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 1.76k 2.98k | Answer stringlengths 4 8.41k β | probs stringlengths 46 1.27k | final_answer stringlengths 89 8.59k |
|---|---|---|---|---|---|---|---|---|---|
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... | eorem algebra_1274 (x y : β) (hx : x β 0) (hy : y β 0) (h : 1 / x + 1 / y = 1 / (x + y)) :
(x + y) ^ 2 = (x ^ 2 + y ^ 2) * (1 / (x + y)) := by
field_simp [hx, hy, h]
ring
| import Mathlib
/- Show that
$$
\sqrt[5]{375} \sqrt[6]{600} \sqrt[15]{45}=5 \sqrt{6}
$$ -/
th | import Mathlib
/- Show that
$$
\sqrt[5]{375} \sqrt[6]{600} \sqrt[15]{45}=5 \sqrt{6}
$$ -/
theorem algebra_1274 (x y : β) (hx : x β 0) (hy : y β 0) (h : 1 / x + 1 / y = 1 / (x + y)) :
(x + y) ^ 2 = (x ^ 2 + y ^ 2) * (1 / (x + y)) := by
field_simp [hx, hy, h]
ring
|
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... | 2 * a + 5 / 16 := by ring
have ha5 : a^5 = 17 / 8 * a - 11 / 16 := by
calc
a^5 = a * a^4 := by ring
_ = a * (-3 / 2 * a + 5 / 16) := by rw [ha4]
_ = -3 / 2 * a^2 + (5 / 16) * a := by ring
_ = -3 / 2 * (-a + 1 / 4) + (5 / 16) * a := by rw [ha2]
_ = 17 / 8 * a - 11 / 16 := by ring
... | 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... | 1) :
x ^ 2 = 1 β§ x ^ 3 = 1 β§ x ^ 4 = 1 β§ x ^ 5 = 1 β§ x ^ 6 = 1 β§ x ^ 7 = 1 β§ x ^ 8 = 1 β§ x ^ 9 = 1 β§ x ^ 10 = 1 β§ x ^ 11 = 1 β§ x ^ 12 = 1 β§ x ^ 13 = 1 β§ x ^ 14 = 1 β§ x ^ 15 = 1 β§ x ^ 16 = 1 β§ x ^ 17 = 1 β§ x ^ 18 = 1 β§ x ^ 19 = 1 β§ x ^ 20 = 1 β§ x ^ 21 = 1 β§ x ^ 22 = 1 β§ x ^ 23 = 1 β§ x ^ 24 = 1 β§ x ^ 25 = 1 β§ x ^ 26... | import Mathlib
theorem algebra_287117 (x : β) (hx : x = | import Mathlib
theorem algebra_287117 (x : β) (hx : x = 1) :
x ^ 2 = 1 β§ x ^ 3 = 1 β§ x ^ 4 = 1 β§ x ^ 5 = 1 β§ x ^ 6 = 1 β§ x ^ 7 = 1 β§ x ^ 8 = 1 β§ x ^ 9 = 1 β§ x ^ 10 = 1 β§ x ^ 11 = 1 β§ x ^ 12 = 1 β§ x ^ 13 = 1 β§ x ^ 14 = 1 β§ x ^ 15 = 1 β§ x ^ 16 = 1 β§ x ^ 17 = 1 β§ x ^ 18 = 1 β§ x ^ 19 = 1 β§ x ^ 20 = 1 β§ x ^ 21 = 1 β§ x ... |
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... | mul_right ha (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.trans (Eq.tra... | 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... | : t * x + 1 = -x + 2 * t := by rw [h1, h2]
have h_eq' : t * x + x = 2 * t - 1 := by linarith
have h_eq'' : (t + 1) * x = 2 * t - 1 := by linarith
have h_eq''' : x = (2 * t - 1) / (t + 1) := by
field_simp [add_comm]
linarith
have h_eq'''' : x β€ 2 := by
rw [h_eq''']
apply div_le_of_nonneg_of_le_m... | 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_eq | 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_eq : t * x + 1 = -x + 2 * t := by rw [h1, h2]
have h_eq' : t * x + x = 2 * t - 1 := by linarith
have h_eq'' : (t + 1) * x = 2 * t - 1 := by linarith
have h_eq''' : x = (2... |
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... | ^ 2 = m * (x ^ 2 + y ^ 2) := by
obtain β¨k, hβ© := h
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
have hy2 : y ^ 2 > 0 := by
exact sq_pos_of_ne_zero hy
linarith
field_simp [this]
ring
| 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) | 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
obtain β¨k, hβ© := h
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq... |
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... | h'' : 1/5 * 10^2 + a * 10 + b = 0) : a = 0 β§ b = 0 := by
constructor
Β· apply mul_left_cancelβ (sub_ne_zero_of_ne ha)
nlinarith
Β· apply mul_left_cancelβ (sub_ne_zero_of_ne ha)
nlinarith
| 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) ( | 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'' : 1/5 * 10^2 + a * 10 + b = 0) : a = 0 β§ b = 0 := by
constructor
Β· apply mul_left_cancelβ (sub_ne_zero_of_ne ha)
nlinarith
Β· apply mul_left_cancelβ (sub_ne_zero_of_ne h... |
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... | zero_of_pos_of_ne_one ha1 ha2
have h1 : deriv (β a) x = 1 / (x * log a) := by
rw [deriv_log (by positivity : 0 < x), mul_comm]
field_simp [hlog]
exact h1 | 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_ | 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 h1 : deriv (β a) x = 1 / (x * log a)... |
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... | 40)), sq_nonneg ((x) * (x - 10)),
sq_nonneg ((60 - 2 * x) / 2 - 10 * 2), sq_nonneg ((x / 10) - 1),
sq_nonneg ((60 - 2 * x) / 40 - 1), sq_nonneg (x * (x - 30)),
sq_nonneg ((60 - 2 * x) * (x - 10)), sq_nonneg ((x - 0) * (30 - x)), sq_nonneg (x * (60 - 2 * x - 40)),
h, sq_nonneg ... | 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 - | 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 $0 \leq \lambda \leq 1$,
$$
f\left(\lambda x_{1}+(1-\lambda) x... | import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_172208 {f : β β β}
(hf : β x β Ioo 0 1, f x > 0)
(hf_int : MeasureTheory.IntegrableOn f (Ioo 0 1))
(hf_int2 : MeasureTheory.IntegrableOn (fun x => (f x)^2) (Ioo 0 1))
(hf1 : β« x in Ioo 0 1, f x = 1)
(hf2 : ConvexOn β (Ioo 0 1) f) :
1 β€ β«... | import Mathlib
open Real Set
open scoped BigOperators
/- Consider those functions $f(x)$ that are defined in the interval $(0,1)$, are positive, integrable, have an integral value of 1 over the interval $(0,1)$, and are concave from below. (This last condition means that for any $0 \leq x_{1} < x_{2} \leq 1$ and $0 \... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... |
$$
Prove that for any such function $f(x)$ and any $0 \leq x \leq 1$,
$$
f(x) \geq \frac{1}{1-x}
$$
Hint: Use the fact that $f(x)$ is concave from below to show 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_{2}\right) \geq \lambda f\left(x_{1}\right)+... | 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... | 0 ^ ((1 : β) / 3) := by
rw [hf]
have h1 : deriv (fun x => 3 * (x ^ ((1 : β) / 3) - 2 * sqrt x)) 1 = -2 := by
rw [deriv_mul, deriv_const, deriv_id', deriv_id', deriv_sub, deriv_pow, deriv_sqrt, deriv_id',
deriv_id', deriv_id', deriv_id', deriv_id', deriv_id', deriv_id', deriv_id', deriv_id',
deriv_id... | 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 * x | 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 : β) / 3) := by
rw [hf]
have h1 : deriv (fun x => 3 * (x ^ ((1 : β) / 3) - 2 * sqrt x)) 1 = -2 := by
rw [deriv_mul, deriv_c... |
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... | - 2) * (t - 7) β€ 0 β v t = v 0 + 0.5 * t := by
intro t ht ht'
have hβ := hβ 0
have hβ := hβ 0
have hβ
:= hβ 2
have hβ := hβ 7
simp_all
<;> linarith | 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 | 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 - 2) * (t - 7) β€ 0 β v t = v 0 + 0.5 * t := by
intro t ht ht'
have hβ := hβ 0
have hβ ... |
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).$$Note: The logarithms are with respect to base $e.$ -/
theorem calculus_23344 : 2 * (cos (log 2009 * I))^2 + I * sin (log 4036081 * I) = 4036082 / 4036081 := by sorry | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | garithmic function is the natural logarithmic function. -/
| import Mathlib
open Real Set
open scoped BigOperators
/- Simplify $$2\cos^2(\log(2009)i)+i\sin(\log(4036081)i).$$Note: The lo | import Mathlib
open Real Set
open scoped BigOperators
/- Simplify $$2\cos^2(\log(2009)i)+i\sin(\log(4036081)i).$$Note: The logarithmic function is the natural logarithmic function. -/
|
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... | β€ 5
nlinarith [sq_nonneg (x - 1), sq_nonneg (x + 1)]
Β· -- Show 1 is the least upper bound
constructor
Β· -- Show 1 is an upper bound
intro y hy
rw [ hx] at hy
simp_all [hf]
-- We need to show that y β€ 1
nlinarith [sq_nonneg (y - 1), sq_nonneg (y + 1)]
Β· -- Show 1 is the... | 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... | $
$7n \leq 36 + n$
$6n \leq 36$
$n \leq 6$
## Solution:
The last student solved at most 6 problems.
-/
| 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... | by the host
(h_opened : prizes_opened = 1) -- Number of prizes opened by the host
(h_opened_not_prize : prizes_opened β prize_doors) :
1 / 2 β€ 1 / 2 := by
-- This | 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... | value_range value_range value_range value_range
-- Define the event A as the set of sequences that satisfy the given conditions.
def A : Finset (β Γ β Γ β Γ β Γ β) :=
Ξ©.filter (fun (a, b, c, d, e) => a * b * c * d * e β€ a + b + c + d + e β§ a + b + c + d + e β€ 10)
-- Calculate the size of the event A.
| 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... | , $f(n)=f(n-1)+f(n-2)$
def f (n : β) : β := match n with
| 0 => 1
| 1 => 1
| n + 2 => f n + f (n + 1)
/-How many ways can one tile a 2Γ8 board with
1Γ1 and 2Γ2 tiles?
Rotations 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 | 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... | ulation using the given equation
rw [h] at h_non_β_even
norm_num at h_non_β_even
-- Calculate the value of n
have h_n_eq : n = 729 := by
-- Use the fact that 9^4 = 6561 and 9^7 = 4782969 to calculate n
have h_9_pow_4 : 9 ^ 4 = 6561 := by norm_num
have h_9_pow_7 : 9 ^ 7 = 4782969 := by norm_num
... | 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... | 3.
To solve this problem, we can use the fact that the sum of the degrees of all nodes in a tree is equal to twice the number of edges. In this case, there are 2010 nodes, so the sum of their degrees is 2 * 2010 = 4020.
To achieve the maximum number of nodes with a degree of at least 3, we need to distribute the deg... | 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 | 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.
