problem stringlengths 10 5.15k | answer dict |
|---|---|
From the numbers \\(1, 2, \ldots, 100\\) totaling \\(100\\) numbers, three numbers \\(x, y, z\\) are chosen in sequence. The probability that these three numbers satisfy \\(x+z=2y\\) is __________. | {
"answer": "\\dfrac{1}{198}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two vertical towers, \( AB \) and \( CD \), are located \( 16 \mathrm{~m} \) apart on flat ground. Tower \( AB \) is \( 18 \mathrm{~m} \) tall and tower \( CD \) is \( 30 \mathrm{~m} \) tall. Ropes are tied from \( A \) to \( C \) and from \( B \) to \( C \). Assuming the ropes are taut, calculate the total length of rope, in \(\mathrm{m}\). | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $0 < α < \dfrac {π}{2}$, $- \dfrac {π}{2} < β < 0$, $\cos ( \dfrac {π}{4}+α)= \dfrac {1}{3}$, and $\cos ( \dfrac {π}{4}-β)= \dfrac { \sqrt {3}}{3}$, find $\cos (α+β)$. | {
"answer": "\\dfrac {5 \\sqrt {3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, let $P$ be a point on the ellipse such that the projection of $P$ onto the $x$-axis is the left focus $F_1$. Let $A$ be the intersection point of the ellipse with the positive semi-major axis, and let $B$ be the intersection point of the ellipse with the positive semi-minor axis, such that $AB$ is parallel to $OP$ (where $O$ is the origin). Find the eccentricity of the ellipse. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and they satisfy the equation $\sin A + \sin B = [\cos A - \cos (π - B)] \sin C$.
1. Determine whether triangle $ABC$ is a right triangle and explain your reasoning.
2. If $a + b + c = 1 + \sqrt{2}$, find the maximum area of triangle $ABC$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\alpha\) and \(\beta\) be the two real roots of the quadratic equation \(x^{2} - 2kx + k + 20 = 0\). Find the minimum value of \((\alpha+1)^{2} + (\beta+1)^{2}\), and determine the value of \(k\) for which this minimum value is achieved. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \(\triangle PMO\), \(PM = 6\sqrt{3}\), \(PO = 12\sqrt{3}\), and \(S\) is a point on \(MO\) such that \(PS\) is the angle bisector of \(\angle MPO\). Let \(T\) be the reflection of \(S\) across \(PM\). If \(PO\) is parallel to \(MT\), find the length of \(OT\). | {
"answer": "2\\sqrt{183}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, the side \(AB\) of \(\triangle ABC\) is divided into \(n\) equal parts (\(n > 1990\)). Through the \(n-1\) division points, lines parallel to \(BC\) are drawn intersecting \(AC\) at points \(B_i, C_i\) respectively for \(i=1, 2, 3, \cdots, n-1\). What is the ratio of the area of \(\triangle AB_1C_1\) to the area of the quadrilateral \(B_{1989} B_{1990} C_{1990} C_{1989}\)? | {
"answer": "1: 3979",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three three-digit numbers, with all digits except zero being used in their digits, sum up to 1665. In each number, the first digit was swapped with the last digit. What is the sum of the new numbers? | {
"answer": "1665",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a $3 \times 3$ table, numbers are placed such that each number is 4 times smaller than the number in the adjacent cell to the right and 3 times smaller than the number in the adjacent cell above. The sum of all the numbers in the table is 546. Find the number in the central cell. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a table lie 140 different cards with numbers $3, 6, 9, \ldots, 417, 420$ (each card has exactly one number, and each number appears exactly once). In how many ways can you choose 2 cards so that the sum of the numbers on the selected cards is divisible by $7?$ | {
"answer": "1390",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circular poles, with diameters of 8 inches and 24 inches, touch each other at a single point. A wire is wrapped around them such that it goes around the entire configuration including a straight section tangential to both poles. Find the length of the shortest wire that sufficiently encloses both poles.
