problem stringlengths 10 5.15k | answer dict |
|---|---|
A bus arrives randomly between 3:30 pm and 4:30 pm, waits for 40 minutes, and then departs. If Sara also arrives randomly between 3:30 pm and 4:30 pm, what is the probability that the bus will still be there when she arrives? | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M$ be the number of positive integers that are less than or equal to $2048$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cuckoo clock chimes "cuckoo" as many times as the hour indicated by the hour hand (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock at 9:05 and started turning the minute hand until the clock advanced by 7 hours. How many times did the clock chime "cuckoo" during this period? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The high-speed train "Sapsan," approaching a railway station at a speed of \( v = 216 \) km/h, emits a warning sound signal lasting \( \Delta t = 5 \) seconds when it is half a kilometer away from the station. What will be the duration of the signal \( \Delta t_{1} \) from the perspective of passengers standing on the platform? The speed of sound in the air is \( c = 340 \) m/s. | {
"answer": "4.12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the total number of cards in a stack where cards are numbered consecutively from 1 through $2n$ and rearranged such that, after a similar process of splitting into two piles and restacking alternately (starting with pile B), card number 252 retains its original position. | {
"answer": "504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Analyzing the intersection of $y = x^3 - 6x + 2$ and $y = m$ where $-10 < m < 10$, define $L(m)$ as the smallest $x$ coordinate of their intersection points. Calculate the function $r = \frac{L(-m) - L(m)}{m}$ as $m$ approaches zero. Find the limit of this function. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum value that the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\) can take, where \(a, b, c\) are distinct non-zero digits? | {
"answer": "1/203",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $IJKL$ is contained within square $WXYZ$ such that each side of $IJKL$ can be extended to pass through a vertex of $WXYZ$. The side length of square $WXYZ$ is $\sqrt{98}$, and $WI = 2$. What is the area of the inner square $IJKL$?
A) $62$
B) $98 - 4\sqrt{94}$
C) $94 - 4\sqrt{94}$
D) $98$
E) $100$ | {
"answer": "98 - 4\\sqrt{94}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 9 representatives from different countries, with 3 people from each country. They sit randomly around a round table with 9 chairs. What is the probability that each representative has at least one representative from another country sitting next to them? | {
"answer": "41/56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The jury, when preparing versions of the district math olympiad problems for grades $7, 8, 9, 10, 11$, aims to ensure that each version for each grade contains exactly 7 problems, of which exactly 4 do not appear in any other version. What is the maximum number of problems that can be included in the olympiad? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest positive integer \( n \) for which we can find a set of distinct positive integers such that each integer is at most 2002 and if \( a \) and \( b \) are in the set, then \( a^2 \) and \( ab \) are not in the set. | {
"answer": "1958",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the 800-digit integer
$$
234523452345 \cdots 2345 .
$$
The first \( m \) digits and the last \( n \) digits of the above integer are crossed out so that the sum of the remaining digits is 2345. Find the value of \( m+n \). | {
"answer": "130",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of kings that can be placed on a chessboard such that no two of them attack each other? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Perpendiculars $BE$ and $DF$ dropped from vertices $B$ and $D$ of parallelogram $ABCD$ onto sides $AD$ and $BC$, respectively, divide the parallelogram into three parts of equal area. A segment $DG$, equal to segment $BD$, is laid out on the extension of diagonal $BD$ beyond vertex $D$. Line $BE$ intersects segment $AG$ at point $H$. Find the ratio $AH: HG$. | {
"answer": "1:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube is dissected into 6 pyramids by connecting a given point in the interior of the cube with each vertex of the cube, so that each face of the cube forms the base of a pyramid. The volumes of five of these pyramids are 200, 500, 1000, 1100, and 1400. What is the volume of the sixth pyramid? | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Adam, Bendeguz, Cathy, and Dennis all see a positive integer $n$ . Adam says, " $n$ leaves a remainder of $2$ when divided by $3$ ." Bendeguz says, "For some $k$ , $n$ is the sum of the first $k$ positive integers." Cathy says, "Let $s$ be the largest perfect square that is less than $2n$ . Then $2n - s = 20$ ." Dennis says, "For some $m$ , if I have $m$ marbles, there are $n$ ways to choose two of them." If exactly one of them is lying, what is $n$ ? | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, and $c$ be positive real numbers such that $abc = 4$. Find the minimum value of
\[(3a + b)(2b + 3c)(ac + 4).\] | {
"answer": "384",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers, divisible by 4 and less than 1000, do not contain any of the digits 6, 7, 8, 9, or 0? | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the number of games won to the number of games lost by the High School Hurricanes is $7/3$ with 5 games ended in a tie. Determine the percentage of games lost by the Hurricanes, rounded to the nearest whole percent. | {
"answer": "24\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$ and $q$ be constants. Suppose that the equation
\[\frac{(x+p)(x+q)(x-15)}{(x-5)^2} = 0\]
has exactly $3$ distinct roots, while the equation
\[\frac{(x-2p)(x-5)(x+10)}{(x+q)(x-15)} = 0\]
has exactly $2$ distinct roots. Compute $100p + q.$ | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
After lunch, there are dark spots with a total area of $S$ on a transparent square tablecloth. It turns out that if the tablecloth is folded in half along any of the two lines connecting the midpoints of its opposite sides or along one of its two diagonals, the total visible area of the spots becomes $S_{1}$. However, if the tablecloth is folded in half along the other diagonal, the total visible area of the spots remains $S$. What is the smallest possible value of the ratio $S_{1}: S$? | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the system
$$
\left\{\begin{array}{l}
x^{3}+3 y^{3}=11 \\
x^{2} y+x y^{2}=6
\end{array}\right.
