uuid stringlengths 16 16 | formal_statement stringlengths 64 646 | goal_state stringlengths 12 485 |
|---|---|---|
41816ac1d2b2f3e8 | import Mathlib
theorem aime_1991_p1 (x y : ℕ) (h₀ : 0 < x ∧ 0 < y) (h₁ : x * y + (x + y) = 71)
(h₂ : x ^ 2 * y + x * y ^ 2 = 880) : x ^ 2 + y ^ 2 = 146 := by | x y : ℕ
h₀ : 0 < x ∧ 0 < y
h₁ : x * y + (x + y) = 71
h₂ : x ^ 2 * y + x * y ^ 2 = 880
⊢ x ^ 2 + y ^ 2 = 146 |
9b04ff37c3290f6f | import Mathlib
theorem mathd_algebra_43 (a b : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f x = a * x + b) (h₁ : f 7 = 4)
(h₂ : f 6 = 3) : f 3 = 0 := by | a b : ℝ
f : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = a * x + b
h₁ : f 7 = 4
h₂ : f 6 = 3
⊢ f 3 = 0 |
676f38cada79f3fe | import Mathlib
theorem imo_1988_p6 (a b : ℕ) (h₀ : 0 < a ∧ 0 < b) (h₁ : a * b + 1 ∣ a ^ 2 + b ^ 2) :
∃ x : ℕ, (x ^ 2 : ℝ) = (a ^ 2 + b ^ 2) / (a * b + 1) := by | a b : ℕ
h₀ : 0 < a ∧ 0 < b
h₁ : a * b + 1 ∣ a ^ 2 + b ^ 2
⊢ ∃ x, ↑x ^ 2 = (↑a ^ 2 + ↑b ^ 2) / (↑a * ↑b + 1) |
c23251c5e6a7cb6e | import Mathlib
theorem aime_1996_p5 (a b c r s t : ℝ) (f g : ℝ → ℝ)
(h₀ : ∀ x, f x = x ^ 3 + 3 * x ^ 2 + 4 * x - 11) (h₁ : ∀ x, g x = x ^ 3 + r * x ^ 2 + s * x + t)
(h₂ : f a = 0) (h₃ : f b = 0) (h₄ : f c = 0) (h₅ : g (a + b) = 0) (h₆ : g (b + c) = 0)
(h₇ : g (c + a) = 0) (h₈ : List.Pairwise (· ≠ ·) [a, b, c]) :... | a b c r s t : ℝ
f g : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = x ^ 3 + 3 * x ^ 2 + 4 * x - 11
h₁ : ∀ (x : ℝ), g x = x ^ 3 + r * x ^ 2 + s * x + t
h₂ : f a = 0
h₃ : f b = 0
h₄ : f c = 0
h₅ : g (a + b) = 0
h₆ : g (b + c) = 0
h₇ : g (c + a) = 0
h₈ : List.Pairwise (fun x x_1 => x ≠ x_1) [a, b, c]
⊢ t = 23 |
e0ee4dfe5300e231 | import Mathlib
theorem mathd_algebra_55 (q p : ℝ) (h₀ : q = 2 - 4 + 6 - 8 + 10 - 12 + 14)
(h₁ : p = 3 - 6 + 9 - 12 + 15 - 18 + 21) : q / p = 2 / 3 := by | q p : ℝ
h₀ : q = 2 - 4 + 6 - 8 + 10 - 12 + 14
h₁ : p = 3 - 6 + 9 - 12 + 15 - 18 + 21
⊢ q / p = 2 / 3 |
93afb0809e2cd01a | import Mathlib
theorem algebra_sqineq_2at2pclta2c2p41pc (a c : ℝ) :
2 * a * (2 + c) ≤ a ^ 2 + c ^ 2 + 4 * (1 + c) := by | a c : ℝ
⊢ 2 * a * (2 + c) ≤ a ^ 2 + c ^ 2 + 4 * (1 + c) |
3f6233502aa874c9 | import Mathlib
theorem mathd_numbertheory_43 : IsGreatest { n : ℕ | 15 ^ n ∣ 942! } 233 := by | ⊢ IsGreatest {n | 15 ^ n ∣ 942!} 233 |
5bc80dbb56d25164 | import Mathlib
theorem mathd_algebra_214 (a : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f x = a * (x - 2) ^ 2 + 3) (h₁ : f 4 = 4) :
f 6 = 7 := by | a : ℝ
f : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = a * (x - 2) ^ 2 + 3
h₁ : f 4 = 4
⊢ f 6 = 7 |
fdc8aa6d31aa9108 | import Mathlib
theorem mathd_algebra_96 (x y z a : ℝ) (h₀ : 0 < x ∧ 0 < y ∧ 0 < z)
(h₁ : Real.log x - Real.log y = a) (h₂ : Real.log y - Real.log z = 15)
(h₃ : Real.log z - Real.log x = -7) : a = -8 := by | x y z a : ℝ
h₀ : 0 < x ∧ 0 < y ∧ 0 < z
h₁ : x.log - y.log = a
h₂ : y.log - z.log = 15
h₃ : z.log - x.log = -7
⊢ a = -8 |
13a6774394e91f5a | import Mathlib
theorem amc12_2001_p2 (a b n : ℕ) (h₀ : 1 ≤ a ∧ a ≤ 9) (h₁ : 0 ≤ b ∧ b ≤ 9) (h₂ : n = 10 * a + b)
(h₃ : n = a * b + a + b) : b = 9 := by | a b n : ℕ
h₀ : 1 ≤ a ∧ a ≤ 9
h₁ : 0 ≤ b ∧ b ≤ 9
h₂ : n = 10 * a + b
h₃ : n = a * b + a + b
⊢ b = 9 |
a357c1a0150f3b2d | import Mathlib
theorem mathd_algebra_185 (s : Finset ℤ) (f : ℤ → ℤ) (h₀ : ∀ x, f x = abs (x + 4))
(h₁ : ∀ x, x ∈ s ↔ f x < 9) : s.card = 17 := by | s : Finset ℤ