A) $16\sqrt{3} + 32\pi$
B) $16\sqrt{3} + 24\pi$
C) $12\sqrt{3} + 32\pi$
D) $16\sqrt{3} + 16\pi$
E) $8\sqrt{3} + 32\pi$ | {
"answer": "16\\sqrt{3} + 32\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a circle of radius 1 with center $O$ down in the diagram. Points $A$, $B$, $C$, and $D$ lie on the circle such that $\angle AOB = 120^\circ$, $\angle BOC = 60^\circ$, and $\angle COD = 180^\circ$. A point $X$ lies on the minor arc $\overarc{AC}$. If $\angle AXB = 90^\circ$, find the length of $AX$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of the digits of \(10^{2008} - 2008\)? | {
"answer": "18063",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A coordinate system and parametric equations problem (4-4):
In the rectangular coordinate system $x0y$, the parametric equations of line $l$ are given by $\begin{cases} x = \frac{1}{2}t \ y = \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}t \end{cases}$, where $t$ is the parameter. If we establish a polar coordinate system with point $O$ in the rectangular coordinate system $x0y$ as the pole, $0x$ as the polar axis, and the same length unit, the polar equation of curve $C$ is given by $\rho = 2 \cos(\theta - \frac{\pi}{4})$.
(1) Find the slope angle of line $l$.
(2) If line $l$ intersects curve $C$ at points $A$ and $B$, find the length of $AB$. | {
"answer": "\\frac{\\sqrt{10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\frac12$ , and no side parallel to an axis? | {
"answer": "\\sqrt{5} + \\sqrt{2} + \\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Several consecutive natural numbers are written on the board. It is known that \(48\%\) of them are even, and \(36\%\) of them are less than 30. Find the smallest of the written numbers. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the standard equation of the hyperbola $M$ as $\frac{x^{2}}{4}-\frac{y^{2}}{2}=1$. Find the length of the real axis, the length of the imaginary axis, the focal distance, and the eccentricity of the hyperbola $M$. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a computer executes the following program:
(1) Initial values are $x=3, S=0$.
(2) $x=x+2$.
(3) $S=S+x$.
(4) If $S \geqslant 10000$, proceed to step 5; otherwise, go back to step 2.
(5) Print $x$.
(6) Stop.
Then what is the value printed in step 5? | {
"answer": "201",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 23 socks in a drawer: 8 white and 15 black. Every minute, Marina goes to the drawer and pulls out a sock. If at any moment Marina has pulled out more black socks than white ones, she exclaims, "Finally!" and stops the process.
What is the maximum number of socks Marina can pull out before she exclaims, "Finally!"? The last sock Marina pulled out is included in the count. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right cylinder with a height of 8 inches is enclosed inside another cylindrical shell of the same height but with a radius 1 inch greater than the inner cylinder. The radius of the inner cylinder is 3 inches. What is the total surface area of the space between the two cylinders, in square inches? Express your answer in terms of $\pi$. | {
"answer": "16\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 20 points, each pair of adjacent points are equally spaced. By connecting four points with straight lines, you can form a square. Using this method, you can form _ squares. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of participants that could have been in the school drama club if fifth-graders constituted more than $25\%$, but less than $35\%$; sixth-graders more than $30\%$, but less than $40\%$; and seventh-graders more than $35\%$, but less than $45\%$ (there were no participants from other grades)? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An geometric sequence $\{a_n\}$ has 20 terms, where the product of the first four terms is $\frac{1}{128}$, and the product of the last four terms is 512. The product of all terms in this geometric sequence is \_\_\_\_\_\_. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the function $f(x)$ on $\mathbb{R}$ such that $f(0)=0$, $f(x)+f(1-x)=1$, $f\left(\frac{x}{5}\right)=\frac{1}{2}f(x)$, and for $0 \leq x_1 < x_2 \leq 1$, $f(x_1) \leq f(x_2)$. Find the value of $f\left(\frac{1}{2007}\right)$. | {
"answer": "\\frac{1}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Grandpa is twice as strong as Grandma, Grandma is three times as strong as Granddaughter, Granddaughter is four times as strong as Doggie, Doggie is five times as strong as Cat, and Cat is six times as strong as Mouse. Grandpa, Grandma, Granddaughter, Doggie, and Cat together with Mouse can pull up the Turnip, but without Mouse they can't. How many Mice are needed so that they can pull up the Turnip on their own? | {
"answer": "1237",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer $k$ such that, in every convex 1001-gon, the sum of any k diagonals is greater than or equal to the sum of the remaining diagonals? | {
