problem stringlengths 10 5.15k | answer dict |
|---|---|
A school organizes a social practice activity during the summer vacation and needs to allocate 8 grade 10 students to company A and company B evenly. Among these students, two students with excellent English scores cannot be allocated to the same company, and neither can the three students with strong computer skills. How many different allocation schemes are there? (Answer with a number) | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Charlie and Daisy each arrive at a cafe at a random time between 1:00 PM and 3:00 PM. Each stays for 20 minutes. What is the probability that Charlie and Daisy are at the cafe at the same time? | {
"answer": "\\frac{4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The triangle $\triangle ABC$ is an isosceles triangle where $AC = 6$ and $\angle A$ is a right angle. If $I$ is the incenter of $\triangle ABC,$ then what is $BI$? | {
"answer": "6\\sqrt{2} - 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the fourth grade, there are 20 boys and 26 girls. The percentage of the number of boys to the number of girls is %. | {
"answer": "76.9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $AC = \sqrt{7}$, $BC = 2$, and $\angle B = 60^\circ$. What is the area of $\triangle ABC$? | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( f(x) = \sum_{k=1}^{2017} \frac{\cos k x}{\cos^k x} \), find \( f\left(\frac{\pi}{2018}\right) \). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[
\frac{(1 + 23) \left( 1 + \dfrac{23}{2} \right) \left( 1 + \dfrac{23}{3} \right) \dotsm \left( 1 + \dfrac{23}{25} \right)}{(1 + 25) \left( 1 + \dfrac{25}{2} \right) \left( 1 + \dfrac{25}{3} \right) \dotsm \left( 1 + \dfrac{25}{23} \right)}.
\] | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f : [0, 1] \rightarrow \mathbb{R}$ be a monotonically increasing function such that $$ f\left(\frac{x}{3}\right) = \frac{f(x)}{2} $$ $$ f(1 0 x) = 2018 - f(x). $$ If $f(1) = 2018$ , find $f\left(\dfrac{12}{13}\right)$ .
| {
"answer": "2018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a moving point P in the plane, the difference between the distance from point P to point F(1, 0) and the distance from point P to the y-axis is equal to 1.
(Ⅰ) Find the equation of the trajectory C of the moving point P;
(Ⅱ) Draw two lines $l_1$ and $l_2$ through point F, both with defined slopes and perpendicular to each other. Suppose $l_1$ intersects trajectory C at points A and B, and $l_2$ intersects trajectory C at points D and E. Find the minimum value of $\overrightarrow {AD} \cdot \overrightarrow {EB}$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle \( \triangle ABC \), \( \angle C = 90^{\circ} \), and \( CB > CA \). Point \( D \) is on \( BC \) such that \( \angle CAD = 2 \angle DAB \). If \( \frac{AC}{AD} = \frac{2}{3} \) and \( \frac{CD}{BD} = \frac{m}{n} \) where \( m \) and \( n \) are coprime positive integers, then what is \( m + n \)?
(49th US High School Math Competition, 1998) | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \(a, b, c, d,\) and \(e\) respectively. We know that \(a + 2 = b - 2 = 2c = \frac{d}{2} = e^2\).
Gina picks up the card which has the largest integer on its reverse. What number is on the front of Gina's card? | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function \( f(x) \) defined on \( (0, +\infty) \) satisfy \( f(x) > -\frac{3}{x} \) for any \( x \in (0, +\infty) \) and \( f\left(f(x) + \frac{3}{x}\right) = 2 \). Find \( f(5) \). | {
"answer": "\\frac{7}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the distinct prime factors of $7^7 - 7^4$. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let points $A = (0,0)$, $B = (2,3)$, $C = (5,4)$, and $D = (6,0)$. Quadrilateral $ABCD$ is divided into two equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left (\frac{p}{q}, \frac{r}{s} \right )$, where these fractions are in lowest terms. Determine $p + q + r + s$.
