problem stringlengths 10 5.15k | answer dict |
|---|---|
How many distinct sequences of five letters can be made from the letters in FREQUENCY if each sequence must begin with F, end with Y, and no letter can appear in a sequence more than once? Further, the second letter must be a vowel. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola \( C: x^{2} = 2py \) with \( p > 0 \), two tangents \( RA \) and \( RB \) are drawn from the point \( R(1, -1) \) to the parabola \( C \). The points of tangency are \( A \) and \( B \). Find the minimum area of the triangle \( \triangle RAB \) as \( p \) varies. | {
"answer": "3 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among four people, A, B, C, and D, they pass a ball to each other. The first pass is from A to either B, C, or D, and the second pass is from the receiver to any of the other three. This process continues for several passes. Calculate the number of ways the ball can be passed such that it returns to A on the fourth pass. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of the quadrilateral pyramid \( S A B C D \) is a rhombus \( A B C D \) with an acute angle at vertex \( A \). The height of the rhombus is 4, and the point of intersection of its diagonals is the orthogonal projection of vertex \( S \) onto the plane of the base. A sphere with radius 2 touches the planes of all the faces of the pyramid. Find the volume of the pyramid, given that the distance from the center of the sphere to the line \( A C \) is \( \frac{2 \sqrt{2}}{3} A B \). | {
"answer": "8\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $b_0 = \sin^2 \left( \frac{\pi}{30} \right)$ and for $n \geq 0$,
\[ b_{n + 1} = 4b_n (1 - b_n). \]
Find the smallest positive integer $n$ such that $b_n = b_0$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\ln (1+x)- \frac {x(1+λx)}{1+x}$, if $f(x)\leqslant 0$ when $x\geqslant 0$, calculate the minimum value of $λ$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the last initial of Mr. and Mrs. Alpha's baby's monogram is 'A', determine the number of possible monograms in alphabetical order with no letter repeated. | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the product $\frac{1}{3} \cdot \frac{9}{1} \cdot \frac{1}{27} \cdot \frac{81}{1} \dotsm \frac{1}{6561} \cdot \frac{19683}{1}$. | {
"answer": "729",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a bike shed, there are bicycles (two wheels), tricycles, and cars (four wheels). The number of bicycles is four times the number of cars. Several students counted the total number of wheels in the shed, but each of them obtained a different count: $235, 236, 237, 238, 239$. Among these, one count is correct. Smart kid, please calculate the number of different combinations of the three types of vehicles that satisfy the given conditions. (For example, if there are 1 bicycle, 2 tricycles, and 3 cars or 3 bicycles, 2 tricycles, and 1 car, it counts as two different combinations). | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three of the four vertices of a square are $(2, 8)$, $(13, 8)$, and $(13, -3)$. What is the area of the intersection of this square region and the region inside the graph of the equation $(x - 2)^2 + (y + 3)^2 = 16$? | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of times and the positions in which it appears $\frac12$ in the following sequence of fractions: $$ \frac11, \frac21, \frac12 , \frac31 , \frac22 , \frac13 , \frac41,\frac32,\frac23,\frac14,..., \frac{1}{1992} $$ | {
"answer": "664",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the arithmetic sequence $\{a_{n}\}$, given that $a_{3}=-2, a_{n}=\frac{3}{2}, S_{n}=-\frac{15}{2}$, find the value of $a_{1}$. | {
"answer": "-\\frac{19}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate:<br/>$(1)(\sqrt{\frac{1}{3}})^{2}+\sqrt{0．{3}^{2}}-\sqrt{\frac{1}{9}}$;<br/>$(2)(\sqrt{6}-\sqrt{\frac{1}{2}})-(\sqrt{24}+2\sqrt{\frac{2}{3}})$;<br/>$(3)(\frac{\sqrt{32}}{3}-4\sqrt{\frac{1}{2}}+3\sqrt{27})÷2\sqrt{2}$;<br/>$(4)(\sqrt{3}+\sqrt{2}-1)(\sqrt{3}-\sqrt{2}+1)$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The postal department stipulates that for letters weighing up to $100$ grams (including $100$ grams), each $20$ grams requires a postage stamp of $0.8$ yuan. If the weight is less than $20$ grams, it is rounded up to $20$ grams. For weights exceeding $100$ grams, the initial postage is $4$ yuan. For each additional $100$ grams beyond $100$ grams, an extra postage of $2$ yuan is required. In Class 8 (9), there are $11$ students participating in a project to learn chemistry knowledge. If each answer sheet weighs $12$ grams and each envelope weighs $4$ grams, and these $11$ answer sheets are divided into two envelopes for mailing, the minimum total amount of postage required is ____ yuan. | {
