problem stringlengths 10 5.15k | answer dict |
|---|---|
In the Cartesian coordinate system $xoy$, given point $A(0,-2)$, point $B(1,-1)$, and $P$ is a moving point on the circle $x^{2}+y^{2}=2$, then the maximum value of $\dfrac{|\overrightarrow{PB}|}{|\overrightarrow{PA}|}$ is ______. | {
"answer": "\\dfrac{3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Alice is 1.6 meters tall, can reach 50 centimeters above her head, and the ceiling is 3 meters tall, find the minimum height of the stool in centimeters needed for her to reach a ceiling fan switch located 15 centimeters below the ceiling. | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the following cryptarithm ensuring that identical letters correspond to identical digits:
$$
\begin{array}{r}
\text { К O Ш К A } \\
+ \text { К O Ш К A } \\
\text { К O Ш К A } \\
\hline \text { С О Б А К А }
\end{array}
$$ | {
"answer": "50350",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a fixed circle $\odot P$ with a radius of 1, the distance from the center $P$ to a fixed line $l$ is 2. Point $Q$ is a moving point on $l$, and circle $\odot Q$ is externally tangent to circle $\odot P$. Circle $\odot Q$ intersects $l$ at points $M$ and $N$. For any diameter $MN$, there is always a fixed point $A$ on the plane such that the angle $\angle MAN$ is a constant value. Find the degree measure of $\angle MAN$. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are 5 balls in a pocket, among which there are 2 black balls and 3 white balls, calculate the probability that two randomly drawn balls of the same color are both white. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a slightly larger weekend softball tournament, five teams (A, B, C, D, E) are participating. On Saturday, Team A plays Team B, Team C plays Team D, and Team E will automatically advance to the semi-final round. On Sunday, the winners of A vs B and C vs D play each other (including E), resulting in one winner, while the remaining two teams (one from initial losers and Loser of semifinal of E's match) play for third and fourth places. The sixth place is reserved for the loser of the losers' game. One possible ranking of the teams from first place to sixth place at the end of this tournament is the sequence AECDBF. What is the total number of possible six-team ranking sequences at the end of the tournament? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given plane vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $|\vec{a}| = |\vec{b}|$, $\vec{c} = \lambda \vec{a} + \mu \vec{b}$, $|\vec{c}| = 1 + \vec{a} \cdot \vec{b}$, and $(\vec{a} + \vec{b}) \cdot \vec{c} = 1$, find the minimum possible value of $\frac{|\vec{a} - \vec{c}|}{|1 + \mu|}$. | {
"answer": "2 - \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of a trapezoid, whose diagonals are 7 and 8, and whose bases are 3 and 6. | {
"answer": "4 : 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 7 cylindrical logs with a diameter of 5 decimeters each. They are tied together at two places using a rope. How many decimeters of rope are needed at least (excluding the rope length at the knots, and using $\pi$ as 3.14)? | {
"answer": "91.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To obtain the graph of the function $y=\cos \left( \frac{1}{2}x+ \frac{\pi}{6}\right)$, determine the necessary horizontal shift of the graph of the function $y=\cos \frac{1}{2}x$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle of radius $r$ has chords $\overline{AB}$ of length $12$ and $\overline{CD}$ of length $9$. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at point $P$, which is outside of the circle. If $\angle{APD}=90^\circ$ and $BP=10$, determine $r^2$. | {
"answer": "221",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the radius $r$ of a circle inscribed within three mutually externally tangent circles of radii $a = 5$, $b = 10$, and $c = 20$ using the formula:
\[
\frac{1}{r} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 2 \sqrt{\frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc}}.
\] | {
"answer": "1.381",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \(a\) in the following sequence: \(1, 8, 27, 64, a, 216, \ldots \ldots\)
\[1^{3}, 2^{3}, 3^{3}, 4^{3}, a, 6^{3}, \ldots \ldots\] | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to arrange the $6$ permutations of the tuple $(1, 2, 3)$ in a sequence, such that each pair of adjacent permutations contains at least one entry in common?
