problem stringlengths 10 5.15k | answer dict |
|---|---|
Given $\cos \alpha = \frac{1}{7}$ and $\cos (\alpha-\beta) = \frac{13}{14}$, with $0 < \beta < \alpha < \frac{\pi}{2}$, find $\beta$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A novice economist-cryptographer received a cryptogram from a ruler which contained a secret decree about implementing an itemized tax on a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that a greater amount could not be collected in that market. Unfortunately, the economist-cryptographer made an error in decrypting the cryptogram—the digits of the tax revenue amount were identified in the wrong order. Based on erroneous data, a decision was made to introduce an itemized tax on producers of 90 monetary units per unit of goods. It is known that the market demand is represented by \( Q_d = 688 - 4P \), and the market supply is linear. When there are no taxes, the price elasticity of market supply at the equilibrium point is 1.5 times higher than the modulus of the price elasticity of the market demand function. After the tax was introduced, the producer price fell to 64 monetary units.
1) Restore the market supply function.
2) Determine the amount of tax revenue collected at the chosen rate.
3) Determine the itemized tax rate that would meet the ruler's decree.
4) What is the amount of tax revenue specified by the ruler to be collected? | {
"answer": "6480",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area in the plane contained by the graph of
\[|2x + 3y| + |2x - 3y| \le 12.\] | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, it is known that $\cos A= \frac {3}{5},\cos B= \frac {5}{13}$, and $AC=3$. Find the length of $AB$. | {
"answer": "\\frac {14}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the probability that a point $P$ randomly selected within the plane region $N$ defined by the system of inequalities $\begin{cases} 0 \leq x \leq 1 \\ 0 \leq y \leq e \end{cases}$ also lies within the plane region $M$ defined by the system of inequalities $\begin{cases} x + y \geq 1 \\ e^x - y \geq 0 \\ 0 \leq x \leq 1 \end{cases}$. | {
"answer": "1 - \\frac{3}{2e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $\cos A \cos B = \sin A \sin B - \frac{\sqrt{2}}{2}$.
$(1)$ Find the measure of angle $C$.
$(2)$ Given $b=4$ and the area of $\triangle ABC$ is $6$, find:
① the value of side $c$;
② the value of $\cos \left(2B-C\right)$. | {
"answer": "\\frac{\\sqrt{2}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest number, with its digits all different, whose digits add up to 19, and does not contain the digit '0'? | {
"answer": "982",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four of the five vertices of a polygon shaped as a right trapezoid are (5, 11), (16, 11), (16, -2), and (5, -2); it includes a semicircle with a diameter along the bottom base, centered at (10.5, -2) and radius 5.5. Determine the total area bounded by the lower base of the trapezoid and the semicircle. | {
"answer": "15.125\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) When $x \in \left[\frac{\pi}{6}, \frac{7\pi}{6}\right]$, find the maximum value of the function $y = 3 - \sin x - 2\cos^2 x$.
