problem stringlengths 10 5.15k | answer dict |
|---|---|
In $\triangle ABC$ , $AB = 40$ , $BC = 60$ , and $CA = 50$ . The angle bisector of $\angle A$ intersects the circumcircle of $\triangle ABC$ at $A$ and $P$ . Find $BP$ .
*Proposed by Eugene Chen* | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The teacher fills some numbers into the circles in the diagram below (each circle can and must only contain one number). The sum of the three numbers in each of the left and right closed loops is 30, and the sum of the four numbers in each of the top and bottom closed loops is 40. If the number in circle $X$ is 9, then the number in circle $Y$ is $\qquad$ | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a convex quadrilateral \(ABCD\) with \(X\) as the midpoint of the diagonal \(AC\), it turns out that \(CD \parallel BX\). Find \(AD\) if it is known that \(BX = 3\), \(BC = 7\), and \(CD = 6\). | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained by adding the smallest non-zero digit of its decimal representation to the previous number. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th place in this sequence? | {
"answer": "3001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Place parentheses and operation signs in the sequence 22222 so that the result is 24. | {
"answer": "(2+2+2) \\times (2+2)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the real numbers \(a_{1}, a_{2}, \cdots, a_{100}\) satisfy the following conditions: (i) \(a_{1} \geqslant a_{2} \geqslant \cdots \geqslant a_{100} \geqslant 0\); (ii) \(a_{1}+a_{2} \leqslant 100\); (iii) \(a_{3}+a_{4} + \cdots + a_{100} \leqslant 100\). Find the maximum value of \(a_{1}^{2}+a_{2}^{2}+\cdots+a_{100}^{2}\) and the values of \(a_{1}, a_{2}, \cdots, a_{100}\) when the maximum value is reached. | {
"answer": "10000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We have $2022$ $1s$ written on a board in a line. We randomly choose a strictly increasing sequence from ${1, 2, . . . , 2022}$ such that the last term is $2022$ . If the chosen sequence is $a_1, a_2, ..., a_k$ ( $k$ is not fixed), then at the $i^{th}$ step, we choose the first a $_i$ numbers on the line and change the 1s to 0s and 0s to 1s. After $k$ steps are over, we calculate the sum of the numbers on the board, say $S$ . The expected value of $S$ is $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a + b.$ | {
"answer": "1012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Translate the function $f(x) = \sin 2x + \sqrt{3}\cos 2x$ to the left by $\varphi$ ($\varphi > 0$) units. If the resulting graph is symmetric about the y-axis, then the minimum value of $\varphi$ is \_\_\_\_\_. | {
"answer": "\\frac{\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the degree of ionization using the formula:
$$
\alpha=\sqrt{ } K_{\mathrm{HCN}} \mathrm{C}
$$
Given values:
$$
\alpha_{\text {ion }}=\sqrt{ }\left(7,2 \cdot 10^{-10}\right) / 0,1=\sqrt{ } 7,2 \cdot 10^{-9}=8,5 \cdot 10^{-5}, \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$
Alternatively, if the concentration of ions is known, you can calculate $\alpha$ as:
$$
\mathrm{C} \cdot \alpha=[\mathrm{H}^{+}]=[\mathrm{CN}^{-}], [\mathrm{H}^{+}]=[\mathrm{CN}^{-}]=8,5 \cdot 10^{-6} \text{ mol/L}
$$
Then:
$$
\alpha_{\text{ion }}=8,5 \cdot 10^{-6}, 0,1=8,5 \cdot 10^{-5} \text{ or } 8,5 \cdot 10^{-5} \cdot 10^{2}=0,0085\%
$$ | {
"answer": "0.0085",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the book "Nine Chapters on the Mathematical Art," a tetrahedron with all four faces being right-angled triangles is called a "biēnào." Given that tetrahedron $ABCD$ is a "biēnào," $AB\bot $ plane $BCD$, $BC\bot CD$, and $AB=\frac{1}{2}BC=\frac{1}{3}CD$. If the volume of this tetrahedron is $1$, then the surface area of its circumscribed sphere is ______. | {
