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 : β) (ht : 1 < x β§ 1 < y β§ 1 < z) (hw : 0 β€ 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) (ht : 1 < x β§ 1 < y β§ 1 < z) (hw : 0 β€ 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) (ht : 1 < x β§ 1 < y β§ 1 < z) (hw : 0 β€ 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) (ht : 1 < x β§ 1 < y β§ 1 < z) (hw : 0 β€ 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) (ht : 1 < x β§ 1 < y β§ 1 < z) (hw : 0 β€ 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) (ht : 1 < x β§ 1 < y β§ 1 < z) (hw : 0 β€ 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) (ht : 1 < x β§ 1 < y β§ 1 < z) (hw : 0 β€ 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) (ht : 1 < x β§ 1 < y β§ 1 < z) (hw : 0 β€ 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β) < x := by
norm_cast
<;> linarith [ht.1]
have hy : (1 : β) < y := by
norm_cast
<;> linarith [ht.2.1]
have hz : (1 : β) < 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 : β)
(f : β β β)
(hβ : 0 < p β§ p < 15)
(hβ : p β€ x β§ x β€ 15)
(hβ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 β€ f x | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 1 | false | theorem aime_1983_p2
(x p : β)
(f : β β β)
(hβ : 0 < p β§ p < 15)
(hβ : p β€ x β§ x β€ 15)
(hβ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 β€ f x | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 2 | false | theorem aime_1983_p2
(x p : β)
(f : β β β)
(hβ : 0 < p β§ p < 15)
(hβ : p β€ x β§ x β€ 15)
(hβ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 β€ f x | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 3 | false | theorem aime_1983_p2
(x p : β)
(f : β β β)
(hβ : 0 < p β§ p < 15)
(hβ : p β€ x β§ x β€ 15)
(hβ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 β€ f x | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 4 | false | theorem aime_1983_p2
(x p : β)
(f : β β β)
(hβ : 0 < p β§ p < 15)
(hβ : p β€ x β§ x β€ 15)
(hβ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 β€ f x | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 5 | false | theorem aime_1983_p2
(x p : β)
(f : β β β)
(hβ : 0 < p β§ p < 15)
(hβ : p β€ x β§ x β€ 15)
(hβ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 β€ f x | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 := by l... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 6 | false | theorem aime_1983_p2
(x p : β)
(f : β β β)
(hβ : 0 < p β§ p < 15)
(hβ : p β€ x β§ x β€ 15)
(hβ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 β€ f x | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 1 | 7 | false | theorem aime_1983_p2
(x p : β)
(f : β β β)
(hβ : 0 < p β§ p < 15)
(hβ : p β€ x β§ x β€ 15)
(hβ : f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)) :
15 β€ f x | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 :... | by
have hβ : abs (x - p) = x - p := by
have hββ : x - p β₯ 0 := by linarith
rw [abs_of_nonneg hββ]
<;>
linarith
have hβ : abs (x - 15) = 15 - x := by
have hββ : x - 15 β€ 0 := by linarith
rw [abs_of_nonpos hββ]
<;>
linarith
have hβ
: x - p - 15 β€ 0 := by
have hβ
β : x β€ 15 := by l... | 169 | 169 | 0 | 1 | false | 55 | 55 | 0 | false | null |
additive | minif2f | 2 | 0 | false | theorem aime_1983_p9 (x : β) (hβ : 0 < x β§ x < Real.pi) :
12 β€ (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x) | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 1 | false | theorem aime_1983_p9 (x : β) (hβ : 0 < x β§ x < Real.pi) :
12 β€ (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x) | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 2 | false | theorem aime_1983_p9 (x : β) (hβ : 0 < x β§ x < Real.pi) :
12 β€ (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x) | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 3 | true | theorem aime_1983_p9 (x : β) (hβ : 0 < x β§ x < Real.pi) :
12 β€ (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x) | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | 537 | 535 | 2 | 1.0037 | true | 117 | 113 | 4 | false | null |
additive | minif2f | 2 | 4 | false | theorem aime_1983_p9 (x : β) (hβ : 0 < x β§ x < Real.pi) :
12 β€ (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x) | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 5 | false | theorem aime_1983_p9 (x : β) (hβ : 0 < x β§ x < Real.pi) :
12 β€ (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x) | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 6 | false | theorem aime_1983_p9 (x : β) (hβ : 0 < x β§ x < Real.pi) :
12 β€ (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x) | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hββ : 0 ... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 2 | 7 | false | theorem aime_1983_p9 (x : β) (hβ : 0 < x β§ x < Real.pi) :
12 β€ (9 * (x ^ 2 * Real.sin x ^ 2) + 4) / (x * Real.sin x) | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | by