(2) Given that $5\sin\beta = \sin(2\alpha + \beta)$ and $\tan(\alpha + \beta) = \frac{9}{4}$, find $\tan \alpha$. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two lines $l_1: y = 2x$, $l_2: y = -2x$, and a line $l$ passing through point $M(-2, 0)$ intersects $l_1$ and $l_2$ at points $A$ and $B$, respectively, where point $A$ is in the third quadrant, point $B$ is in the second quadrant, and point $N(1, 0)$;
(1) If the area of $\triangle NAB$ is 16, find the equation of line $l$;
(2) Line $AN$ intersects $l_2$ at point $P$, and line $BN$ intersects $l_1$ at point $Q$. If the slopes of line $l$ and $PQ$ both exist, denoted as $k_1$ and $k_2$ respectively, determine whether $\frac {k_{1}}{k_{2}}$ is a constant value? If it is a constant value, find this value; if not, explain why. | {
"answer": "-\\frac {1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain high school, there are 300 freshmen students, including 180 boys and 120 girls. In order to understand the height information of the freshmen students, a stratified random sampling method is used to select samples according to the proportion of the sample size. It is found that the average height of the boys' sample is $\overline{x}=170$ (unit: $cm$) with a variance of ${s}_{1}^{2}=14$, and the average height of the girls' sample is $\overline{y}=160$ (unit: $cm$) with a variance of ${s}_{2}^{2}=24$. Based on the above data, estimate the average height and variance of the freshmen students in this high school. | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a > b > 0$, and $a + b = 2$, find the minimum value of $$\frac {3a-b}{a^{2}+2ab-3b^{2}}$$. | {
"answer": "\\frac {3+ \\sqrt {5}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph of the function y=sin(2x+φ) is shifted to the left by π/6 units along the x-axis, resulting in an even function graph. Determine the value of φ such that the equation 2(x + π/6) + φ = -x + 2πk is satisfied for some integer k. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a, b, c \) be real numbers satisfying \( 9a^2 + 4b^2 + 36c^2 = 4 \). Find the minimum value of \( 3a + 6b + 12c \). | {
"answer": "-2\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the remainder when $2^{2^{2^2}}$ is divided by $500$. | {
"answer": "536",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), given that \( \frac{\cos A}{\sin B} + \frac{\cos B}{\sin A} = 2 \) and the perimeter of \( \triangle ABC \) is 12, find the maximum possible area of the triangle. | {
"answer": "36(3 - 2\\sqrt{2})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x) = 3x^2 + 2x + 1$, if $\int_{-1}^{1} f(x)\,dx = 2f(a)$, then $a = \_\_\_\_\_\_$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos (2x+\varphi)$, where $|\varphi| \leqslant \frac{\pi}{2}$, if $f\left( \frac{8\pi}{3}-x\right)=-f(x)$, determine the horizontal shift required to obtain the graph of $y=\sin 2x$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ellipse C is defined by the equation $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ (a > b > 0), and its four vertices form a rhombus with side length $$\sqrt {3}$$ and area $$2 \sqrt {2}$$.
(1) Find the equation of ellipse C.
(2) A line l passes through point Q(0, -2) and intersects ellipse C at points A and B. If the product of the slopes of OA and OB (where O is the origin) is -1, find the length of segment AB. | {
"answer": "\\frac {4 \\sqrt {21}}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\alpha$ and $\beta$ are two interior angles of an oblique triangle, if $\frac{{\cos \alpha - \sin \alpha}}{{\cos \alpha + \sin \alpha}} = \cos 2\beta$, then the minimum value of $\tan \alpha + \tan \beta$ is ______. | {
"answer": "-\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the product of all constants $t$ such that the quadratic $x^2 + tx - 12$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. | {
"answer": "1936",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, there is a point $P$ on the ellipse such that the distance to the two foci of the ellipse, $F_{1}$ and $F_{2}$, satisfies $|PF_{1}|+|PF_{2}|=10$, and the eccentricity $e= \dfrac {4}{5}$.
$(1)$ Find the standard equation of the ellipse.
$(2)$ If $\angle F_{1}PF_{2}=60^{\circ}$, find the area of $\triangle F_{1}PF_{2}$. | {
"answer": "3 \\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an island, there are only knights, who always tell the truth, and liars, who always lie. One fine day, 30 islanders sat around a round table. Each of them can see everyone except himself and his neighbors. Each person in turn said the phrase: "Everyone I see is a liar." How many liars were sitting at the table? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Rivertown, car plates each contain three symbols: two letters followed by a digit. The first letter is chosen from the set ${A, B, G, H, T}$, the second letter from ${E, I, O, U}$, and the digit from $0$ to $9$.
