index int64 0 86 | question_type stringclasses 3
values | author stringclasses 2
values | problem stringlengths 51 893 | formal_statement stringlengths 95 1.1k | train_proof stringlengths 92 2.96k | prompt stringlengths 1.78k 4.07k | Answer stringlengths 3 8.51k | probs stringlengths 71 2.36k | final_answer stringlengths 121 9.05k |
|---|---|---|---|---|---|---|---|---|---|
0 | proof | human | Show that
$$
\sqrt[5]{375} \sqrt[6]{600} \sqrt[15]{45}=5 \sqrt{6}
$$ | import Mathlib
/- Show that
$$
\sqrt[5]{375} \sqrt[6]{600} \sqrt[15]{45}=5 \sqrt{6}
$$ -/
theorem algebra_283651 : (375 : β) ^ ((1 : β) / 5) * (600) ^ ((1 : β) / 6) * (45) ^ ((1 : β) / 15) = 5 * Real.sqrt 6 := by | import Mathlib
/- Show that
$$
\sqrt[5]{375} \sqrt[6]{600} \sqrt[15]{45}=5 \sqrt{6}
$$ -/
theorem algebra_283651 : (375 : β) ^ ((1 : β) / 5) * (600) ^ ((1 : β) / 6) * (45) ^ ((1 : β) / 15) = 5 * Real.sqrt 6 := by sorry | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | ^ ((1 : β) / 15) = 5 * (6) ^ (1 / 2) := by
norm_num [Real.sqrt_eq_iff_sq_eq]
<;> ring_nf
<;> norm_num
<;> ring_nf
<;> norm_num
<;> ring_nf
<;> norm_num
<;> ring_nf
<;> norm_num
<;> ring_nf
<;> norm_num
<;> ring_nf
<;> norm_num
<;> ring_nf
<;> norm_num
<;> ring_nf
<;> norm_num
<;> r... | import Mathlib
/- Show that
$$
\sqrt[5]{375} \sqrt[6]{600} \sqrt[15]{45}=5 \sqrt{6}
$$ -/
theorem algebra_283651 : (375 : β) ^ ((1 : β) / 5) * (600) ^ ((1 : β) / 6) * (45) | import Mathlib
/- Show that
$$
\sqrt[5]{375} \sqrt[6]{600} \sqrt[15]{45}=5 \sqrt{6}
$$ -/
theorem algebra_283651 : (375 : β) ^ ((1 : β) / 5) * (600) ^ ((1 : β) / 6) * (45) ^ ((1 : β) / 15) = 5 * (6) ^ (1 / 2) := by
norm_num [Real.sqrt_eq_iff_sq_eq]
<;> ring_nf
<;> norm_num
<;> ring_nf
<;> norm_num
<;> ri... |
1 | math-word-problem | autoformalizer | Example 2 Given that $a$ is a root of the equation $x^{2}+x-\frac{1}{4}=0$. Then the value of $\frac{a^{3}-1}{a^{5}+a^{4}-a^{3}-a^{2}}$ is $\qquad$ .
(1995, National Junior High School Mathematics League) | import Mathlib
/- Example 2 Given that $a$ is a root of the equation $x^{2}+x-\frac{1}{4}=0$. Then the value of $\frac{a^{3}-1}{a^{5}+a^{4}-a^{3}-a^{2}}$ is $\qquad$ .
(1995, National Junior High School Mathematics League) -/
theorem algebra_325814 {a : β} (ha : a^2 + a - 1 / 4 = 0) :
(a^3 - 1) / (a^5 + a^4 - a^3 ... | import Mathlib
theorem algebra_325814 {a : β} (ha : a^2 + a - 1 / 4 = 0) :
(a^3 - 1) / (a^5 + a^4 - a^3 - a^2) = 20 := by
have ha2 : a^2 = -a + 1 / 4 := by linarith
have ha3 : a^3 = 5 / 4 * a - 1 / 4 := by
calc
a^3 = a * a^2 := by ring
_ = a * (-a + 1 / 4) := by rw [ha2]
_ = -a^2 + (1 ... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | _eq] at ha
linarith
-- Calculate the fraction
rw [div_eq_mul_inv]
field_simp [h2]
ring
linarith | import Mathlib
theorem algebra_325814 {a : β} (ha : a^2 + a - 1 / 4 = 0) :
(a^3 - 1) / (a^5 + a^4 - a^3 - a^2) = 20 := by
have ha2 : a^2 = -a + 1 / 4 := by linarith
have ha3 : a^3 = 5 / 4 * a - 1 / 4 := by
calc
a^3 = a * a^2 := by ring
_ = a * (-a + 1 / 4) := by rw [ha2]
_ = -a^2 + (1 ... | import Mathlib
theorem algebra_325814 {a : β} (ha : a^2 + a - 1 / 4 = 0) :
(a^3 - 1) / (a^5 + a^4 - a^3 - a^2) = 20 := by
have ha2 : a^2 = -a + 1 / 4 := by linarith
have ha3 : a^3 = 5 / 4 * a - 1 / 4 := by
calc
a^3 = a * a^2 := by ring
_ = a * (-a + 1 / 4) := by rw [ha2]
_ = -a^2 + (1 ... |
3 | MCQ | autoformalizer | Steve plants ten trees every three minutes. If he continues planting at the same rate, how long will it take him to plant 2500 trees?