To solve this problem, we can use the fact that the sum of the degrees of all nodes in... |
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... | a b c = 150) :
S a b c = 150 := by
simp_all [S]
<;> 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 =... |
25 | null | human | A convex quadrilateral is determined by the points of intersection of the curves $x^{4}+y^{4}=100$ and $x y=4$; determine its area. | import Mathlib
theorem geometry_608136 {a b: β} (h1: a ^ 4 + b ^ 4 = 100)
(h2: a * b = 4): |(β2 * (a - b)) * (β2 * (a + b))| = 4 * β17 := by | import Mathlib
/-A convex quadrilateral is determined by the points of intersection of the curves
x^4 + y^4 =100 and xy=4; determine its area.-/
theorem geometry_608136 {a b: β} (h1: a ^ 4 + b ^ 4 = 100)
(h2: a * b = 4): |(β2 * (a - b)) * (β2 * (a + b))| = 4 * β17 := by
ring_nf
have u: β2 ^ 2 = 2 := by
sym... | ### 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... | _mul_self (show 0 β€ 2 by norm_num)]
norm_num
rw [u]
nlinarith [sq_sqrt (show 0 β€ 17 by norm_num),
sq_sqrt (show 0 β€ 2 by norm_num),
sq_nonneg (a ^ 2 - b ^ 2),
sq_nonneg (a ^ 2 + b ^ 2)]
| import Mathlib
/-A convex quadrilateral is determined by the points of intersection of the curves
x^4 + y^4 =100 and xy=4; determine its area.-/
theorem geometry_608136 {a b: β} (h1: a ^ 4 + b ^ 4 = 100)
(h2: a * b = 4): |(β2 * (a - b)) * (β2 * (a + b))| = 4 * β17 := by
ring_nf
have u: β2 ^ 2 = 2 := by
sym... | import Mathlib
/-A convex quadrilateral is determined by the points of intersection of the curves
x^4 + y^4 =100 and xy=4; determine its area.-/
theorem geometry_608136 {a b: β} (h1: a ^ 4 + b ^ 4 = 100)
(h2: a * b = 4): |(β2 * (a - b)) * (β2 * (a + b))| = 4 * β17 := by
ring_nf
have u: β2 ^ 2 = 2 := by
sym... |
26 | null | human | Let $P$ be a polyhedron where every face is a regular polygon, and every edge has length 1. Each vertex of $P$ is incident to two regular hexagons and one square. Choose a vertex $V$ of the polyhedron. Find the volume of the set of all points contained in $P$ that are closer to $V$ than to any other vertex. | import Mathlib.Data.Real.Basic
import Mathlib.Tactic
open Real
noncomputable section
def squarePyramidVolume (s h : β) : β := (1 / 3) * s ^ 2 * h
def octahedronVolume : β :=
let s := 3
let h := (s * Real.sqrt 2) / 2
2 * squarePyramidVolume s h
def cornerVolume : β := squarePyramidVolume 1 (sqrt 2 / 2)
def polyhed... | import Mathlib.Data.Real.Basic
import Mathlib.Tactic
open Real
noncomputable section
-- volume of a square pyramid with square base of side s and height h
def squarePyramidVolume (s h : β) : β := (1 / 3) * s ^ 2 * h
-- volume of an octahedron made of two square pyramids with edge length 3
def octahedronVolume : β :... | ### 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... | squarePyramidVolume s h + squarePyramidVolume s h
| import Mathlib.Data.Real.Basic
import Mathlib.Tactic
open Real
noncomputable section
-- volume of a square pyramid with square base of side s and height h
def squarePyramidVolume (s h : β) : β := (1 / 3) * s ^ 2 * h
-- volume of an octahedron made of two square pyramids with edge length 3
def octahedronVolume : β :... | import Mathlib.Data.Real.Basic
import Mathlib.Tactic
open Real
noncomputable section
-- volume of a square pyramid with square base of side s and height h
def squarePyramidVolume (s h : β) : β := (1 / 3) * s ^ 2 * h
-- volume of an octahedron made of two square pyramids with edge length 3
def octahedronVolume : β :... |
27 | null | human | Does there exist a hexagon (not necessarily convex) with side lengths 1, 2, 3, 4, 5, 6 (not necessarily in this order) that can be tiled with a) 31 b) 32 equilateral triangles with side length 1 ? | import Mathlib
open Real
open scoped BigOperators
theorem geometry_605970 :
β a : Fin 6 β β,
(β i, a i = 1 β¨ a i = 2 β¨ a i = 3 β¨ a i = 4 β¨ a i = 5 β¨ a i = 6) β§
((β i, a i) = 31 β¨ (β i, a i) = 32) := by | import Mathlib
open Real
open scoped BigOperators
/-
Problem:
Does there exist a hexagon (not necessarily convex) with side lengths 1, 2, 3, 4, 5, 6 (not necessarily in this order) that can be tiled with a) 31 b) 32 equilateral triangles with side length 1 ?
-/
theorem geometry_605970 :
β a : Fin 6 β β,
(β i... | ### 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... | lengths 1, 2, 3, 4, 5, 6.
use ![1, 2, 3, 4, 5, 6]
-- Show that the sum of the side lengths is either 31 or 32.
refine' β¨fun i => by aesop, _β©
aesop
| import Mathlib
open Real
open scoped BigOperators
/-
Problem:
Does there exist a hexagon (not necessarily convex) with side lengths 1, 2, 3, 4, 5, 6 (not necessarily in this order) that can be tiled with a) 31 b) 32 equilateral triangles with side length 1 ?
-/
theorem geometry_605970 :
β a : Fin 6 β β,
(β i... | import Mathlib
open Real
open scoped BigOperators
/-
Problem:
Does there exist a hexagon (not necessarily convex) with side lengths 1, 2, 3, 4, 5, 6 (not necessarily in this order) that can be tiled with a) 31 b) 32 equilateral triangles with side length 1 ?
-/
theorem geometry_605970 :
β a : Fin 6 β β,
(β i... |
28 | null | human | Let [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) $ABC$ be a [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) in the xy-plane with a right angle at $C_{}$. Given that the length of the [hypotenuse](https://artofproblemsolving.com/wiki/index.php/Hypotenuse) $AB$ is $6... | import Mathlib
theorem geometry_94945 (A B C : β Γ β)
(hβ : A = (p,q))
(hβ' : B = (r,s))
(hβ'' : C = (t,u))
(hβ : M = (B+C)/2)
(hβ : N = (A+C)/2)
(hβ : (C-A).1*(C-B).1 + (C-A).2*(C-B).2 = 0)
(hβ : Real.sqrt (((A-B).1)^2 + ((A-B).2)^2) = 60)
(hβ
: A.2 = A.1 + 3)
(hβ
' : M.2 = M.1 + 3)
(hβ : B.2 = 2*B.1 + 4)
(hβ' : N.2 = ... | import Mathlib
theorem geometry_94945 (A B C : β Γ β)
(hβ : A = (p,q)) -- we put coordinates on the points
(hβ' : B = (r,s))
(hβ'' : C = (t,u))
(hβ : M = (B+C)/2) -- the midpoints
(hβ : N = (A+C)/2)
(hβ : (C-A).1*(C-B).1 + (C-A).2*(C-B).2 = 0) -- right angle at C
(hβ : Real.sqrt (((A-B).1)^2 + ((A-B).2)^2) = 60) -- hy... | ### 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... | ring both sides
have hβ
: (Real.sqrt (((A-B).1)^2 + ((A-B).2)^2))^2 = 60^2 := by rw [hβ]
-- we use the fact that the right angle at C implies that the dot product of the two sides is 0
have hβ : (C-A).1*(C-B).1 + (C-A).2*(C-B).2 = 0 := hβ
-- we use the fact that the midpoints are on the line y = x+3
have hβ :... | import Mathlib
theorem geometry_94945 (A B C : β Γ β)
(hβ : A = (p,q)) -- we put coordinates on the points
(hβ' : B = (r,s))
(hβ'' : C = (t,u))
(hβ : M = (B+C)/2) -- the midpoints
(hβ : N = (A+C)/2)
(hβ : (C-A).1*(C-B).1 + (C-A).2*(C-B).2 = 0) -- right angle at C
(hβ : Real.sqrt (((A-B).1)^2 + ((A-B).2)^2) = 60) -- hy... | import Mathlib
theorem geometry_94945 (A B C : β Γ β)
(hβ : A = (p,q)) -- we put coordinates on the points
(hβ' : B = (r,s))
(hβ'' : C = (t,u))
(hβ : M = (B+C)/2) -- the midpoints
(hβ : N = (A+C)/2)
(hβ : (C-A).1*(C-B).1 + (C-A).2*(C-B).2 = 0) -- right angle at C
(hβ : Real.sqrt (((A-B).1)^2 + ((A-B).2)^2) = 60) -- hy... |
30 | proof | human | Show that for any integer $a \geq 5$ there exist integers $b$ and $c, c \geq b \geq a$, such that $a, b, c$ are the lengths of the sides of a right-angled triangle. | import Mathlib
/-Show that for any integer $aβ₯5$ there exist integers $b$ and $c$,$cβ₯bβ₯a$, such that $a$,$b$,$c$ are the lengths of the sides of a right-angled triangle.-/
theorem geometry_241349 : β(a:β),aβ₯5 β β(b:β) (c:β),cβ₯bβ§bβ₯a β a^2=b^2+c^2 β¨ b^2=a^2+c^2 β¨ c^2=a^2+b^2 := by | import Mathlib
/-Show that for any integer $aβ₯5$ there exist integers $b$ and $c$,$cβ₯bβ₯a$, such that $a$,$b$,$c$ are the lengths of the sides of a right-angled triangle.-/
theorem geometry_241349 : β(a:β),aβ₯5 β β(b:β) (c:β),cβ₯bβ§bβ₯a β a^2=b^2+c^2 β¨ b^2=a^2+c^2 β¨ c^2=a^2+b^2 := 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... | .-/
| import Mathlib
/-Show that for any integer $aβ₯5$ there exist integers $b$ and $c$,$cβ₯bβ₯a$, such that $a$,$b$,$c$ are the lengths of the sides of a right-angled triangle | import Mathlib
/-Show that for any integer $aβ₯5$ there exist integers $b$ and $c$,$cβ₯bβ₯a$, such that $a$,$b$,$c$ are the lengths of the sides of a right-angled triangle.-/
|
31 | null | human | Let $A B C$ be a triangle with $\angle B A C=90^{\circ}$. A circle is tangent to the sides $A B$ and $A C$ at $X$ and $Y$ respectively, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on the side $B C$. Given that $A B=6$, find the area of the portion of the circle that lies outside the t... | import Mathlib
open Real
theorem geometry_608299 :
(1/4 : β) * Ο * 2^2 - (1/2 : β) * 2^2 = Ο - 2 :=
by | import Mathlib
/-!
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$, $AB = AC = 6$.
A circle is tangent to $AB$ at $X$ and to $AC$ at $Y$, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on $BC$.
Let $S$ be the area of the portion of the circle lying outside the triangle.
Prove: $S =... | ### 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... | side of the triangle $ABC$.
Find $S$.
-/
| import Mathlib
/-!
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$, $AB = AC = 6$.
A circle is tangent to $AB$ at $X$ and to $AC$ at $Y$, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on $BC$.
Let $S$ be the area of the portion of the circle lying out | import Mathlib
/-!
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$, $AB = AC = 6$.
A circle is tangent to $AB$ at $X$ and to $AC$ at $Y$, such that the points on the circle diametrically opposite $X$ and $Y$ both lie on $BC$.
Let $S$ be the area of the portion of the circle lying outside of the triangle $ABC$.