To accommodate an increase in the number of cars, Rivertown decides to expand each set by adding new symbols. Rivertown adds two new letters and one new digit. These additional symbols can be added entirely to one set or distributed among the sets. Determine the largest number of ADDITIONAL car plates that can be made possible by the most advantageous distribution of these new symbols. | {
"answer": "130",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chloe wants to purchase a jacket that costs $\$45.50$. She checks her purse and finds she has four $\$10$ bills, ten quarters, and some nickels and dimes. What is the minimum number of dimes that she must have if she has 15 nickels? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If one vertex and the two foci of an ellipse form an equilateral triangle, determine the eccentricity of this ellipse. | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, given the parametric equations of circle $C_{1}$ as $\left\{\begin{array}{l}x=2+2\cos\alpha\\ y=1+2\sin\alpha\end{array}\right.$ ($\alpha$ is the parameter), establish a polar coordinate system with the coordinate origin as the pole and the positive x-axis as the polar axis.<br/>$(1)$ Find the polar equation of $C_{1}$;<br/>$(2)$ If the polar equation of line $C_{2}$ is $θ=\frac{π}{4}$ ($ρ∈R$), and the intersection points of $C_{2}$ and $C_{1}$ are $P$ and $Q$, find the area of $\triangle C_{1}PQ$. | {
"answer": "\\frac{\\sqrt{7}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the polar coordinate equation of curve C is $\rho = \sqrt{3}$, and the parametric equations of line l are $x = 1 + \frac{\sqrt{2}}{2}t$, $y = \frac{\sqrt{2}}{2}t$ (where t is the parameter), and the parametric equations of curve M are $x = \cos \theta$, $y = \sqrt{3} \sin \theta$ (where $\theta$ is the parameter).
1. Write the Cartesian coordinate equation for curve C and line l.
2. If line l intersects curve C at points A and B, and P is a moving point on curve M, find the maximum area of triangle ABP. | {
"answer": "\\frac{3\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The points $A$ , $B$ , $C$ , $D$ , and $E$ lie in one plane and have the following properties: $AB = 12, BC = 50, CD = 38, AD = 100, BE = 30, CE = 40$ .
Find the length of the segment $ED$ . | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the share of gold in the currency structure of the National Welfare Fund (NWF) as of December 1, 2022, using one of the following methods:
First Method:
a) Find the total amount of NWF funds allocated in gold as of December 1, 2022:
\[GOLD_{22} = 1388.01 - 41.89 - 2.77 - 478.48 - 309.72 - 0.24 = 554.91 \, (\text{billion rubles})\]
b) Determine the share of gold in the currency structure of NWF funds as of December 1, 2022:
\[\alpha_{22}^{GOLD} = \frac{554.91}{1388.01} \approx 39.98\% \]
c) Calculate by how many percentage points and in which direction the share of gold in the currency structure of NWF funds changed over the review period:
\[\Delta \alpha^{GOLD} = \alpha_{22}^{GOLD} - \alpha_{21}^{GOLD} = 39.98 - 31.8 = 8.18 \approx 8.2 \, (\text{p.p.})\]
Second Method:
a) Determine the share of euro in the currency structure of NWF funds as of December 1, 2022:
\[\alpha_{22}^{EUR} = \frac{41.89}{1388.01} \approx 3.02\% \]
b) Determine the share of gold in the currency structure of NWF funds as of December 1, 2022:
\[\alpha_{22}^{GOLD} = 100 - 3.02 - 0.2 - 34.47 - 22.31 - 0.02 = 39.98\%\]
c) Calculate by how many percentage points and in which direction the share of gold in the currency structure of NWF funds changed over the review period:
\[\Delta \alpha^{GOLD} = \alpha_{22}^{GOLD} - \alpha_{21}^{GOLD} = 39.98 - 31.8 = 8.18 \approx 8.2 \, (\text{p.p.})\] | {
"answer": "8.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a factory, a total of $100$ parts were produced. Among them, A produced $0$ parts, with $35$ being qualified. B produced $60$ parts, with $50$ being qualified. Let event $A$ be "Selecting a part from the $100$ parts at random, and the part is qualified", and event $B$ be "Selecting a part from the $100$ parts at random, and the part is produced by A". Then, the probability of $A$ given $B$ is \_\_\_\_\_\_. | {
"answer": "\\dfrac{7}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x) = x - 3$ and $g(x) = x/2$. Compute \[f(g^{-1}(f^{-1}(g(f^{-1}(g(f(23))))))).\] | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\alpha $, $\beta \in (0, \frac{π}{2})$, $\sin \alpha = \frac{{\sqrt{5}}}{5}$, $\cos \beta = \frac{1}{{\sqrt{10}}}$, find the value of $\alpha - \beta$. | {
"answer": "-\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$: $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ with its left and right foci being $F_{1}$ and $F_{2}$ respectively, and its eccentricity is $\dfrac {1}{2}$. Let $M$ be any point on the ellipse and the perimeter of $\triangle MF_{1}F_{2}$ equals $6$.