"answer": "5.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many four-digit numbers, without repeating digits, that can be formed using the digits 0, 1, 2, 3, 4, 5, are divisible by 25? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A set \( \mathcal{T} \) of distinct positive integers has the following property: for every integer \( y \) in \( \mathcal{T}, \) the arithmetic mean of the set of values obtained by deleting \( y \) from \( \mathcal{T} \) is an integer. Given that 2 belongs to \( \mathcal{T} \) and that 1024 is the largest element of \( \mathcal{T}, \) what is the greatest number of elements that \( \mathcal{T} \) can have? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest number \( n > 1980 \) such that the number
$$
\frac{x_{1} + x_{2} + x_{3} + \ldots + x_{n}}{5}
$$
is an integer for any given integer values \( x_{1}, x_{2}, x_{3}, \ldots, x_{n} \), none of which is divisible by 5. | {
"answer": "1985",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the point F(0,1) is the focus of the parabola $x^2=2py$,
(1) Find the equation of the parabola C;
(2) Points A, B, and C are three points on the parabola such that $\overrightarrow{FA} + \overrightarrow{FB} + \overrightarrow{FC} = \overrightarrow{0}$, find the maximum value of the area of triangle ABC. | {
"answer": "\\frac{3\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the remainder when $1 + 11 + 11^2 + \cdots + 11^{1024}$ is divided by $500$. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ be a triangle with $AB=5, BC=6, CA=7$ . Suppose $P$ is a point inside $\triangle ABC$ such that $\triangle BPA\sim \triangle APC$ . If $AP$ intersects $BC$ at $X$ , find $\frac{BX}{CX}$ .
[i]Proposed by Nathan Ramesh | {
"answer": "25/49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer $n$ such that $\sqrt{n}-\sqrt{n-1}<0.02$?
A) 624
B) 625
C) 626
D) 627
E) 628 | {
"answer": "626",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that 20% of the birds are geese, 40% are swans, 10% are herons, and 20% are ducks, and the remaining are pigeons, calculate the percentage of the birds that are not herons and are ducks. | {
"answer": "22.22\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A factory received a task to process 6000 pieces of part P and 2000 pieces of part Q. The factory has 214 workers. Each worker spends the same amount of time processing 5 pieces of part P as they do processing 3 pieces of part Q. The workers are divided into two groups to work simultaneously on different parts. In order to complete this batch of tasks in the shortest time, the number of people processing part P is \_\_\_\_\_\_. | {
"answer": "137",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, the equation of circle C is given by ρ = 2sinθ, and the equation of line l is given by $ρsin(θ+ \frac {π}{3})=a$. If line l is tangent to circle C, find the value of the real number a. | {
"answer": "- \\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in $\triangle ABC$, $\sin A + 2 \sin B \cos C = 0$, find the maximum value of $\tan A$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the triangle formed by the axis of the parabola $y^{2}=8x$ and the two asymptotes of the hyperbola $(C)$: $\frac{x^{2}}{8}-\frac{y^{2}}{4}=1$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A metal bar at a temperature of $20^{\circ} \mathrm{C}$ is placed in water at a temperature of $100^{\circ} \mathrm{C}$. After thermal equilibrium is established, the temperature becomes $80^{\circ} \mathrm{C}$. Then, without removing the first bar, another identical metal bar also at $20^{\circ} \mathrm{C}$ is placed in the water. What will be the temperature of the water after thermal equilibrium is established? | {
"answer": "68",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of an equilateral triangle ABC is 36. Points P, Q, R are located on BC, AB, and CA respectively, such that BP = 1/3 BC, AQ = QB, and PR is perpendicular to AC. Find the area of triangle PQR. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number $1$ , an arc between a red and a blue point is assigned a number $2$ , and an arc between a blue and a green point is assigned a number $3$ . The arcs between two points of the same colour are assigned a number $0$ . What is the greatest possible sum of all the numbers assigned to the arcs? | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product \( 29 \cdot 11 \), and the numbers 1059, 1417, and 2312, are each divided by \( d \). If the remainder is always \( r \), where \( d \) is an integer greater than 1, what is \( d - r \) equal to? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice celebrated her birthday on Friday, March 15 in the year 2012. Determine the next year when her birthday will next fall on a Monday.