A) 50
B) 58
C) 62
D) 66
E) 70 | {
"answer": "58",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all real numbers $x$ such that the product $(x + 2i)((x + 1) + 2i)((x + 2) + 2i)((x + 3) + 2i)$ is purely imaginary. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( b = \frac{\pi}{2010} \). Find the smallest positive integer \( m \) such that
\[
2\left[\cos(b) \sin(b) + \cos(4b) \sin(2b) + \cos(9b) \sin(3b) + \cdots + \cos(m^2b) \sin(mb)\right]
\]
is an integer. | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( p, q, r, s \) be distinct real numbers such that the roots of \( x^2 - 12px - 13q = 0 \) are \( r \) and \( s \), and the roots of \( x^2 - 12rx - 13s = 0 \) are \( p \) and \( q \). Find the value of \( p + q + r + s \). | {
"answer": "2028",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define an ordered quadruple of integers $(a, b, c, d)$ as captivating if $1 \le a < b < c < d \le 15$, and $a+d > 2(b+c)$. How many captivating ordered quadruples are there? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hexagon has its vertices alternately on two concentric circles with radii 3 units and 5 units and centered at the origin on a Cartesian plane. Each alternate vertex starting from the origin extends radially outward to the larger circle. Calculate the area of this hexagon. | {
"answer": "\\frac{51\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $b$ is an even multiple of $7786$, find the greatest common divisor of $8b^2 + 85b + 200$ and $2b + 10$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the set of three-digit numbers that do not contain the digits $0,1,2,3,4,5$, several numbers were written down in such a way that no two numbers could be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have been written? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \le x \le 5$ and $1 \le y \le 3$? | {
"answer": "416",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Every element in $S$ is a positive integer not exceeding 100.
(2) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest common divisor (gcd) of $a + b$ and $c$ is 1.
(3) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the gcd of $a + b$ and $c$ is greater than 1. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle XYZ$ have side lengths $XY=15$, $XZ=20$, and $YZ=25$. Inside $\angle XYZ$, there are two circles: one is tangent to the rays $\overline{XY}$, $\overline{XZ}$, and the segment $\overline{YZ}$, while the other is tangent to the extension of $\overline{XY}$ beyond $Y$, $\overline{XZ}$, and $\overline{YZ}$. Compute the distance between the centers of these two circles. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Geoff and Trevor each roll a fair eight-sided die (the sides are labeled 1 through 8). What is the probability that the product of the numbers they roll is even or a prime number? | {
"answer": "\\frac{7}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sterling draws 6 circles on the plane, which divide the plane into regions (including the unbounded region). What is the maximum number of resulting regions? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, and $|\overrightarrow{a}-\overrightarrow{b}|=\sqrt{7}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cars, $A$ and $B$, depart from one city to another. In the first 5 minutes, they traveled the same distance. Then, due to an engine failure, $B$ had to reduce its speed to 2/5 of its original speed, and thus arrived at the destination 15 minutes after car $A$, which continued at a constant speed. If the failure had occurred 4 km farther from the starting point, $B$ would have arrived only 10 minutes after $A$. How far apart are the two cities? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the original spherical dome has a height of $55$ meters and can be represented as holding $250,000$ liters of air, and Emily's scale model can hold only $0.2$ liters of air, determine the height, in meters, of the spherical dome in Emily's model. | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of straight cuts needed to cut a cake in 100 pieces? The pieces do not need to be the same size or shape but cannot be rearranged between cuts. You may assume that the cake is a large cube and may be cut from any direction. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $n\neq 2004$ for which there exists a polynomial $f\in\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions. | {
"answer": "(1002!)^2 + 2004",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( g_{1}(x) = \sqrt{2-x} \), and for integers \( n \geq 2 \), define
\[ g_{n}(x) = g_{n-1}\left(\sqrt{(n+1)^2 - x}\right) \].