(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) With $M$ as the center and $MF_{1}$ as the radius, draw a circle $M$. When circle $M$ and the line $l$: $x=4$ have common points, find the maximum area of $\triangle MF_{1}F_{2}$. | {
"answer": "\\dfrac { \\sqrt {15}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Construct a new shape by adding an eighth unit cube to the previously described configuration of seven cubes. Place this new cube adjacent to one of the six outlying cubes from the central cube. What is the ratio of the volume in cubic units to the surface area in square units for this new configuration?
A) $\frac{8}{31}$
B) $\frac{8}{32}$
C) $\frac{8}{33}$
D) $\frac{8}{34}$ | {
"answer": "\\frac{8}{33}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $(2x-1)^{2015} = a_{0} + a_{1}x + a_{2}x^{2} + \ldots + a_{2015}x^{2015}$ ($x \in \mathbb{R}$), evaluate the expression $\frac {1}{2}+ \frac {a_{2}}{2^{2}a_{1}}+ \frac {a_{3}}{2^{3}a_{1}}+\ldots+ \frac {a_{2015}}{2^{2015}a_{1}}$. | {
"answer": "\\frac {1}{4030}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y=kx+1$ intersects the parabola $C: x^2=4y$ at points $A$ and $B$, and a line $l$ is parallel to $AB$ and is tangent to the parabola $C$ at point $P$, find the minimum value of the area of triangle $PAB$ minus the length of $AB$. | {
"answer": "-\\frac{64}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $P$ is a moving point on the parabola $y^{2}=4x$. The minimum value of the sum of the distances from point $P$ to point $A(0,-1)$ and from point $P$ to the line $x=-1$ is ______. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \( AB \) and \( CD \) intersect at \( E \). If \(\triangle BCE\) is equilateral and \(\triangle ADE\) is a right-angled triangle, what is the value of \( x \)? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\begin{align*}
4a + 2b + 5c + 8d &= 67 \\
4(d+c) &= b \\
2b + 3c &= a \\
c + 1 &= d \\
\end{align*}
Given the above system of equations, find \(a \cdot b \cdot c \cdot d\). | {
"answer": "\\frac{1201 \\times 572 \\times 19 \\times 124}{105^4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, if median $\overline{AD}$ makes an angle of $30^\circ$ with side $\overline{BC},$ then find the value of $|\cot B - \cot C|.$ | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \\(\{x_{1},x_{2},x_{3},x_{4}\} \subseteq \{x | (x-3) \cdot \sin \pi x = 1, x > 0\}\\), find the minimum value of \\(x_{1}+x_{2}+x_{3}+x_{4}\\). | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many whole numbers between 1 and 500 do not contain the digit 2? | {
"answer": "323",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle $PQR$, where $\angle P = 90^\circ$, suppose $\cos Q = \frac{4}{5}$. The length of side $PQ$ (adjacent to $\angle Q$) is $12$. What is the length of $PR$ (hypotenuse)? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a triangle \(ABC\) and two lines \(AD\) and \(BE\), where \(D\) is the midpoint of \(BC\) and \(E\) lies on \(CA\). The lines \(AD\) and \(BE\) meet at \(Z\), the midpoint of \(AD\). What is the ratio of the length \(CE\) to the length \(EA\)? | {
"answer": "2:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 6 by 5 grid, how many 10-step paths are there from $W$ to $X$ that must pass through a point $H$? Assume $W$ is located at the top-left corner, $X$ at the bottom-right corner, and $H$ is three squares to the right and two squares down from $W$. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $E$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, its eccentricity is $\frac{\sqrt{2}}{2}$, point $F$ is the left focus of the ellipse, point $A$ is the right vertex, and point $B$ is the upper vertex. Additionally, $S_{\triangle ABF} = \frac{\sqrt{2}+1}{2}$.