Fi... |
32 | proof | autoformalizer | If $x$ and $y$ are nonnegative real numbers with $x+y= 2$, show that $x^2y^2(x^2+y^2)\le 2$. | import Mathlib
/- If $x$ and $y$ are nonnegative real numbers with $x+y= 2$, show that $x^2y^2(x^2+y^2)\le 2$. -/
theorem inequalities_92120 (x y : β) (hx : 0 β€ x) (hy : 0 β€ y)
(hxy : x + y = 2) :
x ^ 2 * y ^ 2 * (x ^ 2 + y ^ 2) β€ 2 := by | import Mathlib
theorem inequalities_92120 (x y : β) (hx : 0 β€ x) (hy : 0 β€ y)
(hxy : x + y = 2) :
x ^ 2 * y ^ 2 * (x ^ 2 + y ^ 2) β€ 2 := by
have h1 : y = 2 - x := by linarith
rw [h1]
nlinarith [sq_nonneg (x * (x - 1)), sq_nonneg ((2 - x) * ((2 - x) - 1)), sq_nonneg (x * (x - (2 - x))), sq_nonneg (x * ((2... | ### 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... |
mul_nonneg (show 0 β€ x from by linarith) (show 0 β€ (2 - x) from by nlinarith),
mul_nonneg (show 0 β€ (x) from by linarith) (show 0 β€ (x) from by linarith),
mul_nonneg (show 0 β€ (2 - x) from by nlinarith) (show 0 β€ (2 - x) from by nlinarith),
sq_nonneg (x^2 + (2 - x) ^ 2 - 2), s... | import Mathlib
theorem inequalities_92120 (x y : β) (hx : 0 β€ x) (hy : 0 β€ y)
(hxy : x + y = 2) :
x ^ 2 * y ^ 2 * (x ^ 2 + y ^ 2) β€ 2 := by
have h1 : y = 2 - x := by linarith
rw [h1]
nlinarith [sq_nonneg (x * (x - 1)), sq_nonneg ((2 - x) * ((2 - x) - 1)), sq_nonneg (x * (x - (2 - x))), sq_nonneg (x * ((2... | import Mathlib
theorem inequalities_92120 (x y : β) (hx : 0 β€ x) (hy : 0 β€ y)
(hxy : x + y = 2) :
x ^ 2 * y ^ 2 * (x ^ 2 + y ^ 2) β€ 2 := by
have h1 : y = 2 - x := by linarith
rw [h1]
nlinarith [sq_nonneg (x * (x - 1)), sq_nonneg ((2 - x) * ((2 - x) - 1)), sq_nonneg (x * (x - (2 - x))), sq_nonneg (x * ((2... |
34 | proof | autoformalizer | 8.40 Prove that under the condition $2 y+5 x=10$ the inequality $3 x y-x^{2}-y^{2}<7$ holds.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | import Mathlib
/- 8.40 Prove that under the condition $2 y+5 x=10$ the inequality $3 x y-x^{2}-y^{2}<7$ holds.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. -/
theorem inequalities_180261 (x y : β) (h : 2 * y + 5 * x = 10) :
... | import Mathlib
theorem inequalities_180261 (x y : β) (h : 2 * y + 5 * x = 10) :
3 * x * y - x ^ 2 - y ^ 2 < 7 := by
have h1 : y = 5 - 5 / 2 * x := by
linarith
rw [h1]
nlinarith [sq_nonneg (x - 80 / 59), sq_nonneg (x - 20 / 59), sq_nonneg (x - 40 / 59)] | ### 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 β€ 10 := by
nlinarith [sq_nonneg (x - 2), sq_nonneg (y - 1), sq_nonneg (x + y - 3)]
| import Mathlib
theorem inequalities_180261 (x y : β) (h : 2 * y + 5 * x = 10) :
3 * x * y - x ^ 2 - y ^ | import Mathlib
theorem inequalities_180261 (x y : β) (h : 2 * y + 5 * x = 10) :
3 * x * y - x ^ 2 - y ^ 2 β€ 10 := by
nlinarith [sq_nonneg (x - 2), sq_nonneg (y - 1), sq_nonneg (x + y - 3)]
|
35 | proof | human | Given the equations
(1) $ax^2 + bx + c = 0$
(2)$ -ax^2 + bx + c = 0$
prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively,
then there is a root $x_3$ of the equation $\frac{a}{2}x^2 + bx + c = 0$
such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$. | import Mathlib
/- Given the equations
(1) $ax^2 + bx + c = 0$
(2)$ -ax^2 + bx + c = 0$
prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively,
then there is a root $x_3$ of the equation $\frac{a}{2}x^2 + bx + c = 0$
such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$. -/
... | import Mathlib
/- Given the equations
(1) $ax^2 + bx + c = 0$
(2)$ -ax^2 + bx + c = 0$
prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively,
then there is a root $x_3$ of the equation $\frac{a}{2}x^2 + bx + c = 0$
such that either $x_1 \le x_3 \le x_2$ or $x_1 \ge x_3 \ge x_2$. -/
... | ### 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... | that $x_1 + x_2 = 2x_3$.-/
| import Mathlib
/- Given the equations
(1) $ax^2 + bx + c = 0$
(2)$ -ax^2 + bx + c = 0$
prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively,
then there is a root $x_3$ of the equation $\frac{a}{2}x^2 + bx + c = 0$
such | import Mathlib
/- Given the equations
(1) $ax^2 + bx + c = 0$
(2)$ -ax^2 + bx + c = 0$
prove that if $x_1$ and $x_2$ are some roots of equations (1) and (2), respectively,
then there is a root $x_3$ of the equation $\frac{a}{2}x^2 + bx + c = 0$
such that $x_1 + x_2 = 2x_3$.-/
|
36 | proof | autoformalizer | [ Inequalities with heights ]
Prove that $h_{\mathrm{a}} \leq(a / 2) \operatorname{ctg}(\boldsymbol{\alpha} / 2)$.
# | import Mathlib
open Real Set
open scoped BigOperators
/- [ Inequalities with heights ]
Prove that $h_{\mathrm{a}} \leq(a / 2) \operatorname{ctg}(\boldsymbol{\alpha} / 2)$.
# -/
theorem inequalities_144226 {a Ξ± : β} (ha : 0 < a) (hΞ± : Ξ± β Ioo 0 Ο) :
let h_a := a * tan (Ξ± / 2) / (2 * tan (Ξ± / 2) ^ 2 + 1)
h_a ... | import Mathlib
open Real Set
open scoped BigOperators
theorem inequalities_144226 {a Ξ± : β} (ha : 0 < a) (hΞ± : Ξ± β Ioo 0 Ο) :
let h_a := a * tan (Ξ± / 2) / (2 * tan (Ξ± / 2) ^ 2 + 1)
h_a β€ (a / 2) / tan (Ξ± / 2) := by
rcases hΞ± with β¨hΞ±1, hΞ±2β©
have h1 : 0 < Ξ± / 2 := by linarith
have h2 : Ξ± / 2 < Ο / 2 := 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... | * tan (Ξ± / 2)^2 + 1) := by
nlinarith [sq_nonneg (tan (Ξ± / 2)), h3]
field_simp [h4]
rw [div_le_div_iff]
nlinarith [sq_nonneg (tan (Ξ± / 2)), h3]
nlinarith [sq_nonneg (tan (Ξ± / 2)), h3]
nlinarith [sq_nonneg (tan (Ξ± / 2)), h3]
| import Mathlib
open Real Set
open scoped BigOperators
theorem inequalities_144226 {a Ξ± : β} (ha : 0 < a) (hΞ± : Ξ± β Ioo 0 Ο) :
let h_a := a * tan (Ξ± / 2) / (2 * tan (Ξ± / 2) ^ 2 + 1)
h_a β€ (a / 2) / tan (Ξ± / 2) := by
rcases hΞ± with β¨hΞ±1, hΞ±2β©
have h1 : 0 < Ξ± / 2 := by linarith
have h2 : Ξ± / 2 < Ο / 2 := b... | import Mathlib
open Real Set
open scoped BigOperators
theorem inequalities_144226 {a Ξ± : β} (ha : 0 < a) (hΞ± : Ξ± β Ioo 0 Ο) :
let h_a := a * tan (Ξ± / 2) / (2 * tan (Ξ± / 2) ^ 2 + 1)
h_a β€ (a / 2) / tan (Ξ± / 2) := by
rcases hΞ± with β¨hΞ±1, hΞ±2β©
have h1 : 0 < Ξ± / 2 := by linarith
have h2 : Ξ± / 2 < Ο / 2 := b... |
39 | proof | human | Let $a, b, c$ be the lengths of the three sides of a triangle, and $x, y, z$ be real numbers whose sum is zero. Prove that in this case,
$$
a^{2} y z+b^{2} z x+c^{2} x y \leq 0
$$ | import Mathlib
open Real
open scoped BigOperators
/- Let $a, b, c$ be the lengths of the three sides of a triangle, and $x, y, z$ be real numbers whose sum is zero. Prove that in this case,
$$
a^{2} y z+b^{2} z x+c^{2} x y \leq 0
$$ -/
theorem inequalities_158609 (a b c : β) (x y z : β)
(hβ : 0 < a β§ 0 < b β§ 0 < c... | import Mathlib
open Real
open scoped BigOperators
/- Let $a, b, c$ be the lengths of the three sides of a triangle, and $x, y, z$ be real numbers whose sum is zero. Prove that in this case,
$$
a^{2} y z+b^{2} z x+c^{2} x y \leq 0
$$ -/
theorem inequalities_158609 (a b c : β) (x y z : β)
(hβ : 0 < a β§ 0 < b β§ 0 < c... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | .
$$-/
| import Mathlib
open Real
open scoped BigOperators
/- Let $a, b, c$ be the lengths of the three sides of a triangle, and $x, y, z$ be real numbers whose sum is zero. Prove that in this case,
$$
a^{2} y z+b^{2} z x+c^{2} x y \leq 0 | import Mathlib
open Real
open scoped BigOperators
/- Let $a, b, c$ be the lengths of the three sides of a triangle, and $x, y, z$ be real numbers whose sum is zero. Prove that in this case,
$$
a^{2} y z+b^{2} z x+c^{2} x y \leq 0.