f : ℤ → ℤ
h₀ : ∀ (x : ℤ), f x = |x + 4|
h₁ : ∀ (x : ℤ), x ∈ s ↔ f x < 9
⊢ s.card = 17 |
2c298468e7f04db7 | import Mathlib
theorem algebra_binomnegdiscrineq_10alt28asqp1 (a : ℝ) : 10 * a ≤ 28 * a ^ 2 + 1 := by | a : ℝ
⊢ 10 * a ≤ 28 * a ^ 2 + 1 |
1e4a1b970d57931f | import Mathlib
theorem mathd_numbertheory_284 (a b : ℕ) (h₀ : 1 ≤ a ∧ a ≤ 9 ∧ b ≤ 9)
(h₁ : 10 * a + b = 2 * (a + b)) : 10 * a + b = 18 := by | a b : ℕ
h₀ : 1 ≤ a ∧ a ≤ 9 ∧ b ≤ 9
h₁ : 10 * a + b = 2 * (a + b)
⊢ 10 * a + b = 18 |
9b6671244ff4f393 | import Mathlib
theorem amc12a_2009_p2 : 1 + 1 / (1 + 1 / (1 + 1)) = (5 : ℚ) / 3 := by | ⊢ 1 + 1 / (1 + 1 / (1 + 1)) = 5 / 3 |
cf45fc92da783205 | import Mathlib
theorem mathd_numbertheory_709 (n : ℕ) (h₀ : 0 < n) (h₁ : Finset.card (Nat.divisors (2 * n)) = 28)
(h₂ : Finset.card (Nat.divisors (3 * n)) = 30) : Finset.card (Nat.divisors (6 * n)) = 35 := by | n : ℕ
h₀ : 0 < n
h₁ : (2 * n).divisors.card = 28
h₂ : (3 * n).divisors.card = 30
⊢ (6 * n).divisors.card = 35 |
994edf3410f504db | import Mathlib
theorem amc12a_2013_p8 (x y : ℝ) (h₀ : x ≠ 0) (h₁ : y ≠ 0) (h₂ : x ≠ y)
(h₃ : x + 2 / x = y + 2 / y) : x * y = 2 := by | x y : ℝ
h₀ : x ≠ 0
h₁ : y ≠ 0
h₂ : x ≠ y
h₃ : x + 2 / x = y + 2 / y
⊢ x * y = 2 |
baae31e11f7be935 | import Mathlib
theorem mathd_numbertheory_461 (n : ℕ)
(h₀ : n = Finset.card (Finset.filter (fun x => Nat.gcd x 8 = 1) (Finset.Icc 1 7))) :
3 ^ n % 8 = 1 := by | n : ℕ
h₀ : n = (Finset.filter (fun x => x.gcd 8 = 1) (Finset.Icc 1 7)).card
⊢ 3 ^ n % 8 = 1 |
5d4a6824ed3e6d5b | import Mathlib
theorem mathd_algebra_59 (b : ℝ) (h₀ : (4 : ℝ) ^ b + 2 ^ 3 = 12) : b = 1 := by | b : ℝ
h₀ : 4 ^ b + 2 ^ 3 = 12
⊢ b = 1 |
dc0b0989c234539f | import Mathlib
theorem mathd_algebra_234 (d : ℝ) (h₀ : 27 / 125 * d = 9 / 25) : 3 / 5 * d ^ 3 = 25 / 9 := by | d : ℝ
h₀ : 27 / 125 * d = 9 / 25
⊢ 3 / 5 * d ^ 3 = 25 / 9 |
e3c85982301953ab | import Mathlib
theorem imo_1973_p3 (a b : ℝ) (h₀ : ∃ x, x ^ 4 + a * x ^ 3 + b * x ^ 2 + a * x + 1 = 0) :
4 / 5 ≤ a ^ 2 + b ^ 2 := by | a b : ℝ
h₀ : ∃ x, x ^ 4 + a * x ^ 3 + b * x ^ 2 + a * x + 1 = 0
⊢ 4 / 5 ≤ a ^ 2 + b ^ 2 |
61db31417a3bebad | import Mathlib
theorem amc12b_2020_p5 (a b : ℕ) (h₀ : (5 : ℚ) / 8 * b = 2 / 3 * a + 7)
(h₁ : (b : ℚ) - 5 / 8 * b = a - 2 / 3 * a + 7) : a = 42 := by | a b : ℕ
h₀ : 5 / 8 * ↑b = 2 / 3 * ↑a + 7
h₁ : ↑b - 5 / 8 * ↑b = ↑a - 2 / 3 * ↑a + 7
⊢ a = 42 |
7b2c51c0f77db0ff | import Mathlib
theorem numbertheory_sqmod3in01d (a : ℤ) : a ^ 2 % 3 = 0 ∨ a ^ 2 % 3 = 1 := by | a : ℤ
⊢ a ^ 2 % 3 = 0 ∨ a ^ 2 % 3 = 1 |
372f318766f4f467 | import Mathlib
theorem mathd_algebra_131 (a b : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f x = 2 * x ^ 2 - 7 * x + 2)
(h₁ : f a = 0) (h₂ : f b = 0) (h₃ : a ≠ b) : 1 / (a - 1) + 1 / (b - 1) = -1 := by | a b : ℝ
f : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = 2 * x ^ 2 - 7 * x + 2
h₁ : f a = 0
h₂ : f b = 0
h₃ : a ≠ b
⊢ 1 / (a - 1) + 1 / (b - 1) = -1 |
6973c3d480f8fd25 | import Mathlib
theorem amc12b_2003_p17 (x y : ℝ) (h₀ : 0 < x ∧ 0 < y) (h₁ : Real.log (x * y ^ 3) = 1)
(h₂ : Real.log (x ^ 2 * y) = 1) : Real.log (x * y) = 3 / 5 := by | x y : ℝ
h₀ : 0 < x ∧ 0 < y
h₁ : (x * y ^ 3).log = 1
h₂ : (x ^ 2 * y).log = 1
⊢ (x * y).log = 3 / 5 |
7c8817c5ae1159ff | import Mathlib
theorem mathd_algebra_536 : ↑3! * ((2 : ℝ) ^ 3 + Real.sqrt 9) / 2 = (33 : ℝ) := by | ⊢ ↑3! * (2 ^ 3 + √9) / 2 = 33 |