"answer": "249750",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five students, A, B, C, D, and E, participated in a labor skills competition. A and B asked about the results. The respondent told A, "Unfortunately, neither you nor B got first place." To B, the respondent said, "You certainly are not the worst." Determine the number of different possible rankings the five students could have. | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron on the plane containing this edge. | {
"answer": "\\frac{9\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expression \((x+y+z)^{2034}+(x-y-z)^{2034}\), the brackets were expanded, and like terms were combined. How many monomials of the form \(x^{a} y^{b} z^{c}\) have a non-zero coefficient? | {
"answer": "1036324",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radius of a sphere that touches all the edges of a regular tetrahedron is 1. Find the edge length of the tetrahedron. | {
"answer": "2 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Barney Schwinn noted that his bike's odometer showed a reading of $2332$, a palindrome. After riding for $5$ hours one day and $4$ hours the next day, he observed that the odometer displayed another palindrome, $2552$. Calculate Barney's average riding speed during this period. | {
"answer": "\\frac{220}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $A=\{a^{2},a+1,-3\}$ and $B=\{a-3,3a-1,a^{2}+1\}$, if $A∩B=\{-3\}$, find the value of the real number $a$. | {
"answer": "- \\frac {2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \) be nonnegative real numbers whose sum is 300. Let \( M \) be the maximum of the four numbers \( x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, \) and \( x_{4} + x_{5} \). Find the least possible value of \( M \). | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three concentric circles have radii $5$ meters, $15$ meters, and $25$ meters respectively. Calculate the total distance a beetle travels, which starts at a point $P$ on the outer circle, moves inward along a radius to the middle circle, traces a one-third arc of the middle circle, then travels radially to the inner circle, follows a half arc of the inner circle, and finally moves directly through the center to the opposite point on the outer circle. | {
"answer": "15\\pi + 70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that cosα + 2cos(α + $$\frac{π}{3}$$) = 0, find tan(α + $$\frac{π}{6}$$). | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest \( n \) so that the number of integers less than or equal to \( n \) and divisible by 3 equals the number divisible by 5 or 7 (or both). | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\cos (\sqrt{3}x+\varphi)$, where $\varphi \in (-\pi, 0)$. If the function $g(x)=f(x)+f'(x)$ (where $f'(x)$ is the derivative of $f(x)$) is an even function, determine the value of $\varphi$. | {
"answer": "-\\frac{\\pi }{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{\sin \left(\pi\left(\frac{x}{2}+1\right)\right)}$$ | {
"answer": "-\\frac{8}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square floor is tiled with a large number of regular hexagonal tiles, which are either blue or white. Each blue tile is surrounded by 6 white tiles, and each white tile is surrounded by 3 white and 3 blue tiles. Determine the ratio of the number of blue tiles to the number of white tiles, ignoring part tiles. | {
"answer": "1: 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Ivanov family consists of three people: a father, a mother, and a daughter. Today, on the daughter's birthday, the mother calculated the sum of the ages of all family members and got 74 years. It is known that 10 years ago, the total age of the Ivanov family members was 47 years. How old is the mother now if she gave birth to her daughter at the age of 26? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, there is a point $M(2,1)$ inside it. Two lines $l_1$ and $l_2$ passing through $M$ intersect the ellipse $E$ at points $A$, $C$ and $B$, $D$ respectively, and satisfy $\overrightarrow{AM}=\lambda \overrightarrow{MC}, \overrightarrow{BM}=\lambda \overrightarrow{MD}$ (where $\lambda > 0$, and $\lambda \neq 1$). If the slope of $AB$ is always $- \frac{1}{2}$ when $\lambda$ changes, find the eccentricity of the ellipse $E$. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain fruit store deals with two types of fruits, A and B. The situation of purchasing fruits twice is shown in the table below:
| Purchase Batch | Quantity of Type A Fruit ($\text{kg}$) | Quantity of Type B Fruit ($\text{kg}$) | Total Cost ($\text{元}$) |
|----------------|---------------------------------------|---------------------------------------|------------------------|
| First | $60$ | $40$ | $1520$ |
| Second | $30$ | $50$ | $1360$ |
$(1)$ Find the purchase prices of type A and type B fruits.
