problem stringlengths 10 5.15k | answer dict |
|---|---|
Mia and Tom jog on a circular track. Mia jogs counterclockwise and completes a lap every 96 seconds, while Tom jogs clockwise and completes a lap every 75 seconds. They both start from the same point at the same time. If a photographer positioned inside the track takes a snapshot sometime between 12 minutes and 13 minutes after they begin jogging, capturing one-third of the track centered on the starting point, what is the probability that both Mia and Tom are in the picture?
A) $\frac{1}{6}$
B) $\frac{1}{4}$
C) $\frac{2}{3}$
D) $\frac{5}{6}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$, given that $a^{2} + b^{2} - 3c^{2} = 0$, where $c$ is the semi-latus rectum, find the value of $\frac{a + c}{a - c}$. | {
"answer": "3 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four cyclists. Four identical circles represent four tracks. The four cyclists start from the center at noon. Each moves along their track at speeds: the first at 6 km/h, the second at 9 km/h, the third at 12 km/h, and the fourth at 15 km/h. They agreed to ride until they all meet again in the center for the fourth time. The length of each circular track is $\frac{1}{3}$ km.
When will they meet again? | {
"answer": "12:26:40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the smallest positive integer $n$ such that $(n + i), (n + i)^3,$ and $(n + i)^4$ are the vertices of a triangle in the complex plane whose area is greater than 3000. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest natural number consisting of distinct digits such that the product of its digits equals 2016. | {
"answer": "876321",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a polynomial \( P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_1 x + a_0 \) where each coefficient \( a_i \) belongs to the interval \([100,101]\), what is the smallest natural number \( n \) such that this polynomial can have a real root? | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the vertices of a unit square, perpendiculars are erected to its plane. On them, on one side of the plane of the square, points are taken at distances of 3, 4, 6, and 5 from this plane (in order of traversal). Find the volume of the polyhedron whose vertices are the specified points and the vertices of the square. | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Seven people are seated together around a circular table. Each one will toss a fair coin. If the coin shows a head, then the person will stand. Otherwise, the person will remain seated. The probability that after all of the tosses, no two adjacent people are both standing, can be written in the form \( p / q \), where \( p \) and \( q \) are relatively prime positive integers. What is \( p+q \)? | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain kingdom, the workforce consists only of a clan of dwarves and a clan of elves. Historically, in this kingdom, dwarves and elves have always worked separately, and no enterprise has ever allowed itself to hire both at the same time. The aggregate labor supply of dwarves is given by the function \( w_{\text{dwarves}}^{S} = 1 + \frac{L}{3} \), and the aggregate labor supply of elves is \( w_{\text{elves}}^{S} = 3 + L \). The inverse function of aggregate labor demand for dwarves is \( w_{\text{dwarves}}^{D} = 10 - \frac{2L}{3} \), and the inverse function of aggregate labor demand for elves is \( w_{\text{elves}}^{D} = 18 - 2L \). Recently, a newly enthroned king became very concerned that the wage rates of his subjects are different, so he issued a law stating that the wages of elves and dwarves must be equal and that workers should not be discriminated against based on their clan affiliation. The king believes that regulatory intervention in wage rates will negatively impact the kingdom's economy overall and mandates that all his subjects behave in a perfectly competitive manner. By how many times will the wage of the group of workers whose wage was lower before the king's intervention increase if the firms in the kingdom are indifferent to hiring elves or dwarves? | {