602d45f9146f2367 | import Mathlib
theorem mathd_algebra_22 : Real.logb (5 ^ 2) (5 ^ 4) = 2 := by | ⊢ (5 ^ 2).logb (5 ^ 4) = 2 |
e130b2da7b3e2446 | import Mathlib
theorem numbertheory_xsqpysqintdenomeq (x y : ℚ) (h₀ : (x ^ 2 + y ^ 2).den = 1) : x.den = y.den := by | x y : ℚ
h₀ : (x ^ 2 + y ^ 2).den = 1
⊢ x.den = y.den |
a99fb5dca86f950f | import Mathlib
theorem aimeII_2001_p3 (x : ℕ → ℤ) (h₀ : x 1 = 211) (h₂ : x 2 = 375) (h₃ : x 3 = 420)
(h₄ : x 4 = 523) (h₆ : ∀ n ≥ 5, x n = x (n - 1) - x (n - 2) + x (n - 3) - x (n - 4)) :
x 531 + x 753 + x 975 = 898 := by | x : ℕ → ℤ
h₀ : x 1 = 211
h₂ : x 2 = 375
h₃ : x 3 = 420
h₄ : x 4 = 523
h₆ : ∀ n ≥ 5, x n = x (n - 1) - x (n - 2) + x (n - 3) - x (n - 4)
⊢ x 531 + x 753 + x 975 = 898 |
4980a50044476074 | import Mathlib
theorem mathd_numbertheory_22 (b : ℕ) (h₀ : b < 10)
(h₁ : Nat.sqrt (10 * b + 6) * Nat.sqrt (10 * b + 6) = 10 * b + 6) : b = 3 ∨ b = 1 := by | b : ℕ
h₀ : b < 10
h₁ : (10 * b + 6).sqrt * (10 * b + 6).sqrt = 10 * b + 6
⊢ b = 3 ∨ b = 1 |
aaee02069b219fc1 | import Mathlib
theorem aime_1987_p8 :
IsGreatest { n : ℕ | 0 < n ∧ ∃! k : ℕ, (8 : ℝ) / 15 < n / (n + k) ∧ (n : ℝ) / (n + k) < 7 / 13 } 112 := by | ⊢ IsGreatest {n | 0 < n ∧ ∃! k, 8 / 15 < ↑n / (↑n + ↑k) ∧ ↑n / (↑n + ↑k) < 7 / 13} 112 |
2d04daef09402101 | import Mathlib
theorem mathd_numbertheory_136 (n : ℕ) (h₀ : 123 * n + 17 = 39500) : n = 321 := by | n : ℕ
h₀ : 123 * n + 17 = 39500
⊢ n = 321 |
e1dd0fcdb741e756 | import Mathlib
theorem amc12_2000_p11 (a b : ℝ) (h₀ : a ≠ 0 ∧ b ≠ 0) (h₁ : a * b = a - b) :
a / b + b / a - a * b = 2 := by | a b : ℝ
h₀ : a ≠ 0 ∧ b ≠ 0
h₁ : a * b = a - b
⊢ a / b + b / a - a * b = 2 |
890af5c3805013ca | import Mathlib
theorem amc12b_2003_p9 (a b : ℝ) (f : ℝ → ℝ) (h₀ : ∀ x, f x = a * x + b) (h₁ : f 6 - f 2 = 12) :
f 12 - f 2 = 30 := by | a b : ℝ
f : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = a * x + b
h₁ : f 6 - f 2 = 12
⊢ f 12 - f 2 = 30 |
9bd8ef8ce5b9699e | import Mathlib
theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : ℂ) :
x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) := by | x : ℂ
⊢ x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) |
ba23df85d9d1f9dd | import Mathlib
theorem mathd_numbertheory_198 : 5 ^ 2005 % 100 = 25 := by | ⊢ 5 ^ 2005 % 100 = 25 |
554d56d947bcdf18 | import Mathlib
theorem mathd_algebra_149 (f : ℝ → ℝ) (h₀ : ∀ x < -5, f x = x ^ 2 + 5)
(h₁ : ∀ x ≥ -5, f x = 3 * x - 8) (h₂ : Fintype (f ⁻¹' {10})) :
(∑ k in (f ⁻¹' {10}).toFinset, k) = 6 := by | f : ℝ → ℝ
h₀ : ∀ x < -5, f x = x ^ 2 + 5
h₁ : ∀ x ≥ -5, f x = 3 * x - 8
h₂ : Fintype ↑(f ⁻¹' {10})
⊢ ∑ k ∈ (f ⁻¹' {10}).toFinset, k = 6 |
ff5700132868ff6c | import Mathlib
theorem mathd_algebra_132 (x : ℝ) (f g : ℝ → ℝ) (h₀ : ∀ x, f x = x + 2) (h₁ : ∀ x, g x = x ^ 2)
(h₂ : f (g x) = g (f x)) : x = -1 / 2 := by | x : ℝ
f g : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = x + 2
h₁ : ∀ (x : ℝ), g x = x ^ 2
h₂ : f (g x) = g (f x)
⊢ x = -1 / 2 |
88705136662a0d81 | import Mathlib
theorem mathd_numbertheory_37 : Nat.lcm 9999 100001 = 90900909 := by | ⊢ Nat.lcm 9999 100001 = 90900909 |
e5c3d3a9c24f4449 | import Mathlib
theorem aime_1983_p9 (x : ℝ) (h₀ : 0 < x ∧ x < Real.pi) :
12 ≤ (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x) := by | x : ℝ
h₀ : 0 < x ∧ x < π
⊢ 12 ≤ (9 * (x ^ 2 * x.sin ^ 2) + 4) / (x * x.sin) |
e6500c1129f093d0 | import Mathlib