have hβ : Real.sin x > 0 := by
apply Real.sin_pos_of_pos_of_lt_pi
Β· exact hβ.1
Β· exact hβ.2
have hβ : x * Real.sin x > 0 := by
have hββ : 0 < x := hβ.1
have hββ : 0 < Real.sin x := hβ
positivity
have hβ : β (y : β), y > 0 β 12 β€ (9 * y ^ 2 + 4) / y := by
intro y hy
have hβ... | 537 | 537 | 0 | 1 | false | 117 | 117 | 0 | false | null |
additive | minif2f | 3 | 0 | false | theorem aime_1984_p1
(u : β β β)
(hβ : β n, u (n + 1) = u n + 1)
(hβ : β k in Finset.range 98, u k.succ = 137) :
β k in Finset.range 49, u (2 * k.succ) = 93 | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
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 : β β β)
(hβ : β n, u (n + 1) = u n + 1)
(hβ : β k in Finset.range 98, u k.succ = 137) :
β k in Finset.range 49, u (2 * k.succ) = 93 | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
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 : β β β)
(hβ : β n, u (n + 1) = u n + 1)
(hβ : β k in Finset.range 98, u k.succ = 137) :
β k in Finset.range 49, u (2 * k.succ) = 93 | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
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 : β β β)
(hβ : β n, u (n + 1) = u n + 1)
(hβ : β k in Finset.range 98, u k.succ = 137) :
β k in Finset.range 49, u (2 * k.succ) = 93 | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
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 : β β β)
(hβ : β n, u (n + 1) = u n + 1)
(hβ : β k in Finset.range 98, u k.succ = 137) :
β k in Finset.range 49, u (2 * k.succ) = 93 | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
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 : β β β)
(hβ : β n, u (n + 1) = u n + 1)
(hβ : β k in Finset.range 98, u k.succ = 137) :
β k in Finset.range 49, u (2 * k.succ) = 93 | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
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 | 6 | true | theorem aime_1984_p1
(u : β β β)
(hβ : β n, u (n + 1) = u n + 1)
(hβ : β k in Finset.range 98, u k.succ = 137) :
β k in Finset.range 49, u (2 * k.succ) = 93 | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | 869 | 819 | 50 | 1.0611 | true | 275.5 | 246.5 | 29 | false | null |
additive | minif2f | 3 | 7 | false | theorem aime_1984_p1
(u : β β β)
(hβ : β n, u (n + 1) = u n + 1)
(hβ : β k in Finset.range 98, u k.succ = 137) :
β k in Finset.range 49, u (2 * k.succ) = 93 | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
simp [ih]
<;> ring_nf at *
<;> norm_num at *
<;> linarith
exact by
... | by
have hβ : β n : β, u n = u 0 + n := by
intro n
induction n with
| zero =>
simp
| succ n ih =>
have hβ := hβ n
have hβ : u (n + 1) = u 0 + (n + 1 : β) := by
rw [hβ]
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 | 4 | 0 | true | theorem aime_1988_p4 (n : β) (a : β β β) (hβ : β n, abs (a n) < 1)
(hβ : (β k in Finset.range n, abs (a k)) = 19 + abs (β k in Finset.range n, a k)) : 20 β€ n | by
by_contra! h
have hβ : n β€ 19 := by linarith
-- We will show that if n β€ 19, the equation cannot hold.
have hβ : β k in Finset.range n, abs (a k) < n := by
-- Since each |a_k| < 1, the sum of absolute values is less than n.
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
... | by
by_contra! h
have hβ : n β€ 19 := by linarith
have hβ : β k in Finset.range n, abs (a k) < n := by
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
exact hβ k
calc
β k in Finset.range n, abs (a k) < β k in Finset.range n, (1 : β) := Finset.sum_lt_sum (fun i hi => by l... | 319 | 246 | 73 | 1.2967 | true | 46.5 | 38.5 | 8 | false | null |
additive | minif2f | 4 | 1 | false | theorem aime_1988_p4 (n : β) (a : β β β) (hβ : β n, abs (a n) < 1)
(hβ : (β k in Finset.range n, abs (a k)) = 19 + abs (β k in Finset.range n, a k)) : 20 β€ n | by
by_contra! h
have hβ : n β€ 19 := by linarith
-- We will show that if n β€ 19, the equation cannot hold.
have hβ : β k in Finset.range n, abs (a k) < n := by
-- Since each |a_k| < 1, the sum of absolute values is less than n.
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
... | by
by_contra! h
have hβ : n β€ 19 := by linarith
have hβ : β k in Finset.range n, abs (a k) < n := by
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
exact hβ k
calc
β k in Finset.range n, abs (a k) < β k in Finset.range n, (1 : β) := Finset.sum_lt_sum (fun i hi => by l... | 319 | 246 | 73 | 1.2967 | true | 46.5 | 38.5 | 8 | false | null |
additive | minif2f | 4 | 2 | false | theorem aime_1988_p4 (n : β) (a : β β β) (hβ : β n, abs (a n) < 1)
(hβ : (β k in Finset.range n, abs (a k)) = 19 + abs (β k in Finset.range n, a k)) : 20 β€ n | by
by_contra! h
have hβ : n β€ 19 := by linarith
-- We will show that if n β€ 19, the equation cannot hold.
have hβ : β k in Finset.range n, abs (a k) < n := by
-- Since each |a_k| < 1, the sum of absolute values is less than n.