(I) Find the equation of the ellipse $E$;
(II) If the line $l$: $x - 2y - 1 = 0$ intersects the ellipse $E$ at points $P$ and $Q$, find the perimeter and area of $\triangle FPQ$. | {
"answer": "\\frac{\\sqrt{10}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | {
"answer": "12\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the moving point $P$ is on the line $y=x+1$, and the moving point $Q$ is on the curve $x^{2}=-2y$, calculate the minimum value of $|PQ|$. | {
"answer": "\\frac{\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A solid is formed by rotating a triangle with sides of lengths 3, 4, and 5 around the line containing its shortest side. Find the surface area of this solid. | {
"answer": "36\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate \(14 \cdot 31\) and \(\left\lfloor\frac{2+\sqrt{2}}{2}\right\rfloor + \left\lfloor\frac{3+\sqrt{3}}{3}\right\rfloor + \left\lfloor\frac{4+\sqrt{4}}{4}\right\rfloor + \cdots + \left\lfloor\frac{1989+\sqrt{1989}}{1989}\right\rfloor + \left\lfloor\frac{1990+\sqrt{1990}}{1990}\right\rfloor\). | {
"answer": "1989",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tim plans a weeklong prank to repeatedly steal Nathan's fork during lunch. He involves different people each day:
- On Monday, he convinces Joe to do it.
- On Tuesday, either Betty or John could undertake the prank.
- On Wednesday, there are only three friends from whom he can seek help, as Joe, Betty, and John are not available.
- On Thursday, neither those involved earlier in the week nor Wednesday's helpers are willing to participate, but four new individuals are ready to help.
- On Friday, Tim decides he could either do it himself or get help from one previous assistant who has volunteered again.
How many different combinations of people could be involved in the prank over the week? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A road of 1500 meters is being repaired. In the first week, $\frac{5}{17}$ of the total work was completed, and in the second week, $\frac{4}{17}$ was completed. What fraction of the total work was completed in these two weeks? And what fraction remains to complete the entire task? | {
"answer": "\\frac{8}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given functions $f(x)=xe^x$ and $g(x)=-\frac{lnx}{x}$, if $f(x_{1})=g(x_{2})=t\left( \gt 0\right)$, find the maximum value of $\frac{{x}_{1}}{{x}_{2}{e}^{t}}$. | {
"answer": "\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0, b>0$) with a point C on it, a line passing through the center of the hyperbola intersects the hyperbola at points A and B. Let the slopes of the lines AC and BC be $k_1$ and $k_2$ respectively. Find the eccentricity of the hyperbola when $\frac{2}{k_1 k_2} + \ln{k_1} + \ln{k_2}$ is minimized. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence of positive terms $\{a\_n\}$, with $a\_1=2$, $(a\_n+1)a_{n+2}=1$, and $a\_2=a\_6$, find the value of $a_{11}+a_{12}$. | {
"answer": "\\frac{1}{9}+\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lead used to make twelve solid lead balls, each with a radius of 2 cm, is reused to form a single larger solid lead sphere. What is the radius of this larger sphere? | {
"answer": "\\sqrt[3]{96}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin\alpha= \frac {2 \sqrt {2}}{3}$, $\cos(\alpha+\beta)=- \frac {1}{3}$, and $\alpha, \beta\in(0, \frac {\pi}{2})$, determine the value of $\sin(\alpha-\beta)$. | {
"answer": "\\frac {10 \\sqrt {2}}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $\triangle ABC$ has a right angle at $C$, $\angle A = 45^\circ$, and $AC=12$. Find the radius of the incircle of $\triangle ABC$. | {
"answer": "6 - 3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an opaque bag, there are four identical balls labeled with numbers $3$, $4$, $5$, and $6$ respectively. Outside the bag, there are two balls labeled with numbers $3$ and $6$. Determine the probability that a triangle with the drawn ball and the numbers on the two balls outside the bag forms an isosceles triangle. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify and write the result as a common fraction: $$\sqrt[4]{\sqrt[3]{\sqrt{\frac{1}{65536}}}}$$ | {
"answer": "\\frac{1}{\\sqrt[3]{4}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two fixed points on the plane, \\(A(-2,0)\\) and \\(B(2,0)\\), and a moving point \\(T\\) satisfying \\(|TA|+|TB|=2 \sqrt {6}\\).