Let \( M \) be the largest value of \( n \) for which the domain of \( g_n \) is non-empty. For this value of \( M \), the domain of \( g_M \) consists of a single point \(\{d\}\). Compute \( d \). | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the area of the crescent moon enclosed by the portion of the circle of radius 4 centered at (0,0) that lies in the first quadrant, the portion of the circle with radius 2 centered at (0,1) that lies in the first quadrant, and the line segment from (0,0) to (4,0). | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Juan takes a number, adds 3 to it, squares the result, then multiplies the answer by 2, subtracts 3 from the result, and finally divides that number by 2. If his final answer is 49, what was the original number? | {
"answer": "\\sqrt{\\frac{101}{2}} - 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve tiles numbered $1$ through $12$ are turned up at random, and an 8-sided die (sides numbered from 1 to 8) is rolled. Calculate the probability that the product of the numbers on the tile and the die will be a square. | {
"answer": "\\frac{7}{48}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane rectangular coordinate system $xOy$, it is known that $MN$ is a chord of the circle $C: (x-2)^{2} + (y-4)^{2} = 2$, and satisfies $CM\perp CN$. Point $P$ is the midpoint of $MN$. As the chord $MN$ moves on the circle $C$, there exist two points $A$ and $B$ on the line $2x-y-3=0$, such that $\angle APB \geq \frac{\pi}{2}$ always holds. Find the minimum value of the length of segment $AB$. | {
"answer": "\\frac{6\\sqrt{5}}{5} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $0 < \beta < \frac{\pi}{2} < \alpha < \pi$ and $\cos\left(\alpha - \frac{\beta}{2}\right) = -\frac{1}{9}$, $\sin\left(\frac{\alpha}{2} - \beta\right) = \frac{2}{3}$, find the value of $\cos(\alpha + \beta)$. | {
"answer": "-\\frac{239}{729}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If vectors $\mathbf{a} = (1,0)$, $\mathbf{b} = (0,1)$, $\mathbf{c} = k\mathbf{a} + \mathbf{b}$ ($k \in \mathbb{R}$), $\mathbf{d} = \mathbf{a} - \mathbf{b}$, and $\mathbf{c} \parallel \mathbf{d}$, calculate the value of $k$ and the direction of vector $\mathbf{c}$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the coordinates of the foci of an ellipse are $F_{1}(-1,0)$, $F_{2}(1,0)$, and a line perpendicular to the major axis through $F_{2}$ intersects the ellipse at points $P$ and $Q$, with $|PQ|=3$.
$(1)$ Find the equation of the ellipse;
$(2)$ A line $l$ through $F_{2}$ intersects the ellipse at two distinct points $M$ and $N$. Does the area of the incircle of $\triangle F_{1}MN$ have a maximum value? If it exists, find this maximum value and the equation of the line at this time; if not, explain why. | {
"answer": "\\frac {9}{16}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 100 points located on a line. Mark the midpoints of all possible segments with endpoints at these points. What is the minimum number of marked points that can result? | {
"answer": "197",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the digits of the integer equal to \( 777777777777777^2 - 222222222222223^2 \) can be found by evaluating the expression. | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Given $$x^{ \frac {1}{2}}+x^{- \frac {1}{2}}=3$$, find the value of $x+x^{-1}$;
(2) Calculate $$( \frac {1}{8})^{- \frac {1}{3}}-3^{\log_{3}2}(\log_{3}4)\cdot (\log_{8}27)+2\log_{ \frac {1}{6}} \sqrt {3}-\log_{6}2$$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Through the vertices \( A, C, D \) of the parallelogram \( ABCD \) with sides \( AB = 7 \) and \( AD = 4 \), a circle is drawn that intersects the line \( BD \) at point \( E \), and \( DE = 13 \). Find the length of diagonal \( BD \). | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are equal to 8 and 18, respectively. It is known that the circumcircle of triangle \(ABD\) is tangent to lines \(BC\) and \(CD\). Find the perimeter of the trapezoid. | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $|\vec{a}|=1$, $|\vec{b}|=6$, and $\vec{a}\cdot(\vec{b}-\vec{a})=2$, calculate the angle between $\vec{a}$ and $\vec{b}$. | {
"answer": "\\dfrac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point. | {
"answer": "2 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The amplitude, period, frequency, phase, and initial phase of the function $y=3\sin \left( \frac {1}{2}x- \frac {\pi}{6}\right)$ are ______, ______, ______, ______, ______, respectively. | {
"answer": "- \\frac {\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can you rearrange the letters of ‘Alejandro’ such that it contains one of the words ‘ned’ or ‘den’? | {
"answer": "40320",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(9^{-1} \equiv 90 \pmod{101}\), find \(81^{-1} \pmod{101}\), as a residue modulo 101. (Give an answer between 0 and 100, inclusive.) | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F$ is the focus of the parabola $C_{1}$: $y^{2}=2ρx (ρ > 0)$, and point $A$ is a common point of one of the asymptotes of the hyperbola $C_{2}$: $\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0, b > 0)$ and $AF \perp x$-axis, find the eccentricity of the hyperbola. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integer divisors of 294, except 1, are arranged around a circle so that every pair of adjacent integers has a common factor greater than 1. What is the sum of the two integers adjacent to 21? | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