"answer": "1.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pyramid \( S A B C D \) has a trapezoid \( A B C D \) as its base, with bases \( B C \) and \( A D \). Points \( P_1, P_2, P_3 \) lie on side \( B C \) such that \( B P_1 < B P_2 < B P_3 < B C \). Points \( Q_1, Q_2, Q_3 \) lie on side \( A D \) such that \( A Q_1 < A Q_2 < A Q_3 < A D \). Let \( R_1, R_2, R_3, \) and \( R_4 \) be the intersection points of \( B Q_1 \) with \( A P_1 \); \( P_2 Q_1 \) with \( P_1 Q_2 \); \( P_3 Q_2 \) with \( P_2 Q_3 \); and \( C Q_3 \) with \( P_3 D \) respectively. It is known that the sum of the volumes of the pyramids \( S R_1 P_1 R_2 Q_1 \) and \( S R_3 P_3 R_4 Q_3 \) equals 78. Find the minimum value of
\[ V_{S A B R_1}^2 + V_{S R_2 P_2 R_3 Q_2}^2 + V_{S C D R_4}^2 \]
and give the closest integer to this value. | {
"answer": "2028",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Line segments \( AB \) and \( CD \) are situated between two parallel planes \( \alpha \) and \( \beta \). \( AC \subset \alpha \) and \( BD \subset \beta \). Given \( AB \perp \alpha \), \( AC = BD = 5 \), \( AB = 12 \), and \( CD = 13 \). Points \( E \) and \( F \) divide \( AB \) and \( CD \) in the ratio \( 1:2 \) respectively. Find the length of the line segment \( EF \). | {
"answer": "\\frac{5}{3} \\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of digits in the value of $2^{15} \times 5^{12} - 10^5$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six IMO competitions are hosted sequentially by two Asian countries, two European countries, and two African countries, where each country hosts once but no continent can host consecutively. How many such arrangements are possible? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using the trapezoidal rule with an accuracy of 0.01, calculate $\int_{2}^{3} \frac{d x}{x-1}$. | {
"answer": "0.6956",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ordered quadruples \((a, b, c, d)\) of positive odd integers are there that satisfy the equation \(a + b + c + 2d = 15?\) | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. If $a^{2}=b^{2}+4bc\sin A$ and $\tan A \cdot \tan B=2$, then $\tan B-\tan A=$ ______. | {
"answer": "-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathrm{O}$ be the intersection point of the diagonals of a convex quadrilateral $A B C D$, and let $P, Q, R$, and $S$ be the centroids of triangles $A O B$, $B O C$, $C O D$, and $D O A$, respectively. Find the ratio of the areas of the quadrilateral $P Q R S$ to that of $A B C D$. | {
"answer": "\\frac{2}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangles $\triangle DEF$ and $\triangle D'E'F'$ are in the coordinate plane with vertices $D(2,2)$, $E(2,14)$, $F(18,2)$, $D'(32,26)$, $E'(44,26)$, $F'(32,10)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\triangle DEF$ to $\triangle D'E'F'$. Find $n+u+v$. | {
"answer": "124",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kolya, after walking one-fourth of the way from home to school, realized that he forgot his problem book. If he does not go back for it, he will arrive at school 5 minutes before the bell rings, but if he goes back, he will be 1 minute late. How long (in minutes) does it take to get to school? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Primes like $2, 3, 5, 7$ are natural numbers greater than 1 that can only be divided by 1 and themselves. We split 2015 into the sum of 100 prime numbers, requiring that the largest of these prime numbers be as small as possible. What is this largest prime number? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Grandma told her grandchildren: "Today I am 60 years and 50 months and 40 weeks and 30 days old."