A) 2021
B) 2022
C) 2023
D) 2024
E) 2025 | {
"answer": "2025",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain teacher received $10$, $6$, $8$, $5$, $6$ letters from Monday to Friday, then the variance of this data set is $s^{2}=$____. | {
"answer": "3.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest number, with its digits all different and none of them being zero, whose digits add up to 20? | {
"answer": "9821",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In square $ABCD$, a point $P$ is chosen at random. The probability that $\angle APB < 90^{\circ}$ is ______. | {
"answer": "1 - \\frac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The rules of table tennis competition stipulate: In a game, before the opponent's score reaches 10-all, one side serves twice consecutively, then the other side serves twice consecutively, and so on. Each serve, the winning side scores 1 point, and the losing side scores 0 points. In a game between player A and player B, the probability of the server scoring 1 point on each serve is 0.6, and the outcomes of each serve are independent of each other. Player A serves first in a game.
(1) Find the probability that the score is 1:2 in favor of player B at the start of the fourth serve;
(2) Find the probability that player A is leading in score at the start of the fifth serve. | {
"answer": "0.3072",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ellipse whose axes are parallel to the coordinate axes is tangent to the $x$-axis at $(6, 0)$ and tangent to the $y$-axis at $(0, 2)$. Find the distance between the foci of the ellipse. | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that \( ABCDEF \) is a regular hexagon with sides of length 6. Each interior angle of \( ABCDEF \) is equal to \( 120^{\circ} \).
(a) A circular arc with center \( D \) and radius 6 is drawn from \( C \) to \( E \). Determine the area of the shaded sector.
(b) A circular arc with center \( D \) and radius 6 is drawn from \( C \) to \( E \), and a second arc with center \( A \) and radius 6 is drawn from \( B \) to \( F \). These arcs are tangent (touch) at the center of the hexagon. Line segments \( BF \) and \( CE \) are also drawn. Determine the total area of the shaded regions.
(c) Along each edge of the hexagon, a semi-circle with diameter 6 is drawn. Determine the total area of the shaded regions; that is, determine the total area of the regions that lie inside exactly two of the semi-circles. | {
"answer": "18\\pi - 27\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, with the origin as the pole and the positive semi-axis of $x$ as the polar axis, establish a polar coordinate system using the same unit of length. The parametric equations of line $l$ are given by $\begin{cases}x=2+t \\ y=1+t\end{cases} (t \text{ is a parameter})$, and the polar coordinate equation of circle $C$ is given by $\rho=4 \sqrt{2}\sin (θ+ \dfrac{π}{4})$.
(1) Find the standard equation of line $l$ and the Cartesian coordinate equation of circle $C$;
(2) Let points $A$ and $B$ be the intersections of curve $C$ and line $l$, and point $P$ has Cartesian coordinates $(2,1)$. Find the value of $||PA|-|PB||$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $b\cos C=3a\cos B-c\cos B$, $\overrightarrow{BA}\cdot \overrightarrow{BC}=2$, find the area of $\triangle ABC$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equilateral triangle has sides of \(2x\) and \(x+15\) as shown. Find the perimeter of the triangle in terms of \(x\). | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Mindy made four purchases for $2.96, 6.57, 8.49, and 12.38. Each amount needs to be rounded up to the nearest dollar except the amount closest to a whole number, which should be rounded down. Calculate the total rounded amount. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a line $y=-x+1$ and an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), they intersect at points A and B.
(1) If the eccentricity of the ellipse is $\frac{\sqrt{3}}{3}$ and the focal distance is 2, find the length of the segment AB.
(2) (For Liberal Arts students) If segment OA is perpendicular to segment OB (where O is the origin), find the value of $\frac{1}{a^2} + \frac{1}{b^2}$.
(3) (For Science students) If segment OA is perpendicular to segment OB (where O is the origin), and when the eccentricity of the ellipse $e$ lies in the interval $\left[ \frac{1}{2}, \frac{\sqrt{2}}{2} \right]$, find the maximum length of the major axis of the ellipse. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, it is known that $\sqrt{3}\sin{2B} = 1 - \cos{2B}$.