"answer": "14\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a, b, c \) be pairwise distinct positive integers such that \( a+b, b+c \) and \( c+a \) are all square numbers. Find the smallest possible value of \( a+b+c \). | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the set \(\left\{-3, -\frac{5}{4}, -\frac{1}{2}, 0, \frac{1}{3}, 1, \frac{4}{5}, 2\right\}\), two numbers are drawn without replacement. Find the concept of the two numbers being the slopes of perpendicular lines. | {
"answer": "3/28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system xOy, it is known that 0 < α < 2π. Point P, with coordinates $(1 - \tan{\frac{\pi}{12}}, 1 + \tan{\frac{\pi}{12}})$, lies on the terminal side of angle α. Determine the value of α. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum constant \( k \) such that for \( x, y, z \in \mathbb{R}_+ \), the following inequality holds:
$$
\sum \frac{x}{\sqrt{y+z}} \geqslant k \sqrt{\sum x},
$$
where " \( \sum \) " denotes a cyclic sum. | {
"answer": "\\sqrt{\\frac{3}{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To celebrate her birthday, Ana is going to prepare pear and apple pies. In the market, an apple weighs $300 \text{ g}$ and a pear weighs $200 \text{ g}$. Ana's bag can hold a maximum weight of $7 \text{ kg}$. What is the maximum number of fruits she can buy to make pies with both types of fruits? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many non-similar regular 1200-pointed stars are there, considering the definition of a regular $n$-pointed star provided in the original problem? | {
"answer": "160",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two alloys of copper and zinc. In the first alloy, there is twice as much copper as zinc, and in the second alloy, there is five times less copper than zinc. In what ratio should these alloys be combined to obtain a new alloy in which zinc is twice as much as copper? | {
"answer": "1 : 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( z_{1} \) and \( z_{2} \) are complex numbers with \( \left|z_{1}\right|=3 \), \( \left|z_{2}\right|=5 \), and \( \left|z_{1}+z_{2}\right|=7 \), find the value of \( \arg \left(\left(\frac{z_{2}}{z_{1}}\right)^{3}\right) \). | {
"answer": "\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A satellite is launched vertically from the Earth's pole with the first cosmic velocity. What is the maximum distance the satellite will reach from the Earth's surface? (The gravitational acceleration at the Earth's surface is $g = 10 \, \mathrm{m/s^2}$, and the Earth's radius is $R = 6400 \, \mathrm{km}$). | {
"answer": "6400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lateral surface area of a regular triangular pyramid is 3 times the area of its base. The area of the circle inscribed in the base is numerically equal to the radius of this circle. Find the volume of the pyramid. | {
"answer": "\\frac{2 \\sqrt{6}}{\\pi^3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of triangle \(ABC\), if \(AC = 3\), \(BC = 4\), and the medians \(AK\) and \(BL\) are mutually perpendicular. | {
"answer": "\\sqrt{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express the given data "$20$ nanoseconds" in scientific notation. | {
"answer": "2 \\times 10^{-8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the real numbers \(a\) and \(b\), it holds that \(a^{2} + 4b^{2} = 4\). How large can \(3a^{5}b - 40a^{3}b^{3} + 48ab^{5}\) be? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a survey of $150$ employees at a tech company, it is found that:
- $90$ employees are working on project A.
- $50$ employees are working on project B.
- $30$ employees are working on both project A and B.