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
... | by
by_contra! h
have hβ : n β€ 19 := by linarith
have hβ : β k in Finset.range n, abs (a k) < n := by
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
exact hβ k
calc
β k in Finset.range n, abs (a k) < β k in Finset.range n, (1 : β) := Finset.sum_lt_sum (fun i hi => by l... | 319 | 246 | 73 | 1.2967 | true | 46.5 | 31 | 15.5 | false | null |
additive | minif2f | 4 | 3 | false | theorem aime_1988_p4 (n : β) (a : β β β) (hβ : β n, abs (a n) < 1)
(hβ : (β k in Finset.range n, abs (a k)) = 19 + abs (β k in Finset.range n, a k)) : 20 β€ n | by
by_contra! h
have hβ : n β€ 19 := by linarith
-- We will show that if n β€ 19, the equation cannot hold.
have hβ : β k in Finset.range n, abs (a k) < n := by
-- Since each |a_k| < 1, the sum of absolute values is less than n.
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
... | by
by_contra! h
have hβ : n β€ 19 := by linarith
have hβ : β k in Finset.range n, abs (a k) < n := by
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
exact hβ k
calc
β k in Finset.range n, abs (a k) < β k in Finset.range n, (1 : β) := Finset.sum_lt_sum (fun i hi => by l... | 319 | 246 | 73 | 1.2967 | true | 46.5 | 38.5 | 8 | false | null |
additive | minif2f | 4 | 4 | false | theorem aime_1988_p4 (n : β) (a : β β β) (hβ : β n, abs (a n) < 1)
(hβ : (β k in Finset.range n, abs (a k)) = 19 + abs (β k in Finset.range n, a k)) : 20 β€ n | by
by_contra! h
have hβ : n β€ 19 := by linarith
-- We will show that if n β€ 19, the equation cannot hold.
have hβ : β k in Finset.range n, abs (a k) < n := by
-- Since each |a_k| < 1, the sum of absolute values is less than n.
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
... | by
by_contra! h
have hβ : n β€ 19 := by linarith
have hβ : β k in Finset.range n, abs (a k) < n := by
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
exact hβ k
calc
β k in Finset.range n, abs (a k) < β k in Finset.range n, (1 : β) := Finset.sum_lt_sum (fun i hi => by l... | 319 | 246 | 73 | 1.2967 | true | 46.5 | 38.5 | 8 | false | null |
additive | minif2f | 4 | 5 | false | theorem aime_1988_p4 (n : β) (a : β β β) (hβ : β n, abs (a n) < 1)
(hβ : (β k in Finset.range n, abs (a k)) = 19 + abs (β k in Finset.range n, a k)) : 20 β€ n | by
by_contra! h
have hβ : n β€ 19 := by linarith
-- We will show that if n β€ 19, the equation cannot hold.
have hβ : β k in Finset.range n, abs (a k) < n := by
-- Since each |a_k| < 1, the sum of absolute values is less than n.
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
... | by
by_contra! h
have hβ : n β€ 19 := by linarith
have hβ : β k in Finset.range n, abs (a k) < n := by
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
exact hβ k
calc
β k in Finset.range n, abs (a k) < β k in Finset.range n, (1 : β) := Finset.sum_lt_sum (fun i hi => by l... | 319 | 246 | 73 | 1.2967 | true | 46.5 | 38.5 | 8 | false | null |
additive | minif2f | 4 | 6 | false | theorem aime_1988_p4 (n : β) (a : β β β) (hβ : β n, abs (a n) < 1)
(hβ : (β k in Finset.range n, abs (a k)) = 19 + abs (β k in Finset.range n, a k)) : 20 β€ n | by
by_contra! h
have hβ : n β€ 19 := by linarith
-- We will show that if n β€ 19, the equation cannot hold.
have hβ : β k in Finset.range n, abs (a k) < n := by
-- Since each |a_k| < 1, the sum of absolute values is less than n.
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
... | by
by_contra! h
have hβ : n β€ 19 := by linarith
have hβ : β k in Finset.range n, abs (a k) < n := by
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
exact hβ k
calc
β k in Finset.range n, abs (a k) < β k in Finset.range n, (1 : β) := Finset.sum_lt_sum (fun i hi => by l... | 319 | 246 | 73 | 1.2967 | true | 46.5 | 38.5 | 8 | false | null |
additive | minif2f | 4 | 7 | false | theorem aime_1988_p4 (n : β) (a : β β β) (hβ : β n, abs (a n) < 1)
(hβ : (β k in Finset.range n, abs (a k)) = 19 + abs (β k in Finset.range n, a k)) : 20 β€ n | by
by_contra! h
have hβ : n β€ 19 := by linarith
-- We will show that if n β€ 19, the equation cannot hold.
have hβ : β k in Finset.range n, abs (a k) < n := by
-- Since each |a_k| < 1, the sum of absolute values is less than n.