$$-/
|
42 | math-word-problem | human | Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$. Find $\rho^2.$ | import Mathlib
open Real
/-- Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$.Β Find $\rho^2.$-/
the... | import Mathlib
open Real
/-- Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$.Β Find $\rho^2.$-/
the... | ### Example 1:
import Mathlib
theorem algebra_1274 {{x y : β}} (hx : x β 0) (hy : y β 0) (hxy : x + y β 0)
(h : β k, 1 / x + 1 / y = k / (x + y)) :
β m, (x + y) ^ 2 = m * (x ^ 2 + y ^ 2) := by
use (x + y) ^ 2 / (x ^ 2 + y ^ 2)
have : x ^ 2 + y ^ 2 β 0 := by
have hx2 : x ^ 2 > 0 := by
exact sq_po... | a-x)^2 + (b-y)^2) β§ (a/b)^2 = 4/3):= by
refine' β¨fun a b hab h => _, _β©
Β· have h1 : (a/b)^2 β€ 4/3 := by
by_cases h2 : b = 0 <;>
simp_all [sq, div_eq_mul_inv, mul_inv_rev, mul_assoc]
<;>
nlinarith [sq_nonneg (a - b), sq_nonneg (a + b)]
exact h1
Β· use 2
use 1
constructor
Β· nl... | import Mathlib
open Real
/-- Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$.Β Find $\rho^2.$-/
the... | import Mathlib
open Real
/-- Let $a$ and $b$ be positive real numbers with $a\ge b$. Let $\rho$ be the maximum possible value of $\frac {a}{b}$ for which the system of equations $$
a^2 + y^2 = b^2 + x^2 = (a - x)^2 + (b - y)^2
$$has a solution in $(x,y)$ satisfying $0\le x < a$ and $0\le y < b$.Β Find $\rho^2.$-/
the... |
43 | math-word-problem | human | Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 5$ and $xy + xz + yz = 8.$ Find the largest possible value of $x.$ | import Mathlib
/-- Auxiliary lemma : for any real numbers $x$ and $y$, $(x+y)^2 \leq 2*(x^2 + y^2)$. -/
lemma p93' {x y : β} : (x + y)^2 β€ 2*(x^2 + y^2) := by
let f := fun i : β => match i with
| 0 => x
| _ => y
let s := Finset.range 2
have : β i β s, f i = x + y := by
dsimp [f,s]
rw [β (show {0... | import Mathlib
/-- Auxiliary lemma : for any real numbers $x$ and $y$, $(x+y)^2 \leq 2*(x^2 + y^2)$. -/
lemma p93' {x y : β} : (x + y)^2 β€ 2*(x^2 + y^2) := by
let f := fun i : β => match i with
| 0 => x
| _ => y
let s := Finset.range 2
have : β i β s, f i = x + y := by
dsimp [f,s]
rw [β (show {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... | by
simp [f,s]
rw [β (show {0, 1} = Finset.range 2 by simp),Finset.sum_pair (by norm_num)]
rw [β this]
have : β i β s, f i * f i β€ β i β s, (f i) ^ 2 := by
apply Finset.sum_le_sum
intro i hi
nlinarith
nlinarith
| import Mathlib
/-- Auxiliary lemma : for any real numbers $x$ and $y$, $(x+y)^2 \leq 2*(x^2 + y^2)$. -/
lemma p93' {x y : β} : (x + y)^2 β€ 2*(x^2 + y^2) := by
let f := fun i : β => match i with
| 0 => x
| _ => y
let s := Finset.range 2
have : β i β s, f i = x + y := by
dsimp [f,s]
rw [β (show {0... | import Mathlib
/-- Auxiliary lemma : for any real numbers $x$ and $y$, $(x+y)^2 \leq 2*(x^2 + y^2)$. -/
lemma p93' {x y : β} : (x + y)^2 β€ 2*(x^2 + y^2) := by
let f := fun i : β => match i with
| 0 => x
| _ => y
let s := Finset.range 2
have : β i β s, f i = x + y := by
dsimp [f,s]
rw [β (show {0... |
47 | math-word-problem | human | There exists a constant $k$ so that the minimum value of
\[4x^2 - 6kxy + (3k^2 + 2) y^2 - 4x - 4y + 6\]over all real numbers $x$ and $y$ is 0. Find $k.$ | import Mathlib
open Real
/-- `f k x y` denotes that $4*x^2 - 6*k*x*y + (3*k^2+2)*y^2 - 4*x - 4*y + 6
$-/
def f_4098 (k : β) x y := 4*x^2 - 6*k*x*y + (3*k^2+2)*y^2 - 4*x - 4*y + 6
/-- simplify the expression `f k x y`. -/
lemma p86' : β (x y k : β),
f_4098 k x y = (x-2)^2 + 2*(y-1)^2 + 3*(x-k*y)^2 := by
i... | import Mathlib
open Real
/-- `f k x y` denotes that $4*x^2 - 6*k*x*y + (3*k^2+2)*y^2 - 4*x - 4*y + 6
$-/
def f_4098 (k : β) x y := 4*x^2 - 6*k*x*y + (3*k^2+2)*y^2 - 4*x - 4*y + 6
/-- simplify the expression `f k x y`. -/
lemma p86' : β (x y k : β),
f_4098 k x y = (x-2)^2 + 2*(y-1)^2 + 3*(x-k*y)^2 := by
i... | ### 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... | 6']
norm_num
Β· -- If `f k x y` is nonegative for all $x$ and $y$, then $k=2$.
rintro β¨h1, β¨x, y, h2β©β©
have h3 := h1 2 1
have h4 := h1 2 2
have h5 := h1 1 1
have h6 := h1 1 2
have h7 := h1 0 1
have h8 := h1 0 2
have h9 := h1 (-1) 1
have h10 := h1 (-1) 2
have h11 := h1 (-2)... | import Mathlib
open Real
/-- `f k x y` denotes that $4*x^2 - 6*k*x*y + (3*k^2+2)*y^2 - 4*x - 4*y + 6
$-/
def f_4098 (k : β) x y := 4*x^2 - 6*k*x*y + (3*k^2+2)*y^2 - 4*x - 4*y + 6
/-- simplify the expression `f k x y`. -/
lemma p86' : β (x y k : β),
f_4098 k x y = (x-2)^2 + 2*(y-1)^2 + 3*(x-k*y)^2 := by
i... | import Mathlib
open Real
/-- `f k x y` denotes that $4*x^2 - 6*k*x*y + (3*k^2+2)*y^2 - 4*x - 4*y + 6
$-/
def f_4098 (k : β) x y := 4*x^2 - 6*k*x*y + (3*k^2+2)*y^2 - 4*x - 4*y + 6
/-- simplify the expression `f k x y`. -/
lemma p86' : β (x y k : β),
f_4098 k x y = (x-2)^2 + 2*(y-1)^2 + 3*(x-k*y)^2 := by
i... |
51 | null | human | There are five guys named Alan, Bob, Casey, Dan, and Eric. Each one either always tells the truth or always lies. You overhear the following discussion between them:
```
Alan: "All of us are truth-tellers."
Bob: "No, only Alan and I are truth-tellers."
Casey: "You are both liars."
Dan: "If Casey is a truth-teller, then... | import Mathlib
theorem logic_and_puzzles_608575 (a b c d e : Bool)
(ha : a = (a && b && c && d && e))
(hb : b = (a && b && !c && !d && !e))
(hc : c = (!a && !b))
(hd : d = (cond c e true))
(he : e = (List.count false [a, b, c, d, e] % 2 == 1)) :
!a && !b && c && !d && !e := by | import Mathlib
/- There are five guys named Alan, Bob, Casey, Dan, and Eric. Each one either always tells the truth or always lies. You overhear the following discussion between them:
```
Alan: "All of us are truth-tellers."
Bob: "No, only Alan and I are truth-tellers."
Casey: "You are both liars."
Dan: "If Casey is 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... | constructor <;>
(try simp_all) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop) <;>
(try aesop... | import Mathlib
/- There are five guys named Alan, Bob, Casey, Dan, and Eric. Each one either always tells the truth or always lies. You overhear the following discussion between them:
```
Alan: "All of us are truth-tellers."
Bob: "No, only Alan and I are truth-tellers."
Casey: "You are both liars."
Dan: "If Casey is a... | import Mathlib
/- There are five guys named Alan, Bob, Casey, Dan, and Eric. Each one either always tells the truth or always lies. You overhear the following discussion between them:
```
Alan: "All of us are truth-tellers."
Bob: "No, only Alan and I are truth-tellers."
Casey: "You are both liars."
Dan: "If Casey is a... |
52 | null | human | A broken calculator has the + and $\times$ keys switched. For how many ordered pairs $(a, b)$ of integers will it correctly calculate $a+b$ using the labelled + key? | import Mathlib
theorem number_theory_611121 : {(a, b) : β€ Γ β€ | a + b = a * b}.encard = 2 := by | import Mathlib
/- A broken calculator has the + and $\times$ keys switched. For how many ordered pairs
$(a, b)$ of integers will it correctly calculate $a+b$ using the labelled + key? -/
theorem number_theory_611121 : {(a, b) : β€ Γ β€ | a + b = a * b}.encard = 2 := by
transitivity ({(0, 0), (2, 2)} : Set (β€ Γ β€)).e... | ### 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 - 1) = 1`. -/
have h1 : a + b = a * b := h
have h2 : a * b - a - b = 0 := by linarith
have h3 : (a - 1) * (b - 1) = 1 := by
nlinarith
exact h3
Β· /- From `(b - 1) (a - 1) = 1`, we prove that `a + b = ab`. -/
have h1 : (b - 1) * (a - 1) = 1 := h
have h2 : a * b - a - b ... | import Mathlib
/- A broken calculator has the + and $\times$ keys switched. For how many ordered pairs
$(a, b)$ of integers will it correctly calculate $a+b$ using the labelled + key? -/
theorem number_theory_611121 : {(a, b) : β€ Γ β€ | a + b = a * b}.encard = 2 := by
transitivity ({(0, 0), (2, 2)} : Set (β€ Γ β€)).e... | import Mathlib
/- A broken calculator has the + and $\times$ keys switched. For how many ordered pairs
$(a, b)$ of integers will it correctly calculate $a+b$ using the labelled + key? -/
theorem number_theory_611121 : {(a, b) : β€ Γ β€ | a + b = a * b}.encard = 2 := by
transitivity ({(0, 0), (2, 2)} : Set (β€ Γ β€)).e... |
53 | null | human | A malfunctioning digital clock shows the time $9: 57 \mathrm{AM}$; however, the correct time is $10: 10 \mathrm{AM}$. There are two buttons on the clock, one of which increases the time displayed by 9 minutes, and another which decreases the time by 20 minutes. What is the minimum number of button presses necessary to ... | import Mathlib
theorem logic_and_puzzles_609734 (currentTime correctTime : β) (h1 : currentTime = 9 * 60 + 57) (h2 : correctTime = 10 * 60 + 10) : IsLeast { n | β a b, currentTime + 9 * a - 20 * b = correctTime β§ a + b = n } 24 := by | import Mathlib
theorem logic_and_puzzles_609734 (currentTime correctTime : β) (h1 : currentTime = 9 * 60 + 57) (h2 : correctTime = 10 * 60 + 10) : IsLeast { n | β a b, currentTime + 9 * a - 20 * b = correctTime β§ a + b = n } 24 := by
constructor
Β· -- supply witnesses
simp only [Set.mem_setOf_eq]
apply Exist... | ### 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 = a * 60 + b β§ a β€ 10 β§ b β€ 60} correctTime β§ Β¬ Even correctTime β§ correctTime β 0 β§ correctTime β 10 * 60 + 10 := by
constructor
Β· use 10, 10
simp [h1, h2]
Β· constructor
Β· intro h
simp_all [Nat.even_iff, Nat.mod_eq_of_lt]
Β· constructor
Β· intro h
simp_all [Nat.even_iff, Nat.mod_... | import Mathlib
theorem logic_and_puzzles_609734 (currentTime correctTime : β) (h1 : currentTime = 9 * 60 + 57) (h2 : correctTime = 10 * 60 + 10) : IsLeast { n | β a b, | import Mathlib
theorem logic_and_puzzles_609734 (currentTime correctTime : β) (h1 : currentTime = 9 * 60 + 57) (h2 : correctTime = 10 * 60 + 10) : IsLeast { n | β a b, n = a * 60 + b β§ a β€ 10 β§ b β€ 60} correctTime β§ Β¬ Even correctTime β§ correctTime β 0 β§ correctTime β 10 * 60 + 10 := by
constructor
Β· use 10, 10
... |
56 | math-word-problem | autoformalizer | Matt has somewhere between $1000$ and $2000$ pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2$, $3$, $4$, $5$, $6$, $7$, and $8$ piles but ends up with one sheet left over each time. How many piles does he need? | import Mathlib
/- Matt has somewhere between $1000$ and $2000$ pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2$, $3$, $4$, $5$, $6$, $7$, and $8$ piles but ends up with one sheet left over each time. How many piles does he need? -/
the... | import Mathlib
theorem number_theory_67977 {n : β} (hn : n β₯ 1000 β§ n < 2000)
(h1 : n % 2 = 1) (h2 : n % 3 = 1) (h3 : n % 4 = 1) (h4 : n % 5 = 1)
(h5 : n % 6 = 1) (h6 : n % 7 = 1) (h7 : n % 8 = 1) :
n % 41 = 0 := by
rcases hn with β¨hn1, hn2β©
interval_cases n <;> omega | ### 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) (h3 : n % 5 = 3) (h4 : n % 7 = 4) (h5 : n % 11 = 5) :
n = 1847 := by
have h6 : n % 2 = 1 := h1
have h7 : n % 3 = 2 := h2
have h8 : n % 5 = 3 := h3
have h9 : n % 7 = 4 := h4
have h10 : n % 11 = 5 := h5
have h11 : n < 2000 := by linarith
have h12 : n β₯ 1000 := by linarith
have h13 : n % 2 = 1 := h... | import Mathlib
theorem number_theory_67977 {n : β} (hn : n β₯ 1000 β§ n < 2000)
(h1 : n % 2 = 1) (h2 : n % 3 = | import Mathlib
theorem number_theory_67977 {n : β} (hn : n β₯ 1000 β§ n < 2000)
(h1 : n % 2 = 1) (h2 : n % 3 = 2) (h3 : n % 5 = 3) (h4 : n % 7 = 4) (h5 : n % 11 = 5) :
n = 1847 := by
have h6 : n % 2 = 1 := h1
have h7 : n % 3 = 2 := h2
have h8 : n % 5 = 3 := h3
have h9 : n % 7 = 4 := h4
have h10 : n % 1... |
57 | math-word-problem | autoformalizer | Let $f(x) = x^2 + 6x + 7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x.$ | import Mathlib
open Real Set
open scoped BigOperators
/- Let $f(x) = x^2 + 6x + 7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x.$ -/
theorem calculus_75601 {f : β β β} (hf : f = Ξ» x => x ^ 2 + 6 * x + 7) :
IsLeast {y | β x, y = f (f (f (f x)))} 23 :=
| import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_75601 {f : β β β} (hf : f = Ξ» x => x ^ 2 + 6 * x + 7) :
IsLeast {y | β x, y = f (f (f (f x)))} 23 := by
constructor
Β· -- First part of IsLeast: Show 23 is in the set.