theorem mathd_algebra_37 (x y : ℝ) (h₀ : x + y = 7) (h₁ : 3 * x + y = 45) : x ^ 2 - y ^ 2 = 217 := by | x y : ℝ
h₀ : x + y = 7
h₁ : 3 * x + y = 45
⊢ x ^ 2 - y ^ 2 = 217 |
4ba887f36913a626 | import Mathlib
theorem mathd_numbertheory_458 (n : ℕ) (h₀ : n % 8 = 7) : n % 4 = 3 := by | n : ℕ
h₀ : n % 8 = 7
⊢ n % 4 = 3 |
21ecd380b264d9ef | import Mathlib
theorem amc12a_2008_p15 (k : ℕ) (h₀ : k = 2008 ^ 2 + 2 ^ 2008) : (k ^ 2 + 2 ^ k) % 10 = 6 := by | k : ℕ
h₀ : k = 2008 ^ 2 + 2 ^ 2008
⊢ (k ^ 2 + 2 ^ k) % 10 = 6 |
3e9626034f9dbff9 | import Mathlib
theorem mathd_numbertheory_301 (j : ℕ) (h₀ : 0 < j) : 3 * (7 * ↑j + 3) % 7 = 2 := by | j : ℕ
h₀ : 0 < j
⊢ 3 * (7 * j + 3) % 7 = 2 |
2f12e7c3d2489abc | import Mathlib
theorem amc12a_2009_p15 (n : ℕ) (h₀ : 0 < n)
(h₁ : (∑ k in Finset.Icc 1 n, ↑k * Complex.I ^ k) = 48 + 49 * Complex.I) : n = 97 := by | n : ℕ
h₀ : 0 < n
h₁ : ∑ k ∈ Finset.Icc 1 n, ↑k * Complex.I ^ k = 48 + 49 * Complex.I
⊢ n = 97 |
5f7a682a830e6633 | import Mathlib
theorem algebra_sqineq_36azm9asqle36zsq (z a : ℝ) : 36 * (a * z) - 9 * a ^ 2 ≤ 36 * z ^ 2 := by | z a : ℝ
⊢ 36 * (a * z) - 9 * a ^ 2 ≤ 36 * z ^ 2 |
a7b03d73fa7e9484 | import Mathlib
theorem amc12a_2013_p7 (s : ℕ → ℝ) (h₀ : ∀ n, s (n + 2) = s (n + 1) + s n) (h₁ : s 9 = 110)
(h₂ : s 7 = 42) : s 4 = 10 := by | s : ℕ → ℝ
h₀ : ∀ (n : ℕ), s (n + 2) = s (n + 1) + s n
h₁ : s 9 = 110
h₂ : s 7 = 42
⊢ s 4 = 10 |
d47b48d91ab524f0 | import Mathlib
theorem mathd_algebra_104 (x : ℝ) (h₀ : 125 / 8 = x / 12) : x = 375 / 2 := by | x : ℝ
h₀ : 125 / 8 = x / 12
⊢ x = 375 / 2 |
0daf13dd7b43e0fa | import Mathlib
theorem mathd_numbertheory_252 : 7! % 23 = 3 := by | ⊢ 7! % 23 = 3 |
ff17c8da959c989a | import Mathlib
theorem amc12a_2020_p21 (S : Finset ℕ)
(h₀ : ∀ n : ℕ, n ∈ S ↔ 5 ∣ n ∧ Nat.lcm 5! n = 5 * Nat.gcd 10! n) : S.card = 48 := by | S : Finset ℕ
h₀ : ∀ (n : ℕ), n ∈ S ↔ 5 ∣ n ∧ 5!.lcm n = 5 * 10!.gcd n
⊢ S.card = 48 |
aa5331923fa7cf46 | import Mathlib
theorem mathd_algebra_493 (f : ℝ → ℝ) (h₀ : ∀ x, f x = x ^ 2 - 4 * Real.sqrt x + 1) :
f (f 4) = 70 := by | f : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = x ^ 2 - 4 * √x + 1
⊢ f (f 4) = 70 |
b266d5ed27bb491a | import Mathlib
theorem numbertheory_nckeqnm1ckpnm1ckm1 (n k : ℕ) (h₀ : 0 < n ∧ 0 < k) (h₁ : k ≤ n) :
Nat.choose n k = Nat.choose (n - 1) k + Nat.choose (n - 1) (k - 1) := by | n k : ℕ
h₀ : 0 < n ∧ 0 < k
h₁ : k ≤ n
⊢ n.choose k = (n - 1).choose k + (n - 1).choose (k - 1) |
546bf91f52acd12d | import Mathlib
theorem algebra_3rootspoly_amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta (b c d a : ℂ) :
(a - d) * (a - c) * (a - b) =
-((a ^ 2 - (b + c) * a + c * b) * d) + (a ^ 2 - (b + c) * a + c * b) * a := by | b c d a : ℂ
⊢ (a - d) * (a - c) * (a - b) = -((a ^ 2 - (b + c) * a + c * b) * d) + (a ^ 2 - (b + c) * a + c * b) * a |
b4081863878342c9 | import Mathlib
theorem mathd_numbertheory_403 : (∑ k in Nat.properDivisors 198, k) = 270 := by | ⊢ ∑ k ∈ properDivisors 198, k = 270 |
540709036db25b3d | import Mathlib
theorem mathd_algebra_190 : ((3 : ℝ) / 8 + 7 / 8) / (4 / 5) = 25 / 16 := by | ⊢ (3 / 8 + 7 / 8) / (4 / 5) = 25 / 16 |
bf88d97b7460c4de | import Mathlib
theorem mathd_numbertheory_269 : (2005 ^ 2 + 2005 ^ 0 + 2005 ^ 0 + 2005 ^ 5) % 100 = 52 := by | ⊢ (2005 ^ 2 + 2005 ^ 0 + 2005 ^ 0 + 2005 ^ 5) % 100 = 52 |
8313e831a928db6b | import Mathlib
theorem aime_1990_p2 :