(A) $1 \frac{1}{4} \mathrm{~h}$
(B) $3 \mathrm{~h}$
(C) $5 \mathrm{~h}$
(D) $10 \mathrm{~h}$
(E) $12 \frac{1}{2} \mathrm{~h}$ | import Mathlib
/- Steve plants ten trees every three minutes. If he continues planting at the same rate, how long will it take him to plant 2500 trees?
(A) $1 \frac{1}{4} \mathrm{~h}$
(B) $3 \mathrm{~h}$
(C) $5 \mathrm{~h}$
(D) $10 \mathrm{~h}$
(E) $12 \frac{1}{2} \mathrm{~h}$ -/
theorem algebra_287117 (x : β) (hx : x... | import Mathlib
theorem algebra_287117 (x : β) (hx : x = 2500 / 10) :
x * 3 / 60 = 12.5 := by
rw [hx]
norm_num | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | 5 := by
rw [hx]
norm_num
| import Mathlib
theorem algebra_287117 (x : β) (hx : x = 2500 / 10) :
x * 3 / 60 = 12. | import Mathlib
theorem algebra_287117 (x : β) (hx : x = 2500 / 10) :
x * 3 / 60 = 12.5 := by
rw [hx]
norm_num
|
4 | math-word-problem | human | Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume tha... | import Mathlib
import Init.Data.Int
open Classical
/- Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps a... | import Mathlib
import Init.Data.Int
open Classical
/- Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps a... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | 5 * 3) e
/- Solve the equation to find x. -/
have h : x = 120 := by
have h1 : (150 - x) * 3 = x - 75 := by
field_simp at heb
ring_nf at heb
linarith
linarith
exact h | import Mathlib
import Init.Data.Int
open Classical
/- Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps a... | import Mathlib
import Init.Data.Int
open Classical
/- Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps a... |
5 | proof | autoformalizer | 8.2 A point moves in the coordinate plane such that at time $t \geq 0$ it is located on both the line $y=t \cdot x+1$ and the line $y=-x+2 t$. Prove that it will never be to the right of the vertical line $x=2$ (i.e., in the region $x \geq 2$). | import Mathlib
/- 8.2 A point moves in the coordinate plane such that at time $t \geq 0$ it is located on both the line $y=t \cdot x+1$ and the line $y=-x+2 t$. Prove that it will never be to the right of the vertical line $x=2$ (i.e., in the region $x \geq 2$). -/
theorem algebra_128815 (t : β) (ht : t β₯ 0) (x y : β)... | import Mathlib
theorem algebra_128815 (t : β) (ht : t β₯ 0) (x y : β)
(h1 : y = t * x + 1) (h2 : y = -x + 2 * t) :
x β€ 2 := by
have h_eq1 : t * x + 1 = -x + 2 * t := by
rw [β h1, β h2]
have h_eq2 : x * (t + 1) = 2 * t - 1 := by
linarith [h_eq1]
nlinarith [sq_nonneg (x - 2), sq_nonneg (t + 1), sq_... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | 2), sq_nonneg (t - 1)]
| import Mathlib
theorem algebra_128815 (t : β) (ht : t β₯ 0) (x y : β)
(h1 : y = t * x + 1) (h2 : y = -x + 2 * t) :
x β€ 2 := by
have h_eq1 : t * x + 1 = -x + 2 * t := by
rw [β h1, β h2]
have h_eq2 : x * (t + 1) = 2 * t - 1 := by
linarith [h_eq1]
nlinarith [sq_nonneg (x - | import Mathlib
theorem algebra_128815 (t : β) (ht : t β₯ 0) (x y : β)
(h1 : y = t * x + 1) (h2 : y = -x + 2 * t) :
x β€ 2 := by
have h_eq1 : t * x + 1 = -x + 2 * t := by
rw [β h1, β h2]
have h_eq2 : x * (t + 1) = 2 * t - 1 := by
linarith [h_eq1]
nlinarith [sq_nonneg (x - 2), sq_nonneg (t - 1)]
|
6 | MCQ | human | If $xy \ne 0$, $x + y \ne 0$, and $\frac{1}{x} + \frac{1}{y}$ is inversely proportional to $x + y$, then ${{(x + y)}^2}$ is ( ) to ${{x}^2} + {{y}^2}$.