(1) Find the value of angle $B$;
(2) If $BC = 2$ and $A = \frac{\pi}{4}$, find the area of triangle $ABC$. | {
"answer": "\\frac{3 + \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given any point $P$ on the line $l: x-y+4=0$, two tangent lines $AB$ are drawn to the circle $O: x^{2}+y^{2}=4$ with tangent points $A$ and $B$. The line $AB$ passes through a fixed point ______; let the midpoint of segment $AB$ be $Q$. The minimum distance from point $Q$ to the line $l$ is ______. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles \( C_{1} \) and \( C_{2} \) touch each other externally and the line \( l \) is a common tangent. The line \( m \) is parallel to \( l \) and touches the two circles \( C_{1} \) and \( C_{3} \). The three circles are mutually tangent. If the radius of \( C_{2} \) is 9 and the radius of \( C_{3} \) is 4, what is the radius of \( C_{1} \)? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( x \) and \( y \) are non-zero real numbers and they satisfy \(\frac{x \sin \frac{\pi}{5} + y \cos \frac{\pi}{5}}{x \cos \frac{\pi}{5} - y \sin \frac{\pi}{5}} = \tan \frac{9 \pi}{20}\),
1. Find the value of \(\frac{y}{x}\).
2. In \(\triangle ABC\), if \(\tan C = \frac{y}{x}\), find the maximum value of \(\sin 2A + 2 \cos B\). | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Originally, there were 5 books on the bookshelf. If 2 more books are added, but the relative order of the original books must remain unchanged, then there are $\boxed{\text{different ways}}$ to place the books. | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle XYZ$ be a triangle in the plane, and let $W$ be a point outside the plane of $\triangle XYZ$, so that $WXYZ$ is a pyramid whose faces are all triangles.
Suppose that the edges of $WXYZ$ have lengths of either $24$ or $49$, and no face of $WXYZ$ is equilateral. Determine the surface area of the pyramid $WXYZ$. | {
"answer": "48 \\sqrt{2257}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The total number of toothpicks used to build a rectangular grid 15 toothpicks high and 12 toothpicks wide, with internal diagonal toothpicks, is calculated by finding the sum of the toothpicks. | {
"answer": "567",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $ABCD$ has $AB = CD = 3$ and $BC = DA = 5$. The rectangle is first rotated $90^\circ$ clockwise around vertex $D$, then it is rotated $90^\circ$ clockwise around the new position of vertex $C$ (after the first rotation). What is the length of the path traveled by point $A$?
A) $\frac{3\pi(\sqrt{17} + 6)}{2}$
B) $\frac{\pi(\sqrt{34} + 5)}{2}$
C) $\frac{\pi(\sqrt{30} + 5)}{2}$
D) $\frac{\pi(\sqrt{40} + 5)}{2}$ | {
"answer": "\\frac{\\pi(\\sqrt{34} + 5)}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a > 0$, $b > 0$, $c > 1$, and $a + b = 1$, find the minimum value of $( \frac{a^{2}+1}{ab} - 2) \cdot c + \frac{\sqrt{2}}{c - 1}$. | {
"answer": "4 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A real number $a$ is chosen randomly and uniformly from the interval $[-10, 15]$. Find the probability that the roots of the polynomial
\[ x^4 + 3ax^3 + (3a - 3)x^2 + (-5a + 4)x - 3 \]
are all real. | {
"answer": "\\frac{23}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Juan rolls a fair regular dodecahedral die marked with the numbers 1 through 12. Then Amal rolls a fair eight-sided die. What is the probability that the product of the two rolls is a multiple of 4? | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be a point not on line $XY$, and $Q$ a point on line $XY$ such that $PQ \perp XY$. Meanwhile, $R$ is a point on line $PY$ such that $XR \perp PY$. Given $XR = 6$, $PQ = 12$, and $XY = 7$, find the length of $PY$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $x$ such that $\lceil x \rceil \cdot x = 210$. Express $x$ as a decimal. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p, q, r,$ and $s$ be the roots of the polynomial $3x^4 - 8x^3 - 15x^2 + 10x - 2 = 0$. Find $pqrs$. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parabola $C:y^2=2px (p > 0)$, the sum of the distances from any point $Q$ on the parabola to a point inside it, $P(3,1)$, and the focus $F$, has a minimum value of $4$.