$$
Calculate the values of the expression $\frac{x_{k}}{y_{k}}$ for each solution $\left(x_{k}, y_{k}\right)$ of the system and find the smallest among them. If necessary, round your answer to two decimal places. | {
"answer": "-1.31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The geometric sequence $\left\{a_{n}\right\}$ has the first term $a_{1}=1536$, and common ratio $q=-\frac{1}{2}$. Let $\Pi_{n}$ denote the product of its first $n$ terms $\left(n \in \mathbf{N}^{*}\right)$. Find the value of $n$ that maximizes $\Pi_{n}\$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A gives 24 apples to B and C, and each of the three people has at least two apples. How many different ways are there to distribute the apples? | {
"answer": "190",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $O$ as the circumcenter of $\triangle ABC$ and $D$ as the midpoint of $BC$. If $\overrightarrow{AO} \cdot \overrightarrow{AD}=4$ and $BC=2 \sqrt{6}$, then find the length of $AD$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, a rectangular ceiling \( P Q R S \) measures \( 6 \mathrm{~m} \) by \( 4 \mathrm{~m} \) and is to be completely covered using 12 rectangular tiles, each measuring \( 1 \mathrm{~m} \) by \( 2 \mathrm{~m} \). If there is a beam, \( T U \), that is positioned so that \( P T = S U = 2 \mathrm{~m} \) and that cannot be crossed by any tile, then the number of possible arrangements of tiles is: | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an isosceles triangle \( \triangle ABC \) with base angles \( \angle ABC = \angle ACB = 50^\circ \), points \( D \) and \( E \) lie on \( BC \) and \( AC \) respectively. Lines \( AD \) and \( BE \) intersect at point \( P \). Given \( \angle ABE = 30^\circ \) and \( \angle BAD = 50^\circ \), find \( \angle BED \). | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, 3, \ldots, 100\} \). Find the smallest positive integer \( n \) such that every \( n \)-element subset of \( S \) contains 4 pairwise coprime numbers. | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If Person B trades all their chairs for the same number of tables as Person A, Person B needs to pay an additional 320 yuan. If Person B does not pay the extra money, they would receive 5 fewer tables. It is known that the price of 3 tables is 48 yuan less than the price of 5 chairs. How many chairs does Person B originally have? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the following image, there is a hexagon $ABEFGD$. Quadrilaterals $ABCD$ and $EFGC$ are congruent rectangles, and quadrilateral $BEGD$ is also a rectangle. Determine the ratio of the areas of the white and shaded parts of the hexagon, given that $|AB| = 5 \text{ cm}$ and triangle $BEC$ is equilateral. | {
"answer": "2:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a tetrahedron \(ABCD\), \(\angle ADB = \angle BDC = \angle CDA = 60^\circ\). The areas of \(\triangle ADB\), \(\triangle BDC\), and \(\triangle CDA\) are \(\frac{\sqrt{3}}{2}\), \(2\), and \(1\) respectively. What is the volume of the tetrahedron? | {
"answer": "\\frac{2\\sqrt{6}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Hangzhou Asian Games are underway, and table tennis, known as China's "national sport," is receiving a lot of attention. In table tennis matches, each game is played to 11 points, with one point awarded for each winning shot. In a game, one side serves two balls first, followed by the other side serving two balls, and the service alternates every two balls. The winner of a game is the first side to reach 11 points with a lead of at least 2 points. If the score is tied at 10-10, the service order remains the same, but the service alternates after each point until one side wins by a margin of 2 points. In a singles table tennis match between players A and B, assuming player A serves first, the probability of player A scoring when serving is $\frac{2}{3}$, and the probability of player A scoring when player B serves is $\frac{1}{2}$. The outcomes of each ball are independent.