Determine what percent of the employees surveyed are not working on either project. | {
"answer": "26.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four points are randomly chosen from the vertices of a regular 12-sided polygon. Find the probability that the four chosen points form a rectangle (including square). | {
"answer": "1/33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive real t such that
\[ x_1 + x_3 = 2tx_2, \]
\[ x_2 + x_4 = 2tx_3, \]
\[ x_3 + x_5 = 2tx_4 \]
has a solution \( x_1, x_2, x_3, x_4, x_5 \) in non-negative reals, not all zero. | {
"answer": "\\frac{1}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), \(\cos \alpha = \frac{4}{5}\), \(\tan (\alpha - \beta) = -\frac{1}{3}\), find \(\cos \beta\). | {
"answer": "\\frac{9 \\sqrt{10}}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the complex plane, let the vertices \( A \) and \( B \) of triangle \( \triangle AOB \) correspond to the complex numbers \( \alpha \) and \( \beta \), respectively, and satisfy the conditions: \( \beta = (1 + i)\alpha \) and \( |\alpha - 2| = 1 \). \( O \) is the origin. Find the maximum value of the area \( S \) of the triangle \( \triangle OAB \). | {
"answer": "9/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $|\sqrt{3}-2|+(\pi -\sqrt{10})^{0}-\sqrt{12}$. | {
"answer": "3-3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F_{1}$ and $F_{2}$ are two foci of the hyperbola $C: x^{2}-\frac{{y}^{2}}{3}=1$, $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $\angle PF_{2}Q=120^{\circ}$, find the area of quadrilateral $PF_{1}QF_{2}$. | {
"answer": "6\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = 2^x + \ln x$, if $a_n = 0.1n$ ($n \in \mathbb{N}^*$), find the value of $n$ that minimizes $|f(a_n) - 2012|$. | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There were no more than 70 mushrooms in the basket, among which 52% were white. If you throw out the three smallest mushrooms, the white mushrooms will become half of the total.
How many mushrooms are in the basket? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
ABCDEF is a six-digit number. All its digits are different and arranged in ascending order from left to right. This number is a perfect square.
Determine what this number is. | {
"answer": "134689",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $\sqrt{3}a\cos C=(2b-\sqrt{3}c)\cos A$
(Ⅰ) Find the magnitude of angle $A$;
(Ⅱ) If $a=2$, find the maximum area of $\Delta ABC$. | {
"answer": "2+ \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Victor was driving to the airport in a neighboring city. Half an hour into the drive at a speed of 60 km/h, he realized that if he did not change his speed, he would be 15 minutes late. So he increased his speed, covering the remaining distance at an average speed of 80 km/h, and arrived at the airport 15 minutes earlier than planned initially. What is the distance from Victor's home to the airport? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that 9 cups of tea cost less than 10 rubles, and 10 cups of tea cost more than 11 rubles. How much does one cup of tea cost? | {
"answer": "111",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ilya takes a triplet of numbers and transforms it following the rule: at each step, each number is replaced by the sum of the other two. What is the difference between the largest and the smallest numbers in the triplet after the 1989th application of this rule, if the initial triplet of numbers was \(\{70, 61, 20\}\)? If the question allows for multiple solutions, list them all as a set. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On each side of an equilateral triangle, a point is taken. The sides of the triangle with vertices at these points are perpendicular to the sides of the original triangle. In what ratio does each of these points divide the side of the original triangle? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex quadrilateral \(ABCD\), the midpoint of side \(AD\) is marked as point \(M\). Segments \(BM\) and \(AC\) intersect at point \(O\). It is known that \(\angle ABM = 55^\circ\), \(\angle AMB = 70^\circ\), \(\angle BOC = 80^\circ\), and \(\angle ADC = 60^\circ\). How many degrees is \(\angle BCA\)? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), \( AB = \sqrt{2} \), \( AC = \sqrt{3} \), and \( \angle BAC = 30^\circ \). Let \( P \) be an arbitrary point in the plane containing \( \triangle ABC \). Find the minimum value of \( \mu = \overrightarrow{PA} \cdot \overrightarrow{PB} + \overrightarrow{PB} \cdot \overrightarrow{PC} + \overrightarrow{PC} \cdot \overrightarrow{PA} \). | {
"answer": "\\frac{\\sqrt{2}}{2} - \\frac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a new game, Jane and her brother each spin a spinner once. The spinner has six congruent sectors labeled from 1 to 6. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a common fraction. | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest eight-digit positive integer that has exactly four digits which are 4? | {
"answer": "10004444",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cafe "Burattino" operates 6 days a week with a day off on Mondays. Kolya made two statements: "from April 1 to April 20, the cafe worked 18 days" and "from April 10 to April 30, the cafe also worked 18 days." It is known that he made a mistake once. How many days did the cafe work from April 1 to April 27? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the convex quadrilateral \(ABCD\),
\[
\angle BAD = \angle BCD = 120^\circ, \quad BC = CD = 10.