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
... | by
by_contra! h
have hβ : n β€ 19 := by linarith
have hβ : β k in Finset.range n, abs (a k) < n := by
have hβ : β k, k β Finset.range n β abs (a k) < 1 := by
intro k hk
exact hβ k
calc
β k in Finset.range n, abs (a k) < β k in Finset.range n, (1 : β) := Finset.sum_lt_sum (fun i hi => by l... | 319 | 246 | 73 | 1.2967 | true | 46.5 | 38.5 | 8 | false | null |
additive | minif2f | 5 | 0 | false | theorem aime_1989_p8
(a b c d e f g : β)
(hβ : a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1)
(hβ : 4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12)
(hβ : 9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123) :
16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 ... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | 180 | 180 | 0 | 1 | false | 13.5 | 13.5 | 0 | false | null |
additive | minif2f | 5 | 1 | false | theorem aime_1989_p8
(a b c d e f g : β)
(hβ : a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1)
(hβ : 4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12)
(hβ : 9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123) :
16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 ... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | 180 | 180 | 0 | 1 | false | 13.5 | 13.5 | 0 | false | null |
additive | minif2f | 5 | 2 | false | theorem aime_1989_p8
(a b c d e f g : β)
(hβ : a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1)
(hβ : 4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12)
(hβ : 9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123) :
16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 ... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | 180 | 180 | 0 | 1 | false | 13.5 | 13.5 | 0 | false | null |
additive | minif2f | 5 | 3 | false | theorem aime_1989_p8
(a b c d e f g : β)
(hβ : a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1)
(hβ : 4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12)
(hβ : 9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123) :
16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 ... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | 180 | 180 | 0 | 1 | false | 13.5 | 13.5 | 0 | false | null |
additive | minif2f | 5 | 4 | false | theorem aime_1989_p8
(a b c d e f g : β)
(hβ : a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1)
(hβ : 4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12)
(hβ : 9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123) :
16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 ... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | 180 | 180 | 0 | 1 | false | 13.5 | 13.5 | 0 | false | null |
additive | minif2f | 5 | 5 | false | theorem aime_1989_p8
(a b c d e f g : β)
(hβ : a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1)
(hβ : 4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12)
(hβ : 9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123) :
16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 ... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | 180 | 180 | 0 | 1 | false | 13.5 | 13.5 | 0 | false | null |
additive | minif2f | 5 | 6 | false | theorem aime_1989_p8
(a b c d e f g : β)
(hβ : a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1)
(hβ : 4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12)
(hβ : 9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123) :
16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 ... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | 180 | 180 | 0 | 1 | false | 13.5 | 13.5 | 0 | false | null |
additive | minif2f | 5 | 7 | false | theorem aime_1989_p8
(a b c d e f g : β)
(hβ : a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1)
(hβ : 4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12)
(hβ : 9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123) :
16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 ... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | by
have h_main : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334 := by
have hβ : 16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 1 * (a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g) + (-3) * (4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g) + 3 * (9 * a + 16 *... | 180 | 180 | 0 | 1 | false | 13.5 | 13.5 | 0 | false | null |
additive | minif2f | 6 | 0 | false | theorem aime_1990_p15
(a b x y : β)
(hβ : a * x + b * y = 3)
(hβ : a * x^2 + b * y^2 = 7)
(hβ : a * x^3 + b * y^3 = 16)
(hβ : a * x^4 + b * y^4 = 42) :
a * x^5 + b * y^5 = 20 | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | 652 | 652 | 0 | 1 | false | 133 | 133 | 0 | false | null |
additive | minif2f | 6 | 1 | false | theorem aime_1990_p15
(a b x y : β)
(hβ : a * x + b * y = 3)
(hβ : a * x^2 + b * y^2 = 7)
(hβ : a * x^3 + b * y^3 = 16)
(hβ : a * x^4 + b * y^4 = 42) :