For example, a valid such sequence is given by $(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, 3) - (1, 3, 2) - (3, 1, 2)$ . | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)$ be an even function defined on $\mathbb{R}$, which satisfies $f(x+1) = f(x-1)$ for any $x \in \mathbb{R}$. If $f(x) = 2^{x-1}$ for $x \in [0,1]$, then determine the correctness of the following statements:
(1) 2 is a period of the function $f(x)$.
(2) The function $f(x)$ is increasing on the interval (2, 3).
(3) The maximum value of the function $f(x)$ is 1, and the minimum value is 0.
(4) The line $x=2$ is an axis of symmetry for the function $f(x)$.
Identify the correct statements. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The specific heat capacity of a body with mass \( m = 3 \) kg depends on the temperature in the following way: \( c = c_{0}(1 + \alpha t) \), where \( c_{0} = 200 \) J/kg·°C is the specific heat capacity at \( 0^{\circ} \mathrm{C} \), \( \alpha = 0.05 \,^{\circ} \mathrm{C}^{-1} \) is the temperature coefficient, and \( t \) is the temperature in degrees Celsius. Determine the amount of heat that needs to be transferred to this body to heat it from \( 30^{\circ} \mathrm{C} \) to \( 80^{\circ} \mathrm{C} \). | {
"answer": "112.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of cells that need to be marked in a $7 \times 7$ grid so that in each vertical or horizontal $1 \times 4$ strip there is at least one marked cell? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board, the product of the numbers $\overline{\text{IKS}}$ and $\overline{\text{KSI}}$ is written, where the letters correspond to different non-zero decimal digits. This product is a six-digit number and ends with S. Vasya erased all the zeros from the board, after which only IKS remained. What was written on the board? | {
"answer": "100602",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are many ways in which the list \(0,1,2,3,4,5,6,7,8,9\) can be separated into groups. For example, this list could be separated into the four groups \(\{0,3,4,8\}\), \(\{1,2,7\}\), \{6\}, and \{5,9\}. The sum of the numbers in each of these four groups is \(15\), \(10\), \(6\), and \(14\), respectively. In how many ways can the list \(0,1,2,3,4,5,6,7,8,9\) be separated into at least two groups so that the sum of the numbers in each group is the same? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the three-digit numbers formed by the digits 1, 2, 3, 4, 5, 6 with repetition allowed, how many three-digit numbers have exactly two different even digits (for example: 124, 224, 464, …)? (Answer with a number). | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x) = x^4 + 20x^3 + 150x^2 + 500x + 625$. Let $z_1, z_2, z_3, z_4$ be the four roots of $f$. Find the smallest possible value of $|z_a^2 + z_b^2 + z_cz_d|$ where $\{a, b, c, d\} = \{1, 2, 3, 4\}$. | {
"answer": "1875",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
50 people, consisting of 30 people who all know each other, and 20 people who know no one, are present at a conference. Determine the number of handshakes that occur among the individuals who don't know each other. | {
"answer": "1170",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\tan 5105^\circ$. | {
"answer": "11.430",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$ For constant $a$ such that $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$ , evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a flower bouquet contains pink roses, red roses, pink tulips, and red tulips, and that one fourth of the pink flowers are roses, one third of the red flowers are tulips, and seven tenths of the flowers are red, calculate the percentage of the flowers that are tulips. | {
"answer": "46\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, four circles with centers $P$, $Q$, $R$, and $S$ each have a radius of 2. These circles are tangent to one another and to the sides of $\triangle ABC$ as shown. The circles centered at $P$ and $Q$ are tangent to side $AB$, the circle at $R$ is tangent to side $BC$, and the circle at $S$ is tangent to side $AC$. Determine the perimeter of $\triangle ABC$ if $\triangle ABC$ is isosceles with $AB = AC$. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the instantaneous velocity of the robot at the moment $t=2$ given the robot's motion equation $s = t + \frac{3}{t}$? | {