\]
Find \(AC.\) | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( \triangle ABC \) be an acute triangle, with \( M \) being the midpoint of \( \overline{BC} \), such that \( AM = BC \). Let \( D \) and \( E \) be the intersection of the internal angle bisectors of \( \angle AMB \) and \( \angle AMC \) with \( AB \) and \( AC \), respectively. Find the ratio of the area of \( \triangle DME \) to the area of \( \triangle ABC \). | {
"answer": "\\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. How many years did he live? | {
"answer": "79",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $PQRS$ is a square. A circle with center $S$ has arc $PXC$. A circle with center $R$ has arc $PYC$. If $PQ = 3$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the nearest tenth. | {
"answer": "5.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If for any real number \( x \), the function
\[ f(x)=x^{2}-2x-|x-1-a|-|x-2|+4 \]
always yields a non-negative real number, then the minimum value of the real number \( a \) is . | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( a, b, c \) are positive integers, and the parabola \( y = ax^2 + bx + c \) intersects the x-axis at two distinct points \( A \) and \( B \). If the distances from \( A \) and \( B \) to the origin are both less than 1, find the minimum value of \( a + b + c \). | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( ABCD \) be a square with side length \( 5 \), and \( E \) be a point on \( BC \) such that \( BE = 3 \) and \( EC = 2 \). Let \( P \) be a variable point on the diagonal \( BD \). Determine the length of \( PB \) if \( PE + PC \) is minimized. | {
"answer": "\\frac{15 \\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each day, John ate 30% of the chocolates that were in his box at the beginning of that day. At the end of the third day, 28 chocolates remained. How many chocolates were in the box originally? | {
"answer": "82",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows the ellipse whose equation is \(x^{2}+y^{2}-xy+x-4y=12\). The curve cuts the \(y\)-axis at points \(A\) and \(C\) and cuts the \(x\)-axis at points \(B\) and \(D\). What is the area of the inscribed quadrilateral \(ABCD\)? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the 4040 integers ranging from -2020 to 2019, three numbers are randomly chosen and multiplied together. Let the smallest possible product be $m$ and the largest possible product be $n$. What is the value of $\frac{m}{n}$? Provide the answer in simplest fraction form. | {
"answer": "-\\frac{2020}{2017}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the banks of an island, which has the shape of a circle (viewed from above), there are the cities $A, B, C,$ and $D$. A straight asphalt road $AC$ divides the island into two equal halves. A straight asphalt road $BD$ is shorter than road $AC$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h. The island also has straight dirt roads $AB, BC, CD,$ and $AD$, on which the cyclist's speed is the same. The cyclist travels from point $B$ to each of points $A, C,$ and $D$ along a straight road in 2 hours. Find the area enclosed by the quadrilateral $ABCD$. | {
"answer": "450",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle \(ABC\) has \(\angle A = 90^\circ\), side \(BC = 25\), \(AB > AC\), and area 150. Circle \(\omega\) is inscribed in \(ABC\), with \(M\) as its point of tangency on \(AC\). Line \(BM\) meets \(\omega\) a second time at point \(L\). Find the length of segment \(BL\). | {
"answer": "\\frac{45\\sqrt{17}}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In \(\triangle ABC\), \(a, b, c\) are the sides opposite angles \(A, B, C\) respectively. Given \(a+c=2b\) and \(A-C=\frac{\pi}{3}\), find the value of \(\sin B\). | {
"answer": "\\frac{\\sqrt{39}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \( f(x) = \frac{x^{2}}{8} + x \cos x + \cos (2x) \) (for \( x \in \mathbf{R} \)) has a minimum value of ___ | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A deck of fifty-two cards consists of four $1$'s, four $2$'s, ..., four $13$'s. Two matching pairs (two sets of two cards with the same number) are removed from the deck. After removing these cards, find the probability, represented as a fraction $m/n$ in simplest form, where $m$ and $n$ are relatively prime, that two randomly selected cards from the remaining cards also form a pair. Find $m + n$. | {
"answer": "299",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The length of one side of the square \(ABCD\) is 4 units. A circle is drawn tangent to \(\overline{BC}\) and passing through the vertices \(A\) and \(D\). Find the area of the circle. | {
"answer": "\\frac{25 \\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point $P(2,-1)$,
(1) Find the general equation of the line that passes through point $P$ and has a distance of 2 units from the origin.