use -3
norm_num [hf]
Β· -- Second part of IsLeast: Show no val... | ### 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... | use 2
rw [hf]
norm_num
ring
Β· -- Second part of
intro y hy
use 2
rw [hf]
norm_num
ring
| import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_75601 {f : β β β} (hf : f = Ξ» x => x ^ 2 + 6 * x + 7) :
IsLeast {y | β x, y = f (f (f (f x)))} 23 := by
constructor
Β· -- First part of IsLeast: Show 23 is in the set.
| import Mathlib
open Real Set
open scoped BigOperators
theorem calculus_75601 {f : β β β} (hf : f = Ξ» x => x ^ 2 + 6 * x + 7) :
IsLeast {y | β x, y = f (f (f (f x)))} 23 := by
constructor
Β· -- First part of IsLeast: Show 23 is in the set.
use 2
rw [hf]
norm_num
ring
Β· -- Second part of
... |
58 | math-word-problem | autoformalizer | Find all pairs $(x, y)$ of positive integers such that $x^{2}=y^{2}+7 y+6$.
| import Mathlib
/- Find all pairs $(x, y)$ of positive integers such that $x^{2}=y^{2}+7 y+6$.
-/
theorem algebra_116978 (x y : β€) (hx : 0 < x) (hy : 0 < y) :
x^2 = y^2 + 7 * y + 6 β (x, y) = (6, 3) := by | import Mathlib
theorem algebra_116978 (x y : β€) (hx : 0 < x) (hy : 0 < y) :
x^2 = y^2 + 7 * y + 6 β (x, y) = (6, 3) := by
constructor
Β· -- First direction: xΒ² = yΒ² + 7y + 6 β (x, y) = (6, 3)
intro h
have h1 : 4 * x ^ 2 = (2 * y + 7) ^ 2 - 25 := by
nlinarith
have h2 : (2 * y + 7 - 2 * x) * (2... | ### 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... | num) h9
linarith
omega
cases' h4 with h4 h4 <;>
simp_all [Int.mul_emod, Int.add_emod, Int.sub_emod]
<;>
omega
Β· -- Second direction: (x, y) = (6, 3) β xΒ² = yΒ² + 7y + 6
intro h
rw [h]
norm_num | import Mathlib
theorem algebra_116978 (x y : β€) (hx : 0 < x) (hy : 0 < y) :
x^2 = y^2 + 7 * y + 6 β (x, y) = (6, 3) := by
constructor
Β· -- First direction: xΒ² = yΒ² + 7y + 6 β (x, y) = (6, 3)
intro h
have h1 : 4 * x ^ 2 = (2 * y + 7) ^ 2 - 25 := by
nlinarith
have h2 : (2 * y + 7 - 2 * x) * (2... | import Mathlib
theorem algebra_116978 (x y : β€) (hx : 0 < x) (hy : 0 < y) :
x^2 = y^2 + 7 * y + 6 β (x, y) = (6, 3) := by
constructor
Β· -- First direction: xΒ² = yΒ² + 7y + 6 β (x, y) = (6, 3)
intro h
have h1 : 4 * x ^ 2 = (2 * y + 7) ^ 2 - 25 := by
nlinarith
have h2 : (2 * y + 7 - 2 * x) * (2... |
59 | math-word-problem | autoformalizer | 11. Find the smallest positive integer $n$ such that $1^{2}+2^{2}+3^{2}+4^{2}+\cdots+n^{2}$ is divisible by 100 . | import Mathlib
/- 11. Find the smallest positive integer $n$ such that $1^{2}+2^{2}+3^{2}+4^{2}+\cdots+n^{2}$ is divisible by 100 . -/
theorem number_theory_188669 :
IsLeast {n : β | 0 < n β§ 100 β£ β i in Finset.Icc 1 n, i^2} 24 := by | import Mathlib
theorem number_theory_188669 :
IsLeast {n : β | 0 < n β§ 100 β£ β i in Finset.Icc 1 n, i^2} 24 := by
constructor
Β· -- Show that 24 is in the set
constructor
Β· norm_num
Β· -- Check divisibility
native_decide
Β· -- Show that any element in the set is β₯ 24
intro n hn
rcases ... | ### 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... | - Show that 24 is divisible by 100
norm_num
Β· -- Show that 24 is the smallest number in the set
intro n
cases n
Β· -- Show that 0 is not in the set
intro h
cases h
Β· -- Show that 1 is not in the set
intro h
cases h
Β· -- Show that 2 is not in the set
intro h
... | import Mathlib
theorem number_theory_188669 :
IsLeast {n : β | 0 < n β§ 100 β£ β i in Finset.Icc 1 n, i^2} 24 := by
constructor
Β· -- Show that 24 is in the set
constructor
Β· norm_num
Β· - | import Mathlib
theorem number_theory_188669 :
IsLeast {n : β | 0 < n β§ 100 β£ β i in Finset.Icc 1 n, i^2} 24 := by
constructor
Β· -- Show that 24 is in the set
constructor
Β· norm_num
Β· -- Show that 24 is divisible by 100
norm_num
Β· -- Show that 24 is the smallest number in the set
intro n... |
60 | math-word-problem | autoformalizer | A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$. Let $f$ be a function such that:
$f(n)=0$, if n is perfect
$f(n)=0$, if the last digit of n is 4
$f(a.b)=f(a)+f(b)$
Find $f(1998)$ | import Mathlib
def is_perfect (n : β) := β d in n.divisors.erase n, d = n
/- A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$. Let $f$ be a function such that:
$f(n)=0$, if n is perfect
$f(n)=0$, if the last digit of n is 4
$f(a.b)=f(a)+f(b)$
Find $f(1998)$ -/
theorem number_theory_6975... | import Mathlib
def is_perfect (n : β) := β d in n.divisors.erase n, d = n
theorem number_theory_69752 {f : β β β€} (hf1 : β n, is_perfect n β f n = 0)
(hf2 : β n, n % 10 = 4 β f n = 0) (hf3 : β a b, f (a * b) = f a + f b) :
f 1998 = 0 := by
have h1 : f 1998 = f 6 + f 333 := by
calc
f 1998 = f (6 * ... | ### 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... | f 3 + f 111 := by rw [hf3 3 111]
have h6 : f 3 = 0 := by
have h7 : is_perfect 3 := by
unfold is_perfect
native_decide
have h8 : f 3 = 0 := hx1 3 h7
exact h8
have h9 : f 111 = f 3 + f 37 := by
calc
f 111 = f (3 * 37) := by norm_num
_ = f 3 + f 37 := by rw [hf3 3 37]
have h1... | import Mathlib
def is_perfect (n : β) := β d in n.divisors.erase n, d = n
theorem number_theory_69752 {f : β β β€} (hf1 : β n, is_perfect n β f n = 0)
(hf2 : β n, n % 10 = 4 β f n = 0) (hf3 : β a b, f (a * b) = f a + f b) :
f 1998 = 0 := by
have h1 : f 1998 = f 6 + f 333 := by
calc
f 1998 = f (6 * ... | import Mathlib
def is_perfect (n : β) := β d in n.divisors.erase n, d = n
theorem number_theory_69752 {f : β β β€} (hf1 : β n, is_perfect n β f n = 0)
(hf2 : β n, n % 10 = 4 β f n = 0) (hf3 : β a b, f (a * b) = f a + f b) :
f 1998 = 0 := by
have h1 : f 1998 = f 6 + f 333 := by
calc
f 1998 = f (6 * ... |
61 | math-word-problem | autoformalizer | 5. Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations
$$
\frac{x}{\sqrt{x^{2}+y^{2}}}-\frac{1}{x}=7 \text { and } \frac{y}{\sqrt{x^{2}+y^{2}}}+\frac{1}{y}=4 .
$$
The final answer is $ \boxed{(-\frac{13}{96},\frac{13}{40})} $ | import Mathlib
/- 5. Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations
$$
\frac{x}{\sqrt{x^{2}+y^{2}}}-\frac{1}{x}=7 \text { and } \frac{y}{\sqrt{x^{2}+y^{2}}}+\frac{1}{y}=4 .