"answer": "\\frac{13}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute \[\sum_{k=2}^{31} \log_3\left(1 + \frac{2}{k}\right) \log_k 3 \log_{k+1} 3.\] | {
"answer": "\\frac{0.5285}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(x, y\) are real numbers, \(z_{1}=x+\sqrt{11}+yi\), \(z_{6}=x-\sqrt{11}+yi\) (where \(i\) is the imaginary unit). Find \(|z_{1}| + |z_{6}|\). | {
"answer": "30(\\sqrt{2} + 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the function $g$ on the set of integers such that \[g(n)= \begin{cases} n-4 & \mbox{if } n \geq 2000 \\ g(g(n+6)) & \mbox{if } n < 2000. \end{cases}\] Determine $g(172)$. | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $\cos(-\pi - \alpha)$ given a point $P(-3, 4)$ on the terminal side of angle $\alpha$. | {
"answer": "-\\dfrac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In space, there are four spheres with radii 2, 2, 3, and 3. Each sphere is externally tangent to the other three spheres. Additionally, there is a smaller sphere that is externally tangent to all four of these spheres. Find the radius of this smaller sphere. | {
"answer": "\\frac{6}{2 - \\sqrt{26}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the focus of the parabola $y^{2}=ax$ coincides with the left focus of the ellipse $\frac{x^{2}}{6}+ \frac{y^{2}}{2}=1$, find the value of $a$. | {
"answer": "-16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a, b\), and \(c\) be the roots of the cubic polynomial \(3x^3 - 4x^2 + 100x - 3\). Compute \[(a+b+2)^2 + (b+c+2)^2 + (c+a+2)^2.\] | {
"answer": "119.888...",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let (a,b,c,d) be an ordered quadruple of not necessarily distinct integers, each one of them in the set {0,1,2,3,4}. Determine the number of such quadruples that make the expression $a \cdot d - b \cdot c + 1$ even. | {
"answer": "136",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A store sold an air conditioner for 2000 yuan and a color TV for 2000 yuan. The air conditioner made a 30% profit, while the color TV incurred a 20% loss. Could you help the store owner calculate whether the store made a profit or a loss on this transaction, and by how much? | {
"answer": "38.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
George is trying to find the Fermat point $P$ of $\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,2)$, and $C$ is at $(5,4)$. He guesses that the point is at $P = (3,1)$. Compute the sum of the distances from $P$ to the vertices of $\triangle ABC$. If he obtains $x + y\sqrt{z}$, where $x$, $y$, and $z$ are integers, what is $x + y + z$? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a \(10 \times 10\) grid (where the sides of the cells have a unit length), \(n\) cells are selected, and a diagonal is drawn in each of them with an arrow pointing in one of two directions. It turns out that for any two arrows, either the end of one coincides with the beginning of the other, or the distance between their ends is at least 2. What is the largest possible \(n\)? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a$ and $b$ are positive integers such that $ab - 7a + 6b = 559$, what is the minimal possible value of $|a - b|$? | {
"answer": "587",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven frogs are sitting in a row. They come in four colors: two green, two red, two yellow, and one blue. Green frogs refuse to sit next to red frogs, and yellow frogs refuse to sit next to blue frogs. In how many ways can the frogs be positioned respecting these restrictions? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During a survey of 500 people, it was found that $46\%$ of the respondents like strawberry ice cream, $71\%$ like vanilla ice cream, and $85\%$ like chocolate ice cream. Are there at least six respondents who like all three types of ice cream? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the coordinate plane, a point is called a $\text{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12,84)$ . Find the number of right angled triangles $ABC$ in the coordinate plane $B$ and $C$ are lattice points, having a right angle at vertex $A$ and whose incenter is at the origin $(0,0)$ . | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In isosceles $\triangle ABC$ with $AB=AC$, let $D$ be the midpoint of $AC$ and $BD=1$. Find the maximum area of $\triangle ABC$. | {
