problem stringlengths 10 5.15k | answer dict |
|---|---|
The teacher plans to give children a problem of the following type. He will tell them that he has thought of a polynomial \( P(x) \) of degree 2017 with integer coefficients, whose leading coefficient is 1. Then he will tell them \( k \) integers \( n_{1}, n_{2}, \ldots, n_{k} \), and separately he will provide the value of the expression \( P\left(n_{1}\right) P\left(n_{2}\right) \ldots P\left(n_{k}\right) \). Based on this information, the children must find the polynomial that the teacher might have in mind. What is the smallest possible \( k \) for which the teacher can compose a problem of this type such that the polynomial found by the children will necessarily match the intended one? | {
"answer": "2017",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\), and \(\boldsymbol{c}\) such that
\[
|a|=|b|=3, |c|=4, \boldsymbol{a} \cdot \boldsymbol{b}=-\frac{7}{2},
\boldsymbol{a} \perp \boldsymbol{c}, \boldsymbol{b} \perp \boldsymbol{c}
\]
Find the minimum value of the expression
\[
|x \boldsymbol{a} + y \boldsymbol{b} + (1-x-y) \boldsymbol{c}|
\]
for real numbers \(x\) and \(y\). | {
"answer": "\\frac{4 \\sqrt{33}}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c, respectively. Let S be the area of triangle ABC. If 3a² = 2b² + c², find the maximum value of $\frac{S}{b^{2}+2c^{2}}$. | {
"answer": "\\frac{\\sqrt{14}}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The operation $\star$ is defined as $a \star b = a^2 \div b$. For how many negative integer values of $x$ will the value of $12 \star x$ be a positive integer? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the following equation and provide its root. If the equation has multiple roots, provide their product.
\[ \sqrt{2 x^{2} + 8 x + 1} - x = 3 \] | {
"answer": "-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sin x\cos (x+ \frac {π}{3})+ \frac { \sqrt {3}}{2}$.
(I) Find the interval(s) where the function $f(x)$ is monotonically decreasing;
(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac {π}{2}]$. | {
"answer": "-\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex pentagon \(ABCDE\), \(AB = BC\), \(CD = DE\), \(\angle ABC = 100^\circ\), \(\angle CDE = 80^\circ\), and \(BD^2 = \frac{100}{\sin 100^\circ}\). Find the area of the pentagon. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the intercept on the $x$-axis of the line that is perpendicular to the line $3x-4y-7=0$ and forms a triangle with both coordinate axes having an area of $6$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest integer $B$ such that there exists a sequence of consecutive integers, which must include at least one odd and one even number, adding up to 2023. | {
"answer": "1011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The side of a square is increased by $20\%$. To keep the area of the square unchanged, what percentage must the other side be reduced? | {
"answer": "16.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $r(x)$ be a monic quintic polynomial such that $r(1) = 3$, $r(2) = 7$, $r(3) = 13$, $r(4) = 21$, and $r(5) = 31$. Find $r(6)$. | {
"answer": "158",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the following pair of equations:
\[120x^4 + ax^3 + bx^2 + cx + 18 = 0\] and
\[18x^5 + dx^4 + ex^3 + fx^2 + gx + 120 = 0\]
These equations have a common rational root $k$ which is not an integer and is positive. Determine $k$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Based on the data provided, what were the average daily high and low temperatures in Addington from September 15th, 2008 to September 21st, 2008, inclusive? The high temperatures in degrees Fahrenheit for these days are 49, 62, 58, 57, 46, 55, 60 and the corresponding low temperatures are 40, 47, 45, 41, 39, 42, 44. Express your answer as a decimal to the nearest tenth. | {
"answer": "42.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the coefficient of \( x^{1992} \) in the power series \( (1 + x)^{\alpha} = 1 + \alpha x + \dots \) be \( C(\alpha) \). Find \( \int_{0}^{1} C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \, dy \). | {
"answer": "1992",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive number $\lambda$ such that for any triangle with side lengths $a, b, c$, if $a \geqslant \frac{b+c}{3}$, then the following inequality holds:
$$
ac + bc - c^2 \leqslant \lambda \left( a^2 + b^2 + 3c^2 + 2ab - 4bc \right).
$$ | {
"answer": "\\frac{2\\sqrt{2} + 1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(\alpha\) and \(\beta\) are acute angles, and the following equations hold:
$$
\left\{\begin{array}{l}
3 \sin ^{2} \alpha + 2 \sin ^{2} \beta = 1, \\
3 \sin 2 \alpha - 2 \sin 2 \beta = 0.