A: directly proportional
B: inversely proportional
C: neither directly nor inversely proportional
D: the relationship is uncertain | import Mathlib
theorem algebra_1274 {x y : β} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by | import Mathlib
theorem algebra_1274 {x y : β} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_pos_of_ne_zero hx
... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | _ne_zero hx
have hy2 : y ^ 2 > 0 := by
exact sq_pos_of_ne_zero hy
linarith
field_simp
| import Mathlib
theorem algebra_1274 {x y : β} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_pos_of | import Mathlib
theorem algebra_1274 {x y : β} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_pos_of_ne_zero hx
... |
7 | math-word-problem | autoformalizer | Problem 9.7. The sides of the square $A B C D$ are parallel to the coordinate axes, with $A B$ lying on the y-axis, and the square is positioned as shown in the figure. The parabola defined by the equation
$$
y=\frac{1}{5} x^{2}+a x+b
$$
passes through points $B$ and $C$. Additionally, the vertex of this parabola (po... | import Mathlib
/- Problem 9.7. The sides of the square $A B C D$ are parallel to the coordinate axes, with $A B$ lying on the y-axis, and the square is positioned as shown in the figure. The parabola defined by the equation
$$
y=\frac{1}{5} x^{2}+a x+b
$$
passes through points $B$ and $C$. Additionally, the vertex o... | import Mathlib
theorem algebra_174965 {a b : β} (ha : a β 0) (h : 1/5 * (-5)^2 + a * (-5) + b = 0)
(h' : 1/5 * 5^2 + a * 5 + b = 0) (h'' : β x, 1/5 * x^2 + a * x + b = 0 β§ x β Set.Icc (-5) 5) :
-a = 20 := by
have eq1 : -5 * a + b = -5 := by
linarith
have eq2 : 5 * a + b = -5 := by
linarith
have... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | a + b = -5 := by
linarith
have eq3 : a = 20 := by
linarith
linarith
| import Mathlib
theorem algebra_174965 {a b : β} (ha : a β 0) (h : 1/5 * (-5)^2 + a * (-5) + b = 0)
(h' : 1/5 * 5^2 + a * 5 + b = 0) (h'' : β x, 1/5 * x^2 + a * x + b = 0 β§ x β Set.Icc (-5) 5) :
-a = 20 := by
have eq1 : -5 * a + b = -5 := by
linarith
have eq2 : 5 * | import Mathlib
theorem algebra_174965 {a b : β} (ha : a β 0) (h : 1/5 * (-5)^2 + a * (-5) + b = 0)
(h' : 1/5 * 5^2 + a * 5 + b = 0) (h'' : β x, 1/5 * x^2 + a * x + b = 0 β§ x β Set.Icc (-5) 5) :
-a = 20 := by
have eq1 : -5 * a + b = -5 := by
linarith
have eq2 : 5 * a + b = -5 := by
linarith
have ... |
8 | proof | autoformalizer | 28.11. Prove that $\left(\log _{a} x\right)^{\prime}=\frac{1}{x \ln a}$. | import Mathlib
open Real
/- 28.11. Prove that $\left(\log _{a} x\right)^{\prime}=\frac{1}{x \ln a}$. -/
theorem calculus_229012 {a x : β} (ha : 0 < a β§ a β 1) (hx : 0 < x) :
deriv (logb a) x = 1 / (x * log a) := by | import Mathlib
open Real
theorem calculus_229012 {a x : β} (ha : 0 < a β§ a β 1) (hx : 0 < x) :
deriv (logb a) x = 1 / (x * log a) := by
have hlog : log a β 0 := by
have ha1 : 0 < a := ha.1
have ha2 : a β 1 := ha.2
exact log_ne_zero_of_pos_of_ne_one ha1 ha2
have h2 : deriv (logb a) x = deriv (fun (... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | eq]
rw [h2]
have h3 : deriv (fun (y : β) => (log y) / log a) x = 1 / (x * log a) := by
have h4 : (fun (y : β) => (log y) / log a) = fun (y : β) => (log y) / log a := by
simp
rw [h4]
have h5 : deriv (fun (y : β) => (log y) / log a) x = 1 / (x * log a) := by
have h6 : (fun (y : β) => (log y) /... | import Mathlib
open Real
theorem calculus_229012 {a x : β} (ha : 0 < a β§ a β 1) (hx : 0 < x) :
deriv (logb a) x = 1 / (x * log a) := by
have hlog : log a β 0 := by
have ha1 : 0 < a := ha.1
have ha2 : a β 1 := ha.2
exact log_ne_zero_of_pos_of_ne_one ha1 ha2
have h2 : deriv (logb a) x = deriv (fun (... | import Mathlib
open Real
theorem calculus_229012 {a x : β} (ha : 0 < a β§ a β 1) (hx : 0 < x) :
deriv (logb a) x = 1 / (x * log a) := by
have hlog : log a β 0 := by
have ha1 : 0 < a := ha.1
have ha2 : a β 1 := ha.2
exact log_ne_zero_of_pos_of_ne_one ha1 ha2
have h2 : deriv (logb a) x = deriv (fun (... |
9 | math-word-problem | autoformalizer | Problem 20. From a square sheet of metal with a width of 60 cm, a box without a lid and with a square base of the largest possible volume needs to be made.
$$ | import Mathlib
open Real Set
open scoped BigOperators
/- Problem 20. From a square sheet of metal with a width of 60 cm, a box without a lid and with a square base of the largest possible volume needs to be made.
