run large_stringclasses 1
value | benchmark large_stringclasses 1
value | problem_id int64 0 346 | k_index int64 0 7 | is_best_in_problem bool 2
classes | statement large_stringclasses 347
values | original_proof large_stringclasses 347
values | candidate_body large_stringlengths 8 7.19k | len_orig int64 9 1.95k | len_new int64 2 1.85k | d_len int64 -692 729 | ratio float64 0.62 90 | is_shorter bool 2
classes | tier_orig float64 4 476 | tier_new float64 1 606 | d_tier float64 -180 188 | tier_collapse bool 2
classes | verified null |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
additive | minif2f | 0 | 0 | true | theorem aime_1983_p1 (x y z w : ℕ) (ht : 1 < x ∧ 1 < y ∧ 1 < z) (hw : 0 ≤ w)
(h0 : Real.log w / Real.log x = 24) (h1 : Real.log w / Real.log y = 40)
(h2 : Real.log w / Real.log (x * y * z) = 12) : Real.log w / Real.log z = 60 | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | 1,307 | 1,271 | 36 | 1.0283 | true | 302.5 | 292.5 | 10 | false | null |
additive | minif2f | 0 | 1 | false | theorem aime_1983_p1 (x y z w : ℕ) (ht : 1 < x ∧ 1 < y ∧ 1 < z) (hw : 0 ≤ w)
(h0 : Real.log w / Real.log x = 24) (h1 : Real.log w / Real.log y = 40)
(h2 : Real.log w / Real.log (x * y * z) = 12) : Real.log w / Real.log z = 60 | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | 1,307 | 1,307 | 0 | 1 | false | 302.5 | 302.5 | 0 | false | null |
additive | minif2f | 0 | 2 | false | theorem aime_1983_p1 (x y z w : ℕ) (ht : 1 < x ∧ 1 < y ∧ 1 < z) (hw : 0 ≤ w)
(h0 : Real.log w / Real.log x = 24) (h1 : Real.log w / Real.log y = 40)
(h2 : Real.log w / Real.log (x * y * z) = 12) : Real.log w / Real.log z = 60 | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | 1,307 | 1,307 | 0 | 1 | false | 302.5 | 302.5 | 0 | false | null |
additive | minif2f | 0 | 3 | false | theorem aime_1983_p1 (x y z w : ℕ) (ht : 1 < x ∧ 1 < y ∧ 1 < z) (hw : 0 ≤ w)
(h0 : Real.log w / Real.log x = 24) (h1 : Real.log w / Real.log y = 40)
(h2 : Real.log w / Real.log (x * y * z) = 12) : Real.log w / Real.log z = 60 | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | 1,307 | 1,307 | 0 | 1 | false | 302.5 | 302.5 | 0 | false | null |
additive | minif2f | 0 | 4 | false | theorem aime_1983_p1 (x y z w : ℕ) (ht : 1 < x ∧ 1 < y ∧ 1 < z) (hw : 0 ≤ w)
(h0 : Real.log w / Real.log x = 24) (h1 : Real.log w / Real.log y = 40)
(h2 : Real.log w / Real.log (x * y * z) = 12) : Real.log w / Real.log z = 60 | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by positivity) h... | 1,307 | 1,296 | 11 | 1.0085 | true | 302.5 | 300.5 | 2 | false | null |
additive | minif2f | 0 | 5 | false | theorem aime_1983_p1 (x y z w : ℕ) (ht : 1 < x ∧ 1 < y ∧ 1 < z) (hw : 0 ≤ w)
(h0 : Real.log w / Real.log x = 24) (h1 : Real.log w / Real.log y = 40)
(h2 : Real.log w / Real.log (x * y * z) = 12) : Real.log w / Real.log z = 60 | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by positivity) h... | 1,307 | 1,296 | 11 | 1.0085 | true | 302.5 | 300.5 | 2 | false | null |
additive | minif2f | 0 | 6 | false | theorem aime_1983_p1 (x y z w : ℕ) (ht : 1 < x ∧ 1 < y ∧ 1 < z) (hw : 0 ≤ w)
(h0 : Real.log w / Real.log x = 24) (h1 : Real.log w / Real.log y = 40)
(h2 : Real.log w / Real.log (x * y * z) = 12) : Real.log w / Real.log z = 60 | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by positivity) h... | 1,307 | 1,296 | 11 | 1.0085 | true | 302.5 | 300.5 | 2 | false | null |
additive | minif2f | 0 | 7 | false | theorem aime_1983_p1 (x y z w : ℕ) (ht : 1 < x ∧ 1 < y ∧ 1 < z) (hw : 0 ≤ w)
(h0 : Real.log w / Real.log x = 24) (h1 : Real.log w / Real.log y = 40)
(h2 : Real.log w / Real.log (x * y * z) = 12) : Real.log w / Real.log z = 60 | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by posi... | by
have hx : (1 : ℝ) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : ℝ) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : ℝ) < z := by
norm_cast
<;> linarith [ht.2.2]
have hlogx : Real.log x > 0 := by
have h : Real.log x > Real.log 1 := Real.log_lt_log (by positivity) h... | 1,307 | 1,271 | 36 | 1.0283 | true | 302.5 | 292.5 | 10 | false | null |
additive | minif2f | 1 | 0 | false | theorem aime_1983_p2
(x p : ℝ)
(f : ℝ → ℝ)
(h₀ : 0 < p ∧ p < 15)
(h₁ : p ≤ x ∧ x ≤ 15)
(h₂ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 ≤ f x | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 1 | false | theorem aime_1983_p2
(x p : ℝ)
(f : ℝ → ℝ)
(h₀ : 0 < p ∧ p < 15)
(h₁ : p ≤ x ∧ x ≤ 15)
(h₂ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 ≤ f x | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 2 | false | theorem aime_1983_p2
(x p : ℝ)
(f : ℝ → ℝ)
(h₀ : 0 < p ∧ p < 15)
(h₁ : p ≤ x ∧ x ≤ 15)
(h₂ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 ≤ f x | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 3 | false | theorem aime_1983_p2
(x p : ℝ)
(f : ℝ → ℝ)
(h₀ : 0 < p ∧ p < 15)
(h₁ : p ≤ x ∧ x ≤ 15)
(h₂ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 ≤ f x | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 4 | false | theorem aime_1983_p2
(x p : ℝ)
(f : ℝ → ℝ)
(h₀ : 0 < p ∧ p < 15)
(h₁ : p ≤ x ∧ x ≤ 15)