(2) Find the general equation of the line that passes through point $P$ and has the maximum distance from the origin. Calculate the maximum distance. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twenty people, including \( A, B, \) and \( C \), sit randomly at a round table. What is the probability that at least two of \( A, B, \) and \( C \) sit next to each other? | {
"answer": "17/57",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x \sim N(4, 1)$ and $f(x < 3) = 0.0187$, then $f(x < 5) = \_\_\_\_\_\_$. | {
"answer": "0.9813",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ages of Daisy's four cousins are distinct single-digit positive integers, and the product of two of the ages is $24$ while the product of the other two ages is $35$, find the sum of the ages of Daisy's four cousins. | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Elective 4-4: Coordinate System and Parametric Equations
In the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\begin{cases}x=2+t \\ y=kt\end{cases}$ (where $t$ is the parameter), and the parametric equation of line $l_2$ is $\begin{cases}x=-2+m \\ y= \frac{m}{k}\end{cases}$ (where $m$ is the parameter). Let the intersection point of $l_1$ and $l_2$ be $P$. When $k$ changes, the trajectory of $P$ is curve $C$.
(1) Write the general equation of $C$;
(2) Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. Let line $l_3: \rho(\cos \theta +\sin \theta)− \sqrt{2} =0$, and $M$ be the intersection point of $l_3$ and $C$. Find the polar radius of $M$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The letters L, K, R, F, O and the digits 1, 7, 8, 9 are "cycled" separately and put together in a numbered list. Determine the line number on which LKRFO 1789 will appear for the first time. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(\alpha, \beta, \gamma\) satisfy \(0<\alpha<\beta<\gamma<2 \pi\), and for any \(x \in \mathbf{R}\), \(\cos (x+\alpha) + \cos (x+\beta) + \cos (x+\gamma) = 0\), determine the value of \(\gamma - \alpha\). | {
"answer": "\\frac{4\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the spring round of the 2000 Cities Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, but no two students together solved all six problems. What is the minimum possible number of high school students in country $N$ who participated in the spring round? | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A semicircle and a circle each have a radius of 5 units. A square is inscribed in each. Calculate the ratio of the perimeter of the square inscribed in the semicircle to the perimeter of the square inscribed in the circle. | {
"answer": "\\frac{\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right triangle \(ABC\) with \(\angle B = 90^\circ\), \(P\) is a point on the angle bisector of \(\angle A\) inside \(\triangle ABC\). Point \(M\) (distinct from \(A\) and \(B\)) lies on the side \(AB\). The lines \(AP\), \(CP\), and \(MP\) intersect sides \(BC\), \(AB\), and \(AC\) at points \(D\), \(E\), and \(N\) respectively. Given that \(\angle MPB = \angle PCN\) and \(\angle NPC = \angle MBP\), find \(\frac{S_{\triangle APC}}{S_{ACDE}}\). | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integers are grouped as follows:
\( A_1 = \{1\}, A_2 = \{2, 3, 4\}, A_3 = \{5, 6, 7, 8, 9\} \), and so on.