$(1)$ Find the probability that player A scores 3 points after the first 4 balls in a game.
$(2)$ If the game is tied at 10-10, and the match ends after X additional balls are played, find the probability of the event "X ≤ 4." | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest number of consecutive non-negative integers whose sum is $120$? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $ABCD$ is constructed along diameter $AB$ of a semicircle, where both the square and semicircle are coplanar. Line segment $AB$ has a length of 8 centimeters. If point $M$ is the midpoint of arc $AB$, what is the length of segment $MD$? | {
"answer": "4\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of solutions in natural numbers for the equation \(\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1\). | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $I=\{1,2,3,4,5\}$. Choose two non-empty subsets $A$ and $B$ from $I$ such that the smallest number in $B$ is greater than the largest number in $A$. The number of different ways to choose such subsets $A$ and $B$ is ______. | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
With four standard six-sided dice in play, Vivian rolls all four and can choose to reroll any subset of them. To win, Vivian needs the sum of the four dice after possibly rerolling some of them to be exactly 12. Vivian plays optimally to maximize her chances of winning. What is the probability that she chooses to reroll exactly three of the dice?
**A)** $\frac{1}{72}$
**B)** $\frac{1}{12}$
**C)** $\frac{1}{10}$
**D)** $\frac{1}{8}$
**E)** $\frac{1}{6}$ | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the number of digits is $2015!$ . Your score will be given by $\max\{\lfloor125(\min\{\tfrac{A}{C},\tfrac{C}{A}\}-\tfrac{1}{5})\rfloor,0\}$ , where $A$ is your answer and $C$ is the actual answer. | {
"answer": "5787",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tetrahedron \( P-ABC \) has edge lengths \( PA = BC = \sqrt{6} \), \( PB = AC = \sqrt{8} \), and \( PC = AB = \sqrt{10} \). Find the radius of the circumsphere of this tetrahedron. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a bag, there are $5$ balls of the same size, including $3$ red balls and $2$ white balls.<br/>$(1)$ If one ball is drawn with replacement each time, and this process is repeated $3$ times, with the number of times a red ball is drawn denoted as $X$, find the probability distribution and expectation of the random variable $X$;<br/>$(2)$ If one ball is drawn without replacement each time, and the color is recorded before putting it back into the bag, the process continues until two red balls are drawn, with the number of draws denoted as $Y$, find the probability of $Y=4$. | {
"answer": "\\frac{108}{625}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the sides \(AB\) and \(AD\) of square \(ABCD\), points \(E\) and \(F\) are marked such that \(BE : EA = AF : FD = 2022 : 2023\). Segments \(EC\) and \(FC\) intersect the diagonal \(BD\) of the square at points \(G\) and \(H\), respectively. Find the ratio \(GH : BD\). | {
"answer": "\\frac{12271519}{36814556}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Positive integers $a$, $b$, $c$, and $d$ are such that $a<b<c<d$, and the system of equations
\[ 2x + y = 2007 \quad\text{and}\quad y = |x-a| + |x-b| + |x-c| + |x-d| \]
has exactly one solution. What is the minimum value of $d$? | {
"answer": "504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For which values of \( x \) and \( y \) the number \(\overline{x x y y}\) is a square of a natural number? | {
"answer": "7744",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is randomly thrown onto the segment $[11, 18]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}+2k-99\right)x^{2}+(3k-7)x+2=0$ satisfy the condition $x_{1} \leq 2x_{2}$. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a class that includes Petya and Vanya, there are 31 students. In how many ways can a football team be selected from the class? | {
"answer": "2 \\binom{29}{10} + \\binom{29}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(a\) copies of a right-angled isosceles triangle with hypotenuse \(\sqrt{2} \, \mathrm{cm}\) can be assembled to form a trapezium with perimeter equal to \(b \, \mathrm{cm}\), find the least possible value of \(b\). (Give the answer in surd form.) | {
"answer": "4 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$ and $B$ be two subsets of the set $\{1,2, \cdots, 20\}$ such that $A \cap B = \varnothing$, and if $n \in A$, then $2n + 2 \in B$. Let $M(A)$ be the sum of the elements in $A$. Find the maximum value of $M(A)$. | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$ be a point on the circle $x^2 + y^2 + 4x - 4y + 4 = 0$, and let $B$ be a point on the parabola $y^2 = 8x$. Find the smallest possible distance $AB$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \) and \( b \) be positive integers. The quotient of \( a^{2} + b^{2} \) divided by \( a + b \) is \( q \), and the remainder is \( r \), such that \( q^{2} + r = 2010 \). Find the value of \( ab \). | {
"answer": "1643",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What are the last three digits of \(2003^N\), where \(N = 2002^{2001}\)? | {