$(2)$ After selling all the fruits purchased in the first two batches, the fruit store decides to reward customers by launching a promotion. In the third purchase, a total of $200$ $\text{kg}$ of type A and type B fruits are bought, and the capital invested does not exceed $3360$ $\text{元}$. Of these, $m$ $\text{kg}$ of type A fruit and $3m$ $\text{kg}$ of type B fruit are sold at the purchase price, while the remaining type A fruit is sold at $17$ $\text{元}$ per $\text{kg}$ and type B fruit is sold at $30$ $\text{元}$ per $\text{kg}$. If all $200$ $\text{kg}$ of fruits purchased in the third batch are sold, and the maximum profit obtained is not less than $800$ $\text{元}$, find the maximum value of the positive integer $m$. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Egorov decided to open a savings account to buy a car worth 900,000 rubles. The initial deposit is 300,000 rubles. Every month, Egorov plans to add 15,000 rubles to his account. The bank offers a monthly interest rate of $12\%$ per annum. The interest earned each month is added to the account balance, and the interest for the following month is calculated on the new balance. After how many months will there be enough money in the account to buy the car? | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The greatest common divisor of natural numbers \( m \) and \( n \) is 1. What is the greatest possible value of \(\text{GCD}(m + 2000n, n + 2000m) ?\) | {
"answer": "3999999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many different three-letter sets of initials are possible using the letters $A$ through $J$, where no letter is repeated in any set? | {
"answer": "720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parabola $y = ax^2 + bx + c$ ($a \neq 0$) with its axis of symmetry on the left side of the y-axis, where $a, b, c \in \{-3, -2, -1, 0, 1, 2, 3\}$. Let the random variable $X$ represent the value of $|a-b|$. Calculate the expected value $E(X)$. | {
"answer": "\\frac{8}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The extensions of a telephone exchange have only 2 digits, from 00 to 99. Not all extensions are in use. By swapping the order of two digits of an extension in use, you either get the same number or the number of an extension not in use. What is the highest possible number of extensions in use?
(a) Less than 45
(b) 45
(c) Between 45 and 55
(d) More than 55
(e) 55 | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left(x+ \frac {1}{2}\right)= \frac {2x^{4}+x^{2}\sin x+4}{x^{4}+2}$, calculate the value of $f\left( \frac {1}{2017}\right)+f\left( \frac {2}{2017}\right)+\ldots+f\left( \frac {2016}{2017}\right)$. | {
"answer": "4032",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the complex numbers \(z\) and \(w\) satisfy \(|z| = 3\) and \((z + \bar{w})(\bar{z} - w) = 7 + 4i\), where \(i\) is the imaginary unit and \(\bar{z}\), \(\bar{w}\) denote the conjugates of \(z\) and \(w\) respectively. Find the modulus of \((z + 2\bar{w})(\bar{z} - 2w)\). | {
"answer": "\\sqrt{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chord intercepted on the circle $x^{2}+y^{2}=4$ by the line $\begin{cases} x=2-\frac{1}{2}t \\ y=-1+\frac{1}{2}t \end{cases} (t\text{ is the parameter})$ has a length of $\_\_\_\_\_\_\_.$ | {
"answer": "\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During a break between voyages, a sailor turned 20 years old. All six crew members gathered in the cabin to celebrate. "I am twice the age of the cabin boy and 6 years older than the engineer," said the helmsman. "And I am as much older than the cabin boy as I am younger than the engineer," noted the boatswain. "In addition, I am 4 years older than the sailor." "The average age of the crew is 28 years," reported the captain. How old is the captain? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A two-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that according to regulations, a stove should not be located closer than 50 cm to the main gas pipe. | {
"answer": "1/16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Acme Corporation has released a new version of its vowel soup where each vowel (A, E, I, O, U) appears six times, and additionally, each bowl contains one wildcard character that can represent any vowel. How many six-letter "words" can be formed from a bowl of this new Acme Enhanced Vowel Soup? | {