$$ -/
theorem algebra_183620 {x y : β} (hx : x^2 + y^2 β 0) (hy : y β 0)
(h1 : x / Real.sqrt (x^2... | import Mathlib
theorem algebra_183620 {x y : β} (hx : x^2 + y^2 β 0) (hy : y β 0)
(h1 : x / Real.sqrt (x^2 + y^2) - 1 / x = 7) (h2 : y / Real.sqrt (x^2 + y^2) + 1 / y = 4) :
(x, y) = (-13 / 96, 13 / 40) := by
have hx0 : x β 0 := by
by_contra h
rw [h] at h1
field_simp [hy] at h1
have hS1 : 0 < R... | ### 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), Real.sq_sqrt (show x^2 + y^2 β₯ 0 by positivity)]
have Eq5 : x = -13 / 96 := by
nlinarith [Real.sqrt_nonneg (x^2 + y^2), Real.sq_sqrt (show x^2 + y^2 β₯ 0 by positivity)]
have Eq6 : y = 13 / 40 := by
nlinarith [Real.sqrt_nonneg (x^2 + y^2), Real.sq_sqrt (show x^2 + y^2 β₯ 0 by positivity)]
exact β¨Eq5, Eq... | import Mathlib
theorem algebra_183620 {x y : β} (hx : x^2 + y^2 β 0) (hy : y β 0)
(h1 : x / Real.sqrt (x^2 + y^2) - 1 / x = 7) (h2 : y / Real.sqrt (x^2 + y^2) + 1 / y = 4) :
(x, y) = (-13 / 96, 13 / 40) := by
have hx0 : x β 0 := by
by_contra h
rw [h] at h1
field_simp [hy] at h1
have hS1 : 0 < R... | import Mathlib
theorem algebra_183620 {x y : β} (hx : x^2 + y^2 β 0) (hy : y β 0)
(h1 : x / Real.sqrt (x^2 + y^2) - 1 / x = 7) (h2 : y / Real.sqrt (x^2 + y^2) + 1 / y = 4) :
(x, y) = (-13 / 96, 13 / 40) := by
have hx0 : x β 0 := by
by_contra h
rw [h] at h1
field_simp [hy] at h1
have hS1 : 0 < R... |
62 | math-word-problem | autoformalizer | 23. [12] The number $989 \cdot 1001 \cdot 1007+320$ can be written as the product of three distinct primes $p, q, r$ with $p<q<r$. Find $(p, q, r)$. | import Mathlib
/- 23. [12] The number $989 \cdot 1001 \cdot 1007+320$ can be written as the product of three distinct primes $p, q, r$ with $p<q<r$. Find $(p, q, r)$. -/
theorem number_theory_163044 : β p q r : β, p.Prime β§ q.Prime β§ r.Prime β§ p < q β§ q < r β§ p * q * r = 989 * 1001 * 1007 + 320 := by | import Mathlib
theorem number_theory_163044 : β p q r : β, p.Prime β§ q.Prime β§ r.Prime β§ p < q β§ q < r β§ p * q * r = 989 * 1001 * 1007 + 320 := by
use 991, 997, 1009
all_goals 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... | β§ p β q β§ p β r β§ q β r β§ p * q * r = 2016 := by
use 2, 3, 11
constructor
decide
decide
decide
decide
decide
decide
decide
decide
| import Mathlib
theorem number_theory_163044 : β p q r : β, p.Prime β§ q.Prime β§ r.Prime | import Mathlib
theorem number_theory_163044 : β p q r : β, p.Prime β§ q.Prime β§ r.Prime β§ p β q β§ p β r β§ q β r β§ p * q * r = 2016 := by
use 2, 3, 11
constructor
decide
decide
decide
decide
decide
decide
decide
decide
|
64 | null | human | Prove that for all positive integers $n$ there are positive integers $a,b$ such that $$n\mid 4a^2+9b^2-1.$$ | import Mathlib
theorem number_theory_88038 (n : β) (hn : n > 0) :
β a b : β, a > 0 β§ b > 0 β§ n β£ 4 * a ^ 2 + 9 * b ^ 2 - 1 := by | import Mathlib
/- Prove that for all positive integers $n$ there are positive integers $a,b$ such that $$n\mid 4a^2+9b^2-1.$$ -/
theorem number_theory_88038 (n : β) (hn : n > 0) :
β a b : β, a > 0 β§ b > 0 β§ n β£ 4 * a ^ 2 + 9 * b ^ 2 - 1 := 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... | null | import Mathlib
/- Prove that for all positive integers $n$ there are positive integers $a,b$ such that $$n\mid 4a^2+9b^2-1.$$ -/
| import Mathlib
/- Prove that for all positive integers $n$ there are positive integers $a,b$ such that $$n\mid 4a^2+9b^2-1.$$ -/
|
65 | proof | autoformalizer | Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$.
[i]A. Golovanov[/i] | import Mathlib
open Nat
/- Let $P(a)$ be the largest prime positive divisor of $a^2 + 1$. Prove that exist infinitely many positive integers $a, b, c$ such that $P(a)=P(b)=P(c)$.
[i]A. Golovanov[/i] -/
theorem number_theory_81521 {P : β β β}
(hP : β a, P a = sSup {x | Nat.Prime x β§ x β£ a ^ 2 + 1}) :
β n, β a... | import Mathlib
open Nat
theorem number_theory_81521 {P : β β β}
(hP : β a, P a = sSup {x | Nat.Prime x β§ x β£ a ^ 2 + 1}) :
β n, β a b c, a > n β§ b > n β§ c > n β§ P a = P b β§ P b = P c := by
-- Let's think about how to solve this in Lean 4. We need to prove that given any natural number n, there exists a, 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... | er than n and P(a) = P(b) = P(c).
--
-- First, we need to define the function P. In this case, P(a) is the set of prime numbers that divide a^2 + 1.
--
-- Next, we need to show that there are a, b, c such that each is greater than n and P(a) = P(b) = P(c).
--
-- To do this, we can use the fact that there ar... | import Mathlib
open Nat
theorem number_theory_81521 {P : β β β}
(hP : β a, P a = sSup {x | Nat.Prime x β§ x β£ a ^ 2 + 1}) :
β n, β a b c, a > n β§ b > n β§ c > n β§ P a = P b β§ P b = P c := by
-- Let's think about how to solve this in Lean 4. We need to prove that given any natural number n, there exists a, b, ... | import Mathlib
open Nat
theorem number_theory_81521 {P : β β β}
(hP : β a, P a = sSup {x | Nat.Prime x β§ x β£ a ^ 2 + 1}) :
β n, β a b c, a > n β§ b > n β§ c > n β§ P a = P b β§ P b = P c := by
-- Let's think about how to solve this in Lean 4. We need to prove that given any natural number n, there exists a, b, ... |
67 | proof | human | 2. Prove that any integer $r>2$ is composite if and only if at least one of the following two statements is true:
a) $r=2^{s}$ for some $s \in\{2,3, \ldots\}$,
b) $r=\frac{u}{2}(2 v-u+1)$ for some $u, v \in\{3,4, \ldots\},(u \leq v)$. | import Mathlib
/-2. Prove that any integer $r>2$ is composite if and only if at least one of the following two statements is true:
a) $r=2^{s}$ for some $s \in\{2,3, \ldots\}$,
b) $r=\frac{u}{2}(2 v-u+1)$ for some $u, v \in\{3,4, \ldots\},(u \leq v)$.-/
theorem number_theory_245291 (rgt : 2 < (r : β)) : Β¬ r.Prime β (β... | import Mathlib
/-2. Prove that any integer $r>2$ is composite if and only if at least one of the following two statements is true:
a) $r=2^{s}$ for some $s \in\{2,3, \ldots\}$,
b) $r=\frac{u}{2}(2 v-u+1)$ for some $u, v \in\{3,4, \ldots\},(u \leq v)$.-/
theorem number_theory_245291 (rgt : 2 < (r : β)) : Β¬ r.Prime β (β... | ### 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... | than $1$
have lpos : 0 < l := by omega
have kpos : 0 < k := by omega
use 2 * l + 1, 2 * k + 1
constructor
Β· omega
Β· constructor
Β· omega
Β· constructor
Β· omega
Β· rw [mul_comm]
ring_nf
omega
| import Mathlib
/-2. Prove that any integer $r>2$ is composite if and only if at least one of the following two statements is true:
a) $r=2^{s}$ for some $s \in\{2,3, \ldots\}$,
b) $r=\frac{u}{2}(2 v-u+1)$ for some $u, v \in\{3,4, \ldots\},(u \leq v)$.-/
theorem number_theory_245291 (rgt : 2 < (r : β)) : Β¬ r.Prime β (β... | import Mathlib
/-2. Prove that any integer $r>2$ is composite if and only if at least one of the following two statements is true:
a) $r=2^{s}$ for some $s \in\{2,3, \ldots\}$,
b) $r=\frac{u}{2}(2 v-u+1)$ for some $u, v \in\{3,4, \ldots\},(u \leq v)$.-/
theorem number_theory_245291 (rgt : 2 < (r : β)) : Β¬ r.Prime β (β... |
69 | math-word-problem | human | 3. Find the quotient and remainder in the division algorithm with divisor 17 and dividend
a) 100
c) -44
b) 289
d) -100 . | import Mathlib
/-
3. Find the quotient and remainder using the division algorithm with a divisor of 17 and the following dividends:
a) 100
-/
theorem number_theory_4693_1 : 100 / 17 = 5 β§ 100 % 17 = 15 := by
-- verify by computation
decide
/-
3. Find the quotient and remainder using the division algorithm with a ... | import Mathlib
theorem number_theory_4693_1 : 100 / 17 = 5 β§ 100 % 17 = 15 := by
-- verify by computation
decide
theorem number_theory_4693_2 : 289 / 17 = 17 β§ 289 % 17 = 0 := by
-- verify by computation
decide
theorem number_theory_4693_3 : -44 / 17 = -3 β§ -44 % 17 = 7 := by
-- verify by computation
dec... | ### 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§ 289 % 17 = 0 := by
-- verify by computation
decide
| import Mathlib
theorem number_theory_4693_1 : 100 / 17 = 5 β§ 100 % 17 = 15 := by
-- verify by computation
decide
theorem number_theory_4693_2 : 289 / 17 = 17 | import Mathlib
theorem number_theory_4693_1 : 100 / 17 = 5 β§ 100 % 17 = 15 := by
-- verify by computation
decide
theorem number_theory_4693_2 : 289 / 17 = 17 β§ 289 % 17 = 0 := by
-- verify by computation
decide
|
70 | proof | autoformalizer | 725. Prove that two segments are commensurable if and only if the ratio of their lengths is rational (expressed as a fraction with integer numerator and denominator). | import Mathlib
/- 725. Prove that two segments are commensurable if and only if the ratio of their lengths is rational (expressed as a fraction with integer numerator and denominator). -/
theorem number_theory_288690 {a b : β} (ha : 0 < a) (hb : 0 < b) :
β q : β, a * q = b β β c d : β€, 0 < c β§ 0 < d β§ a * c = b * ... | import Mathlib
theorem number_theory_288690 {a b : β} (ha : 0 < a) (hb : 0 < b) :
β q : β, a * q = b β β c d : β€, 0 < c β§ 0 < d β§ a * c = b * d := by
use (b : β) / (a : β)
constructor
Β· -- Assume a * q = b, prove β c d : β€, 0 < c β§ 0 < d β§ a * c = b * d
intro h
have h31 : (βa : β) * (βb / βa) = (βb :... | ### 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§ a * c = b * d
intro h
use 1, 1
simp_all
Β· -- Assume β c d : β€, 0 < c β§ 0 < d β§ a * c = b * d, prove a * (b / a) = b
rintro β¨c, d, h1, h2, h3β©
rw [mul_comm] at h3
field_simp [ha, hb] at h3 β¨h1, h2, h3β©
<;> linarith | import Mathlib
theorem number_theory_288690 {a b : β} (ha : 0 < a) (hb : 0 < b) :
β q : β, a * q = b β β c d : β€, 0 < c β§ 0 < d β§ a * c = b * d := by
use (b : β) / (a : β)
constructor
Β· -- Assume a * q = b, prove β c d : β€, 0 < c β§ 0 < d | import Mathlib
theorem number_theory_288690 {a b : β} (ha : 0 < a) (hb : 0 < b) :
β q : β, a * q = b β β c d : β€, 0 < c β§ 0 < d β§ a * c = b * d := by
use (b : β) / (a : β)
constructor
Β· -- Assume a * q = b, prove β c d : β€, 0 < c β§ 0 < d β§ a * c = b * d
intro h
use 1, 1
simp_all
Β· -- Assume β c... |
73 | MCQ | autoformalizer | 6. What time is it 2017 minutes after $20: 17$ ?