\end{array}\right.
$$
Determine \(\alpha + 2\beta\). | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AB > AC$, $\angle BAC = 45^\circ$. Point $E$ is the intersection of the external angle bisector of $\angle BAC$ with the circumcircle of $\triangle ABC$. Point $F$ is on $AB$ such that $EF \perp AB$. Given $AF = 1$ and $BF = 5$, find the area of $\triangle ABC$. | {
"answer": "6 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$AM$ is the angle bisector of triangle $ABC$. $BM = 2$, $CM = 3$, and $D$ is the point where the extension of $AM$ intersects the circumcircle of the triangle. If $MD = 2$, find $AB$. | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Uncle Wang has some chickens, rabbits, and geese. Geese stand on two legs during the day and on one leg at night; chickens tuck their heads under their wings when sleeping. Careful Yue Yue discovered that the difference between the number of legs and the number of heads is always the same, regardless of whether it is day or night. If Yue Yue counts 56 legs during the day, how many heads will be counted at night? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line passing through one focus F_1 of the ellipse 4x^{2}+2y^{2}=1 intersects the ellipse at points A and B. Then, points A, B, and the other focus F_2 of the ellipse form ∆ABF_2. Calculate the perimeter of ∆ABF_2. | {
"answer": "2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A metallic weight has a mass of 20 kg and is an alloy of four metals. The first metal in this alloy is one and a half times the amount of the second metal. The mass of the second metal relates to the mass of the third metal as $3:4$, and the mass of the third metal to the mass of the fourth metal as $5:6$. Determine the mass of the fourth metal. Give your answer in kilograms, rounding to the nearest hundredth if necessary. | {
"answer": "5.89",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set \( S = \{1, 2, \cdots, 2005\} \), and a subset \( A \subseteq S \) such that the sum of any two numbers in \( A \) is not divisible by 117, determine the maximum value of \( |A| \). | {
"answer": "1003",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $ (a_n)$ is given by $ a_1\equal{}1,a_2\equal{}0$ and:
$ a_{2k\plus{}1}\equal{}a_k\plus{}a_{k\plus{}1}, a_{2k\plus{}2}\equal{}2a_{k\plus{}1}$ for $ k \in \mathbb{N}.$
Find $ a_m$ for $ m\equal{}2^{19}\plus{}91.$ | {
"answer": "91",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height of a triangle, equal to 2, divides the angle of the triangle in the ratio 2:1, and the base of the triangle into parts, the smaller of which is equal to 1. Find the area of the triangle. | {
"answer": "11/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a_{1}, a_{2}, \cdots, a_{n}\) be an increasing sequence of positive integers. For a positive integer \(m\), define
\[b_{m}=\min \left\{n \mid a_{n} \geq m\right\} (m=1,2, \cdots),\]
that is, \(b_{m}\) is the smallest index \(n\) such that \(a_{n} \geq m\). Given \(a_{20}=2019\), find the maximum value of \(S=\sum_{i=1}^{20} a_{i}+\sum_{i=1}^{2019} b_{i}\). | {
"answer": "42399",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the integers \( a, b, c \) that satisfy \( a + b + c = 2 \), and
\[
S = (2a + bc)(2b + ca)(2c + ab) > 200,
\]
find the minimum value of \( S \). | {
"answer": "256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alex and Katy play a game on an \(8 \times 8\) grid made of 64 unit cells. They take turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On Katy's turn, she writes 'K' in two empty cells that share an edge. The game ends when one player cannot move. Katy's score is the number of Ks on the grid at the end of the game. What is the highest score Katy can be sure to get if she plays well, no matter what Alex does? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a company of 100 children, some children are friends with each other (friendship is always mutual). It is known that if any one child is excluded, the remaining 99 children can be divided into 33 groups of three people such that all members in each group are mutually friends. Find the smallest possible number of pairs of children who are friends. | {
"answer": "198",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many six-digit numbers are there in which each subsequent digit is smaller than the previous one? | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Distribute 16 identical books among 4 students so that each student gets at least one book, and each student gets a different number of books. How many distinct ways can this be done? (Answer with a number.) | {
"answer": "216",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, the circle has radius \(\sqrt{5}\). Rectangle \(ABCD\) has \(C\) and \(D\) on the circle, \(A\) and \(B\) outside the circle, and \(AB\) tangent to the circle. What is the area of \(ABCD\) if \(AB = 4AD\)? | {
"answer": "16/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a set of five distinct lines in a plane, there are exactly $M$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $M$? | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the region of the \( xy \)-plane defined by the inequality \( |x|+|y|+|x+y| \leq 1 \). | {
"answer": "3/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the multiplication of the two numbers $1,002,000,000,000,000,000$ and $999,999,999,999,999,999$. Calculate the number of digits in the product of these two numbers. | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the polynomial
\[ p(x) = 985 x^{2021} + 211 x^{2020} - 211, \]
let its 2021 complex roots be \( x_1, x_2, \cdots, x_{2021} \). Calculate