In which group does 2009 belong? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\pi} 2^{4} \sin ^{4}\left(\frac{x}{2}\right) \cos ^{4}\left(\frac{x}{2}\right) d x
$$ | {
"answer": "\\frac{3\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x) = e^{-x}$, calculate the limit $$\lim_{\Delta x \to 0} \frac{f(1 + \Delta x) - f(1 - 2\Delta x)}{\Delta x}$$. | {
"answer": "-\\frac{3}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( A \) is a positive integer such that \( \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \cdots + \frac{1}{(A+1)(A+3)} = \frac{12}{25} \), find the value of \( A \). | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A building has three different staircases, all starting at the base of the building and ending at the top. One staircase has 104 steps, another has 117 steps, and the other has 156 steps. Whenever the steps of the three staircases are at the same height, there is a floor. How many floors does the building have? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum value of \( N \) such that \( N! \) has exactly 2013 trailing zeros? | {
"answer": "8069",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the integer \(n\), such that \(-180 < n < 180\), for which \(\tan n^\circ = \tan 276^\circ.\) | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$ ). The expected value of $N'$ is equal to $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $10^6$ .
*Proposed by Yannick Yao* | {
"answer": "555556",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( ABC - A_1B_1C_1 \) is a right prism with \(\angle BAC = 90^\circ\), points \( D_1 \) and \( F_1 \) are the midpoints of \( A_1B_1 \) and \( B_1C_1 \), respectively. If \( AB = CA = AA_1 \), find the cosine of the angle between \( BD_1 \) and \( CF_1 \). | {
"answer": "\\frac{\\sqrt{30}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the number 79777 has the digit 9 crossed out, the result is the number 7777. How many different five-digit numbers exist from which 7777 can be obtained by crossing out one digit? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cuboid has a diagonal $A A^{\prime}$. In what ratio does the plane passing through the endpoints $B, C, D$ of the edges originating from vertex $A$ divide the $A A^{\prime}$ diagonal? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle XYZ$, the ratio $XZ:ZY$ is $5:3$. The bisector of the exterior angle at $Z$ intersects $YX$ extended at $Q$ ($Y$ is between $Q$ and $X$). Find the ratio $QY:YX$. | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the slant height of a cone is 2, and its net is a semicircle, what is the area of the cross section of the axis of the cone? | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $A,B,C$ and $D$ lie on a line in that order, with $AB = CD$ and $BC = 16$. Point $E$ is not on the line, and $BE = CE = 13$. The perimeter of $\triangle AED$ is three times the perimeter of $\triangle BEC$. Find $AB$.