(52 + 6 * Real.sqrt 43) ^ ((3 : ℝ) / 2) - (52 - 6 * Real.sqrt 43) ^ ((3 : ℝ) / 2) = 828 := by | ⊢ (52 + 6 * √43) ^ (3 / 2) - (52 - 6 * √43) ^ (3 / 2) = 828 |
41dc225412fbd888 | import Mathlib
theorem mathd_numbertheory_101 : 17 * 18 % 4 = 2 := by | ⊢ 17 * 18 % 4 = 2 |
d325f5acae7a8110 | import Mathlib
theorem algebra_sqineq_4bap1lt4bsqpap1sq (a b : ℝ) : 4 * b * (a + 1) ≤ 4 * b ^ 2 + (a + 1) ^ 2 := by | a b : ℝ
⊢ 4 * b * (a + 1) ≤ 4 * b ^ 2 + (a + 1) ^ 2 |
5a16df1e4c4000bf | import Mathlib
theorem mathd_numbertheory_156 (n : ℕ) (h₀ : 0 < n) : Nat.gcd (n + 7) (2 * n + 1) ≤ 13 := by | n : ℕ
h₀ : 0 < n
⊢ (n + 7).gcd (2 * n + 1) ≤ 13 |
d56f75a0261fb6c6 | import Mathlib
theorem mathd_algebra_451 (σ : Equiv ℝ ℝ) (h₀ : σ.2 (-15) = 0) (h₁ : σ.2 0 = 3) (h₂ : σ.2 3 = 9)
(h₃ : σ.2 9 = 20) : σ.1 (σ.1 9) = 0 := by | σ : ℝ ≃ ℝ
h₀ : σ.invFun (-15) = 0
h₁ : σ.invFun 0 = 3
h₂ : σ.invFun 3 = 9
h₃ : σ.invFun 9 = 20
⊢ σ.toFun (σ.toFun 9) = 0 |
0016907e76ae2a4c | import Mathlib
theorem mathd_algebra_144 (a b c d : ℕ) (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d) (h₀ : (c : ℤ) - b = d)
(h₁ : (b : ℤ) - a = d) (h₂ : a + b + c = 60) (h₃ : a + b > c) : d < 10 := by | a b c d : ℕ
h₀✝ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d
h₀ : ↑c - ↑b = ↑d
h₁ : ↑b - ↑a = ↑d
h₂ : a + b + c = 60
h₃ : a + b > c
⊢ d < 10 |
1541c3fb926d1461 | import Mathlib
theorem mathd_algebra_282 (f : ℝ → ℝ) (h₀ : ∀ x : ℝ, ¬ (Irrational x) → f x = abs (Int.floor x))
(h₁ : ∀ x, Irrational x → f x = (Int.ceil x) ^ 2) :
f (8 ^ (1 / 3)) + f (-Real.pi) + f (Real.sqrt 50) + f (9 / 2) = 79 := by | f : ℝ → ℝ
h₀ : ∀ (x : ℝ), ¬Irrational x → f x = ↑|⌊x⌋|
h₁ : ∀ (x : ℝ), Irrational x → f x = ↑⌈x⌉ ^ 2
⊢ f (8 ^ (1 / 3)) + f (-π) + f √50 + f (9 / 2) = 79 |
b26d4fc504bb7cf3 | import Mathlib
theorem mathd_algebra_410 (x y : ℝ) (h₀ : y = x ^ 2 - 6 * x + 13) : 4 ≤ y := by | x y : ℝ
h₀ : y = x ^ 2 - 6 * x + 13
⊢ 4 ≤ y |
fa6ad5959af24e7a | import Mathlib
theorem mathd_numbertheory_232 (x y z : ZMod 31) (h₀ : x = 3⁻¹) (h₁ : y = 5⁻¹)
(h₂ : z = (x + y)⁻¹) : z = 29 := by | x y z : ZMod 31
h₀ : x = 3⁻¹
h₁ : y = 5⁻¹
h₂ : z = (x + y)⁻¹
⊢ z = 29 |
b98361c9b3d81a59 | import Mathlib
theorem mathd_algebra_77 (a b : ℝ) (f : ℝ → ℝ) (h₀ : a ≠ 0 ∧ b ≠ 0) (h₁ : a ≠ b)
(h₂ : ∀ x, f x = x ^ 2 + a * x + b) (h₃ : f a = 0) (h₄ : f b = 0) : a = 1 ∧ b = -2 := by | a b : ℝ
f : ℝ → ℝ
h₀ : a ≠ 0 ∧ b ≠ 0
h₁ : a ≠ b
h₂ : ∀ (x : ℝ), f x = x ^ 2 + a * x + b
h₃ : f a = 0
h₄ : f b = 0
⊢ a = 1 ∧ b = -2 |
059131f52887b089 | import Mathlib
theorem imo_1974_p5 (a b c d s : ℝ) (h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d)
(h₁ : s = a / (a + b + d) + b / (a + b + c) + c / (b + c + d) + d / (a + c + d)) :
1 < s ∧ s < 2 := by | a b c d s : ℝ
h₀ : 0 < a ∧ 0 < b ∧ 0 < c ∧ 0 < d
h₁ : s = a / (a + b + d) + b / (a + b + c) + c / (b + c + d) + d / (a + c + d)
⊢ 1 < s ∧ s < 2 |
90a280bcf781cacd | import Mathlib
theorem aime_1988_p3 (x : ℝ) (h₀ : 0 < x)
(h₁ : Real.logb 2 (Real.logb 8 x) = Real.logb 8 (Real.logb 2 x)) : Real.logb 2 x ^ 2 = 27 := by | x : ℝ
h₀ : 0 < x
h₁ : logb 2 (logb 8 x) = logb 8 (logb 2 x)
⊢ logb 2 x ^ 2 = 27 |
d60dec9a3ccfed16 | import Mathlib
theorem mathd_numbertheory_530 (n k : ℕ) (h₀ : 0 < n ∧ 0 < k) (h₀ : (n : ℝ) / k < 6)
(h₁ : (5 : ℝ) < n / k) : 22 ≤ Nat.lcm n k / Nat.gcd n k := by | n k : ℕ
h₀✝ : 0 < n ∧ 0 < k
h₀ : ↑n / ↑k < 6
h₁ : 5 < ↑n / ↑k
⊢ 22 ≤ n.lcm k / n.gcd k |