\[ \sum_{k=1}^{2021} \frac{1}{x_{k}^{2} + 1} = \]
| {
"answer": "2021",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A group of cows and horses are randomly divided into two equal rows. Each animal in one row is directly opposite an animal in the other row. If 75 of the animals are horses and the number of cows opposite cows is 10 more than the number of horses opposite horses, determine the total number of animals in the group. | {
"answer": "170",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number that is a multiple of 80, in which two of its distinct digits can be rearranged so that the resulting number is also a multiple of 80. | {
"answer": "1520",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive divisors do 9240 and 13860 have in common? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider all prisms whose base is a convex 2015-gon.
What is the maximum number of edges of such a prism that can be intersected by a plane not passing through its vertices? | {
"answer": "2017",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $l:3x-\sqrt{3}y-6=0$, and the curve $C:\rho -4\sin \theta =0$ in the polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.
(I) Write the line $l$ in parametric form $\begin{cases} x=2+t\cos \alpha \\ y=t\sin \alpha \end{cases}(t$ is the parameter, $\alpha \in \left[ 0,\pi \right))$ and find the rectangular coordinate equation of the curve $C$;
(II) Draw a line with a slope of $30^{\circ}$ through any point $P$ on the curve $C$, intersecting $l$ at point $A$, and find the maximum value of $|AP|$. | {
"answer": "6+2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Say that an integer $A$ is yummy if there exist several consecutive non-negative integers, including $A$, that add up to 2023. What is the smallest yummy integer? | {
"answer": "1011",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of sides of a convex polygon that can be divided into right triangles with acute angles measuring 30 and 60 degrees? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
With all angles measured in degrees, calculate the product $\prod_{k=1}^{30} \csc^2(3k)^\circ$, and express the result as $m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, \ldots, 2005\} \). If any set of \( n \) pairwise co-prime numbers in \( S \) always contains at least one prime number, what is the minimum value of \( n \)? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the side lengths of a convex quadrilateral are $a=4, b=5, c=6, d=7$, find the radius $R$ of the circumscribed circle around this quadrilateral. Provide the integer part of $R^{2}$ as the answer. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the complex plane, the points corresponding to the complex number $1+ \sqrt {3}i$ and $- \sqrt {3}+i$ are $A$ and $B$, respectively, with $O$ as the coordinate origin. Calculate the measure of $\angle AOB$. | {
"answer": "\\dfrac {\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On Ming's way to the swimming pool, there are 200 trees. On his round trip, Ming marked some trees with red ribbons. On his way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, and so on, marking every 4th tree. On his way back, he marked the 1st tree he encountered, the 9th tree, the 17th tree, and so on, marking every 7th tree. How many trees are unmarked when he returns home? | {
"answer": "140",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles trapezoid \(ABCD\) (\(BC \parallel AD\)), the angles \(ABD\) and \(DBC\) are \(135^{\circ}\) and \(15^{\circ}\) respectively, and \(BD = \sqrt{6}\). Find the perimeter of the trapezoid. | {
"answer": "9 - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the side AB of triangle ABC with a $100^{\circ}$ angle at vertex C, points P and Q are taken such that $AP = BC$ and $BQ = AC$. Let M, N, and K be the midpoints of segments AB, CP, and CQ respectively. Find the angle $NMK$. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence \(\{a_n\}\) such that: \(a_1 = 1\) and \(a_{n+1} = \frac{a_n}{(n+1)(a_n + 1)}\) for \(n \in \mathbb{Z^+}\), find the value of \(\lim_{n \rightarrow +\infty} n! \cdot a_n\). | {
"answer": "\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the mass of the body $\Omega$ with density $\mu=z$, bounded by the surfaces
$$
x^{2} + y^{2} = 4, \quad z=0, \quad z=\frac{x^{2} + y^{2}}{2}
$$ | {
"answer": "\\frac{16\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two cars, Car A and Car B, start from point A and point B, respectively, and move towards each other simultaneously. They meet after 3 hours, after which Car A turns back towards point A and Car B continues moving forward. Once Car A reaches point A, it turns towards point B again and meets Car B half an hour later. How many minutes does Car B take to travel from point A to point B? | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a quiz, every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions on the quiz. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school is arranging for 5 trainee teachers, including Xiao Li, to be placed in Class 1, Class 2, and Class 3 for teaching practice. If at least one teacher must be assigned to each class and Xiao Li is to be placed in Class 1, the number of different arrangement schemes is ________ (answer with a number only). | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A leak formed in the hold of a ship. A pump was immediately switched on to remove the water, but it couldn't keep up, and after 10 minutes, the water level rose by 20 cm. Then, a second pump of equal power was turned on, and after 5 minutes, the water level dropped by 10 cm. The leak was then sealed.