"answer": "241",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A(2,0)$ be a fixed point in the plane, and let $P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right)$ be a moving point. Find the area swept by the line segment $AP$ as $t$ changes from $15^{\circ}$ to $45^{\circ}$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
According to national regulations, only adults aged between 18 and 70 are eligible to apply for a motor vehicle driver's license. A sixth-grade student, Li Ming, says, "My dad has a driver's license. His age equals the product of the month and day of his birth, and that product is 2975." How old is Li Ming's father? | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a \(10 \times 10\) grid, there are 11 horizontal grid lines and 11 vertical grid lines. The line segments connecting adjacent nodes on the same line are called "links." What is the minimum number of links that must be removed so that at each node, there are at most 3 remaining links? | {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set
$$
T=\left\{n \mid n=5^{a}+5^{b}, 0 \leqslant a \leqslant b \leqslant 30, a, b \in \mathbf{Z}\right\},
$$
if a number is randomly selected from set $T$, what is the probability that the number is a multiple of 9? | {
"answer": "5/31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\( \mathrm{n} \) is a positive integer not greater than 100 and not less than 10, and \( \mathrm{n} \) is a multiple of the sum of its digits. How many such \( \mathrm{n} \) are there? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of cells in an $8 \times 8$ square that can be colored such that the centers of any four colored cells do not form the vertices of a rectangle with sides parallel to the edges of the square? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of triangles with vertices' coordinates $(x, y)$ that satisfy $1 \leqslant x \leqslant 4, 1 \leqslant y \leqslant 4$, and where $x$ and $y$ are integers is $\qquad$ . | {
"answer": "516",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each cell of a $100 \times 100$ board is painted in either blue or white. We call a cell balanced if it has an equal number of blue and white neighboring cells. What is the maximum number of balanced cells that can be found on the board? (Cells are considered neighbors if they share a side.) | {
"answer": "9608",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance from \(A\) to \(B\) is 999 km. Along the road, there are kilometer markers indicating the distances to \(A\) and \(B\): 0।999, 1।998, \(\ldots, 999।0. How many of these markers have only two different digits? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(ABCD\) be a square of side length 1. \(P\) and \(Q\) are two points on the plane such that \(Q\) is the circumcentre of \(\triangle BPC\) and \(D\) is the circumcentre of \(\triangle PQA\). Find the largest possible value of \(PQ^2\). Express the answer in the form \(a + \sqrt{b}\) or \(a - \sqrt{b}\), where \(a\) and \(b\) are rational numbers. | {
"answer": "2 + \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the first site, higher-class equipment was used, while at the second site, first-class equipment was used, with higher-class being less than first-class. Initially, 30% of the equipment from the first site was transferred to the second site. Then, 10% of the equipment at the second site was transferred to the first site, with half of the transferred equipment being first-class. After this, the amount of higher-class equipment at the first site exceeded that at the second site by 6 units, and the total amount of equipment at the second site increased by more than 2% compared to the initial amount. Find the total amount of first-class equipment. | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the center of a circular field, there is a house of geologists. Eight straight roads emanate from the house, dividing the field into 8 equal sectors. Two geologists set off on a journey from their house at a speed of 4 km/h choosing any road randomly. Determine the probability that the distance between them after an hour will be more than 6 km. | {
"answer": "0.375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a football championship, 16 teams participated. A team receives 2 points for a win; in case of a draw in regular time, both teams shoot penalty kicks, and the team that scores more goals receives one point. After 16 rounds, all teams have accumulated a total of 222 points. How many matches ended in a draw in regular time? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $l: \sqrt{3}x-y-4=0$, calculate the slope angle of line $l$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of all divisors \(d=2^a \cdot 3^b\) (where \(a, b > 0\)) of \(N=19^{88}-1\). | {
"answer": "744",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many kings can be placed on an $8 \times 8$ chessboard without putting each other in check? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dylan has a \( 100 \times 100 \) square, and wants to cut it into pieces of area at least 1. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make? | {
"answer": "9999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During the manufacture of a steel cable, it was found that the cable has the same length as the curve given by the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=8 \\
x y+y z+x z=14
\end{array}\right.