\[\frac{\lfloor \sqrt{1} \rfloor \cdot \lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{3} \rfloor \cdot \lfloor \sqrt{5} \rfloor \dotsm \lfloor \sqrt{15} \rfloor}{\lfloor \sqrt{2} \rfloor \cdot \lfloor \sqrt{4} \rfloor \cdot \lfloor \sqrt{6} \rfloor \dotsm \lfloor \sqrt{16} \rfloor}.\] | {
"answer": "\\frac{3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least integer whose square is 36 more than three times its value? | {
"answer": "-6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of all four-digit numbers that can be formed using the digits 0, 1, 2, 3, and 4, with no repeated digits. | {
"answer": "259980",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cards in a stack are numbered consecutively from 1 to $2n$ from top to bottom. The top $n$ cards are removed to form pile $A$ and the remaining cards form pile $B$. The cards are restacked by alternating cards from pile $B$ and $A$, starting with a card from $B$. Given this process, find the total number of cards ($2n$) in the stack if card number 201 retains its original position. | {
"answer": "402",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence of positive integers \(a_{1}, a_{2}, \ldots\) is such that for each \(m\) and \(n\) the following holds: if \(m\) is a divisor of \(n\) and \(m < n\), then \(a_{m}\) is a divisor of \(a_{n}\) and \(a_{m} < a_{n}\). Find the least possible value of \(a_{2000}\). | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circle, points $A,B,C,D$ lie counterclockwise in this order. Let the orthocenters of $ABC,BCD,CDA,DAB$ be $H,I,J,K$ respectively. Let $HI=2$ , $IJ=3$ , $JK=4$ , $KH=5$ . Find the value of $13(BD)^2$ . | {
"answer": "169",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An electronic clock displays time from 00:00:00 to 23:59:59. How much time throughout the day does the clock show a number that reads the same forward and backward? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is a moving point on the parabola $y^{2}=2x$, find the minimum value of the sum of the distance from point $P$ to point $(0,2)$ and the distance from $P$ to the directrix of the parabola. | {
"answer": "\\dfrac { \\sqrt {17}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( P Q R S \) is a square, and point \( O \) is on the line \( R Q \) such that the distance from \( O \) to point \( P \) is 1, what is the maximum possible distance from \( O \) to point \( S \)? | {
"answer": "\\frac{1 + \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Monica is renovating the floor of her 15-foot by 20-foot dining room. She plans to place two-foot by two-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. Additionally, there is a 1-foot by 1-foot column at the center of the room that needs to be tiled as well. Determine the total number of tiles she will use. | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number obtained from the last two nonzero digits of $80!$ is equal to $n$. Find the value of $n$. | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the perpendicular bisectors of sides \(AB\) and \(AC\) are drawn, intersecting lines \(AC\) and \(AB\) at points \(N\) and \(M\) respectively. The length of segment \(NM\) is equal to the length of side \(BC\) of the triangle. The angle at vertex \(C\) of the triangle is \(40^\circ\). Find the angle at vertex \(B\) of the triangle. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many four-digit numbers contain one even digit and three odd digits, with no repeated digits? | {
"answer": "1140",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circle, points \(B\) and \(D\) are located on opposite sides of the diameter \(AC\). It is known that \(AB = \sqrt{6}\), \(CD = 1\), and the area of triangle \(ABC\) is three times the area of triangle \(BCD\). Find the radius of the circle. | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a set of points \( M \) on a plane and seven different circles \( C_{1}, C_{2}, \dots, C_{7} \). Circle \( C_{7} \) passes through exactly 7 points in \( M \); circle \( C_{6} \) passes through exactly 6 points in \( M \); ..., circle \( C_{1} \) passes through exactly 1 point in \( M \). Determine the minimum number of points in set \( M \). | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows a smaller rectangle made from three squares, each of area \(25 \ \mathrm{cm}^{2}\), inside a larger rectangle. Two of the vertices of the smaller rectangle lie on the mid-points of the shorter sides of the larger rectangle. The other two vertices of the smaller rectangle lie on the other two sides of the larger rectangle. What is the area, in \(\mathrm{cm}^{2}\), of the larger rectangle? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When evaluated, the sum of the digits of the integer equal to \(10^{2021} - 2021\) is: | {
"answer": "18185",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangle has a perimeter of 80 inches and an area greater than 240 square inches. How many non-congruent rectangles meet these criteria? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ is a real number, let $\lfloor x \rfloor$ be the greatest integer that is less than or equal to $x$ . If $n$ is a positive integer, let $S(n)$ be defined by
\[
S(n)
= \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
+ 10 \left( n - 10^{\lfloor \log n \rfloor}
\cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
\right) \, .