(I) Find the equation of the parabola;
(II) Through the focus $F$, draw a line $l$ that intersects the parabola $C$ at points $A$ and $B$. Find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find a four-digit number such that the square of the sum of the two-digit number formed by its first two digits and the two-digit number formed by its last two digits is exactly equal to the four-digit number itself. | {
"answer": "2025",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( O \) is the circumcenter of \( \triangle ABC \), and the equation
\[
\overrightarrow{A O} \cdot \overrightarrow{B C} + 2 \overrightarrow{B O} \cdot \overrightarrow{C A} + 3 \overrightarrow{C O} \cdot \overrightarrow{A B} = 0,
\]
find the minimum value of \( \frac{1}{\tan A} + \frac{1}{\tan C} \). | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, touch any part of the opposite 1500-meter wall, and stop at $B$. What is the minimum distance a participant must run? Assume that $A$ is 400 meters directly south of the first wall, and that $B$ is 600 meters directly north of the second wall. The two walls are parallel and are 1500 meters apart. Express your answer to the nearest meter. | {
"answer": "2915",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $A$ , $B$ , $C$ , and $D$ form a rectangle in that order. Point $X$ lies on $CD$ , and segments $\overline{BX}$ and $\overline{AC}$ intersect at $P$ . If the area of triangle $BCP$ is 3 and the area of triangle $PXC$ is 2, what is the area of the entire rectangle? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $|5 - e|$ where $e$ is the base of the natural logarithm. | {
"answer": "2.28172",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can 9 distinct items be distributed into three boxes so that one box contains 3 items, another contains 2 items, and the third contains 4 items? | {
"answer": "7560",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Form a three-digit number using the digits 0, 1, 2, 3. Repeating digits is not allowed.
① How many three-digit numbers can be formed?
② If the three-digit numbers from ① are sorted in ascending order, what position does 230 occupy?
③ If repeating digits is allowed, how many of the formed three-digit numbers are divisible by 3? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square has vertices \( P, Q, R, S \) labelled clockwise. An equilateral triangle is constructed with vertices \( P, T, R \) labelled clockwise. What is the size of angle \( \angle RQT \) in degrees? | {
"answer": "135",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sequence {a<sub>n</sub>} is an arithmetic sequence, a<sub>1</sub> < 0, a<sub>8</sub> + a<sub>9</sub> > 0, a<sub>8</sub> • a<sub>9</sub> < 0. Find the smallest value of n for which S<sub>n</sub> > 0. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | {
"answer": "0.294",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and the vectors $\overrightarrow{m}=({\cos C, \cos({\frac{\pi}{2}-B})})$, $\overrightarrow{n}=({\cos({-4\pi+B}), -\sin C})$, and $\overrightarrow{m} \cdot \overrightarrow{n}=-\frac{\sqrt{2}}{2}$. <br/>$(1)$ Find the measure of angle $A$; <br/>$(2)$ If the altitude on side $AC$ is $2$, $a=3$, find the perimeter of $\triangle ABC$. | {
"answer": "5 + 2\\sqrt{2} + \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest value x such that, given any point inside an equilateral triangle of side 1, we can always choose two points on the sides of the triangle, collinear with the given point and a distance x apart. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$, its foci are equal to the minor axis length of the ellipse $Ω:x^{2}+ \frac{y^{2}}{4}=1$, and the major axis lengths of C and Ω are equal.
(1) Find the equation of ellipse C;
(2) Let $F_1$, $F_2$ be the left and right foci of ellipse C, respectively. A line l that does not pass through $F_1$ intersects ellipse C at two distinct points A and B. If the slopes of lines $AF_1$ and $BF_1$ form an arithmetic sequence, find the maximum area of △AOB. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If four departments A, B, C, and D select from six tourist destinations, calculate the total number of ways in which at least three departments have different destinations. | {
"answer": "1080",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. On a semicircle with AB as the diameter, besides points A and B, there are 6 other points. Since there are also 4 other points on AB, making a total of 12 points, how many quadrilaterals can be formed with these 12 points as vertices?
2. On one side of angle A, there are five points (excluding A), and on the other side, there are four points (excluding A). With these ten points (including A), how many triangles can be formed?