How much time will it take for the pumps to remove the remaining water? | {
"answer": "1.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a supermarket with 100 customers, the table shows the number of customers in each age group and payment method category. Determine the number of customers aged between 40 and 60 years old who do not use mobile payment. | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane, point $P$ is a moving point on the line $x=-1$, point $F(1,0)$, point $Q$ is the midpoint of $PF$, point $M$ satisfies $MQ \perp PF$ and $\overrightarrow{MP}=\lambda \overrightarrow{OF}$, and the tangent line is drawn through point $M$ on the circle $(x-3)^{2}+y^{2}=2$ with tangent points $A$ and $B$, respectively. Find the minimum value of $|AB|$. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a zoo, oranges, bananas, and coconuts were brought to feed three monkeys, with an equal number of each type of fruit. The first monkey was fed only oranges and bananas, with the number of bananas being 40% more than the number of oranges. The second monkey was fed only bananas and coconuts, with the number of coconuts being 25% more than the number of bananas. The third monkey was fed only coconuts and oranges, with the number of oranges being twice that of the coconuts. The monkeys ate all the fruits brought.
Let the first monkey eat $a$ oranges, and the third monkey eat $b$ oranges. Find $a / b$. | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $(x, y)$ moves on the circle $x^{2}+(y-1)^{2}=1$.
(1) Find the maximum and minimum values of $\frac{y-1}{x-2}$;
(2) Find the maximum and minimum values of $2x+y$. | {
"answer": "1 - \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can the natural numbers from 1 to 10 (each used exactly once) be arranged in a $2 \times 5$ table so that the sum of the numbers in each of the five columns is odd? | {
"answer": "460800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four distinct real numbers \( a, b, c, d \) such that \(\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4\) and \( ac = bd \), find the maximum value of \(\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \). | {
"answer": "-12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When $\frac{1}{10101}$ is expressed as a decimal, what is the sum of the first 100 digits after the decimal point? | {
"answer": "450",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If two distinct members of the set $\{ 4, 10, 15, 24, 30, 40, 60 \}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 120? Express your answer as a common fraction. | {
"answer": "\\frac{3}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( |z|=2 \) and \( u=\left|z^{2}-z+1\right| \), find the minimum value of \( u \) where \( z \in \mathbf{C} \). | {
"answer": "\\frac{3}{2} \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two circles $x^2+y^2=a^2$ and $x^2+y^2+ay-6=0$ have a common chord with a length of $2\sqrt{3}$, find the value of $a$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a tetrahedron \(ABCD\), in what ratio does the plane passing through the intersection points of the medians of the faces \(ABC\), \(ABD\), and \(BCD\) divide the edge \(BD\)? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | {
"answer": "481^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the possible value of $x + y$ given that $x^3 + 6x^2 + 16x = -15$ and $y^3 + 6y^2 + 16y = -17$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To reach the Solovyov family's dacha from the station, one must first travel 3 km on the highway and then 2 km on a path. Upon arriving at the station, the mother called her son Vasya at the dacha and asked him to meet her on his bicycle. They started moving towards each other at the same time. The mother walks at a constant speed of 4 km/h, while Vasya rides at a speed of 20 km/h on the path and 22 km/h on the highway. At what distance from the station did Vasya meet his mother? Give the answer in meters. | {
"answer": "800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given sets $A=\{0, |x|\}$ and $B=\{1, 0, -1\}$. If $A \subseteq B$, then find $A \cap B$, $A \cup B$, and the complement of $A$ in $B$, denoted by $\complement_B A$. | {
"answer": "\\{-1\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an island, there are 100 knights and 100 liars. Every resident has at least one friend. Knights always tell the truth, while liars always lie. One morning, each resident said either the phrase "All my friends are knights" or the phrase "All my friends are liars," with exactly 100 people saying each phrase. Find the minimum possible number of pairs of friends where one is a knight and the other is a liar. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the angles $\angle B = 30^\circ$ and $\angle A = 90^\circ$ are known. Point $K$ is marked on side $AC$, and points $L$ and $M$ are marked on side $BC$ such that $KL = KM$ (point $L$ is on segment $BM$).