\]
(All the logarithms are base 10.) How many integers $n$ from 1 to 2011 (inclusive) satisfy $S(S(n)) = n$ ? | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | {
"answer": "8.8\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many lattice points lie on the hyperbola \( x^2 - y^2 = 1800^2 \)? | {
"answer": "250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\theta$ be the angle between the line
\[\frac{x - 2}{4} = \frac{y + 1}{5} = \frac{z - 4}{7}\]
and the plane $3x + 4y - 7z = 5.$ Find $\sin \theta.$ | {
"answer": "\\frac{17}{\\sqrt{6660}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Druv has a $33 \times 33$ grid of unit squares, and he wants to color each unit square with exactly one of three distinct colors such that he uses all three colors and the number of unit squares with each color is the same. However, he realizes that there are internal sides, or unit line segments that have exactly one unit square on each side, with these two unit squares having different colors. What is the minimum possible number of such internal sides? | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest positive integer $n$ such that there exist $n$ distinct positive integers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying
$$
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=2017.
$$ | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are $12$ different cards, with $3$ cards each of red, yellow, green, and blue, select $3$ cards such that they cannot all be of the same color and there can be at most $1$ blue card. | {
"answer": "189",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pair of natural numbers is called "good" if one of the numbers is divisible by the other. The numbers from 1 to 30 are divided into 15 pairs. What is the maximum number of good pairs that could be formed? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a$, $b$, and $c$ are positive numbers such that $ab = 24\sqrt{3}$, $ac = 30\sqrt{3}$, and $bc = 40\sqrt{3}$, find the value of $abc$. | {
"answer": "120\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \frac{2\sqrt{3}}{3})$, with its left and right foci being $F_1$ and $F_2$ respectively. The chord formed by the intersection of the circle $x^2 + y^2 = 2$ and the line $x + y + b = 0$ has a length of $2$.
(Ⅰ) Find the standard equation of the ellipse $C$;
(Ⅱ) Let $Q$ be a moving point on the ellipse $C$ that is not on the $x$-axis, with $Q$ being the origin. A line parallel to $OQ$ passing through $F_2$ intersects the ellipse $C$ at two distinct points $M$ and $N$.
(1) Investigate whether $\frac{|MN|}{|OQ|^2}$ is a constant. If it is, find the constant; if not, explain why.
(2) Let the area of $\triangle QF_2M$ be $S_1$ and the area of $\triangle OF_2N$ be $S_2$, and let $S = S_1 + S_2$. Find the maximum value of $S$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | {
"answer": "20\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right and then subtracting $n$ from the resulting number. For example, $b_4 = 1234 - 4 = 1230$ and $b_{12} = 123456789101112 - 12 = 123456789101100$. For $1 \le k \le 100$, how many $b_k$ are divisible by 9? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer with exactly $12$ positive factors? | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the function $f(x)=4\sin \left(2x+\frac{\pi }{3}\right)$, the following propositions are given:
$(1)$ From $f(x_1)=f(x_2)$, it can be concluded that $x_1-x_2$ is an integer multiple of $\pi$; $(2)$ The expression for $f(x)$ can be rewritten as $f(x)=4\cos \left(2x-\frac{\pi }{6}\right)$; $(3)$ The graph of $f(x)$ is symmetric about the point $\left(-\frac{\pi }{6},0\right)$; $(4)$ The graph of $f(x)$ is symmetric about the line $x=-\frac{\pi }{6}$; Among these, the correct propositions are ______________. | {
"answer": "(2)(3)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Hexagon $ABCDEF$ is divided into five rhombuses, $P, Q, R, S,$ and $T$ , as shown. Rhombuses $P, Q, R,$ and $S$ are congruent, and each has area $\sqrt{2006}.$ Let $K$ be the area of rhombus $T$ . Given that $K$ is a positive integer, find the number of possible values for $K.$ [asy] // TheMathGuyd size(8cm); pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6); draw(A--B--C--D--EE--F--cycle); draw(F--G--(2.1,0)); draw(C--H--(2.1,0)); draw(G--(2.1,-3.2)); draw(H--(2.1,-3.2)); label("$\mathcal{T}$",(2.1,-1.6)); label("$\mathcal{P}$",(0,-1),NE); label("$\mathcal{Q}$",(4.2,-1),NW); label("$\mathcal{R}$",(0,-2.2),SE); label("$\mathcal{S}$",(4.2,-2.2),SW); [/asy] | {
"answer": "89",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $$f(x)= \begin{cases} a^{x}, x<0 \\ ( \frac {1}{4}-a)x+2a, x\geq0\end{cases}$$ such that for any $x\_1 \neq x\_2$, the inequality $$\frac {f(x_{1})-f(x_{2})}{x_{1}-x_{2}}<0$$ holds true. Determine the range of values for the real number $a$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(3,3)$. Find the area of the triangle enclosed by these two lines and the line $x+y=12$. | {
"answer": "8.625",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $(C)$: $\frac{x^{2}}{3m} + \frac{y^{2}}{m} = 1 (m > 0)$ with the length of its major axis being $2\sqrt{6}$, and $O$ is the coordinate origin.