3. Suppose there are 3 equally spaced parallel lines intersecting with another set of 4 equally spaced parallel lines. How many triangles can be formed with these intersection points as vertices? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bag contains $5$ small balls of the same shape and size, with $2$ red balls and $3$ white balls. Three balls are randomly drawn from the bag.<br/>$(1)$ Find the probability of drawing exactly one red ball.<br/>$(2)$ If the random variable $X$ represents the number of red balls drawn, find the distribution of the random variable $X$. | {
"answer": "\\frac{3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the encoded equality $A B + A B + A B + A B + A B + A B + A B + A B + A B = A A B$, digits are replaced with letters: the same digits with the same letter, and different digits with different letters. Find all possible decipherings. (I. Rubanov) | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given M be the greatest five-digit number whose digits have a product of 36, determine the sum of the digits of M. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the asymptotic line of the hyperbola $\frac{x^2}{a}+y^2=1$ has a slope of $\frac{5π}{6}$, determine the value of $a$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the three-digit integer in the decimal system that satisfies the following properties:
1. When the digits in the tens and units places are swapped, the resulting number can be represented in the octal system as the original number.
2. When the digits in the hundreds and tens places are swapped, the resulting number is 16 less than the original number when read in the hexadecimal system.
3. When the digits in the hundreds and units places are swapped, the resulting number is 18 more than the original number when read in the quaternary system. | {
"answer": "139",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\cos x,\sin x)$ and $\overrightarrow{b}=(3,-\sqrt{3})$, with $x\in[0,\pi]$.
$(1)$ If $\overrightarrow{a}\parallel\overrightarrow{b}$, find the value of $x$; $(2)$ Let $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$, find the maximum and minimum values of $f(x)$ and the corresponding values of $x$. | {
"answer": "-2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $y=f(x)$ has an inverse function $y=f^{-1}(x)$, and $y=f(x+2)$ and $y=f^{-1}(x-1)$ are inverse functions of each other, then $f^{-1}(2007)-f^{-1}(1)=$ . | {
"answer": "4012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The smaller square has an area of 16 and the grey triangle has an area of 1. What is the area of the larger square?
A) 17
B) 18
C) 19
D) 20
E) 21 | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation: $4x^2 - (x^2 - 2x + 1) = 0$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a sequence of complex numbers $\left\{z_{n}\right\}$ defined as "interesting" if $\left|z_{1}\right|=1$ and for every positive integer $n$, the following holds:
$$
4 z_{n+1}^{2}+2 z_{n} z_{n+1}+z_{n}^{2}=0.
$$
Find the largest constant $C$ such that for any "interesting" sequence $\left\{z_{n}\right\}$ and any positive integer $m$, the inequality below holds:
$$
\left|z_{1}+z_{2}+\cdots+z_{m}\right| \geqslant C.
$$ | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the $y$-intercepts of the following system of equations:
1. $2x - 3y = 6$
2. $x + 4y = -8$ | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 such that its bases are parallel to the base of the hemisphere and the top of the cylinder touches the top of the hemisphere. What is the height of the cylinder? | {
"answer": "2\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An entrepreneur took out a discounted loan of 12 million HUF with a fixed annual interest rate of 8%. What will be the debt after 10 years if they can repay 1.2 million HUF annually? | {
"answer": "8523225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given three positive numbers \( a, b, \mathrm{and} c \) satisfying \( a \leq b+c \leq 3a \) and \( 3b^2 \leq a(a+c) \leq 5b^2 \), what is the minimum value of \(\frac{b-2c}{a}\)? | {
"answer": "-\\frac{18}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five people can paint a house in 10 hours. How many hours would it take four people to paint the same house and mow the lawn if mowing the lawn takes an additional 3 hours per person, assuming that each person works at the same rate for painting and different rate for mowing? | {
"answer": "15.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Ming's family has three hens. The first hen lays one egg every day, the second hen lays one egg every two days, and the third hen lays one egg every three days. Given that all three hens laid eggs on January 1st, how many eggs did these three hens lay in total in the 31 days of January? | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let us call a number \( \mathrm{X} \) "50-podpyirayushchim" if for any 50 real numbers \( a_{1}, \ldots, a_{50} \) whose sum is an integer, there exists at least one \( a_i \) such that \( \left|a_{i}-\frac{1}{2}\right| \geq X \).