"answer": "\\frac{2\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The minimum value of $\frac{b^{2}+1}{\sqrt{3}a}$ is what occurs when the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is 2. | {
"answer": "\\frac {4 \\sqrt {3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the surface area of a cone is $3\pi$, and its lateral surface unfolds into a semicircle, then the diameter of the base of the cone is ___. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The net change in the population over these four years is a 20% increase, then a 30% decrease, then a 20% increase, and finally a 30% decrease. Calculate the net change in the population over these four years. | {
"answer": "-29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let a $9$ -digit number be balanced if it has all numerals $1$ to $9$ . Let $S$ be the sequence of the numerals which is constructed by writing all balanced numbers in increasing order consecutively. Find the least possible value of $k$ such that any two subsequences of $S$ which has consecutive $k$ numerals are different from each other. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest prime divisor of \( 16^2 + 81^2 \). | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular decagon `ABCDEFGHIJ` has its center at `K`. Each of the vertices and the center are to be associated with one of the digits `1` through `10`, with each digit used once, in such a way that the sums of the numbers on the lines `AKE`, `BKF`, `CKG`, `DLH` and `EJI` are all equal. In how many ways can this be done? | {
"answer": "3840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ and $c$ be positive real numbers such that $ab^2c^3 = 256$. Find the minimum value of
\[a^2 + 8ab + 16b^2 + 2c^5.\] | {
"answer": "768",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a convex pentagon $FGHIJ$ where $\angle F = \angle G = 100^\circ$. Let $FI = IJ = JG = 3$ and $GH = HF = 5$. Calculate the area of pentagon $FGHIJ$. | {
"answer": "\\frac{9\\sqrt{3}}{4} + 24.62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle PQR$ have side lengths $PQ=13$, $PR=15$, and $QR=14$. Inside $\angle QPR$ are two circles: one is tangent to rays $\overline{PQ}$, $\overline{PR}$, and segment $\overline{QR}$; the other is tangent to the extensions of $\overline{PQ}$ and $\overline{PR}$ beyond $Q$ and $R$, and also tangent to $\overline{QR}$. Compute the distance between the centers of these two circles. | {
"answer": "5\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equation of one of the axes of symmetry for the graph of the function $f(x)=\sin \left(x- \frac {\pi}{4}\right)$ $(x\in\mathbb{R})$ can be found. | {
"answer": "-\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In acute triangle \( ABC \), \( M \) and \( N \) are the midpoints of sides \( AB \) and \( BC \), respectively. The tangents to the circumcircle of triangle \( BMN \) at \( M \) and \( N \) meet at \( P \). Suppose that \( AP \) is parallel to \( BC \), \( AP = 9 \), and \( PN = 15 \). Find \( AC \). | {
"answer": "20\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $ABCD$ is a parallelogram with an area of $50$ square units. Points $P$ and $Q$ are located on sides $AB$ and $CD$ respectively, such that $AP = \frac{1}{3}AB$ and $CQ = \frac{2}{3}CD$. What is the area of triangle $APD$? | {