A 05:54
B 09:54
C 16:34
D 20:34
E 23:34 | import Mathlib
/- 6. What time is it 2017 minutes after $20: 17$ ?
A 05:54
B 09:54
C 16:34
D 20:34
E 23:34 -/
theorem other_229758 : (20 * 60 + 17 + 2017) % (24 * 60) = 5 * 60 + 54 := by | import Mathlib
theorem other_229758 : (20 * 60 + 17 + 2017) % (24 * 60) = 5 * 60 + 54 := by
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... | 10 * 10) / 10 = 130 := by
norm_num
| import Mathlib
theorem other_229758 : (20 * 60 + | import Mathlib
theorem other_229758 : (20 * 60 + 10 * 10) / 10 = 130 := by
norm_num
|
74 | math-word-problem | autoformalizer | 7. (15 points) A uniformly charged sphere of radius $R$ is fixed on a horizontal surface at point $A$. The charge of the sphere is $Q$. At point $C$, which is located at a distance $L$ from the surface of the sphere, a charged sphere of radius $r$ and mass $m$ is floating. Its charge is $q$. It is known that $r<<R$. De... | import Mathlib
open Real
/- 7. (15 points) A uniformly charged sphere of radius $R$ is fixed on a horizontal surface at point $A$. The charge of the sphere is $Q$. At point $C$, which is located at a distance $L$ from the surface of the sphere, a charged sphere of radius $r$ and mass $m$ is floating. Its charge is $q... | import Mathlib
open Real
theorem other_166925 {R r L S : β} (hR : R > 0) (hr : r > 0)
(hL : L > 0) (hS : S > 0) (hRr : r < R) :
let g := 9.81
let Q := 4 * Ο * R^3 * g
let q := 4 * Ο * r^3 * g
let m := 4 * Ο * r^3
let k := m * r / (4 * Ο * R^2)
let a := k * r^3 / (R^3 * (L + 2 * R - S)^2)
... | ### 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... | * (R - r) / (R + r)
g β€ 9.81 * (R - r) / R := by
have h : 0 < R + r := by linarith
have h2 : 0 < R * (R + r) := by positivity
have h3 : 0 < R * (R + r) * (R - r) := by positivity
field_simp [h, h2, h3]
rw [div_le_div_iff (by positivity) (by positivity)]
nlinarith [mul_self_nonneg (R - r), mul_self_nonn... | import Mathlib
open Real
theorem other_166925 {R r L S : β} (hR : R > 0) (hr : r > 0)
(hL : L > 0) (hS : S > 0) (hRr : r < R) :
let g := 9.81 | import Mathlib
open Real
theorem other_166925 {R r L S : β} (hR : R > 0) (hr : r > 0)
(hL : L > 0) (hS : S > 0) (hRr : r < R) :
let g := 9.81 * (R - r) / (R + r)
g β€ 9.81 * (R - r) / R := by
have h : 0 < R + r := by linarith
have h2 : 0 < R * (R + r) := by positivity
have h3 : 0 < R * (R + r) * (R -... |
75 | math-word-problem | human | 12. (NET 1) ${ }^{\mathrm{IMO} 2}$ Find two positive integers $a, b$ such that none of the numbers $a, b, a+b$ is divisible by 7 and $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$. | import Mathlib
/- 12. (NET 1) ${ }^{\mathrm{IMO} 2}$ Find two positive integers $a, b$ such that none of the numbers $a, b, a+b$ is divisible by 7 and $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$. -/
theorem number_theory_23912 :
β a b : β, a > 0 β§ b > 0 β§ Β¬(7 β£ a) β§ Β¬(7 β£ b) β§ Β¬(7 β£ a + b) β§
7^7 β£ (a + b... | import Mathlib
/- 12. (NET 1) ${ }^{\mathrm{IMO} 2}$ Find two positive integers $a, b$ such that none of the numbers $a, b, a+b$ is divisible by 7 and $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$. -/
theorem number_theory_23912 :
β a b : β, a > 0 β§ b > 0 β§ Β¬(7 β£ a) β§ Β¬(7 β£ b) β§ Β¬(7 β£ a + b) β§
7^7 β£ (a + 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... | _12_12 (a b : β€) (hβ : a % 7 β 0) (hβ : b % 7 β 0) (hβ : (a + b) % 7 β 0) (hβ : (a + b)^7 - a^7 - b^7 β‘ 0 [ZMOD 7^7]) : False := by
/-
Given positive integers \( a \) and \( b \), we need to find values for \( a \) and \( b \) such that none of the numbers \( a \), \( b \), or \( a + b \) is divisible by 7, and \( ... | import Mathlib
/- 12. (NET 1) ${ }^{\mathrm{IMO} 2}$ Find two positive integers $a, b$ such that none of the numbers $a, b, a+b$ is divisible by 7 and $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$. -/
theorem number_theory | import Mathlib
/- 12. (NET 1) ${ }^{\mathrm{IMO} 2}$ Find two positive integers $a, b$ such that none of the numbers $a, b, a+b$ is divisible by 7 and $(a+b)^{7}-a^{7}-b^{7}$ is divisible by $7^{7}$. -/
theorem number_theory_12_12 (a b : β€) (hβ : a % 7 β 0) (hβ : b % 7 β 0) (hβ : (a + b) % 7 β 0) (hβ : (a + b)^7 - a^7... |
77 | proof | human | Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \le 1\}$. | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
/- Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \le 1\}$. -/
theorem other_43390 {n : β} (hn : 0 < n)... | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
/- Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \le 1\}$. -/
theorem other_43390 {n : β} (hn : 0 < n)... | ### 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... | in the unit circle. -/
| import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
/- Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}$. Prove that $f_n$ has no roots | import Mathlib
open Complex Filter Function Metric Finset
open scoped BigOperators Topology
/- Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}$. Prove that $f_n$ has no roots in the unit circle. -/
|
78 | math-word-problem | human | 26. (KOR 4) Let $n$ be a positive integer and let $a, b$ be given real numbers. Determine the range of $x_{0}$ for which $$ \sum_{i=0}^{n} x_{i}=a \quad \text { and } \quad \sum_{i=0}^{n} x_{i}^{2}=b $$ where $x_{0}, x_{1}, \ldots, x_{n}$ are real variables. | import Mathlib
open Real
/-26. (KOR 4) Let $n$ be a positive integer and let $a, b$ be given real numbers. Determine the range of $x_{0}$ for which $$ \sum_{i=0}^{n} x_{i}=a \quad \text { and } \quad \sum_{i=0}^{n} x_{i}^{2}=b $$ where $x_{0}, x_{1}, \ldots, x_{n}$ are real variables.-/
theorem other_24234 (n : β) (... | import Mathlib
open Real
/-26. (KOR 4) Let $n$ be a positive integer and let $a, b$ be given real numbers. Determine the range of $x_{0}$ for which $$ \sum_{i=0}^{n} x_{i}=a \quad \text { and } \quad \sum_{i=0}^{n} x_{i}^{2}=b $$ where $x_{0}, x_{1}, \ldots, x_{n}$ are real variables.-/
theorem other_24234 (n : β) (... | ### 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_abs, Finset.sum_range_succ, Finset.sum_range_one,
Finset.sum_range_zero, Finset.sum_range_succ, Finset.sum_range_one,
Finset.sum_range_zero, Finset.sum_range_succ, Finset.sum_range_one,
Finset.sum_range_zero, Finset.sum_range_succ, Finset.sum_range_one,
Finset.sum_range_zero, Finset.sum_range... | import Mathlib
open Real
/-26. (KOR 4) Let $n$ be a positive integer and let $a, b$ be given real numbers. Determine the range of $x_{0}$ for which $$ \sum_{i=0}^{n} x_{i}=a \quad \text { and } \quad \sum_{i=0}^{n} x_{i}^{2}=b $$ where $x_{0}, x_{1}, \ldots, x_{n}$ are real variables.-/
theorem other_24234 (n : β) (... | import Mathlib
open Real
/-26. (KOR 4) Let $n$ be a positive integer and let $a, b$ be given real numbers. Determine the range of $x_{0}$ for which $$ \sum_{i=0}^{n} x_{i}=a \quad \text { and } \quad \sum_{i=0}^{n} x_{i}^{2}=b $$ where $x_{0}, x_{1}, \ldots, x_{n}$ are real variables.-/
theorem other_24234 (n : β) (... |
80 | MCQ | autoformalizer | Definition: The annual production growth rate of a factory is:
$\frac{\text{current year's output value} - \text{previous year's output value}}{\text{previous year's output value}} \times 100\%$
If the factory's output value in the year $2000$ is to be $1.44$ times that of the year $1998$, and the annual production g... | import Mathlib
/- Definition: The annual production growth rate of a factory is:
$\frac{\text{current year's output value} - \text{previous year's output value}}{\text{previous year's output value}} \times 100\%$
If the factory's output value in the year $2000$ is to be $1.44$ times that of the year $1998$, and the ... | import Mathlib
theorem algebra_2662 {x : β} (hx : 0 < x) (h : (1 + x / 100) ^ 2 = 1.44) : x = 20 := by
have h1 : (1 + x / 100 : β) ^ 2 - (1.44 : β) = 0 := by
linarith
have h2 : ((1 + x / 100 : β) - 1.2) * ((1 + x / 100 : β) + 1.2) = 0 := by
ring_nf at h1 β’
norm_num at h1 β’
nlinarith
cases' (mul_e... | ### 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... |
nlinarith
have h3 : (1 + x / 100 : β) - 1.2 = 0 β¨ (1 + x / 100 : β) + 1.2 = 0 := by
apply eq_zero_or_eq_zero_of_mul_eq_zero
exact h2
cases h3 with
| simp h3 =>
nlinarith
| simp h3 =>
nlinarith
| import Mathlib
theorem algebra_2662 {x : β} (hx : 0 < x) (h : (1 + x / 100) ^ 2 = 1.44) : x = 20 := by
have h1 : (1 + x / 100 : β) ^ 2 - (1.44 : β) = 0 := by