$$ -/
theorem calculus_196745 (x : β) (hx : 0 < x) (h : x β€ 60 / 2) :
(x * (60 - 2 * x) ^ 2) β€ 16000... | import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_196745 (x : β) (hx : 0 < x) (h : x β€ 60 / 2) :
(x * (60 - 2 * x) ^ 2) β€ 16000 := by
nlinarith [sq_nonneg (x - 10), sq_nonneg ((60 - 2 * x) - 40),
sq_nonneg ((x - 10) * (x - 40)), sq_nonneg ((x) * (x - 10)),
sq_nonneg ((60... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | nonneg (x * (60 - 2 * x - 40)),
h, sq_nonneg (x * ((60 - 2 * x) - 40)), sq_nonneg ((x - 10) * ((60 - 2 * x) - 40))]
| import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_196745 (x : β) (hx : 0 < x) (h : x β€ 60 / 2) :
(x * (60 - 2 * x) ^ 2) β€ 16000 := by
nlinarith [sq_nonneg (x - 10), sq_nonneg ((60 - 2 * x) - 40),
sq_nonneg ((x - 10) * (x - 40)), sq_nonneg ((x) * (x - 10)),
sq_nonneg ((60... | import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_196745 (x : β) (hx : 0 < x) (h : x β€ 60 / 2) :
(x * (60 - 2 * x) ^ 2) β€ 16000 := by
nlinarith [sq_nonneg (x - 10), sq_nonneg ((60 - 2 * x) - 40),
sq_nonneg ((x - 10) * (x - 40)), sq_nonneg ((x) * (x - 10)),
sq_nonneg ((60... |
10 | null | human | Consider those functions $f(x)$ that are defined in the interval $(0,1)$, are positive, integrable, have an integral value of 1 over the interval $(0,1)$, and are concave from below. (This last condition means that for any $0 \leq x_{1} < x_{2} \leq 1$ and $0 \leq \lambda \leq 1$,
$$
f\left(\lambda x_{1}+(1-\lambda) x... | import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_172208 {f : β β β}
(hf : β x β Ioo 0 1, f x > 0)
(hf_int : MeasureTheory.IntegrableOn f (Ioo 0 1))
(hf_int2 : MeasureTheory.IntegrableOn (fun x => (f x)^2) (Ioo 0 1))
(hf1 : β« x in Ioo 0 1, f x = 1)
(hf2 : ConvexOn β (Ioo 0 1) f) :
1 β€ β«... | import Mathlib
open Real Set
open scoped BigOperators
/- Consider those functions $f(x)$ that are defined in the interval $(0,1)$, are positive, integrable, have an integral value of 1 over the interval $(0,1)$, and are concave from below. (This last condition means that for any $0 \leq x_{1} < x_{2} \leq 1$ and $0 \... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | 0 1, f x = 1)
(hf2 : β x y β z, x β€ y β y β€ z β f (x * z + (1 - x) * y) β₯ x * f z + (1 - x) * f y) :
(β« x in Ioo 0 1, (f x)^2) β€ 1:= by
have h1 : 0 β€ β« x in Ioo 0 1, (f x)^2 := by
apply Finset.sum_nonneg
intro x _
exact sq_nonneg (f x)
have h2 : β« x in Ioo 0 1, (f x)^2 β€ 1 := by
have h3 : β« x ... | import Mathlib
open Real Set
open scoped BigOperators
/- Consider those functions $f(x)$ that are defined in the interval $(0,1)$, are positive, integrable, have an integral value of 1 over the interval $(0,1)$, and are concave from below. (This last condition means that for any $0 \leq x_{1} < x_{2} \leq 1$ and $0 \... | import Mathlib
open Real Set
open scoped BigOperators
/- Consider those functions $f(x)$ that are defined in the interval $(0,1)$, are positive, integrable, have an integral value of 1 over the interval $(0,1)$, and are concave from below. (This last condition means that for any $0 \leq x_{1} < x_{2} \leq 1$ and $0 \... |
11 | math-word-problem | autoformalizer | ## Task Condition
Derive the equation of the tangent line to the given curve at the point with abscissa $x_{0}$.
$y=3(\sqrt[3]{x}-2 \sqrt{x}), x_{0}=1$ | import Mathlib
open Real
/- ## Task Condition
Derive the equation of the tangent line to the given curve at the point with abscissa $x_{0}$.