"answer": "46656",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The price of an item is decreased by 20%. To bring it back to its original value and then increase it by an additional 10%, the price after restoration must be increased by what percentage. | {
"answer": "37.5\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. Additionally, for this four-digit number, one prime factor minus five times another prime factor is equal to two times the third prime factor. What is this number? | {
"answer": "1221",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A "fifty percent mirror" is a mirror that reflects half the light shined on it back and passes the other half of the light onward. Two "fifty percent mirrors" are placed side by side in parallel, and a light is shined from the left of the two mirrors. How much of the light is reflected back to the left of the two mirrors? | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest constant $C$ such that for all real numbers $x, y, z$ satisfying $x + y + z = -1$, the following inequality holds:
$$
\left|x^3 + y^3 + z^3 + 1\right| \leqslant C \left|x^5 + y^5 + z^5 + 1\right|.
$$ | {
"answer": "\\frac{9}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a trapezoid $ABCD$ with $\angle A = \angle B = 90^{\circ}$, $|AB| = 5 \text{cm}$, $|BC| = 1 \text{cm}$, and $|AD| = 4 \text{cm}$, point $M$ is taken on side $AB$ such that $2 \angle BMC = \angle AMD$. Find the ratio $|AM| : |BM|$. | {
"answer": "3/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any positive integer \( n \), let \( f(n) \) represent the last digit of \( 1 + 2 + 3 + \cdots + n \). For example, \( f(1) = 1 \), \( f(2) = 3 \), \( f(5) = 5 \), and so on. Find the value of \( f(2) + f(4) + f(6) + \cdots + f(2012) \). | {
"answer": "3523",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right triangle has legs of lengths 3 and 4. Find the volume of the solid formed by revolving the triangle about its hypotenuse. | {
"answer": "\\frac{48\\pi}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a regular dodecagon and a square, whose vertices are also vertices of the dodecagon. What is the value of the ratio of the area of the square to the area of the dodecagon? | {
"answer": "2:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the right focus of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ is $F(\sqrt{6},0)$, a line $l$ passing through $F$ intersects the ellipse at points $A$ and $B$. If the midpoint of chord $AB$ has coordinates $(\frac{\sqrt{6}}{3},-1)$, calculate the area of the ellipse. | {
"answer": "12\\sqrt{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We are given a triangle $ABC$ . Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$ , with the arrangment of points $D - A - B - E$ . The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$ , and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$ . Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$ . | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the angle between the unit vectors $\overrightarrow{e_{1}}$ and $\overrightarrow{e_{2}}$ is $60^{\circ}$, and $\overrightarrow{a}=2\overrightarrow{e_{1}}-\overrightarrow{e_{2}}$, find the projection of $\overrightarrow{a}$ in the direction of $\overrightarrow{e_{1}}$. | {
"answer": "\\dfrac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the real numbers \(x\) and \(y\) satisfy the equation \((x-2)^{2}+(y-1)^{2}=1\), then the minimum value of \(x^{2}+y^{2}\) is \_\_\_\_\_. | {
"answer": "6-2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the subset \( S \) of the set \(\{1,2, \cdots, 15\}\), if a positive integer \( n \) and \( n+|S| \) are both elements of \( S \), then \( n \) is called a "good number" of \( S \). If a subset \( S \) has at least one "good number", then \( S \) is called a "good set". Suppose 7 is a "good number" of a "good set" \( X \). How many such subsets \( X \) are there? | {
"answer": "4096",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n \quad (n \geqslant 1), \quad \text{and} \quad \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n} = 0 \quad (n \geqslant 2).