a * x^5 + b * y^5 = 20 | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | 652 | 652 | 0 | 1 | false | 133 | 133 | 0 | false | null |
additive | minif2f | 6 | 2 | false | theorem aime_1990_p15
(a b x y : β)
(hβ : a * x + b * y = 3)
(hβ : a * x^2 + b * y^2 = 7)
(hβ : a * x^3 + b * y^3 = 16)
(hβ : a * x^4 + b * y^4 = 42) :
a * x^5 + b * y^5 = 20 | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | 652 | 652 | 0 | 1 | false | 133 | 133 | 0 | false | null |
additive | minif2f | 6 | 3 | false | theorem aime_1990_p15
(a b x y : β)
(hβ : a * x + b * y = 3)
(hβ : a * x^2 + b * y^2 = 7)
(hβ : a * x^3 + b * y^3 = 16)
(hβ : a * x^4 + b * y^4 = 42) :
a * x^5 + b * y^5 = 20 | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | 652 | 652 | 0 | 1 | false | 133 | 133 | 0 | false | null |
additive | minif2f | 6 | 4 | false | theorem aime_1990_p15
(a b x y : β)
(hβ : a * x + b * y = 3)
(hβ : a * x^2 + b * y^2 = 7)
(hβ : a * x^3 + b * y^3 = 16)
(hβ : a * x^4 + b * y^4 = 42) :
a * x^5 + b * y^5 = 20 | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | 652 | 652 | 0 | 1 | false | 133 | 133 | 0 | false | null |
additive | minif2f | 6 | 5 | false | theorem aime_1990_p15
(a b x y : β)
(hβ : a * x + b * y = 3)
(hβ : a * x^2 + b * y^2 = 7)
(hβ : a * x^3 + b * y^3 = 16)
(hβ : a * x^4 + b * y^4 = 42) :
a * x^5 + b * y^5 = 20 | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | 652 | 652 | 0 | 1 | false | 133 | 133 | 0 | false | null |
additive | minif2f | 6 | 6 | false | theorem aime_1990_p15
(a b x y : β)
(hβ : a * x + b * y = 3)
(hβ : a * x^2 + b * y^2 = 7)
(hβ : a * x^3 + b * y^3 = 16)
(hβ : a * x^4 + b * y^4 = 42) :
a * x^5 + b * y^5 = 20 | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | 652 | 652 | 0 | 1 | false | 133 | 133 | 0 | false | null |
additive | minif2f | 6 | 7 | false | theorem aime_1990_p15
(a b x y : β)
(hβ : a * x + b * y = 3)
(hβ : a * x^2 + b * y^2 = 7)
(hβ : a * x^3 + b * y^3 = 16)
(hβ : a * x^4 + b * y^4 = 42) :
a * x^5 + b * y^5 = 20 | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | by
have h_s : x + y = -14 := by
have hβ : 7 * (x + y) - 3 * (x * y) = 16 := by
have hββ : a * x ^ 3 + b * y ^ 3 = (x + y) * (a * x ^ 2 + b * y ^ 2) - (x * y) * (a * x + b * y) := by
ring_nf
<;>
(try norm_num) <;>
(try linarith) <;>
(try nlinarith)
rw [hββ] at hβ... | 652 | 652 | 0 | 1 | false | 133 | 133 | 0 | false | null |
additive | minif2f | 7 | 0 | true | theorem aime_1990_p4
(x : β)
(hβ : 0 < x)
(hβ : x^2 - 10 * x - 29 β 0)
(hβ : x^2 - 10 * x - 45 β 0)
(hβ : x^2 - 10 * x - 69 β 0)
(hβ : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0) :
x = 13 | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | 557 | 477 | 80 | 1.1677 | true | 94.5 | 85 | 9.5 | false | null |
additive | minif2f | 7 | 1 | false | theorem aime_1990_p4
(x : β)
(hβ : 0 < x)
(hβ : x^2 - 10 * x - 29 β 0)
(hβ : x^2 - 10 * x - 45 β 0)
(hβ : x^2 - 10 * x - 69 β 0)
(hβ : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0) :
x = 13 | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | 557 | 557 | 0 | 1 | false | 94.5 | 94.5 | 0 | false | null |
additive | minif2f | 7 | 2 | false | theorem aime_1990_p4
(x : β)
(hβ : 0 < x)
(hβ : x^2 - 10 * x - 29 β 0)
(hβ : x^2 - 10 * x - 45 β 0)
(hβ : x^2 - 10 * x - 69 β 0)
(hβ : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0) :
x = 13 | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | 557 | 541 | 16 | 1.0296 | true | 94.5 | 91 | 3.5 | false | null |
additive | minif2f | 7 | 3 | false | theorem aime_1990_p4
(x : β)
(hβ : 0 < x)
(hβ : x^2 - 10 * x - 29 β 0)
(hβ : x^2 - 10 * x - 45 β 0)
(hβ : x^2 - 10 * x - 69 β 0)
(hβ : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0) :
x = 13 | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | 557 | 541 | 16 | 1.0296 | true | 94.5 | 91 | 3.5 | false | null |
additive | minif2f | 7 | 4 | false | theorem aime_1990_p4
(x : β)
(hβ : 0 < x)
(hβ : x^2 - 10 * x - 29 β 0)
(hβ : x^2 - 10 * x - 45 β 0)
(hβ : x^2 - 10 * x - 69 β 0)
(hβ : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0) :
x = 13 | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | 557 | 541 | 16 | 1.0296 | true | 94.5 | 91 | 3.5 | false | null |
additive | minif2f | 7 | 5 | false | theorem aime_1990_p4
(x : β)
(hβ : 0 < x)
(hβ : x^2 - 10 * x - 29 β 0)
(hβ : x^2 - 10 * x - 45 β 0)
(hβ : x^2 - 10 * x - 69 β 0)
(hβ : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0) :
x = 13 | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | 557 | 557 | 0 | 1 | false | 94.5 | 94.5 | 0 | false | null |
additive | minif2f | 7 | 6 | false | theorem aime_1990_p4
(x : β)
(hβ : 0 < x)
(hβ : x^2 - 10 * x - 29 β 0)
(hβ : x^2 - 10 * x - 45 β 0)
(hβ : x^2 - 10 * x - 69 β 0)
(hβ : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0) :