"answer": "16.67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the decimal representation of the even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) participate, and digits may repeat. It is known that the sum of the digits of the number \( 2M \) is 31, and the sum of the digits of the number \( M / 2 \) is 28. What values can the sum of the digits of the number \( M \) have? List all possible answers. | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hyperbola has its center at the origin O, with its foci on the x-axis and two asymptotes denoted as l₁ and l₂. A line perpendicular to l₁ passes through the right focus F intersecting l₁ and l₂ at points A and B, respectively. It is known that the magnitudes of vectors |OA|, |AB|, and |OB| form an arithmetic sequence, and the vectors BF and FA are in the same direction. Determine the eccentricity of the hyperbola. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( a_{n} = \mathrm{C}_{200}^{n} \cdot (\sqrt[3]{6})^{200-n} \cdot \left( \frac{1}{\sqrt{2}} \right)^{n} \) for \( n = 1, 2, \ldots, 95 \), find the number of integer terms in the sequence \(\{a_{n}\}\). | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A truncated right circular cone has a large base radius of 10 cm and a small base radius of 5 cm. The height of the truncated cone is 10 cm. Calculate the volume of this solid. | {
"answer": "583.33\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(x\), \(y\), and \(z\) be positive real numbers such that \(x + y + z = 3.\) Find the maximum value of \(x^3 y^3 z^2.\) | {
"answer": "\\frac{4782969}{390625}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integers between 1 and 3015 are either multiples of 5 or 7 but not multiples of 35? | {
"answer": "948",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the plane vectors $\overrightarrow{a}=(1,0)$ and $\overrightarrow{b}=\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{a}+ \overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC,$ $\angle A = 45^\circ,$ $\angle B = 75^\circ,$ and $AC = 6.$ Find $BC$. | {
"answer": "6\\sqrt{3} - 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a = 1 + 2\binom{20}{1} + 2^2\binom{20}{2} + \ldots + 2^{20}\binom{20}{20}$, and $a \equiv b \pmod{10}$, determine the possible value(s) for $b$. | {
"answer": "2011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(2, 3, 4, 5, 6, 7, 8\) are to be placed, one per square, in the diagram shown so that the sum of the four numbers in the horizontal row equals 21 and the sum of the four numbers in the vertical column also equals 21. In how many different ways can this be done? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{c}{2} = b - a\cos C$,
(1) Determine the measure of angle $A$.
(2) If $a=\sqrt{15}$ and $b=4$, find the length of side $c$. | {
"answer": "2 - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequences $\{a_{n}\}$ and $\{b_{n}\}$, where $a_{n}$ represents the $n$-th digit after the decimal point of $\sqrt{2}=1.41421356237⋯$ (for example, $a_{1}=4$, $a_{6}=3)$, and $b_{1}=a_{1}$, with ${b_{n+1}}={a_{{b_n}}}$ for all $n\in N^{*}$, find all values of $n$ that satisfy $b_{n}=n-2022$. | {
"answer": "675",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If Samuel has a $3 \times 7$ index card and shortening the length of one side by $1$ inch results in an area of $15$ square inches, determine the area of the card in square inches if instead he shortens the length of the other side by $1$ inch. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is inside isosceles triangle $ABC$, with $AB = BC$ and $\angle BPC = 108^{\circ}$. Let $D$ be the midpoint of side $AC$, and let $BD$ intersect $PC$ at point $E$. If $P$ is the incenter of $\triangle ABE$, find $\angle PAC$. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $y=f(x)$ be a quadratic function, and the equation $f(x)=0$ has two equal real roots. Also, $f'(x)=2x+2$.
1. Find the expression for $y=f(x)$.
2. Find the area of the shape enclosed by the graph of $y=f(x)$ and the two coordinate axes.
3. If the line $x=-t$ ($0<t<1$) divides the area enclosed by the graph of $y=f(x)$ and the two coordinate axes into two equal parts, find the value of $t$. | {
"answer": "1-\\frac{1}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
By multiplying a natural number by the number that is one greater than it, the product takes the form $ABCD$, where $A, B, C, D$ are different digits. Starting with the number that is 3 less, the product takes the form $CABD$. Starting with the number that is 30 less, the product takes the form $BCAD$. Determine these numbers. | {
"answer": "8372",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Note that there are exactly three ways to write the integer $4$ as a sum of positive odd integers where the order of the summands matters:
\begin{align*}
1+1+1+1&=4,
1+3&=4,
3+1&=4.