\\((\\)I\\()\\) Find the equation of the trajectory \\(E\\) of point \\(T\\);
\\((\\)II\\()\\) A line passing through point \\(B\\) and having the equation \\(y=k(x-2)\\) intersects the trajectory \\(E\\) at points \\(P\\) and \\(Q\\) \\((k\neq 0)\\). If \\(PQ\\)'s midpoint is \\(N\\) and \\(O\\) is the origin, the line \\(ON\\) intersects the line \\(x=3\\) at point \\(M\\). Find the maximum value of \\( \dfrac {|PQ|}{|MB|}\\). | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the price of a stamp is 45 cents, what is the maximum number of stamps that could be purchased with $50? | {
"answer": "111",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that x > 0, y > 0, and x + 2y = 4, find the minimum value of $$\frac {(x+1)(2y+1)}{xy}$$. | {
"answer": "\\frac {9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f : N \to N$ be a strictly increasing function such that $f(f(n))= 3n$ , for all $n \in N$ . Find $f(2010)$ .
Note: $N = \{0,1,2,...\}$ | {
"answer": "3015",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is currently 3:00:00 PM, as shown on a 12-hour digital clock. In 300 hours, 55 minutes, and 30 seconds, what will the time be and what is the sum of the hours, minutes, and seconds? | {
"answer": "88",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pyramid has a square base $ABCD$ and a vertex $E$. The area of square $ABCD$ is $256$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $120$ and $136$, respectively. The distance from vertex $E$ to the midpoint of side $AB$ is $17$. What is the volume of the pyramid?
- **A)** $1024$
- **B)** $1200$
- **C)** $1280$
- **D)** $1536$
- **E)** $1600$ | {
"answer": "1280",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\overrightarrow{m}=(\sin \omega x,-1)$, $\overrightarrow{n}=(1,- \sqrt {3}\cos \omega x)$ where $x\in\mathbb{R}$, $\omega > 0$, and $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$, and the distance between a certain highest point and its adjacent lowest point on the graph of function $f(x)$ is $5$,
$(1)$ Find the interval of monotonic increase for the function $f(x)$;
$(2)$ If $f\left( \dfrac {3\theta}{\pi}\right)= \dfrac {6}{5}$ where $\theta\in\left(- \dfrac {5\pi}{6}, \dfrac {\pi}{6}\right)$, then find the value of $f\left( \dfrac {6\theta}{\pi}+1\right)$. | {
"answer": "\\dfrac {48}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be triangle such that $|AB| = 5$ , $|BC| = 9$ and $|AC| = 8$ . The angle bisector of $\widehat{BCA}$ meets $BA$ at $X$ and the angle bisector of $\widehat{CAB}$ meets $BC$ at $Y$ . Let $Z$ be the intersection of lines $XY$ and $AC$ . What is $|AZ|$ ? $
\textbf{a)}\ \sqrt{104}
\qquad\textbf{b)}\ \sqrt{145}
\qquad\textbf{c)}\ \sqrt{89}
\qquad\textbf{d)}\ 9
\qquad\textbf{e)}\ 10
$ | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When \( q(x) = Dx^4 + Ex^2 + Fx + 6 \) is divided by \( x - 2 \), the remainder is 14. Find the remainder when \( q(x) \) is divided by \( x + 2 \). | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence ${a_n}$, $a_1=1$ and $a_n a_{n+1} + \sqrt{3}(a_n - a_{n+1}) + 1 = 0$. Determine the value of $a_{2016}$. | {
"answer": "2 - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $xOy$, the ordinary equation of curve $C_1$ is $x^2+y^2-2x=0$. Establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C_2$ is $\rho^{2}= \frac {3}{1+2\sin^{2}\theta }$.