x = 13 | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | 557 | 541 | 16 | 1.0296 | true | 94.5 | 91 | 3.5 | false | null |
additive | minif2f | 7 | 7 | false | theorem aime_1990_p4
(x : β)
(hβ : 0 < x)
(hβ : x^2 - 10 * x - 29 β 0)
(hβ : x^2 - 10 * x - 45 β 0)
(hβ : x^2 - 10 * x - 69 β 0)
(hβ : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0) :
x = 13 | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | by
have hβ
: (x^2 - 10*x - 69)*(x^2 - 10*x - 37) = (x^2 - 10*x - 29)*(x^2 - 10*x - 45) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) = 2 / (x^2 - 10 * x - 69) := by
have hβ
β : 1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45) - 2 / (x^2 - 10 * x - 69) = 0 := hβ
linarith
have h... | 557 | 541 | 16 | 1.0296 | true | 94.5 | 91 | 3.5 | false | null |
additive | minif2f | 8 | 0 | false | theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : β) :
x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | 242 | 242 | 0 | 1 | false | 100.5 | 100.5 | 0 | false | null |
additive | minif2f | 8 | 1 | false | theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : β) :
x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | 242 | 242 | 0 | 1 | false | 100.5 | 100.5 | 0 | false | null |
additive | minif2f | 8 | 2 | false | theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : β) :
x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | 242 | 242 | 0 | 1 | false | 100.5 | 100.5 | 0 | false | null |
additive | minif2f | 8 | 3 | false | theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : β) :
x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | 242 | 242 | 0 | 1 | false | 100.5 | 100.5 | 0 | false | null |
additive | minif2f | 8 | 4 | false | theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : β) :
x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | 242 | 242 | 0 | 1 | false | 100.5 | 100.5 | 0 | false | null |
additive | minif2f | 8 | 5 | false | theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : β) :
x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | 242 | 242 | 0 | 1 | false | 100.5 | 100.5 | 0 | false | null |
additive | minif2f | 8 | 6 | false | theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : β) :
x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | 242 | 242 | 0 | 1 | false | 100.5 | 100.5 | 0 | false | null |
additive | minif2f | 8 | 7 | false | theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i (x : β) :
x ^ 2 + 49 = (x + 7 * Complex.I) * (x + -7 * Complex.I) | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | by
have h_main : (x + 7 * Complex.I) * (x + -7 * Complex.I) = x ^ 2 + 49 := by
calc
(x + 7 * Complex.I) * (x + -7 * Complex.I) = (x + 7 * Complex.I) * (x - 7 * Complex.I) := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
_ = x ^ 2 - (7 * Complex.I) ^ 2 := b... | 242 | 242 | 0 | 1 | false | 100.5 | 100.5 | 0 | false | null |
additive | minif2f | 9 | 0 | true | theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : β) :
(a - 10) * (a + 11) = a ^ 2 + a - 110 | by
have h_main : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by
ring_nf
<;>
simp [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
ring_nf
<;>
norm_num
<;>
simp_all [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
linarith
rw [h_main] | by
ring_nf | 58 | 2 | 56 | 29 | true | 32.5 | 2 | 30.5 | false | null |
additive | minif2f | 9 | 1 | false | theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : β) :
(a - 10) * (a + 11) = a ^ 2 + a - 110 | by
have h_main : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by
ring_nf
<;>
simp [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
ring_nf
<;>
norm_num
<;>
simp_all [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
linarith
rw [h_main] | by
ring_nf | 58 | 2 | 56 | 29 | true | 32.5 | 2 | 30.5 | false | null |
additive | minif2f | 9 | 2 | false | theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : β) :
(a - 10) * (a + 11) = a ^ 2 + a - 110 | by
have h_main : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by
ring_nf
<;>
simp [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
ring_nf
<;>
norm_num
<;>
simp_all [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
linarith
rw [h_main] | by
ring_nf | 58 | 2 | 56 | 29 | true | 32.5 | 2 | 30.5 | false | null |
additive | minif2f | 9 | 3 | false | theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : β) :
(a - 10) * (a + 11) = a ^ 2 + a - 110 | by
have h_main : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by
ring_nf
<;>
simp [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
ring_nf
<;>
norm_num
<;>
simp_all [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
linarith
rw [h_main] | by
ring_nf | 58 | 2 | 56 | 29 | true | 32.5 | 2 | 30.5 | false | null |
additive | minif2f | 9 | 4 | false | theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : β) :
(a - 10) * (a + 11) = a ^ 2 + a - 110 | by