(h₂ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 ≤ f x | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 5 | false | theorem aime_1983_p2
(x p : ℝ)
(f : ℝ → ℝ)
(h₀ : 0 < p ∧ p < 15)
(h₁ : p ≤ x ∧ x ≤ 15)
(h₂ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 ≤ f x | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 := by l... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 6 | false | theorem aime_1983_p2
(x p : ℝ)
(f : ℝ → ℝ)
(h₀ : 0 < p ∧ p < 15)
(h₁ : p ≤ x ∧ x ≤ 15)
(h₂ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 ≤ f x | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 7 | false | theorem aime_1983_p2
(x p : ℝ)
(f : ℝ → ℝ)
(h₀ : 0 < p ∧ p < 15)
(h₁ : p ≤ x ∧ x ≤ 15)
(h₂ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 ≤ f x | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 :... | by
have h₃ : abs (x - p) = x - p := by
have h₃₁ : x - p ≥ 0 := by linarith
rw [abs_of_nonneg h₃₁]
<;>
linarith
have h₄ : abs (x - 15) = 15 - x := by
have h₄₁ : x - 15 ≤ 0 := by linarith
rw [abs_of_nonpos h₄₁]
<;>
linarith
have h₅ : x - p - 15 ≤ 0 := by
have h₅₁ : x ≤ 15 := by l... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 2 | 0 | false | 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
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | by
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 1 | false | 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
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | by
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 2 | false | 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
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | by
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 3 | true | 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
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | by
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | 537 | 535 | 2 | 1.0037 | true | 117 | 113 | 4 | false | null |
additive | minif2f | 2 | 4 | false | 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
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | by
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 5 | false | 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
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | by
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 6 | false | 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
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | by
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃₁ : 0 ... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 7 | false | 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
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | by
have h₁ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
· exact h₀.1
· exact h₀.2
have h₂ : x * Real.sin x > 0 := by
have h₂₁ : 0 < x := h₀.1
have h₂₂ : 0 < Real.sin x := h₁
positivity
have h₃ : ∀ (y : ℝ), y > 0 → 12 ≤ (9 * y ^ 2 + 4) / y := by
intro y hy
have h₃... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 3 | 0 | false | theorem aime_1984_p1
(u : ℕ → ℚ)
(h₀ : ∀ n, u (n + 1) = u n + 1)
(h₁ : ∑ k in Finset.range 98, u k.succ = 137) :
∑ k in Finset.range 49, u (2 * k.succ) = 93 | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | 869 | 869 | 0 | 1 | false | 275.5 | 275.5 | 0 | false | null |
additive | minif2f | 3 | 1 | false | theorem aime_1984_p1
(u : ℕ → ℚ)
(h₀ : ∀ n, u (n + 1) = u n + 1)
(h₁ : ∑ k in Finset.range 98, u k.succ = 137) :
∑ k in Finset.range 49, u (2 * k.succ) = 93 | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | 869 | 869 | 0 | 1 | false | 275.5 | 275.5 | 0 | false | null |
additive | minif2f | 3 | 2 | false | theorem aime_1984_p1
(u : ℕ → ℚ)
(h₀ : ∀ n, u (n + 1) = u n + 1)
(h₁ : ∑ k in Finset.range 98, u k.succ = 137) :
∑ k in Finset.range 49, u (2 * k.succ) = 93 | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | 869 | 869 | 0 | 1 | false | 275.5 | 275.5 | 0 | false | null |
additive | minif2f | 3 | 3 | false | theorem aime_1984_p1
(u : ℕ → ℚ)
(h₀ : ∀ n, u (n + 1) = u n + 1)
(h₁ : ∑ k in Finset.range 98, u k.succ = 137) :
∑ k in Finset.range 49, u (2 * k.succ) = 93 | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | 869 | 833 | 36 | 1.0432 | true | 275.5 | 271.5 | 4 | false | null |
additive | minif2f | 3 | 4 | false | theorem aime_1984_p1
(u : ℕ → ℚ)
(h₀ : ∀ n, u (n + 1) = u n + 1)
(h₁ : ∑ k in Finset.range 98, u k.succ = 137) :
∑ k in Finset.range 49, u (2 * k.succ) = 93 | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | 869 | 869 | 0 | 1 | false | 275.5 | 275.5 | 0 | false | null |
additive | minif2f | 3 | 5 | false | theorem aime_1984_p1
(u : ℕ → ℚ)
(h₀ : ∀ n, u (n + 1) = u n + 1)
(h₁ : ∑ k in Finset.range 98, u k.succ = 137) :
∑ k in Finset.range 49, u (2 * k.succ) = 93 | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have h₂ : ∀ n : ℕ, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have h₃ := h₀ n
have h₄ : u (n + 1) = u 0 + (n + 1 : ℕ) := by
rw [h₃]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | 869 | 869 | 0 | 1 | false | 275.5 | 275.5 | 0 | false | null |
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