\end{align*}
Let $f(n)$ be the number of ways to write a natural number $n$ as a sum of positive odd integers where the order of the summands matters. Find the remainder when $f(2008)$ is divided by $100$ . | {
"answer": "71",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the two-digit integer $MM$, where both digits are equal, when multiplied by the one-digit integer $K$ (different from $M$ and only $1\leq K \leq 9$), it results in a three-digit number $NPK$. Identify the digit pairs $(M, K)$ that yield the highest value of $NPK$. | {
"answer": "891",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the operations:
\( a \bigcirc b = a^{\log _{7} b} \),
\( a \otimes b = a^{\frac{1}{\log ^{6} b}} \), where \( a, b \in \mathbb{R} \).
A sequence \(\{a_{n}\} (n \geqslant 4)\) is given such that:
\[ a_{3} = 3 \otimes 2, \quad a_{n} = (n \otimes (n-1)) \bigcirc a_{n-1}. \]
Then, the integer closest to \(\log _{7} a_{2019}\) is: | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If
\[
\begin{pmatrix}
1 & 3 & b \\
0 & 1 & 5 \\
0 & 0 & 1
\end{pmatrix}^m = \begin{pmatrix}
1 & 33 & 6006 \\
0 & 1 & 55 \\
0 & 0 & 1
\end{pmatrix},
\]
then find $b + m.$ | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the broad money supply $\left(M2\right)$ balance was 2912000 billion yuan, express this number in scientific notation. | {
"answer": "2.912 \\times 10^{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle, two angles measure 45 degrees and 60 degrees. The side opposite the 45-degree angle measures 8 units. Calculate the sum of the lengths of the other two sides. | {
"answer": "19.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
8. Shortening. There is a sequence of 2015 digits. All digits are chosen randomly from the set {0, 9} independently of each other. The following operation is performed on the resulting sequence. If several identical digits go in a row, they are replaced by one such digit. For example, if there was a fragment ...044566667..., then it becomes ...04567...
a) Find the probability that the sequence will shorten by exactly one digit.
b) Find the expected length of the new sequence. | {
"answer": "1813.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $\sin {15}^{{}^\circ }+\cos {15}^{{}^\circ }$. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cao has 6 legs, 3 on each side. A walking pattern for the cao is defined as an ordered sequence of raising and lowering each of the legs exactly once (altogether 12 actions), starting and ending with all legs on the ground. The pattern is safe if at any point, he has at least 3 legs on the ground and not all three legs are on the same side. Estimate $N$, the number of safe patterns.
An estimate of $E>0$ earns $\left\lfloor 20 \min (N / E, E / N)^{4}\right\rfloor$ points. | {
"answer": "1416528",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\tan\alpha= \frac {1}{2}$ and $\tan(\alpha-\beta)=- \frac {2}{5}$, calculate the value of $\tan(2\alpha-\beta)$. | {
"answer": "-\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) be two matrices such that \(\mathbf{A} \mathbf{B} = \mathbf{B} \mathbf{A}\). Assuming \(4b \neq c\), find \(\frac{a - 2d}{c - 4b}\). | {
"answer": "\\frac{3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The same amount of steel used to create six solid steel balls, each with a radius of 2 inches, is used to create one larger steel ball. What is the radius of the larger ball? | {
"answer": "4\\sqrt[3]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $WXYZ$ be a rhombus with diagonals $WY = 20$ and $XZ = 24$. Let $M$ be a point on $\overline{WX}$, such that $WM = MX$. Let $R$ and $S$ be the feet of the perpendiculars from $M$ to $\overline{WY}$ and $\overline{XZ}$, respectively. Find the minimum possible value of $RS$. | {
"answer": "\\sqrt{244}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four points \( K, L, M, N \) that are not coplanar. A sphere touches the planes \( K L M \) and \( K L N \) at points \( M \) and \( N \) respectively. Find the surface area of the sphere, knowing that \( M L = 1 \), \( K M = 2 \), \( \angle M N L = 60^\circ \), and \( \angle K M L = 90^\circ \). | {