$$
Find all possible values of $a_{2021}$. | {
"answer": "2021",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the integer part of $\sqrt{10}$ is $a$ and the decimal part is $b$, then $a=$______, $b=\_\_\_\_\_\_$. | {
"answer": "\\sqrt{10} - 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A group of dancers are arranged in a rectangular formation. When they are arranged in 12 rows, there are 5 positions unoccupied in the formation. When they are arranged in 10 rows, there are 5 positions unoccupied. How many dancers are in the group if the total number is between 200 and 300? | {
"answer": "295",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, it is known that $\cos A= \frac{1}{7}$, $\cos (A-B)= \frac{13}{14}$, and $0 < B < A < \frac{\pi}{2}$. Find the measure of angle $B$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \( y = \cos x + \sin x + \cos x \sin x \) has a maximum value of \(\quad\). | {
"answer": "\\frac{1}{2} + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(100^a = 4\) and \(100^b = 5\), then find \(20^{(1 - a - b)/(2(1 - b))}\). | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a rectangular sheet of paper, a picture in the shape of a "cross" is drawn from two rectangles $ABCD$ and $EFGH$, with sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$. | {
"answer": "52.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the region defined by the equation $x^2 + y^2 - 3 = 6y - 18x + 9$? | {
"answer": "102\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of the angle $α+ \frac {π}{6}$ passes through point P($-1$, $-2\sqrt {2}$), find the value of $\sinα$. | {
"answer": "\\frac{1-2\\sqrt {6}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two square napkins with dimensions \(1 \times 1\) and \(2 \times 2\) are placed on a table so that the corner of the larger napkin falls into the center of the smaller napkin. What is the maximum area of the table that the napkins can cover? | {
"answer": "4.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sine of the angle at the vertex of an isosceles triangle, given that the perimeter of any inscribed rectangle, with two vertices lying on the base, is a constant value. | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and each head always tells the truth or always lies. Additionally, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:
- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."
What is the maximum number of red dragons that could have been at the table? | {
"answer": "176",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A stationery store sells a certain type of pen bag for $18$ yuan each. Xiao Hua went to buy this pen bag. When checking out, the clerk said, "If you buy one more, you can get a 10% discount, which is $36 cheaper than now." Xiao Hua said, "Then I'll buy one more, thank you." According to the conversation between the two, Xiao Hua actually paid ____ yuan at checkout. | {
"answer": "486",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange positive integers that are neither perfect squares nor perfect cubes (excluding 0) in ascending order as 2, 3, 5, 6, 7, 10, ..., and determine the 1000th number in this sequence. | {
"answer": "1039",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, with the origin O as the pole and the positive x-axis as the polar axis, a polar coordinate system is established. The polar coordinate of point P is $(1, \pi)$. Given the curve $C: \rho=2\sqrt{2}a\sin(\theta+ \frac{\pi}{4}) (a>0)$, and a line $l$ passes through point P, whose parametric equation is:
$$
\begin{cases}
x=m+ \frac{1}{2}t \\
y= \frac{\sqrt{3}}{2}t
\end{cases}
$$
($t$ is the parameter), and the line $l$ intersects the curve $C$ at points M and N.
(1) Write the Cartesian coordinate equation of curve $C$ and the general equation of line $l$;
(2) If $|PM|+|PN|=5$, find the value of $a$. | {
"answer": "2\\sqrt{3}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integers from 1 to 1997 have a sum of digits that is divisible by 5? | {
"answer": "399",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $x$ if $\log_8 x = 1.75$. | {
"answer": "32\\sqrt[4]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We write the equation on the board:
$$
(x-1)(x-2) \ldots(x-2016) = (x-1)(x-2) \ldots(x-2016)
$$
We want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimum number of factors that must be erased to achieve this? | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the length of the side opposite to angle $A$ is $2$, and the vectors $\overrightarrow{m} = (2, 2\cos^2\frac{B+C}{2} - 1)$ and $\overrightarrow{n} = (\sin\frac{A}{2}, -1)$.