Find the length of segment $LM$, given that $AK = 4$, $BL = 31$, and $MC = 3$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: \(\left(2 \frac{2}{3} \times\left(\frac{1}{3}-\frac{1}{11}\right) \div\left(\frac{1}{11}+\frac{1}{5}\right)\right) \div \frac{8}{27} = 7 \underline{1}\). | {
"answer": "7 \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A kitten bites off a quarter of a sausage from one end, then a puppy bites off a third of the remaining piece from the opposite end, then the kitten again bites off a quarter from its end, and the puppy bites off a third from its end, and so on. You need to tie a string around the sausage in advance to ensure that no one eats the string. In what ratio should the sausage be divided? | {
"answer": "1:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Maria ordered a certain number of televisions at $R$ \$ 1994.00 each. She noticed that in the total amount to be paid, there are no digits 0, 7, 8, or 9. What was the smallest number of televisions she ordered? | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9)>f(16)$ . The probability that $f(1)>f(16)>f(25)$ can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$ .
Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, f(100)\}$ is a permutation of $\{1,2, \ldots, 100\}$ .
*Proposed by Evan Chen* | {
"answer": "124",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the left focus $F$ of the ellipse $C$: $$\frac {x^{2}}{a^{2}}$$+ $$\frac {y^{2}}{b^{2}}$$\=1 ($a>b>0$), a line is drawn through the upper endpoint $B$ of $C$ and intersects the ellipse at another point $A$. If $|BF|=3|AF|$, find the eccentricity of $C$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The parabola $y^2 = 12x$ and the circle $x^2 + y^2 - 4x - 6y = 0$ intersect at two points $C$ and $D$. Find the distance $CD$. | {
"answer": "3\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$ with the common difference $d$ being an integer, and $a_k=k^2+2$, $a_{2k}=(k+2)^2$, where $k$ is a constant and $k\in \mathbb{N}^*$
$(1)$ Find $k$ and $a_n$
$(2)$ Let $a_1 > 1$, the sum of the first $n$ terms of $\{a_n\}$ is $S_n$, the first term of the geometric sequence $\{b_n\}$ is $l$, the common ratio is $q(q > 0)$, and the sum of the first $n$ terms is $T_n$. If there exists a positive integer $m$, such that $\frac{S_2}{S_m}=T_3$, find $q$． | {
"answer": "\\frac{\\sqrt{13}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the numbers 1, 2, 3, 4, 5, a five-digit number is formed with digits not repeating. What is the probability of randomly selecting a five-digit number $\overline{abcde}$ that satisfies the condition "$a < b > c < d > e$"? | {
"answer": "2/15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many different positions can appear on a chessboard if both players, starting from the initial position, make just one move each? | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If: (1) \( a, b, c, d \) are all elements of \( \{1,2,3,4\} \); (2) \( a \neq b, b \neq c, c \neq d, d \neq a \); (3) \( a \) is the smallest value among \( a, b, c, d \),
then, how many different four-digit numbers \( \overline{abcd} \) can be formed? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Leon has cards with digits from 1 to 7. How many ways are there to combine these cards into two three-digit numbers (one card will not be used) so that each of them is divisible by 9? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the game of preference, each of the three players is dealt 10 cards, and two cards are placed in the kitty. How many different arrangements are possible in this game? (Consider possible distributions without accounting for which 10 cards go to each specific player.) | {
"answer": "\\frac{32!}{(10!)^3 \\cdot 2! \\cdot 3!}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2019\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\). | {
"answer": "1010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle \( \triangle ABC \), if \( \frac{\overrightarrow{AB} \cdot \overrightarrow{BC}}{3} = \frac{\overrightarrow{BC} \cdot \overrightarrow{CA}}{2} = \frac{\overrightarrow{CA} \cdot \overrightarrow{AB}}{1} \), find \( \tan A \). | {