have h_main : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by
ring_nf
<;>
simp [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
ring_nf
<;>
norm_num
<;>
simp_all [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
linarith
rw [h_main] | by
ring_nf | 58 | 2 | 56 | 29 | true | 32.5 | 2 | 30.5 | false | null |
additive | minif2f | 9 | 5 | false | theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : β) :
(a - 10) * (a + 11) = a ^ 2 + a - 110 | by
have h_main : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by
ring_nf
<;>
simp [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
ring_nf
<;>
norm_num
<;>
simp_all [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
linarith
rw [h_main] | by
ring_nf | 58 | 2 | 56 | 29 | true | 32.5 | 2 | 30.5 | false | null |
additive | minif2f | 9 | 6 | false | theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : β) :
(a - 10) * (a + 11) = a ^ 2 + a - 110 | by
have h_main : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by
ring_nf
<;>
simp [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
ring_nf
<;>
norm_num
<;>
simp_all [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
linarith
rw [h_main] | by
ring_nf | 58 | 2 | 56 | 29 | true | 32.5 | 2 | 30.5 | false | null |
additive | minif2f | 9 | 7 | false | theorem algebra_2rootsintpoly_am10tap11eqasqpam110 (a : β) :
(a - 10) * (a + 11) = a ^ 2 + a - 110 | by
have h_main : (a - 10) * (a + 11) = a ^ 2 + a - 110 := by
ring_nf
<;>
simp [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
ring_nf
<;>
norm_num
<;>
simp_all [Complex.ext_iff, pow_two, mul_comm]
<;>
norm_num
<;>
linarith
rw [h_main] | by
ring_nf | 58 | 2 | 56 | 29 | true | 32.5 | 2 | 30.5 | false | null |
additive | minif2f | 10 | 0 | true | theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : β) :
(a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | by
ring_nf | 180 | 2 | 178 | 90 | true | 54 | 2 | 52 | false | null |
additive | minif2f | 10 | 1 | false | theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : β) :
(a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | by
ring_nf | 180 | 2 | 178 | 90 | true | 54 | 2 | 52 | false | null |
additive | minif2f | 10 | 2 | false | theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : β) :
(a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | by
ring_nf | 180 | 2 | 178 | 90 | true | 54 | 2 | 52 | false | null |
additive | minif2f | 10 | 3 | false | theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : β) :
(a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | 180 | 180 | 0 | 1 | false | 54 | 54 | 0 | false | null |
additive | minif2f | 10 | 4 | false | theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : β) :
(a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | by
ring_nf | 180 | 2 | 178 | 90 | true | 54 | 2 | 52 | false | null |
additive | minif2f | 10 | 5 | false | theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : β) :
(a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | 180 | 167 | 13 | 1.0778 | true | 54 | 46 | 8 | false | null |
additive | minif2f | 10 | 6 | false | theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : β) :
(a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2... | 180 | 167 | 13 | 1.0778 | true | 54 | 46 | 8 | false | null |
additive | minif2f | 10 | 7 | false | theorem algebra_2rootspoly_apatapbeq2asqp2ab (a b : β) :
(a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | by
have h_main : (a + a) * (a + b) = 2 * a ^ 2 + 2 * (a * b) := by
have hβ : (a + a) * (a + b) = 2 * a * (a + b) := by
ring_nf
<;>
simp [two_mul]
<;>
ring_nf
rw [hβ]
have hβ : 2 * a * (a + b) = 2 * a * a + 2 * a * b := by
ring_nf
rw [hβ]
have hβ : 2 * a * a + 2 ... | 180 | 165 | 15 | 1.0909 | true | 54 | 44 | 10 | false | null |
additive | minif2f | 11 | 0 | false | theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7
(f z: β)
(hβ : f + 3*z = 11)
(hβ : 3*(f - 1) - 5*z = -68) :
f = -10 β§ z = 7 | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | 801 | 786 | 15 | 1.0191 | true | 319.5 | 316 | 3.5 | false | null |
additive | minif2f | 11 | 1 | false | theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7
(f z: β)
(hβ : f + 3*z = 11)
(hβ : 3*(f - 1) - 5*z = -68) :
f = -10 β§ z = 7 | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | 801 | 801 | 0 | 1 | false | 319.5 | 319.5 | 0 | false | null |
additive | minif2f | 11 | 2 | true | theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7
(f z: β)
(hβ : f + 3*z = 11)
(hβ : 3*(f - 1) - 5*z = -68) :
f = -10 β§ z = 7 | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | 801 | 729 | 72 | 1.0988 | true | 319.5 | 308.5 | 11 | false | null |
additive | minif2f | 11 | 3 | false | theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7
(f z: β)
(hβ : f + 3*z = 11)