(I) Find the parametric equation of $C_1$ and the rectangular equation of $C_2$;
(II) The ray $\theta= \frac {\pi}{3}(\rho\geq0)$ intersects $C_1$ at a point $A$ distinct from the pole, and $C_2$ at point $B$. Find $|AB|$. | {
"answer": "\\frac { \\sqrt {30}}{5}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four steel balls, each with a radius of 1, are to be completely packed into a container shaped as a regular tetrahedron. What is the minimum height of the tetrahedron? | {
"answer": "2 + \\frac{2 \\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the exponential function y=f(x) whose graph passes through the point $\left( \frac{1}{2}, \frac{\sqrt{2}}{2} \right)$, find the value of $\log_2 f(2)$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mayuki walks once around a track shaped with straight sides and semicircular ends at a constant speed daily. The track has a width of \(4\) meters, and it takes her \(24\) seconds longer to walk around the outside edge than the inside edge. Determine Mayuki's speed in meters per second. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In writing the integers from 20 through 199 inclusive, how many times is the digit 7 written? | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If 2023 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the base areas of two cylinders be $S_1$ and $S_2$, and their volumes be $\upsilon_1$ and $\upsilon_2$, respectively. If their lateral areas are equal, and $$\frac {S_{1}}{S_{2}}= \frac {16}{9},$$ then the value of $$\frac {\upsilon_{1}}{\upsilon_{2}}$$ is \_\_\_\_\_\_. | {
"answer": "\\frac {4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $\xi$ follows a normal distribution $N(1,4)$, if $p(\xi > 4)=0.1$, then $p(-2 \leqslant \xi \leqslant 4)=$ _____ . | {
"answer": "0.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $\begin{cases} & x=2\sqrt{3}\cos a \\ & y=2\sin a \end{cases}$, where $a$ is a parameter and $a \in (0, \pi)$. In the polar coordinate system with the origin $O$ as the pole and the positive half axis of $x$ as the polar axis, the polar coordinates of point $P$ are $(4\sqrt{2}, \frac{\pi}{4})$, and the polar equation of line $l$ is $\rho \sin(\theta - \frac{\pi}{4}) + 5\sqrt{2} = 0$.
(1) Find the Cartesian equation of line $l$ and the general equation of curve $C$.
(2) Suppose $Q$ is a moving point on curve $C$, and $M$ is the midpoint of segment $PQ$. Find the maximum distance from point $M$ to the line $l$. | {
"answer": "6\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that angle DEF is a right angle and the sides of triangle DEF are the diameters of semicircles, the area of the semicircle on segment DE equals $18\pi$, and the arc of the semicircle on segment DF has length $10\pi$. Determine the radius of the semicircle on segment EF. | {
"answer": "\\sqrt{136}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with a radius of 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$ | {
"answer": "\\sqrt{59}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square \( ABCD \) has a side length of 12 inches. A segment \( AE \) is drawn where \( E \) is on side \( DC \) and \( DE \) is 5 inches long. The perpendicular bisector of \( AE \) intersects \( AE, AD, \) and \( BC \) at points \( M, P, \) and \( Q \) respectively. The ratio of the segments \( PM \) to \( MQ \) is: | {
"answer": "5:19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perimeter of quadrilateral PQRS, made from two similar right-angled triangles PQR and PRS, is given that the length of PQ is 3, the length of QR is 4, and ∠PRQ = ∠PSR. Find the perimeter of PQRS. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(\alpha)= \dfrac {\sin (\alpha- \dfrac {5\pi}{2})\cos ( \dfrac {3\pi}{2}+\alpha)\tan (\pi-\alpha)}{\tan (-\alpha-\pi)\sin (\pi-\alpha)}$.