$$
Find the length of the cable. | {
"answer": "4\\pi \\sqrt{\\frac{11}{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The angle between the slant height of a cone and the base plane is $30^{\circ}$. The lateral surface area of the cone is $3 \pi \sqrt{3}$ square units. Determine the volume of a regular hexagonal pyramid inscribed in the cone. | {
"answer": "\\frac{27 \\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \in \mathbf{R} \). A complex number is given by \(\omega = 1 + a\mathrm{i}\). A complex number \( z \) satisfies \( \overline{\omega} z - \omega = 0 \). Determine the value of \( a \) such that \(|z^2 - z + 2|\) is minimized, and find this minimum value. | {
"answer": "\\frac{\\sqrt{14}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The integer \( n \) has a total of 10 divisors. These divisors are arranged in ascending order, and the 8th divisor is \( \frac{n}{3} \). Find the maximum value of the integer \( n \). | {
"answer": "162",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 16 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). Everyone said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $y=f(x)$ is an odd function defined on $\mathbb{R}$ and satisfies $f(x-1)=f(x+1)$ for all $x \in \mathbb{R}$. When $x \in (0,1]$ and $x_1 \neq x_2$, we have $\frac{f(x_2) - f(x_1)}{x_2 - x_1} < 0$. Determine the correct statement(s) among the following:
(1) $f(1)=0$
(2) $f(x)$ has 5 zeros in $[-2,2]$
(3) The point $(2014,0)$ is a symmetric center of the function $y=f(x)$
(4) The line $x=2014$ is a symmetry axis of the function $y=f(x)$ | {
"answer": "(1) (2) (3)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
8 distinct nonzero natural numbers are arranged in increasing order. The average of the first 3 numbers is 9, the average of all 8 numbers is 19, and the average of the last 3 numbers is 29. What is the maximum possible difference between the second largest number and the second smallest number? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer \( n \) is said to be 'good' if \( n^2 - 1 \) can be written as the product of three distinct prime numbers. Find the sum of the five smallest 'good' integers. | {
"answer": "104",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = x^3 - 3x^2 - 9x + 1$, find the intervals of monotonicity and the extrema of $f(x)$. | {
"answer": "-26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability of an event occurring in each of 900 independent trials is 0.5. Find the probability that the relative frequency of the event will deviate from its probability by no more than 0.02. | {
"answer": "0.7698",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A novel is recorded onto compact discs, taking a total of 505 minutes to read aloud. Each disc can hold up to 53 minutes of reading. Assuming the smallest possible number of discs is used and each disc contains the same length of reading, calculate the number of minutes of reading each disc will contain. | {
"answer": "50.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dragon has 40 piles of gold coins, with the number of coins in any two piles differing. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the minimum number of different piles of coins that could result? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles triangle \( ABC \), the bisectors \( AD, BE, CF \) are drawn.