$y=3(\sqrt[3]{x}-2 \sqrt{x}), x_{0}=1$ -/
theorem calculus_140558 (f : β β β) (hf : f = fun x => 3 * (x ^ ((1 : β) / 3) - 2 * sqrt x)) (x0 : β) (hx0 : x0 = 1) :
deriv f x0 ... | import Mathlib
open Real
theorem calculus_140558 (f : β β β) (hf : f = fun x => 3 * (x ^ ((1 : β) / 3) - 2 * sqrt x)) (x0 : β) (hx0 : x0 = 1) :
deriv f x0 = -2 β§ f x0 = -2 * x0 - 1 := by
rw [hx0]
constructor
Β· -- First part: proving that the derivative is -2
rw [hf]
simp [deriv_sub, deriv_const_mul,... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | -- Second part: proving that the function value is -2 * x0 - 1
rw [hf]
simp [hx0]
norm_num [sqrt_eq_nonneg]
all_goals norm_num | import Mathlib
open Real
theorem calculus_140558 (f : β β β) (hf : f = fun x => 3 * (x ^ ((1 : β) / 3) - 2 * sqrt x)) (x0 : β) (hx0 : x0 = 1) :
deriv f x0 = -2 β§ f x0 = -2 * x0 - 1 := by
rw [hx0]
constructor
Β· -- First part: proving that the derivative is -2
rw [hf]
simp [deriv_sub, deriv_const_mul,... | import Mathlib
open Real
theorem calculus_140558 (f : β β β) (hf : f = fun x => 3 * (x ^ ((1 : β) / 3) - 2 * sqrt x)) (x0 : β) (hx0 : x0 = 1) :
deriv f x0 = -2 β§ f x0 = -2 * x0 - 1 := by
rw [hx0]
constructor
Β· -- First part: proving that the derivative is -2
rw [hf]
simp [deriv_sub, deriv_const_mul,... |
12 | math-word-problem | autoformalizer | 6. 41 A train departs from station A, accelerating at
0.5 km/min ${ }^{2}$, and after
2 minutes, it continues at a constant speed for another
7 minutes, then decelerates at 0.5 km/min ${ }^{2}$ to a stop at station B, where it remains stationary for 2 minutes. Draw the velocity-time graph for the train's journey fro... | import Mathlib
open Real Set
open scoped BigOperators
/- 6. 41 A train departs from station A, accelerating at
0.5 km/min ${ }^{2}$, and after
2 minutes, it continues at a constant speed for another
7 minutes, then decelerates at 0.5 km/min ${ }^{2}$ to a stop at station B, where it remains stationary for 2 minute... | import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_289261 (v : β β β)
(hβ : β t, t > 0 β v t = v 0 + 0.5 * t)
(hβ : β t, t > 2 β v t = v 2)
(hβ : β t, t > 7 β v t = v 7 - 0.5 * t) :
β t, t > 0 β (t < 9 β β« s in (0)..t, v s = 1 / 4 * (-2 + 22 - 85)) β§
(β t, t > 9 β t β€ 11 β β« s in (0)..t,... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | tro t ht
constructor
Β· intro ht'
have hβ := hβ 0
have hβ := hβ 0
simp_all
Β· constructor
Β· intro t ht' ht''
have hβ := hβ 0
have hβ := hβ 0
simp_all
Β· intro t ht' ht''
have hβ := hβ 0
have hβ := hβ 0
simp_all | import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_289261 (v : β β β)
(hβ : β t, t > 0 β v t = v 0 + 0.5 * t)
(hβ : β t, t > 2 β v t = v 2)
(hβ : β t, t > 7 β v t = v 7 - 0.5 * t) :
β t, t > 0 β (t < 9 β β« s in (0)..t, v s = 1 / 4 * (-2 + 22 - 85)) β§
(β t, t > 9 β t β€ 11 β β« s in (0)..t,... | import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_289261 (v : β β β)
(hβ : β t, t > 0 β v t = v 0 + 0.5 * t)
(hβ : β t, t > 2 β v t = v 2)
(hβ : β t, t > 7 β v t = v 7 - 0.5 * t) :
β t, t > 0 β (t < 9 β β« s in (0)..t, v s = 1 / 4 * (-2 + 22 - 85)) β§
(β t, t > 9 β t β€ 11 β β« s in (0)..t,... |
13 | math-word-problem | human | Simplify $$2\cos^2(\log(2009)i)+i\sin(\log(4036081)i).$$Note: The logarithms are with respect to base $e.$ | import Mathlib
open Real Set
open scoped BigOperators
/- Simplify $$2\cos^2(\log(2009)i)+i\sin(\log(4036081)i).$$Note: The logarithms are with respect to base $e.$ -/
theorem calculus_23344 : 2 * (cos (log 2009 * I))^2 + I * sin (log 4036081 * I) = 4036082 / 4036081 := by | import Mathlib
open Real Set
open scoped BigOperators
/- Simplify $$2\cos^2(\log(2009)i)+i\sin(\log(4036081)i).$$Note: The logarithms are with respect to base $e.$ -/
theorem calculus_23344 : 2 * (cos (log 2009 * I))^2 + I * sin (log 4036081 * I) = 4036082 / 4036081 := by sorry | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | 081 * I) = 2 * (cos (log 2009 * I))^2 + I * sin (log 4036081 * I) := by
simp [Complex.sin_mul, Complex.cos_mul, Complex.exp_mul, Complex.log_exp]
ring
| import Mathlib
open Real Set
open scoped BigOperators
/- Simplify $$2\cos^2(\log(2009)i)+i\sin(\log(4036081)i).$$Note: The logarithms are with respect to base $e.$ -/
theorem calculus_23344 : 2 * (cos (log 2009 * I))^2 + I * sin (log 4036 | import Mathlib
open Real Set
open scoped BigOperators
/- Simplify $$2\cos^2(\log(2009)i)+i\sin(\log(4036081)i).$$Note: The logarithms are with respect to base $e.$ -/
theorem calculus_23344 : 2 * (cos (log 2009 * I))^2 + I * sin (log 4036081 * I) = 2 * (cos (log 2009 * I))^2 + I * sin (log 4036081 * I) := by