"answer": "\\frac{64\\pi}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function $g : \mathbb{R} \to \mathbb{R}$ satisfies
\[g(x) + 3g(1 - x) = 2x^2 + 1\]for all $x.$ Find $g(5).$ | {
"answer": "-9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A car license plate contains three letters and three digits, for example, A123BE. The allowed letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters) and all digits except the combination 000. Kira considers a license plate lucky if the second letter is a vowel, the second digit is odd, and the third digit is even (other symbols have no restrictions). How many license plates does Kira consider lucky? | {
"answer": "359999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If Ravi shortens the length of one side of a $5 \times 7$ index card by $1$ inch, the card would have an area of $24$ square inches. What is the area of the card in square inches if instead he shortens the length of the other side by $1$ inch? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tina is trying to solve the equation by completing the square: $$25x^2+30x-55 = 0.$$ She needs to rewrite the equation in the form \((ax + b)^2 = c\), where \(a\), \(b\), and \(c\) are integers and \(a > 0\). What is the value of \(a + b + c\)? | {
"answer": "-38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $PQRS$ be an isosceles trapezoid with bases $PQ=120$ and $RS=25$. Suppose $PR=QS=y$ and a circle with center on $\overline{PQ}$ is tangent to segments $\overline{PR}$ and $\overline{QS}$. If $n$ is the smallest possible value of $y$, then $n^2$ equals what? | {
"answer": "2850",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \frac {1}{3}x^{3}+x^{2}+ax+1$, and the slope of the tangent line to the curve $y=f(x)$ at the point $(0,1)$ is $-3$.
$(1)$ Find the intervals of monotonicity for $f(x)$;
$(2)$ Find the extrema of $f(x)$. | {
"answer": "-\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a modified sequence whose $n$th term is defined by $a_n = (-1)^n \cdot \lfloor \frac{3n+1}{2} \rfloor$. What is the average of the first $150$ terms of this sequence? | {
"answer": "-\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ $c,$ and $d$ be real numbers such that $ab = 2$ and $cd = 18.$ Find the minimum value of
\[(ac)^2 + (bd)^2.\] | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of $\triangle ABC$ measure 11,20, and 21 units. We fold it along $PQ,QR,RP$ where $P,Q,R$ are the midpoints of its sides until $A,B,C$ coincide. What is the volume of the resulting tetrahedron? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangle \( ABC \) with legs \( AC = 3 \) and \( BC = 4 \). Construct triangle \( A_1 B_1 C_1 \) by successively translating point \( A \) a certain distance parallel to segment \( BC \) to get point \( A_1 \), then translating point \( B \) parallel to segment \( A_1 C \) to get point \( B_1 \), and finally translating point \( C \) parallel to segment \( A_1 B_1 \) to get point \( C_1 \). If it turns out that angle \( A_1 B_1 C_1 \) is a right angle and \( A_1 B_1 = 1 \), what is the length of segment \( B_1 C_1 \)? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\arccos(\cos 9).$ All functions are in radians. | {
"answer": "9 - 2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of triples of natural numbers \( m, n, k \) that are solutions to the equation \( m + \sqrt{n+\sqrt{k}} = 2023 \). | {
"answer": "27575680773",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the nearest integer to $(3 + 2)^6$? | {
"answer": "9794",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $ABC$ . Let $A_1B_1$ , $A_2B_2$ , $ ...$ , $A_{2008}B_{2008}$ be $2008$ lines parallel to $AB$ which divide triangle $ABC$ into $2009$ equal areas. Calculate the value of $$ \left\lfloor \frac{A_1B_1}{2A_2B_2} + \frac{A_1B_1}{2A_3B_3} + ... + \frac{A_1B_1}{2A_{2008}B_{2008}} \right\rfloor $$ | {
"answer": "29985",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[17]{3}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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