7 / x) : x = (2) := by | x : ℝ
h₀ : x ≠ 0
h₁ : 3 + 1 / x = 7 / x
⊢ x = 2 |
41620ccb6deb33e6 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_numbertheory_84 : Int.floor ((9 : ℝ) / 160 * 100) = (5) := by | ⊢ ⌊9 / 160 * 100⌋ = 5 |
96bd620500910323 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_numbertheory_412 (x y : ℕ) (h₀ : x % 19 = 4) (h₁ : y % 19 = 7) :
(x + 1) ^ 2 * (y + 5) ^ 3 % 19 = (13) := by | x y : ℕ
h₀ : x % 19 = 4
h₁ : y % 19 = 7
⊢ (x + 1) ^ 2 * (y + 5) ^ 3 % 19 = 13 |
8731093af59551a5 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_181 (n : ℝ) (h₀ : n ≠ 3) (h₁ : (n + 5) / (n - 3) = 2) : n = (11) := by | n : ℝ
h₀ : n ≠ 3
h₁ : (n + 5) / (n - 3) = 2
⊢ n = 11 |
1932f0ac32331562 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem amc12a_2016_p3 (f : ℝ → ℝ → ℝ)
(h₀ : ∀ x, ∀ y ≠ 0, f x y = x - y * Int.floor (x / y)) :
f (3 / 8) (-(2 / 5)) = (-(1 / 40)) := by | f : ℝ → ℝ → ℝ
h₀ : ∀ (x y : ℝ), y ≠ 0 → f x y = x - y * ↑⌊x / y⌋
⊢ f (3 / 8) (-(2 / 5)) = -(1 / 40) |
a7d895a8aff1a31b | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_247 (t s : ℝ) (n : ℤ) (h₀ : t = 2 * s - s ^ 2) (h₁ : s = n ^ 2 - 2 ^ n + 1)
(h₂ : n = 3) : t = (0) := by | t s : ℝ
n : ℤ
h₀ : t = 2 * s - s ^ 2
h₁ : s = ↑n ^ 2 - 2 ^ n + 1
h₂ : n = 3
⊢ t = 0 |
0505ade09c62714a | import Mathlib
open scoped Nat Real Topology Polynomial
theorem algebra_sqineq_2unitcircatblt1 (a b : ℝ) (h₀ : a ^ 2 + b ^ 2 = 2) : a * b ≤ 1 := by | a b : ℝ
h₀ : a ^ 2 + b ^ 2 = 2
⊢ a * b ≤ 1 |
1b015266d5d26a46 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_numbertheory_629 : IsLeast {t : ℕ | 0 < t ∧ Nat.lcm 12 t ^ 3 = (12 * t) ^ 2} (18) := by | ⊢ IsLeast {t | 0 < t ∧ Nat.lcm 12 t ^ 3 = (12 * t) ^ 2} 18 |
57ffdd4fca44bce0 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem amc12a_2017_p2 (x y : ℝ) (h₀ : x ≠ 0) (h₁ : y ≠ 0) (h₂ : x + y = 4 * (x * y)) :
1 / x + 1 / y = (4) := by | x y : ℝ
h₀ : x ≠ 0
h₁ : y ≠ 0
h₂ : x + y = 4 * (x * y)
⊢ 1 / x + 1 / y = 4 |
00620639a9fa579d | import Mathlib
open scoped Nat Real Topology Polynomial
theorem algebra_amgm_sumasqdivbsqgeqsumbdiva (a b c : ℝ) (h₀ : 0 < a ∧ 0 < b ∧ 0 < c) :
a ^ 2 / b ^ 2 + b ^ 2 / c ^ 2 + c ^ 2 / a ^ 2 ≥ b / a + c / b + a / c := by | a b c : ℝ
h₀ : 0 < a ∧ 0 < b ∧ 0 < c
⊢ a ^ 2 / b ^ 2 + b ^ 2 / c ^ 2 + c ^ 2 / a ^ 2 ≥ b / a + c / b + a / c |
cebb6d5af38738d9 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_numbertheory_202 : (19 ^ 19 + 99 ^ 99) % 10 = (8) := by | ⊢ (19 ^ 19 + 99 ^ 99) % 10 = 8 |
d2d261aad22620e9 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem imo_1979_p1 (p q : ℕ) (h₀ : 0 < q)
(h₁ : ∑ k ∈ Finset.Icc (1 : ℕ) 1319, (-1) ^ (k + 1) * ((1 : ℝ) / k) = p / q) : 1979 ∣ p := by | p q : ℕ
h₀ : 0 < q
h₁ : ∑ k ∈ Finset.Icc 1 1319, (-1) ^ (k + 1) * (1 / ↑k) = ↑p / ↑q
⊢ 1979 ∣ p |
88a9f7e28f1b5659 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_51 (a b : ℝ) (h₀ : 0 < a ∧ 0 < b) (h₁ : a + b = 35) (h₂ : a = 2 / 5 * b) :
b - a = (15) := by | a b : ℝ
h₀ : 0 < a ∧ 0 < b
h₁ : a + b = 35
h₂ : a = 2 / 5 * b
⊢ b - a = 15 |
3706cb101167be58 | import Mathlib
open scoped Nat Real Topology Polynomial
theorem mathd_algebra_10 : abs ((120 : ℝ) / 100 * 30 - 130 / 100 * 20) = (10) := by | ⊢ |120 / 100 * 30 - 130 / 100 * 20| = 10 |
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