Find \( BC \), given that \( AB = AC = 1 \), and the vertex \( A \) lies on the circle passing through the points \( D, E, \) and \( F \). | {
"answer": "\\frac{\\sqrt{17} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The percentage of seventh-grade students participating in the gymnastics section is between 2.9% and 3.1%. Determine the smallest possible number of students in this class. | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, triangles $ABC$ and $CDE$ are isosceles. The perimeter of $\triangle CDE$ is $22,$ the perimeter of $\triangle ABC$ is $24,$ and the length of $CE$ is $9.$ What is the length of $AB?$ | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum positive integer $r$ that satisfies the following condition: For any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ common elements. | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid \(ABCD\), the angles \(A\) and \(D\) at the base \(AD\) are \(60^{\circ}\) and \(30^{\circ}\) respectively. Point \(N\) lies on the base \(BC\) such that \(BN : NC = 2\). Point \(M\) lies on the base \(AD\), the line \(MN\) is perpendicular to the bases of the trapezoid and divides its area in half. Find the ratio \(AM : MD\). | {
"answer": "3:4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Chinese ancient music, the pentatonic scale is the basic scale, hence the idiom "五音不全" (meaning "lacking in musical talent"). The five notes of the pentatonic scale in Chinese ancient music are: 宫 (gong), 商 (shang), 角 (jue), 徵 (zhi), 羽 (yu). If these five notes are all used and arranged into a sequence of five notes, with the condition that 宫 (gong), 角 (jue), and 羽 (yu) cannot be adjacent to each other, the number of different possible sequences is ______. | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve chess players played a round-robin tournament. Each player then wrote 12 lists. In the first list, only the player himself was included, and in the $(k+1)$-th list, the players included those who were in the $k$-th list as well as those whom they defeated. It turned out that each player's 12th list differed from the 11th list. How many draws were there? | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $ABCD$ is a rectangle with sides of length $12$ and $18$ . Let $S$ be the region of points contained in $ABCD$ which are closer to the center of the rectangle than to any of its vertices. Find the area of $S$ . | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The last three digits of \( 1978^n \) and \( 1978^m \) are the same. Find the positive integers \( m \) and \( n \) such that \( m+n \) is minimized (here \( n > m \geq 1 \)). | {
"answer": "106",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the function
$$
f(x)=\sin (x+\sin x)+\sin (x-\sin x)+\left(\frac{\pi}{2}-2\right) \sin (\sin x)
$$ | {
"answer": "\\frac{\\pi - 2}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the probability that two people, A and B, randomly choosing their rooms among 6 different rooms in a family hotel, which has two rooms on each of the three floors, will stay in two rooms on the same floor? | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The villages Arkadino, Borisovo, and Vadimovo are connected by straight roads in pairs. Adjacent to the road between Arkadino and Borisovo is a square field, one side of which completely coincides with this road. Adjacent to the road between Borisovo and Vadimovo is a rectangular field, one side of which completely coincides with this road, and the other side is 4 times longer. Adjacent to the road between Arkadino and Vadimovo is a rectangular forest, one side of which completely coincides with this road, and the other side is 12 km. The area of the forest is 45 square km more than the sum of the areas of the fields. Find the total area of the forest and fields in square km. | {
"answer": "135",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([2, 4]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{5-\sqrt{11}}{2}\right)) \ldots) \). Round your answer to the nearest hundredth if necessary. | {
"answer": "4.16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the following diagram, \(ABCD\) is a square, and \(E\) is the center of the square \(ABCD\). \(P\) is a point on a semi-circle with diameter \(AB\). \(Q\) is a point on a semi-circle with diameter \(AD\). Moreover, \(Q, A,\) and \(P\) are collinear (that is, they are on the same line). Suppose \(QA = 14 \text{ cm}\), \(AP = 46 \text{ cm}\), and \(AE = x \text{ cm}\). Find the value of \(x\). | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{a},$ $\mathbf{b},$ $\mathbf{c},$ $\mathbf{d}$ be four distinct unit vectors in space such that
\[\mathbf{a} \cdot \mathbf{b} = \mathbf{a} \cdot \mathbf{c} = \mathbf{b} \cdot \mathbf{c} = \mathbf{b} \cdot \mathbf{d} = \mathbf{c} \cdot \mathbf{d} = -\frac{1}{7}.\]
Find $\mathbf{a} \cdot \mathbf{d}$. | {
"answer": "-\\frac{19}{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = e^x \cos x - x$.
(I) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$;
(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac{\pi}{2}]$. | {
"answer": "-\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCDEFGH$ is a cube having a side length of 2. $P$ is the midpoint of $EF$, as shown. The area of $\triangle APB$ is: | {
"answer": "$\\sqrt{8}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $|\vec{a}|=1$, $|\vec{b}|=\sqrt{2}$, $\vec{c}=\vec{a}+\vec{b}$, and $\vec{c} \perp \vec{a}$, the angle between vectors $\vec{a}$ and $\vec{b}$ is $\_\_\_\_\_\_\_\_\_.$ | {
"answer": "\\frac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers with less than four digits (from 0 to 9999) are neither divisible by 3, nor by 5, nor by 7? | {
"answer": "4571",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bamboo pole of length 24 meters is leaning against a wall with one end on the ground. If the vertical distance from a certain point on the pole to the wall and the vertical distance from that point to the ground are both 7 meters, find the vertical distance from the end of the pole touching the wall to the ground in meters, or $\qquad$ meters. | {
"answer": "16 - 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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