"answer": "\\sqrt{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-zero vectors and $(\overrightarrow{a} - 6\overrightarrow{b}) \perp \overrightarrow{a}$, $(2\overrightarrow{a} - 3\overrightarrow{b}) \perp \overrightarrow{b}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the sides \(A B, B C, C D\) and \(A D\) of the convex quadrilateral \(A B C D\) are points \(M, N, K\) and \(L\) respectively, such that \(A M: M B = 3: 2\), \(C N: N B = 2: 3\), \(C K = K D\) and \(A L: L D = 1: 2\). Find the ratio of the area of the hexagon \(M B N K D L\) to the area of the quadrilateral \(A B C D\). | {
"answer": "4/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An ant starts at the point \((1,0)\). Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point \((x, y)\) with \(|x|+|y| \geq 2\). What is the probability that the ant ends at the point \((1,1)\)? | {
"answer": "7/24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, it is given that $A B = B C = 4$ and $A A_{1} = 2$. Point $P$ lies on the plane $A_{1} B C$, and it holds that $\overrightarrow{D P} \cdot \overrightarrow{P B} = 0$. Find the area of the plane region formed by all such points $P$ satisfying the given condition. | {
"answer": "\\frac{36\\pi}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange the positive integers in the following number matrix:
\begin{tabular}{lllll}
1 & 2 & 5 & 10 & $\ldots$ \\
4 & 3 & 6 & 11 & $\ldots$ \\
9 & 8 & 7 & 12 & $\ldots$ \\
16 & 15 & 14 & 13 & $\ldots$ \\
$\ldots$ & $\ldots$ & $\ldots$ & $\ldots$ & $\ldots$
\end{tabular}
What is the number in the 21st row and 21st column? | {
"answer": "421",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Yura has unusual clocks with several minute hands, moving in different directions. Yura counted that in one hour, the minute hands coincided exactly 54 times in pairs. What is the maximum number of minute hands that Yura's clock can have? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangular wire frame with side lengths of $13, 14, 15$ is fitted over a sphere with a radius of 10. Find the distance between the plane containing the triangle and the center of the sphere. | {
"answer": "2\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Zhang has three watches. The first watch runs 2 minutes fast every hour, the second watch runs 6 minutes fast, and the third watch runs 16 minutes fast. If the minute hands of the three watches are currently all pointing in the same direction, after how many hours will the three minute hands point in the same direction again? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(x\) and \(y\) be real numbers such that \(2(x^3 + y^3) = x + y\). Find the maximum value of \(x - y\). | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a debate competition with 4 participants, the rules are as follows: each participant must choose one topic from two options, A and B. For topic A, answering correctly earns 100 points, and answering incorrectly results in a loss of 100 points. For topic B, answering correctly earns 90 points, and answering incorrectly results in a loss of 90 points. If the total score of the 4 participants is 0 points, how many different scoring situations are there for these 4 participants? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the total area of a pentagon with sides of lengths 18, 25, 30, 28, and 25 units, assuming it can be divided into a right triangle and a trapezoid. | {
"answer": "995",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Adam filled a $3 \times 3$ table with the numbers from 1 to 9 as follows:
| 7 | 6 | 4 |
| :--- | :--- | :--- |
| 1 | 2 | 8 |
| 9 | 3 | 5 |
For this arrangement, the sum of the numbers along every side of the table remains the same. Adam found that the numbers can be arranged differently, still preserving the property of equal sums along each side.
What is the smallest possible sum for this arrangement? Provide an example of a table with this minimum sum along the sides and explain why it cannot be smaller. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of queens that can be placed on an $8 \times 8$ chessboard so that each queen can attack at least one other queen? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$ , respectively, and meet perpendicularly at $T$ . $Q$ is on $AT$ , $S$ is on $BT$ , and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$ . Determine the radius of the circle. | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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