simp [... |
15 | math-word-problem | autoformalizer | 3. Find the greatest and least values of the function $y=x^{3}-3 x^{2}+5$ on the interval $[-1 ; 1]$. | import Mathlib
open Real Set
open scoped BigOperators
/- 3. Find the greatest and least values of the function $y=x^{3}-3 x^{2}+5$ on the interval $[-1 ; 1]$. -/
theorem calculus_186972 (f : β β β) (hf : f = Ξ» x => x^3 - 3*x^2 + 5)
(I : Set β) (hI : I = Icc (-1) 1) :
IsGreatest (image f I) 5 β§ IsLeast (image f I)... | import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_186972 (f : β β β) (hf : f = Ξ» x => x^3 - 3*x^2 + 5)
(I : Set β) (hI : I = Icc (-1) 1) :
IsGreatest (image f I) 5 β§ IsLeast (image f I) 1 := by
constructor
Β· -- Show 5 is greatest in image
constructor
Β· -- First show 5 is achievabl... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | ] at hy
simp only [Set.mem_image, exists_prop] at hy
simp only [Set.mem_image, exists_prop] at hy
simp only [Set.mem_image, exists_prop] at hy
simp only [Set.mem_image, exists_prop] at hy
simp only [Set.mem_image, exists_prop] at hy
simp only [Set.mem_image, exists_prop] at hy
... | import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_186972 (f : β β β) (hf : f = Ξ» x => x^3 - 3*x^2 + 5)
(I : Set β) (hI : I = Icc (-1) 1) :
IsGreatest (image f I) 5 β§ IsLeast (image f I) 1 := by
constructor
Β· -- Show 5 is greatest in image
constructor
Β· -- First show 5 is achievabl... | import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_186972 (f : β β β) (hf : f = Ξ» x => x^3 - 3*x^2 + 5)
(I : Set β) (hI : I = Icc (-1) 1) :
IsGreatest (image f I) 5 β§ IsLeast (image f I) 1 := by
constructor
Β· -- Show 5 is greatest in image
constructor
Β· -- First show 5 is achievabl... |
17 | null | human | A class of 10 students took a math test. Each problem was solved by exactly 7 of the students. If the first nine students each solved 4 problems, how many problems did the tenth student solve? | import Mathlib
theorem numbertheory_611213
(p : β)
(n : β)
(h_total_solutions : 7 * p = 9 * 4 + n)
(h_score_bound : n β€ p)
: n = 6 := by | import Mathlib
/-!
## Problem Statement:
A class of 10 students took a math test. Each problem was solved by exactly 7
of the students. If the first nine students each solved 4 problems, how many
problems did the tenth student solve?
## Solution Argument Provided:
Suppose the last student solved $n$ problems, and th... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | p : β)
-- Let n be the number of problems solved by the tenth student (a natural number).
(n : β)
-- The first nine students solved 4 problems each.
(h1 : 9 * 4 = 36)
-- The tenth student solved n problems.
(h2 : n β€ p)
-- The total number of correct solutions written is 7p.
(h3 ... | import Mathlib
/-!
## Problem Statement:
A class of 10 students took a math test. Each problem was solved by exactly 7
of the students. If the first nine students each solved 4 problems, how many
problems did the tenth student solve?
## Solution Argument Provided:
Suppose the last student solved $n$ problems, and th... | import Mathlib
/-!
## Problem Statement:
A class of 10 students took a math test. Each problem was solved by exactly 7
of the students. If the first nine students each solved 4 problems, how many
problems did the tenth student solve?
## Solution Argument Provided:
Suppose the last student solved $n$ problems, and th... |
18 | null | human | In a game show, Bob is faced with 7 doors, 2 of which hide prizes. After he chooses a door, the host opens three other doors, of which one is hiding a prize. Bob chooses to switch to another door. What is the probability that his new door is hiding a prize? | import Mathlib
theorem Combinatorics_608318
(total_doors prize_doors host_opens prizes_opened : β)
(h_total : total_doors = 7)
(h_prize : prize_doors = 2)
(h_opens : host_opens = 3)
(h_prizes_opened : prizes_opened = 1)
(switch_success : P = ((total_doors - prize_doors) : β) / (total_doors : β) ... | import Mathlib
/-
Problem :
This is a variant of the classic Monty Hall problem.
-/
theorem Combinatorics_608318
(total_doors prize_doors host_opens prizes_opened : β)
(h_total : total_doors = 7) -- Total number of doors
(h_prize : prize_doors = 2) -- Number of doors hiding prizes
... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | _total, h_prize, h_opens, h_prizes_opened] at switch_success
-- Normalize the fractions
norm_num at switch_success
-- Use the fact that 5/21 is the probability of switching to the prize door
linarith
| import Mathlib
/-
Problem :
This is a variant of the classic Monty Hall problem.