Find the greatest 50-podpyirayushchee \( X \), rounded to the nearest hundredth according to standard mathematical rules. | {
"answer": "0.01",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the ellipse (C) with the equation \\(\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)\\) and an eccentricity of \\(\frac{\sqrt{3}}{3}\\). A line passing through the right focus (F2(c,0)) perpendicular to the x-axis intersects the ellipse at points A and B, such that |AB|=\\(\frac{4\sqrt{3}}{3}\\). Additionally, a line (l) passing through the left focus (F1(-c,0)) intersects the ellipse at point M.
1. Find the equation of the ellipse (C).
2. Given that points A and B on the ellipse (C) are symmetric with respect to line (l), find the maximum area of triangle AOB. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3, \dots, 2020\}$. What is the probability that $abc + ab + a$ is divisible by $4$?
A) $\frac{1}{4}$
B) $\frac{1}{32}$
C) $\frac{8}{32}$
D) $\frac{9}{32}$
E) $\frac{1}{16}$ | {
"answer": "\\frac{9}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\frac {(a+b)^{2}-c^{2}}{3ab}=1$.
$(1)$ Find $\angle C$;
$(2)$ If $c= \sqrt {3}$ and $b= \sqrt {2}$, find $\angle B$ and the area of $\triangle ABC$. | {
"answer": "\\frac {3+ \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Angle ABC is a right angle. The diagram shows four quadrilaterals, where three are squares on each side of triangle ABC, and one square is on the hypotenuse. The sum of the areas of all four squares is 500 square centimeters. What is the number of square centimeters in the area of the largest square? | {
"answer": "\\frac{500}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Person A and Person B decided to go to a restaurant. Due to high demand, Person A arrived first and took a waiting number, while waiting for Person B. After a while, Person B arrived but did not see Person A, so he also took a waiting number. While waiting, Person B saw Person A, and they compared their waiting numbers. They found that the digits of these two numbers are two-digit numbers in reverse order, and the sum of the digits of both numbers is 8. Additionally, Person B's number is 18 greater than Person A's. What is Person A's number? $\qquad$ | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, there are points $P_{1}, P_{2}, \ldots, P_{n-1}, P_{n}, \ldots (n \in \mathbb{N}^{*})$. Let the coordinates of point $P_{n}$ be $(n, a_{n})$, where $a_{n}= \frac {2}{n} (n \in \mathbb{N}^{*})$. The line passing through points $P_{n}$ and $P_{n+1}$ forms a triangle with the coordinate axes, and the area of this triangle is $b_{n}$. Let $S_{n}$ represent the sum of the first $n$ terms of the sequence $\{b_{n}\}$. Then, $S_{5}=$ \_\_\_\_\_\_. | {
"answer": "\\frac {125}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a smaller circle is entirely inside a larger circle, such that the larger circle has a radius $R = 2$, and the areas of the two circles form an arithmetic progression, with the largest area being that of the larger circle, find the radius of the smaller circle. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose convex hexagon $ \text{HEXAGN}$ has $ 120^\circ$ -rotational symmetry about a point $ P$ —that is, if you rotate it $ 120^\circ$ about $ P$ , it doesn't change. If $ PX\equal{}1$ , find the area of triangle $ \triangle{GHX}$ . | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\theta\) be an angle in the second quadrant, and if \(\tan (\theta+ \frac {\pi}{3})= \frac {1}{2}\), calculate the value of \(\sin \theta+ \sqrt {3}\cos \theta\). | {
"answer": "- \\frac {2 \\sqrt {5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T = \{ 1, 2, 3, \dots, 14, 15 \}$ . Say that a subset $S$ of $T$ is *handy* if the sum of all the elements of $S$ is a multiple of $5$ . For example, the empty set is handy (because its sum is 0) and $T$ itself is handy (because its sum is 120). Compute the number of handy subsets of $T$ . | {
"answer": "6560",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To investigate a non-luminous black planet in distant space, Xiao Feitian drives a high-speed spaceship equipped with a powerful light, traveling straight towards the black planet at a speed of 100,000 km/s. When Xiao Feitian had just been traveling for 100 seconds, the spaceship instruments received light reflected back from the black planet. If the speed of light is 300,000 km/s, what is the distance from Xiao Feitian's starting point to the black planet in 10,000 kilometers? | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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