4774956dfe165759 | import Mathlib
theorem mathd_algebra_109 (a b : ℝ) (h₀ : 3 * a + 2 * b = 12) (h₁ : a = 4) : b = 0 := by | a b : ℝ
h₀ : 3 * a + 2 * b = 12
h₁ : a = 4
⊢ b = 0 |
6aa00b3dc62af4b5 | import Mathlib
theorem imo_1967_p3 (k m n : ℕ) (c : ℕ → ℕ) (h₀ : 0 < k ∧ 0 < m ∧ 0 < n)
(h₁ : ∀ s, c s = s * (s + 1)) (h₂ : Nat.Prime (k + m + 1)) (h₃ : n + 1 < k + m + 1) :
(∏ i in Finset.Icc 1 n, c i) ∣ ∏ i in Finset.Icc 1 n, c (m + i) - c k := by | k m n : ℕ
c : ℕ → ℕ
h₀ : 0 < k ∧ 0 < m ∧ 0 < n
h₁ : ∀ (s : ℕ), c s = s * (s + 1)
h₂ : (k + m + 1).Prime
h₃ : n + 1 < k + m + 1
⊢ ∏ i ∈ Finset.Icc 1 n, c i ∣ ∏ i ∈ Finset.Icc 1 n, c (m + i) - c k |
929c2a9e5685f6d6 | import Mathlib
theorem mathd_algebra_11 (a b : ℝ) (h₀ : a ≠ b) (h₁ : a ≠ 2 * b)
(h₂ : (4 * a + 3 * b) / (a - 2 * b) = 5) : (a + 11 * b) / (a - b) = 2 := by | a b : ℝ
h₀ : a ≠ b
h₁ : a ≠ 2 * b
h₂ : (4 * a + 3 * b) / (a - 2 * b) = 5
⊢ (a + 11 * b) / (a - b) = 2 |
ddd9634eeb2cb86d | import Mathlib
theorem amc12a_2003_p1 (u v : ℕ → ℕ) (h₀ : ∀ n, u n = 2 * n + 2) (h₁ : ∀ n, v n = 2 * n + 1) :
((∑ k in Finset.range 2003, u k) - ∑ k in Finset.range 2003, v k) = 2003 := by | u v : ℕ → ℕ
h₀ : ∀ (n : ℕ), u n = 2 * n + 2
h₁ : ∀ (n : ℕ), v n = 2 * n + 1
⊢ ∑ k ∈ Finset.range 2003, u k - ∑ k ∈ Finset.range 2003, v k = 2003 |
665c1bea467bbc94 | import Mathlib
theorem numbertheory_aneqprodakp4_anmsqrtanp1eq2 (a : ℕ → ℝ) (h₀ : a 0 = 1)
(h₁ : ∀ n, a (n + 1) = (∏ k in Finset.range (n + 1), a k) + 4) :
∀ n ≥ 1, a n - Real.sqrt (a (n + 1)) = 2 := by | a : ℕ → ℝ
h₀ : a 0 = 1
h₁ : ∀ (n : ℕ), a (n + 1) = ∏ k ∈ Finset.range (n + 1), a k + 4
⊢ ∀ n ≥ 1, a n - √(a (n + 1)) = 2 |
5e860012203c2726 | import Mathlib
theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : ℂ) :
(a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by | a b : ℂ
⊢ (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) |
960441ab87891ba0 | import Mathlib
theorem induction_sum_odd (n : ℕ) : (∑ k in Finset.range n, 2 * k) + 1 = n ^ 2 := by | n : ℕ
⊢ ∑ k ∈ Finset.range n, 2 * k + 1 = n ^ 2 |
7e0853d5d11d04c0 | import Mathlib
theorem mathd_algebra_568 (a : ℝ) :
(a - 1) * (a + 1) * (a + 2) - (a - 2) * (a + 1) = a ^ 3 + a ^ 2 := by | a : ℝ
⊢ (a - 1) * (a + 1) * (a + 2) - (a - 2) * (a + 1) = a ^ 3 + a ^ 2 |
628df7c7442cc99a | import Mathlib
theorem mathd_algebra_616 (f g : ℝ → ℝ) (h₀ : ∀ x, f x = x ^ 3 + 2 * x + 1)
(h₁ : ∀ x, g x = x - 1) : f (g 1) = 1 := by | f g : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = x ^ 3 + 2 * x + 1
h₁ : ∀ (x : ℝ), g x = x - 1
⊢ f (g 1) = 1 |
b284c3920da9a237 | import Mathlib
theorem mathd_numbertheory_690 :
IsLeast { a : ℕ | 0 < a ∧ a ≡ 2 [MOD 3] ∧ a ≡ 4 [MOD 5] ∧ a ≡ 6 [MOD 7] ∧ a ≡ 8 [MOD 9] } 314 := by | ⊢ IsLeast {a | 0 < a ∧ a ≡ 2 [MOD 3] ∧ a ≡ 4 [MOD 5] ∧ a ≡ 6 [MOD 7] ∧ a ≡ 8 [MOD 9]} 314 |
a6b78fc89c6d558f | import Mathlib
theorem amc12a_2016_p2 (x : ℝ) (h₀ : (10 : ℝ) ^ x * 100 ^ (2 * x) = 1000 ^ 5) : x = 3 := by | x : ℝ
h₀ : 10 ^ x * 100 ^ (2 * x) = 1000 ^ 5
⊢ x = 3 |
a2fd79d556493c64 | import Mathlib
theorem mathd_numbertheory_405 (a b c : ℕ) (t : ℕ → ℕ) (h₀ : t 0 = 0) (h₁ : t 1 = 1)
(h₂ : ∀ n > 1, t n = t (n - 2) + t (n - 1)) (h₃ : a ≡ 5 [MOD 16]) (h₄ : b ≡ 10 [MOD 16])
(h₅ : c ≡ 15 [MOD 16]) : (t a + t b + t c) % 7 = 5 := by | a b c : ℕ
t : ℕ → ℕ
h₀ : t 0 = 0
h₁ : t 1 = 1
h₂ : ∀ n > 1, t n = t (n - 2) + t (n - 1)
h₃ : a ≡ 5 [MOD 16]