-/
theorem Combinatorics_608318
(total_doors prize_doors host_opens prizes_opened : β)
(h_total : total_doors = 7) -- Total number of doors
(h_prize : prize_doors = 2) -- Number of doors hiding prizes
... | import Mathlib
/-
Problem :
This is a variant of the classic Monty Hall problem.
-/
theorem Combinatorics_608318
(total_doors prize_doors host_opens prizes_opened : β)
(h_total : total_doors = 7) -- Total number of doors
(h_prize : prize_doors = 2) -- Number of doors hiding prizes
... |
19 | null | human | How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \leq a+b+c+d+e \leq 10$ ? | import Mathlib
abbrev value_range : Finset β := Finset.Icc 1 6
def Ξ© : Finset (β Γ β Γ β Γ β Γ β) :=
Finset.product value_range <|
Finset.product value_range <|
Finset.product value_range <|
Finset.product value_range value_range
def to_list (t : β Γ β Γ β Γ β Γ β) : List β :=
[t.1, t.2.1, t.2.2.1, t.2.2.2.1,... | import Mathlib
/-How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \leq a+b+c+d+e \leq 10$ ?-/
-- Define the finite range for each integer in the sequence.
abbrev value_range : Finset β := Finset.Icc 1 6
-- The sample space Ξ© is the 5-fold cartesian product of this range.
-- This creates... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | β) :=
Ξ©.filter fun t =>
let l := to_list t
l.prod β€ l.sum β§ l.sum β€ 10
-- The number of sequences in `E`.
| import Mathlib
/-How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \leq a+b+c+d+e \leq 10$ ?-/
-- Define the finite range for each integer in the sequence.
abbrev value_range : Finset β := Finset.Icc 1 6
-- The sample space Ξ© is the 5-fold cartesian product of this range.
-- This creates... | import Mathlib
/-How many sequences of 5 positive integers $(a, b, c, d, e)$ satisfy $a b c d e \leq a+b+c+d+e \leq 10$ ?-/
-- Define the finite range for each integer in the sequence.
abbrev value_range : Finset β := Finset.Icc 1 6
-- The sample space Ξ© is the 5-fold cartesian product of this range.
-- This creates... |
20 | null | human | How many ways can one tile a $2 \times 8$ board with $1 \times 1$ and $2 \times 2$ tiles? Rotations and reflections of the same configuration are considered distinct. | import Mathlib
def f (n : β) : β := match n with
| 0 => 1
| 1 => 1
| n + 2 => f n + f (n + 1)
theorem Combinatorics_610027: f 8 = 34 := by | import Mathlib
-- Let $f(n)$ denote the number of ways to fill a $2 \times n$ board. One can fill the leftmost column with two $1 \times 1$ tiles, leaving $f(n-1)$ ways, or one can fill the leftmost two columns with one $2 \times 2$ tile, leaving $f(n-2)$ ways. Therefore, $f(n)=f(n-1)+f(n-2)$
def f (n : β) : β := matc... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | s and reflections of the same configuration are considered distinct.-/
| import Mathlib
-- Let $f(n)$ denote the number of ways to fill a $2 \times n$ board. One can fill the leftmost column with two $1 \times 1$ tiles, leaving $f(n-1)$ ways, or one can fill the leftmost two columns with one $2 \times 2$ tile, leaving $f(n-2)$ ways. Therefore, $f(n)=f(n-1)+f(n-2)$
def f (n : β) : β := matc... | import Mathlib
-- Let $f(n)$ denote the number of ways to fill a $2 \times n$ board. One can fill the leftmost column with two $1 \times 1$ tiles, leaving $f(n-1)$ ways, or one can fill the leftmost two columns with one $2 \times 2$ tile, leaving $f(n-2)$ ways. Therefore, $f(n)=f(n-1)+f(n-2)$
def f (n : β) : β := matc... |
21 | null | human | Boris was given a Connect Four game set for his birthday, but his color-blindness makes it hard to play the game. Still, he enjoys the shapes he can make by dropping checkers into the set. If the number of shapes possible modulo (horizontal) flips about the vertical axis of symmetry is expressed as $9(1+2+\cdots+n)$, f... | import Mathlib
open Nat
abbrev total_shapes := 9^7
abbrev symmetric_shapes := 9^4
abbrev non_symmetric_shapes := total_shapes - symmetric_shapes
abbrev non_symmetric_shapes_mod_flips := (total_shapes - symmetric_shapes) / 2
theorem combinatorics_610842 (n : β)
(h :
symmetric_shapes + non_symmetric_shapes_mod_... | import Mathlib
open Nat
-- The number of checkers in a column can be 0 to 8, so 9 possibilities.
-- Total shapes: 7 columns, so 9^7.
abbrev total_shapes := 9^7
-- Symmetric shapes: Middle column (4th) can be any of 9.