(hβ : 3*(f - 1) - 5*z = -68) :
f = -10 β§ z = 7 | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | 801 | 786 | 15 | 1.0191 | true | 319.5 | 316 | 3.5 | false | null |
additive | minif2f | 11 | 4 | false | theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7
(f z: β)
(hβ : f + 3*z = 11)
(hβ : 3*(f - 1) - 5*z = -68) :
f = -10 β§ z = 7 | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | 801 | 801 | 0 | 1 | false | 319.5 | 319.5 | 0 | false | null |
additive | minif2f | 11 | 5 | false | theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7
(f z: β)
(hβ : f + 3*z = 11)
(hβ : 3*(f - 1) - 5*z = -68) :
f = -10 β§ z = 7 | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | 801 | 786 | 15 | 1.0191 | true | 319.5 | 316 | 3.5 | false | null |
additive | minif2f | 11 | 6 | false | theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7
(f z: β)
(hβ : f + 3*z = 11)
(hβ : 3*(f - 1) - 5*z = -68) :
f = -10 β§ z = 7 | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | 801 | 801 | 0 | 1 | false | 319.5 | 319.5 | 0 | false | null |
additive | minif2f | 11 | 7 | false | theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7
(f z: β)
(hβ : f + 3*z = 11)
(hβ : 3*(f - 1) - 5*z = -68) :
f = -10 β§ z = 7 | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | by
have hβ : 3*f - 5*z = -65 := by
have hββ : 3*(f - 1) - 5*z = -68 := hβ
have hββ : 3*(f - 1) - 5*z = 3*f - 3 - 5*z := by
ring_nf
<;> simp [Complex.ext_iff, Complex.I_mul_I]
<;> norm_num
<;>
(try ring_nf at * <;>
simp_all [Complex.ext_iff, Complex.I_mul_I] <;>
no... | 801 | 801 | 0 | 1 | false | 319.5 | 319.5 | 0 | false | null |
additive | minif2f | 12 | 0 | false | theorem algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4 (x e : β) (hβ : x + e = 7)
(hβ : 2 * x + e = 3) : e = 11 β§ x = -4 | by
have h_x : x = -4 := by
have hβ : x = -4 := by
-- Subtract the first equation from the second to eliminate e
have hβ : (2 * x + e) - (x + e) = 3 - 7 := by
rw [hβ, hβ]
<;> norm_num
-- Simplify the left side to get x = -4
ring_nf at hβ β’
simp [Complex.ext_iff, pow_tw... | by
have h_x : x = -4 := by
have hβ : x = -4 := by
have hβ : (2 * x + e) - (x + e) = 3 - 7 := by
rw [hβ, hβ]
<;> norm_num
ring_nf at hβ β’
simp [Complex.ext_iff, pow_two] at hβ β’
<;>
(try norm_num at hβ β’) <;>
(try constructor <;> linarith) <;>
(try simp_all... | 245 | 187 | 58 | 1.3102 | true | 78 | 66.5 | 11.5 | false | null |
additive | minif2f | 12 | 1 | false | theorem algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4 (x e : β) (hβ : x + e = 7)
(hβ : 2 * x + e = 3) : e = 11 β§ x = -4 | by
have h_x : x = -4 := by
have hβ : x = -4 := by
-- Subtract the first equation from the second to eliminate e
have hβ : (2 * x + e) - (x + e) = 3 - 7 := by
rw [hβ, hβ]
<;> norm_num
-- Simplify the left side to get x = -4
ring_nf at hβ β’
simp [Complex.ext_iff, pow_tw... | by
have h_x : x = -4 := by
have hβ : x = -4 := by
have hβ : (2 * x + e) - (x + e) = 3 - 7 := by
rw [hβ, hβ]
<;> norm_num
ring_nf at hβ β’
simp [Complex.ext_iff, pow_two] at hβ β’
<;>
(try norm_num at hβ β’) <;>
(try constructor <;> linarith) <;>
(try simp_all... | 245 | 187 | 58 | 1.3102 | true | 78 | 66.5 | 11.5 | false | null |
additive | minif2f | 12 | 2 | false | theorem algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4 (x e : β) (hβ : x + e = 7)
(hβ : 2 * x + e = 3) : e = 11 β§ x = -4 | by
have h_x : x = -4 := by
have hβ : x = -4 := by
-- Subtract the first equation from the second to eliminate e
have hβ : (2 * x + e) - (x + e) = 3 - 7 := by
rw [hβ, hβ]
<;> norm_num
-- Simplify the left side to get x = -4
ring_nf at hβ β’
simp [Complex.ext_iff, pow_tw... | by
have h_x : x = -4 := by
have hβ : (2 * x + e) - (x + e) = 3 - 7 := by
rw [hβ, hβ]
<;> norm_num
ring_nf at hβ β’
simp [Complex.ext_iff, pow_two] at hβ β’
<;>
(try norm_num at hβ β’) <;>
(try constructor <;> linarith) <;>
(try simp_all [Complex.ext_iff, pow_two]) <;>
(try nor... | 245 | 181 | 64 | 1.3536 | true | 78 | 63 | 15 | false | null |
additive | minif2f | 12 | 3 | true | theorem algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4 (x e : β) (hβ : x + e = 7)
(hβ : 2 * x + e = 3) : e = 11 β§ x = -4 | by
have h_x : x = -4 := by
have hβ : x = -4 := by
-- Subtract the first equation from the second to eliminate e
have hβ : (2 * x + e) - (x + e) = 3 - 7 := by
rw [hβ, hβ]
<;> norm_num
-- Simplify the left side to get x = -4
ring_nf at hβ β’
simp [Complex.ext_iff, pow_tw... | by
have h_x : x = -4 := by
have hβ : x = -4 := by
have hβ : (2 * x + e) - (x + e) = 3 - 7 := by
rw [hβ, hβ]
<;> norm_num
ring_nf at hβ β’
<;> simp [Complex.ext_iff, pow_two] at hβ β’
<;> try norm_num at hβ β’
<;> try constructor <;> linarith
exact hβ
have h_e : e =... | 245 | 141 | 104 | 1.7376 | true | 78 | 54.5 | 23.5 | false | null |
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