1. Find the value of angle $A$ when the dot product $\overrightarrow{m} \cdot \overrightarrow{n}$ is at its maximum.
2. Under the conditions of part (1), find the maximum area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an infinite grid where each cell is either red or blue, such that in any \(2 \times 3\) rectangle exactly two cells are red, determine how many red cells are in a \(9 \times 11\) rectangle. | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point \\(A\\) on the terminal side of angle \\(\alpha\\) has coordinates \\(\left( \sqrt{3}, -1\right)\\),
\\((1)\\) Find the set of angle \\(\alpha\\)
\\((2)\\) Simplify the following expression and find its value: \\( \dfrac{\sin (2\pi-\alpha)\tan (\pi+\alpha)\cot (-\alpha-\pi)}{\csc (-\alpha)\cos (\pi-\alpha)\tan (3\pi-\alpha)} \\) | {
"answer": "\\dfrac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P(x,y)$ is a moving point on the circle $x^{2}+y^{2}=2y$,
(1) Find the range of $z=2x+y$;
(2) If $x+y+a\geqslant 0$ always holds, find the range of real numbers $a$;
(3) Find the maximum and minimum values of $x^{2}+y^{2}-16x+4y$. | {
"answer": "6-2\\sqrt{73}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $\frac{x^2}{25}+\frac{y^2}{m^2}=1\left(m \gt 0\right)$ with one focus at $F\left(0,4\right)$, find $m$. | {
"answer": "\\sqrt{41}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the operation "□" as: $a□b=a^2+2ab-b^2$. Let the function $f(x)=x□2$, and the equation related to $x$ is $f(x)=\lg|x+2|$ ($x\neq -2$) has exactly four distinct real roots $x_1$, $x_2$, $x_3$, $x_4$. Find the value of $x_1+x_2+x_3+x_4$. | {
"answer": "-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the polar equation of curve C is $\rho - 6\cos\theta + 2\sin\theta + \frac{1}{\rho} = 0$, with the pole as the origin of the Cartesian coordinate system and the polar axis as the positive x-axis, establish a Cartesian coordinate system in the plane xOy. The line $l$ passes through point P(3, 3) with an inclination angle $\alpha = \frac{\pi}{3}$.
(1) Write the Cartesian equation of curve C and the parametric equation of line $l$;
(2) Suppose $l$ intersects curve C at points A and B, find the value of $|AB|$. | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If 10 people need 45 minutes and 20 people need 20 minutes to repair a dam, how many minutes would 14 people need to repair the dam? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the roots of $\tan^2 x - 8\tan x + \sqrt{2} = 0$ that are between $x=0$ and $x=2\pi$ radians. | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of palindromes between 1000 and 10000 that are multiples of 6. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five brothers equally divided an inheritance from their father. The inheritance included three houses. Since three houses could not be divided into 5 parts, the three older brothers took the houses, and the younger brothers were compensated with money. Each of the three older brothers paid 800 rubles, and the younger brothers shared this money among themselves, so that everyone ended up with an equal share. What is the value of one house? | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and a return to the initial state, represented as a quarter circle in the $P / P_{0}, \rho / \rho_{0}$ coordinates with the center at point $(1,1)$. Determine the efficiency of this cycle, knowing that it is 8 times less than the maximum possible efficiency for the same minimum and maximum gas temperatures as in the given cycle. | {
"answer": "1/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly thrown onto the segment [3, 8] and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-2 k-3\right) x^{2}+(3 k-5) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | {
"answer": "4/15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the numbers 2, 3, 4, 3, 1, 6, 3, 7, determine the sum of the mean, median, and mode of these numbers. | {
"answer": "9.625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the definite integral:
$$
\int_{0}^{\pi} 2^{4} \cdot \sin ^{4} x \cos ^{4} x \, dx
$$ | {
"answer": "\\frac{3\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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