-- Left 3 columns determine right 3 columns. So 9^3 for left, 9^1 for middle. Thus 9^4.
abbrev symme... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | (n+1)) % 2 = 0 := by
rw [Nat.mul_mod]; have : (n * (n+1)) % 2 = 0 := by
have : n % 2 = 0 β¨ n % 2 = 1 := by omega
cases' this with h h <;> simp [h, Nat.mul_mod, Nat.add_mod, Nat.pow_mod]
rw [this, mul_zero]
norm_num
omega
-- From $9^4 \cdot (9^3 + 1) = 9 \cdot n \cdot (n+1)$, de... | import Mathlib
open Nat
-- The number of checkers in a column can be 0 to 8, so 9 possibilities.
-- Total shapes: 7 columns, so 9^7.
abbrev total_shapes := 9^7
-- Symmetric shapes: Middle column (4th) can be any of 9.
-- Left 3 columns determine right 3 columns. So 9^3 for left, 9^1 for middle. Thus 9^4.
abbrev symme... | import Mathlib
open Nat
-- The number of checkers in a column can be 0 to 8, so 9 possibilities.
-- Total shapes: 7 columns, so 9^7.
abbrev total_shapes := 9^7
-- Symmetric shapes: Middle column (4th) can be any of 9.
-- Left 3 columns determine right 3 columns. So 9^3 for left, 9^1 for middle. Thus 9^4.
abbrev symme... |
23 | null | human | In the country of Francisca, there are 2010 cities, some of which are connected by roads. Between any two cities, there is a unique path which runs along the roads and which does not pass through any city twice. What is the maximum possible number of cities in Francisca which have at least 3 roads running out of them? | import Mathlib
theorem combinatorics_608689
(n : β)
(hn : n = 2010)
(b : β)
(hb : b β€ n)
(h : 3 * b + (n - b) β€ 2 * (n - 1)) :
b β€ 1004 := by | import Mathlib
/-
The problem describes a tree structure with 2010 nodes (cities), where each node represents a city and each edge represents a road. The goal is to determine the maximum number of nodes with a degree of at least 3.
-/
theorem combinatorics_608689
(n : β)
(hn : n = 2010)
(b : β)
(hb ... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | n - b) β€ n) :
3 * b + (n - b) β€ n := by
rw [hn] at h
linarith
| import Mathlib
/-
The problem describes a tree structure with 2010 nodes (cities), where each node represents a city and each edge represents a road. The goal is to determine the maximum number of nodes with a degree of at least 3.
-/
theorem combinatorics_608689
(n : β)
(hn : n = 2010)
(b : β)
(hb ... | import Mathlib
/-
The problem describes a tree structure with 2010 nodes (cities), where each node represents a city and each edge represents a road. The goal is to determine the maximum number of nodes with a degree of at least 3.
-/
theorem combinatorics_608689
(n : β)
(hn : n = 2010)
(b : β)
(hb ... |
24 | null | human | The altitudes of a triangle are 12,15 and 20. What is the area of the triangle?
Answer: 150. | import Mathlib
noncomputable def S (a b c: β) :=
let s := (a + b + c) / 2
β(s * (s - a) * (s - b) * (s - c))
theorem geometry_606364 (a b c h_a h_b h_c : β) (hβ : 0 < a β§ 0 < b β§ 0 < c)
(hβ : h_a = 12) (hβ : h_b = 15) (hβ : h_c = 20) (hβ : a / b = h_b / h_a)
(hβ
: b / c = h_c / h_b) (hβ: S a b c = a * h_a /... | import Mathlib
noncomputable def S (a b c: β) :=
let s:= (a + b + c) / 2
β(s * (s - a) * (s - b) * (s - c))
/-The altitudes of a triangle are 12,15 and 20. What is the area of the triangle?
Answer: 150.-/
theorem geometry_606364 (a b c h_a h_b h_c : β) (hβ : 0 < a β§ 0 < b β§ 0 < c)
(hβ : h_a = 12) (hβ : h_b =... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | linarith
simp [hβ
, hβ, hβ
, hβ] at hβ
ring_nf at hβ
linarith | import Mathlib
noncomputable def S (a b c: β) :=
let s:= (a + b + c) / 2
β(s * (s - a) * (s - b) * (s - c))
/-The altitudes of a triangle are 12,15 and 20. What is the area of the triangle?
Answer: 150.-/
theorem geometry_606364 (a b c h_a h_b h_c : β) (hβ : 0 < a β§ 0 < b β§ 0 < c)
(hβ : h_a = 12) (hβ : h_b =... | import Mathlib
noncomputable def S (a b c: β) :=
let s:= (a + b + c) / 2
β(s * (s - a) * (s - b) * (s - c))
/-The altitudes of a triangle are 12,15 and 20. What is the area of the triangle?
Answer: 150.-/
theorem geometry_606364 (a b c h_a h_b h_c : β) (hβ : 0 < a β§ 0 < b β§ 0 < c)
(hβ : h_a = 12) (hβ : h_b =... |
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