h₄ : b ≡ 10 [MOD 16]
h₅ : c ≡ 15 [MOD 16]
⊢ (t a + t b + t c) % 7 = 5 |
eba86e0a72dbfe11 | import Mathlib
theorem mathd_numbertheory_110 (a b : ℕ) (h₀ : 0 < a ∧ 0 < b ∧ b ≤ a) (h₁ : (a + b) % 10 = 2)
(h₂ : (2 * a + b) % 10 = 1) : (a - b) % 10 = 6 := by | a b : ℕ
h₀ : 0 < a ∧ 0 < b ∧ b ≤ a
h₁ : (a + b) % 10 = 2
h₂ : (2 * a + b) % 10 = 1
⊢ (a - b) % 10 = 6 |
edd5ac31e76266de | import Mathlib
theorem amc12a_2003_p25 (a b : ℝ) (f : ℝ → ℝ) (h₀ : 0 < b)
(h₁ : ∀ x, f x = Real.sqrt (a * x ^ 2 + b * x)) (h₂ : { x | 0 ≤ f x } = f '' { x | 0 ≤ f x }) :
a = 0 ∨ a = -4 := by | a b : ℝ
f : ℝ → ℝ
h₀ : 0 < b
h₁ : ∀ (x : ℝ), f x = √(a * x ^ 2 + b * x)
h₂ : {x | 0 ≤ f x} = f '' {x | 0 ≤ f x}
⊢ a = 0 ∨ a = -4 |
144a038e4b247a95 | import Mathlib
theorem amc12a_2010_p10 (p q : ℝ) (a : ℕ → ℝ) (h₀ : ∀ n, a (n + 2) - a (n + 1) = a (n + 1) - a n)
(h₁ : a 1 = p) (h₂ : a 2 = 9) (h₃ : a 3 = 3 * p - q) (h₄ : a 4 = 3 * p + q) : a 2010 = 8041 := by | p q : ℝ
a : ℕ → ℝ
h₀ : ∀ (n : ℕ), a (n + 2) - a (n + 1) = a (n + 1) - a n
h₁ : a 1 = p
h₂ : a 2 = 9
h₃ : a 3 = 3 * p - q
h₄ : a 4 = 3 * p + q
⊢ a 2010 = 8041 |
5e9546527c45710d | import Mathlib
theorem mathd_algebra_509 :
Real.sqrt ((5 / Real.sqrt 80 + Real.sqrt 845 / 9 + Real.sqrt 45) / Real.sqrt 5) = 13 / 6 := by | ⊢ √((5 / √80 + √845 / 9 + √45) / √5) = 13 / 6 |
4bd246078e6c2060 | import Mathlib
theorem mathd_algebra_159 (b : ℝ) (f : ℝ → ℝ)
(h₀ : ∀ x, f x = 3 * x ^ 4 - 7 * x ^ 3 + 2 * x ^ 2 - b * x + 1) (h₁ : f 1 = 1) : b = -2 := by | b : ℝ
f : ℝ → ℝ
h₀ : ∀ (x : ℝ), f x = 3 * x ^ 4 - 7 * x ^ 3 + 2 * x ^ 2 - b * x + 1
h₁ : f 1 = 1
⊢ b = -2 |
b6c6f87409f49650 | import Mathlib
theorem aime_1997_p11 (x : ℝ)
(h₀ :
x =
(∑ n in Finset.Icc (1 : ℕ) 44, Real.cos (n * Real.pi / 180)) /
∑ n in Finset.Icc (1 : ℕ) 44, Real.sin (n * Real.pi / 180)) :
Int.floor (100 * x) = 241 := by | x : ℝ
h₀ : x = (∑ n ∈ Finset.Icc 1 44, (↑n * π / 180).cos) / ∑ n ∈ Finset.Icc 1 44, (↑n * π / 180).sin
⊢ ⌊100 * x⌋ = 241 |
cce6e4189648bccf | import Mathlib
theorem aimeI_2000_p7 (x y z : ℝ) (m : ℚ) (h₀ : 0 < x ∧ 0 < y ∧ 0 < z) (h₁ : x * y * z = 1)
(h₂ : x + 1 / z = 5) (h₃ : y + 1 / x = 29) (h₄ : z + 1 / y = m) (h₅ : 0 < m) :
↑m.den + m.num = 5 := by | x y z : ℝ
m : ℚ
h₀ : 0 < x ∧ 0 < y ∧ 0 < z
h₁ : x * y * z = 1
h₂ : x + 1 / z = 5
h₃ : y + 1 / x = 29
h₄ : z + 1 / y = ↑m
h₅ : 0 < m
⊢ ↑m.den + m.num = 5 |
b5507bc400ed6e13 | import Mathlib
theorem aime_1988_p4 (n : ℕ) (a : ℕ → ℝ) (h₀ : ∀ n, abs (a n) < 1)
(h₁ : (∑ k in Finset.range n, abs (a k)) = 19 + abs (∑ k in Finset.range n, a k)) : 20 ≤ n := by | n : ℕ
a : ℕ → ℝ
h₀ : ∀ (n : ℕ), |a n| < 1
h₁ : ∑ k ∈ Finset.range n, |a k| = 19 + |∑ k ∈ Finset.range n, a k|
⊢ 20 ≤ n |
a9f6e45e316000d3 | import Mathlib
theorem induction_divisibility_9div10tonm1 (n : ℕ) (h₀ : 0 < n) : 9 ∣ 10 ^ n - 1 := by | n : ℕ
h₀ : 0 < n
⊢ 9 ∣ 10 ^ n - 1 |
136a92b6704013d1 | import Mathlib
theorem mathd_numbertheory_126 (x a : ℕ) (h₀ : 0 < x ∧ 0 < a) (h₁ : Nat.gcd a 40 = x + 3)
(h₂ : Nat.lcm a 40 = x * (x + 3))
(h₃ : ∀ b : ℕ, 0 < b → Nat.gcd b 40 = x + 3 ∧ Nat.lcm b 40 = x * (x + 3) → a ≤ b) : a = 8 := by | x a : ℕ
h₀ : 0 < x ∧ 0 < a
h₁ : a.gcd 40 = x + 3
h₂ : a.lcm 40 = x * (x + 3)
h₃ : ∀ (b : ℕ), 0 < b → b.gcd 40 = x + 3 ∧ b.lcm 40 = x * (x + 3) → a ≤ b
⊢ a = 8 |
936c5fc51ce3f0e4 | import Mathlib
theorem mathd_algebra_323 (σ : Equiv ℝ ℝ) (h : ∀ x, σ.1 x = x ^ 3 - 8) : σ.2 (σ.1 (σ.2 19)) = 3 := by | σ : ℝ ≃ ℝ