(I) Find the equation and eccentricity of the ellipse $(C)$;
(II) Let moving line $(l)$ intersect with the $y$-axis at point $B$, and the symmetric point $P(3, 0)$ about line $(l)$ lies on the ellipse $(C)$. Find the minimum value of $|OB|$. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function $f(x)=2x-\cos x$, and $\{a_n\}$ be an arithmetic sequence with a common difference of $\dfrac{\pi}{8}$. If $f(a_1)+f(a_2)+\ldots+f(a_5)=5\pi$, calculate $\left[f(a_3)\right]^2-a_2a_3$. | {
"answer": "\\dfrac{13}{16}\\pi^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \(a\), \(b\), and \(c\) are integers and \(a > 0\). Determine the value of \(a + b + c\). | {
"answer": "86",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S \) be a subset of \(\{1,2,3, \ldots, 199,200\}\). We say that \( S \) is pretty if, for every pair of elements \( a \) and \( b \) in \( S \), the number \( a - b \) is not a prime number. What is the maximum number of elements in a pretty subset of \(\{1,2,3, \ldots, 199,200\}\)? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a subject test, the average score of Xiaofang's four subjects: Chinese, Mathematics, English, and Science, is 88. The average score of the first two subjects is 93, and the average score of the last three subjects is 87. What is Xiaofang's English test score? | {
"answer": "95",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ such that $AKB$, $BLC$, $CMD$, and $DNA$ are isosceles right triangles. If the area of square $ABCD$ is 25, find the area of $KLMN$. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer with exactly $12$ positive factors? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $A$, $B$, and $P$ are three distinct points on the hyperbola ${x^2}-\frac{{y^2}}{4}=1$, and they satisfy $\overrightarrow{PA}+\overrightarrow{PB}=2\overrightarrow{PO}$ (where $O$ is the origin), the slopes of lines $PA$ and $PB$ are denoted as $m$ and $n$ respectively. Find the minimum value of ${m^2}+\frac{{n^2}}{9}$. | {
"answer": "\\frac{8}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We have an equilateral triangle with circumradius $1$ . We extend its sides. Determine the point $P$ inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center $P$ and radius $1$ , is maximum.
(The total distance of the point P from the sides of an equilateral triangle is fixed )
*Proposed by Erfan Salavati* | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the remainder when the value of $m$ is divided by 1000 in the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_6 \le 1500$ such that $a_i-i$ is odd for $1\le i \le 6$. The total number of sequences can be expressed as ${m \choose n}$ for some integers $m>n$. | {
"answer": "752",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points P(-2,-3) and Q(5,3) in the xy-plane; point R(x,m) is such that x=2 and PR+RQ is a minimum. Find m. | {
"answer": "\\frac{3}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In isosceles triangle $\triangle ABC$, $CA=CB=6$, $\angle ACB=120^{\circ}$, and point $M$ satisfies $\overrightarrow{BM}=2 \overrightarrow{MA}$. Determine the value of $\overrightarrow{CM} \cdot \overrightarrow{CB}$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{1260}$ terminate? | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The largest three-digit number divided by an integer, with the quotient rounded to one decimal place being 2.5, will have the smallest divisor as: | {
"answer": "392",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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