A) $\frac{32}{3}$
B) $\frac{34}{3}$
C) $\frac{36}{3}$
D) $\frac{38}{3}$ | {
"answer": "\\frac{34}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain country, there are exactly 2019 cities and between any two of them, there is exactly one direct flight operated by an airline company, that is, given cities $A$ and $B$, there is either a flight from $A$ to $B$ or a flight from $B$ to $A$. Find the minimum number of airline companies operating in the country, knowing that direct flights between any three distinct cities are operated by different companies. | {
"answer": "2019",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point $P(-2,0)$ and the parabola $C$: $y^{2}=4x$, the line passing through $P$ intersects $C$ at points $A$ and $B$, where $|PA|= \frac {1}{2}|AB|$. Determine the distance from point $A$ to the focus of parabola $C$. | {
"answer": "\\frac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $\varphi$ passes through point P(1, -1), and points A($x_1$, $y_1$) and B($x_2$, $y_2$) are any two points on the graph of the function $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$). If $|f(x_1) - f(x_2)| = 2$, the minimum value of $|x_1 - x_2|$ is $\frac{\pi}{3}$. Find the value of $f\left(\frac{\pi}{2}\right)$. | {
"answer": "-\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of one hundred natural numbers $x, x+1, x+2, \cdots, x+99$ has a sum $a$. If the sum of the digits of $a$ is 50, what is the smallest possible value of $x$? | {
"answer": "99950",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tom's algebra notebook consists of 50 pages, 25 sheets of paper. Specifically, page 1 and page 2 are the front and back of the first sheet of paper, page 3 and page 4 are the front and back of the second sheet of paper, and so on. One day, Tom left the notebook on the table while he went out, and his roommate took away several consecutive pages. The average of the remaining page numbers is 19. How many pages did the roommate take away? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(x) = mx^2 + (2n + 1)x - m - 2 \) (where \( m, n \in \mathbb{R} \) and \( m \neq 0 \)) have at least one root in the interval \([3, 4]\). Find the minimum value of \( m^2 + n^2 \). | {
"answer": "0.01",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( k=-\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i} \). In the complex plane, the vertices of \(\triangle ABC\) correspond to the complex numbers \( z_{1}, z_{2}, z_{3} \) which satisfy the equation
\[ z_{1}+k z_{2}+k^{2}\left(2 z_{3}-z_{1}\right)=0 \text {. } \]
Find the radian measure of the smallest interior angle of this triangle. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many possible sequences of the experiment are there, given that 6 procedures need to be implemented in sequence, procedure A can only appear in the first or last step, and procedures B and C must be adjacent when implemented? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of the integers 122 and 78, express both numbers and the resulting sum in base-5. | {
"answer": "1300_5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The chord \( A B \) subtends an arc of the circle equal to \( 120^{\circ} \). Point \( C \) lies on this arc, and point \( D \) lies on the chord \( A B \). Additionally, \( A D = 2 \), \( B D = 1 \), and \( D C = \sqrt{2} \).
Find the area of triangle \( A B C \). | {
"answer": "\\frac{3 \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A 1-liter carton of milk used to cost 80 rubles. Recently, in an effort to cut costs, the manufacturer reduced the carton size to 0.9 liters and increased the price to 99 rubles. By what percentage did the manufacturer's revenue increase? | {
"answer": "37.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The slope angle of the tangent to the curve $y=x^{3}-4x$ at the point $(1,-3)$ is calculated in radians. | {
"answer": "\\frac{3}{4}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with an integer side length was cut into 2020 smaller squares. It is known that the areas of 2019 of these squares are 1, while the area of the 2020th square is not equal to 1. Find all possible values that the area of the 2020th square can take. Provide the smallest of these possible area values in the answer. | {
"answer": "112225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The endpoints of a line segment \\(AB\\) with a fixed length of \\(3\\) move on the parabola \\(y^{2}=x\\). Let \\(M\\) be the midpoint of the line segment \\(AB\\). The minimum distance from \\(M\\) to the \\(y\\)-axis is \_\_\_\_\_\_. | {
"answer": "\\dfrac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system xOy, let there be a line $l$ with an inclination angle $\alpha$ given by $$l: \begin{cases} x=2+t\cos\alpha \\ y= \sqrt {3}+t\sin\alpha \end{cases}$$ (where $t$ is a parameter) that intersects with the curve $$C: \begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases}$$ (where $\theta$ is a parameter) at two distinct points A and B.
(1) If $\alpha= \frac {\pi}{3}$, find the coordinates of the midpoint M of segment AB;
(2) If $|PA|\cdot|PB|=|OP|^2$, where $P(2, \sqrt {3})$, find the slope of the line $l$. | {
"answer": "\\frac { \\sqrt {5}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Choose $n$ numbers from the 2017 numbers $1, 2, \cdots, 2017$ such that the difference between any two chosen numbers is a composite number. What is the maximum value of $n$? | {
"answer": "505",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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