h : ∀ (x : ℝ), σ.toFun x = x ^ 3 - 8
⊢ σ.invFun (σ.toFun (σ.invFun 19)) = 3 |
e4f8f5c95db851a7 | import Mathlib
theorem mathd_algebra_421 (a b c d : ℝ) (h₀ : b = a ^ 2 + 4 * a + 6)
(h₁ : b = 1 / 2 * a ^ 2 + a + 6) (h₂ : d = c ^ 2 + 4 * c + 6) (h₃ : d = 1 / 2 * c ^ 2 + c + 6)
(h₄ : a < c) : c - a = 6 := by | a b c d : ℝ
h₀ : b = a ^ 2 + 4 * a + 6
h₁ : b = 1 / 2 * a ^ 2 + a + 6
h₂ : d = c ^ 2 + 4 * c + 6
h₃ : d = 1 / 2 * c ^ 2 + c + 6
h₄ : a < c
⊢ c - a = 6 |
7922fc270e5f2689 | import Mathlib
theorem imo_1987_p6 (p : ℕ) (f : ℕ → ℕ) (h₀ : ∀ x, f x = x ^ 2 + x + p)
(h₀ : ∀ k : ℕ, k ≤ Nat.floor (Real.sqrt (p / 3)) → Nat.Prime (f k)) :
∀ i ≤ p - 2, Nat.Prime (f i) := by | p : ℕ
f : ℕ → ℕ
h₀✝ : ∀ (x : ℕ), f x = x ^ 2 + x + p
h₀ : ∀ k ≤ ⌊√(↑p / 3)⌋₊, (f k).Prime
⊢ ∀ i ≤ p - 2, (f i).Prime |
70077269d2359264 | import Mathlib
theorem amc12a_2009_p25 (a : ℕ → ℝ) (h₀ : a 1 = 1) (h₁ : a 2 = 1 / Real.sqrt 3)
(h₂ : ∀ n, 1 ≤ n → a (n + 2) = (a n + a (n + 1)) / (1 - a n * a (n + 1))) : abs (a 2009) = 0 := by | a : ℕ → ℝ
h₀ : a 1 = 1
h₁ : a 2 = 1 / √3
h₂ : ∀ (n : ℕ), 1 ≤ n → a (n + 2) = (a n + a (n + 1)) / (1 - a n * a (n + 1))
⊢ |a 2009| = 0 |
eb0b25e86fdeb7bc | import Mathlib
theorem imo_1961_p1 (x y z a b : ℝ) (h₀ : 0 < x ∧ 0 < y ∧ 0 < z) (h₁ : x ≠ y) (h₂ : y ≠ z)
(h₃ : z ≠ x) (h₄ : x + y + z = a) (h₅ : x ^ 2 + y ^ 2 + z ^ 2 = b ^ 2) (h₆ : x * y = z ^ 2) :
0 < a ∧ b ^ 2 < a ^ 2 ∧ a ^ 2 < 3 * b ^ 2 := by | x y z a b : ℝ
h₀ : 0 < x ∧ 0 < y ∧ 0 < z
h₁ : x ≠ y
h₂ : y ≠ z
h₃ : z ≠ x
h₄ : x + y + z = a
h₅ : x ^ 2 + y ^ 2 + z ^ 2 = b ^ 2
h₆ : x * y = z ^ 2
⊢ 0 < a ∧ b ^ 2 < a ^ 2 ∧ a ^ 2 < 3 * b ^ 2 |
204d74c2cda20e9e | import Mathlib
theorem mathd_algebra_31 (x : NNReal) (u : ℕ → NNReal) (h₀ : ∀ n, u (n + 1) = NNReal.sqrt (x + u n))
(h₁ : Filter.Tendsto u Filter.atTop (𝓝 9)) : 9 = NNReal.sqrt (x + 9) := by | x : NNReal
u : ℕ → NNReal
h₀ : ∀ (n : ℕ), u (n + 1) = NNReal.sqrt (x + u n)
h₁ : Filter.Tendsto u Filter.atTop (𝓝 9)
⊢ 9 = NNReal.sqrt (x + 9) |
6a6ee56c8d21ef9c | import Mathlib
theorem algebra_manipexpr_apbeq2cceqiacpbceqm2 (a b c : ℂ) (h₀ : a + b = 2 * c)
(h₁ : c = Complex.I) : a * c + b * c = -2 := by | a b c : ℂ
h₀ : a + b = 2 * c
h₁ : c = Complex.I
⊢ a * c + b * c = -2 |
776833a7b8d29af2 | import Mathlib
theorem mathd_numbertheory_370 (n : ℕ) (h₀ : n % 7 = 3) : (2 * n + 1) % 7 = 0 := by | n : ℕ
h₀ : n % 7 = 3
⊢ (2 * n + 1) % 7 = 0 |
64541dd5fd7a5ace | import Mathlib
theorem mathd_algebra_437 (x y : ℝ) (n : ℤ) (h₀ : x ^ 3 = -45) (h₁ : y ^ 3 = -101) (h₂ : x < n)
(h₃ : ↑n < y) : n = -4 := by | x y : ℝ
n : ℤ
h₀ : x ^ 3 = -45
h₁ : y ^ 3 = -101
h₂ : x < ↑n
h₃ : ↑n < y
⊢ n = -4 |
2cbc0006abb941e7 | import Mathlib
theorem imo_1966_p5 (x a : ℕ → ℝ) (h₀ : a 1 ≠ a 2) (h₁ : a 1 ≠ a 3) (h₂ : a 1 ≠ a 4)
(h₃ : a 2 ≠ a 3) (h₄ : a 2 ≠ a 4) (h₅ : a 3 ≠ a 4) (h₆ : a 1 > a 2) (h₇ : a 2 > a 3)
(h₈ : a 3 > a 4)
(h₉ : abs (a 1 - a 2) * x 2 + abs (a 1 - a 3) * x 3 + abs (a 1 - a 4) * x 4 = 1)
(h₁₀ : abs (a 2 - a 1) * x 1... | x a : ℕ → ℝ
h₀ : a 1 ≠ a 2
h₁ : a 1 ≠ a 3
h₂ : a 1 ≠ a 4
h₃ : a 2 ≠ a 3
h₄ : a 2 ≠ a 4
h₅ : a 3 ≠ a 4
h₆ : a 1 > a 2
h₇ : a 2 > a 3
h₈ : a 3 > a 4
h₉ : |a 1 - a 2| * x 2 + |a 1 - a 3| * x 3 + |a 1 - a 4| * x 4 = 1
h₁₀ : |a 2 - a 1| * x 1 + |a 2 - a 3| * x 3 + |a 2 - a 4| * x 4 = 1
h₁₁ : |a 3 - a 1| * x 1 + |a 3 - a 2| ... |
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