linarith
have h2 : ((1 + x / 100 : β) - 1.2) * ((1 + x / 100 : β) + 1.2) = 0 := by | import Mathlib
theorem algebra_2662 {x : β} (hx : 0 < x) (h : (1 + x / 100) ^ 2 = 1.44) : x = 20 := by
have h1 : (1 + x / 100 : β) ^ 2 - (1.44 : β) = 0 := by
linarith
have h2 : ((1 + x / 100 : β) - 1.2) * ((1 + x / 100 : β) + 1.2) = 0 := by
nlinarith
have h3 : (1 + x / 100 : β) - 1.2 = 0 β¨ (1 + x / 100 :... |
81 | math-word-problem | autoformalizer | $$ 4 ( 15 ^ { 3 } + 14 ^ { 3 } + 13 ^ { 3 } + ... + 1 ^ { 3 } ) + 6 ( 15 ^ { 2 } + 14 ^ { 2 } + 13 ^ { 2 } + ... + 1 ^ { 2 } ) + 4 ( 15 + 14 + 13 + ... + 1 ) + 15 = ?$$ | import Mathlib
/- $$ 4 ( 15 ^ { 3 } + 14 ^ { 3 } + 13 ^ { 3 } + ... + 1 ^ { 3 } ) + 6 ( 15 ^ { 2 } + 14 ^ { 2 } + 13 ^ { 2 } + ... + 1 ^ { 2 } ) + 4 ( 15 + 14 + 13 + ... + 1 ) + 15 = ?$$ -/
theorem algebra_3881 : 4 * β i β Finset.range 15, (i + 1) ^ 3 + 6 * β i β Finset.range 15, (i + 1) ^ 2 + 4 * β i β Finset.range 1... | import Mathlib
theorem algebra_3881 : 4 * β i β Finset.range 15, (i + 1) ^ 3 + 6 * β i β Finset.range 15, (i + 1) ^ 2 + 4 * β i β Finset.range 15, (i + 1) + 15 = 65535 := by
native_decide | ### 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 i β Finset.range 15, (i + 1) ^ 2 + 4 * β i β Finset.range 15, (i + 1) + 1 = 15 ^ 4 := by
simp [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,
Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.su... | import Mathlib
theorem algebra_3881 : 4 * β i β Finset.range 15, (i + 1) ^ 3 + 6 * | import Mathlib
theorem algebra_3881 : 4 * β i β Finset.range 15, (i + 1) ^ 3 + 6 * β i β Finset.range 15, (i + 1) ^ 2 + 4 * β i β Finset.range 15, (i + 1) + 1 = 15 ^ 4 := by
simp [Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ, Finset.sum_range_succ,
Finset.sum_range_s... |
82 | math-word-problem | autoformalizer | Compute $\lfloor \sqrt{n^2 - 10n + 29} \rfloor$ when $n = 19941994.$ | import Mathlib
/- Compute $\lfloor \sqrt{n^2 - 10n + 29} \rfloor$ when $n = 19941994.$ -/
theorem algebra_21564 (n : β) (hn : n = 19941994) : βReal.sqrt (n^2 - 10 * n + 29)β = 19941989 := by | import Mathlib
theorem algebra_21564 (n : β) (hn : n = 19941994) : βReal.sqrt (n^2 - 10 * n + 29)β = 19941989 := by
rw [hn]
have h_eq : ((19941994 : β) ^ 2 - 10 * (19941994 : β) + 29) = (19941989 : β) ^ 2 + 4 := by
norm_num
have h1 : Real.sqrt ((19941994 : β) ^ 2 - 10 * (19941994 : β) + 29) = Real.sqrt ((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... | have h3 : βReal.sqrt ((19941989 : β) ^ 2 + 4)οΏ½ = 19941989 := by
apply Int.floor_eq_iff.mpr
constructor
Β· nlinarith [Real.sqrt_nonneg ((19941989 : β) ^ 2 + 4), Real.sq_sqrt (show (0 : β) β€ ((19941989 : β) ^ 2 + 4) by positivity)]
Β· nlinarith [Real.sqrt_nonneg ((19941989 : β) ^ 2 + 4), Real.sq_sqrt (show... | import Mathlib
theorem algebra_21564 (n : β) (hn : n = 19941994) : βReal.sqrt (n^2 - 10 * n + 29)β = 19941989 := by
rw [hn]
have h_eq : ((19941994 : β) ^ 2 - 10 * (19941994 : β) + 29) = (19941989 : β) ^ 2 + 4 := by
norm_num
have h1 : Real.sqrt ((19941994 : β) ^ 2 - 10 * (19941994 : β) + 29) = Real.sqrt ((1... | import Mathlib
theorem algebra_21564 (n : β) (hn : n = 19941994) : βReal.sqrt (n^2 - 10 * n + 29)β = 19941989 := by
rw [hn]
have h_eq : ((19941994 : β) ^ 2 - 10 * (19941994 : β) + 29) = (19941989 : β) ^ 2 + 4 := by
norm_num
have h1 : Real.sqrt ((19941994 : β) ^ 2 - 10 * (19941994 : β) + 29) = Real.sqrt ((1... |
83 | math-word-problem | autoformalizer | Rationalize the denominator: $\frac{5}{\sqrt{3}}$ . | import Mathlib
/- Rationalize the denominator: $\frac{5}{\sqrt{3}}$ . -/
theorem algebra_21678 : (5 / Real.sqrt 3 : β) = 5 * Real.sqrt 3 / 3 := by | import Mathlib
theorem algebra_21678 : (5 / Real.sqrt 3 : β) = 5 * Real.sqrt 3 / 3 := by
have h1 : Real.sqrt 3 > 0 := by
positivity
field_simp
<;> ring_nf <;> norm_num <;> ring
<;> 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... | 3 := by
field_simp [Real.sqrt_eq_iff_mul_self_eq]
<;> ring
| import Mathlib
theorem algebra_21678 : (5 / Real.sqrt 3 : β) = 5 * Real.sqrt 3 / | import Mathlib
theorem algebra_21678 : (5 / Real.sqrt 3 : β) = 5 * Real.sqrt 3 / 3 := by
field_simp [Real.sqrt_eq_iff_mul_self_eq]
<;> ring
|
84 | math-word-problem | autoformalizer | The total number of digits used to write the page numbers of a math book is $6869$ (for example, a $10$-page book has $9$ one-digit page numbers and $1$ two-digit page number, using a total of $9+2=11$ digits), so the number of pages in this math book isΒ ___Β Β Β Β Β Β Β Β Β Β ___Β . | import Mathlib
def solution := 1994
/- The total number of digits used to write the page numbers of a math book is $6869$ (for example, a $10$-page book has $9$ one-digit page numbers and $1$ two-digit page number, using a total of $9+2=11$ digits), so the number of pages in this math book isΒ ___Β Β Β Β Β Β Β Β Β Β ___Β . -/
th... | import Mathlib
def solution := 1994
theorem number_theory_1544 : β i in Finset.range solution, (Nat.digits 10 (i + 1)).length = 6869 := by
native_decide | ### 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 : β) (hn : n β₯ 2) :
(n ^ 2 - 1) % n = 0 β§ (n ^ 2 - 1) % (n + 1) = 0 β§ (n ^ 2 - 1) % (n - 1) = 0 β§ (n ^ 2 - 1) % (n + 2) = 0 β§ (n ^ 2 - 1) % (n - 2) = 0 β§ (n ^ 2 - 1) % (n + 3) = 0 β§ (n ^ 2 - 1) % (n - 3) = 0 β§ (n ^ 2 - 1) % (n + 4) = 0 β§ (n ^ 2 - 1) % (n - 4) = 0 β§ (n ^ 2 - 1) % (n + 5) = 0 β§ (n ^ 2 - 1) % (n -... | import Mathlib
def solution := 1994
theorem number_theory_1544 | import Mathlib
def solution := 1994
theorem number_theory_1544 (n : β) (hn : n β₯ 2) :
(n ^ 2 - 1) % n = 0 β§ (n ^ 2 - 1) % (n + 1) = 0 β§ (n ^ 2 - 1) % (n - 1) = 0 β§ (n ^ 2 - 1) % (n + 2) = 0 β§ (n ^ 2 - 1) % (n - 2) = 0 β§ (n ^ 2 - 1) % (n + 3) = 0 β§ (n ^ 2 - 1) % (n - 3) = 0 β§ (n ^ 2 - 1) % (n + 4) = 0 β§ (n ^ 2 - 1... |
85 | math-word-problem | autoformalizer | Let $x$ and $y$ be positive real numbers such that $ x ^ { 3 } - y ^ { 3 } + ( x - y ) ^ { 3 } - 3 6 x y = 3 4 5 6 $. Find the value of $ x - y $. | import Mathlib
/- Let $x$ and $y$ be positive real numbers such that $ x ^ { 3 } - y ^ { 3 } + ( x - y ) ^ { 3 } - 3 6 x y = 3 4 5 6 $. Find the value of $ x - y $. -/
theorem algebra_13860 {x y : β} (hx : 0 < x) (hy : 0 < y)
(h : x^3 - y^3 + (x - y)^3 - 36 * x * y = 3456) :
x - y = 12 := by | import Mathlib
theorem algebra_13860 {x y : β} (hx : 0 < x) (hy : 0 < y)
(h : x^3 - y^3 + (x - y)^3 - 36 * x * y = 3456) :
x - y = 12 := by
have h1 : x^3 - y^3 + (x - y)^3 - 36 * x * y = 3456 := h
have h2 : x - y = 12 := by
nlinarith [sq_nonneg (x - y - 12), sq_nonneg (x + y - 24), sq_nonneg (x - 2 * y... | ### 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... | y)^3 = 36 * x * y + 3456 := by linarith
have h2 : (x - y)^3 = 36 * x * y + 3456 - x^3 + y^3 := by linarith
have h3 : x - y = 12 := by
nlinarith [sq_nonneg (x - y), sq_nonneg (x + y), sq_nonneg (x - y - 12)]
exact h3
| import Mathlib
theorem algebra_13860 {x y : β} (hx : 0 < x) (hy : 0 < y)
(h : x^3 - y^3 + (x - y)^3 - 36 * x * y = 3456) :
x - y = 12 := by
have h1 : x^3 - y^3 + (x - | import Mathlib
theorem algebra_13860 {x y : β} (hx : 0 < x) (hy : 0 < y)
(h : x^3 - y^3 + (x - y)^3 - 36 * x * y = 3456) :
x - y = 12 := by
have h1 : x^3 - y^3 + (x - y)^3 = 36 * x * y + 3456 := by linarith
have h2 : (x - y)^3 = 36 * x * y + 3456 - x^3 + y^3 := by linarith
have h3 : x - y = 12 := by
... |
86 | math-word-problem | autoformalizer | Find the remainder when ${{2018}^{2019}}$ is divided by $2019$.
| import Mathlib
/- Find the remainder when ${{2018}^{2019}}$ is divided by $2019$.
-/
theorem number_theory_4606
: 2018 ^ 2019 % 2019 = 2018 := by | import Mathlib
theorem number_theory_4606
: 2018 ^ 2019 % 2019 = 2018 := by
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... | (n : β)
(hβ : 1 < n)
(hβ : β x : β, x β€ n β x β£ n) :
n = 2 := by
have hβ := hβ 1
have hβ := hβ 2
have hβ := hβ 3
have hβ
:= hβ 4
have hβ := hβ 5
have hβ := hβ 6
have hβ := hβ 7
have hββ := hβ 8
have hββ := hβ 9
have hββ := hβ 10
have hββ := hβ 11
have hββ
:= hβ 12
have hββ := hβ 1... | import Mathlib
theorem number_theory_4606
| import Mathlib
theorem number_theory_4606
(n : β)
(hβ : 1 < n)
(hβ : β x : β, x β€ n β x β£ n) :
n = 2 := by
have hβ := hβ 1
have hβ := hβ 2
have hβ := hβ 3
have hβ
:= hβ 4
have hβ := hβ 5
have hβ := hβ 6
have hβ := hβ 7
have hββ := hβ 8
have hββ := hβ 9
have hββ := hβ 10
have hββ :... |
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