(1) Simplify $f(\alpha)$
(2) If $\cos (\alpha+ \dfrac {3\pi}{2})= \dfrac {1}{5}$ and $\alpha$ is an angle in the second quadrant, find the value of $f(\alpha)$. | {
"answer": "\\dfrac{2\\sqrt{6}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest integral value of $n$ such that the quadratic equation
\[3x(nx+3)-2x^2-9=0\]
has no real roots.
A) -2
B) -1
C) 0
D) 1 | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Susie was given $\$1,500$ for her birthday. She decides to invest the money in a bank account that earns $12\%$ interest, compounded quarterly. How much total interest will Susie have earned 4 years later? | {
"answer": "901.55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\cos x\cdot\ln x$, $f(x_{0})=f(x_{1})=0(x_{0}\neq x_{1})$, find the minimum value of $|x_{0}-x_{1}|$ ___. | {
"answer": "\\dfrac {\\pi}{2}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using arithmetic operation signs, write the largest natural number using two twos. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a high school senior year has 12 classes, with exactly 8 classes to be proctored by their own homeroom teachers, find the number of different proctoring arrangements for the math exam. | {
"answer": "4455",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $\overrightarrow{a}^{2}=(5 \overrightarrow{a}-4 \overrightarrow{b})\cdot \overrightarrow{b}$, find the minimum value of $\cos < \overrightarrow{a}, \overrightarrow{b} >$. | {
"answer": "\\frac {4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ottó decided to assign a number to each pair \((x, y)\) and denote it as \((x \circ y)\). He wants the following relationships to hold:
a) \(x \circ y = y \circ x\)
b) \((x \circ y) \circ z = (x \circ z) \circ (y \circ z)\)
c) \((x \circ y) + z = (x + z) \circ (y + z)\).
What number should Ottó assign to the pair \((1975, 1976)\)? | {
"answer": "1975.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer $x$ that, when multiplied by $900$, produces a product that is a multiple of $1152$? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the set of values for parameter \(a\) for which the sum of the cubes of the roots of the equation \(x^{2} + ax + a + 1 = 0\) is equal to 1. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equation of a line is given by $Ax+By=0$. If we choose two different numbers from the set $\{1, 2, 3, 4, 5\}$ to be the values of $A$ and $B$ each time, then the number of different lines that can be obtained is ． | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of all digits used in the numbers 1, 2, 3, ..., 999 is . | {
"answer": "13500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $PQR$ has side lengths $PQ=160, QR=300$, and $PR=240$. Lines $m_P, m_Q$, and $m_R$ are drawn parallel to $\overline{QR}, \overline{RP}$, and $\overline{PQ}$, respectively, such that the intersections of $m_P, m_Q$, and $m_R$ with the interior of $\triangle PQR$ are segments of lengths $75, 60$, and $20$, respectively. Find the perimeter of the triangle whose sides lie on lines $m_P, m_Q$, and $m_R$. | {
"answer": "155",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sara lists the whole numbers from 1 to 50. Lucas copies Sara's numbers, replacing each occurrence of the digit '3' with the digit '2'. Calculate the difference between Sara's sum and Lucas's sum. | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dana goes first and Carl's coin lands heads with probability $\frac{2}{7}$, and Dana's coin lands heads with probability $\frac{3}{8}$. Find the probability that Carl wins the game. | {
"answer": "\\frac{10}{31}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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