How old was Grandma on her last birthday? | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiaopang, Xiaodingding, Xiaoya, and Xiaoqiao have a total of 8 parents and 4 children in their four families. They are going to an amusement park together. The ticket pricing is as follows: Adult tickets are 100 yuan per person, children's tickets are 50 yuan per person. If there are 10 or more people, they can buy group tickets for 70 yuan per person. What is the minimum amount they should pay for the tickets? | {
"answer": "800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain store sells a product, and because the purchase price decreased by 6.4% compared to the original purchase price, the profit margin increased by 8 percentage points. What was the original profit margin for selling this product? | {
"answer": "17\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin (α- \frac {π}{6})= \frac {2}{3}$, $α∈(π, \frac {3π}{2})$, $\cos ( \frac {π}{3}+β)= \frac {5}{13}$, $β∈(0,π)$, find the value of $\cos (β-α)$. | {
"answer": "- \\frac{10+12 \\sqrt{5}}{39}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two Wei Qi teams, $A$ and $B$, each comprising 7 members, compete against each other. Players from each team face off in sequence. The first game is between the first player of each team. The loser is eliminated, and the winner moves on to face the next player of the opposing team. This process continues until one team is entirely eliminated. Find the total number of possible outcomes of the competition. | {
"answer": "3432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 8 blue, 7 red, and 12 white light bulbs. In how many ways can they all be arranged to form a garland such that no two white light bulbs are next to each other? | {
"answer": "11711700",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 2009 numbers arranged in a circle, each of which is either 1 or -1, and not all numbers are the same. Consider all possible consecutive groups of ten numbers. Compute the product of the numbers in each group of ten and sum these products. What is the maximum possible sum? | {
"answer": "2005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\triangle ABC$ is an isosceles right triangle with one leg length of $1$, determine the volume of the resulting geometric solid when $\triangle ABC$ is rotated around one of its sides | {
"answer": "\\frac{\\sqrt{2}\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Olga Ivanovna, the class teacher of Grade 5B, is organizing a "Mathematical Ballet." She wants to arrange boys and girls so that at a distance of 5 meters from each girl there are exactly 2 boys. What is the maximum number of girls that can participate in the ballet, given that 5 boys are participating? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many 5 digit positive integers are there such that each of its digits, except for the last one, is greater than or equal to the next digit? | {
"answer": "715",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the largest natural \( m \), when will the product \( m! \cdot 2022! \) be a factorial of a natural number? | {
"answer": "2022! - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Senya has three straight sticks, each 24 centimeters long. Senya broke one of them into two parts such that with the two pieces of this stick and the two whole sticks, he could form the contour of a right triangle. How many square centimeters is the area of this triangle? | {
"answer": "216",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ denote the value of the sum\[\sum_{n=0}^{432} (-1)^{n} {1500 \choose 3n}\]Determine the remainder obtained when $T$ is divided by $100$. | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all real parameters $a$ for which the equation $x^8 +ax^4 +1 = 0$ has four real roots forming an arithmetic progression. | {
"answer": "-\\frac{82}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular octahedron has a sphere inscribed within it and a sphere circumscribed about it. For each of the eight faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $Q$ is selected at random inside the circumscribed sphere. Determine the probability that $Q$ lies inside one of the nine small spheres. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest prime number that can be represented as the sum of two, three, four, five, and six distinct prime numbers. | {
"answer": "61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
3 red marbles, 4 blue marbles, and 5 green marbles are distributed to 12 students. Each student gets one and only one marble. In how many ways can the marbles be distributed so that Jamy and Jaren get the same color and Jason gets a green marble? | {
"answer": "3150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the trapezium \(ABCD\), the lines \(AB\) and \(DC\) are parallel, \(BC = AD\), \(DC = 2 \times AD\), and \(AB = 3 \times AD\). The angle bisectors of \(\angle DAB\) and \(\angle CBA\) intersect at the point \(E\). What fraction of the area of the trapezium \(ABCD\) is the area of the triangle \(ABE\)? | {
"answer": "3/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence of real numbers \( a_1, a_2, \cdots, a_n, \cdots \) is defined by the following equation: \( a_{n+1} = 2^n - 3a_n \) for \( n = 0, 1, 2, \cdots \).
1. Find an expression for \( a_n \) in terms of \( a_0 \) and \( n \).
2. Find \( a_0 \) such that \( a_{n+1} > a_n \) for any positive integer \( n \). | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the coordinates of the vertices of triangle $\triangle O A B$ are $O(0,0), A(4,4 \sqrt{3}), B(8,0)$, with its incircle center being $I$. Let the circle $C$ pass through points $A$ and $B$, and intersect the circle $I$ at points $P$ and $Q$. If the tangents drawn to the two circles at points $P$ and $Q$ are perpendicular, then the radius of circle $C$ is $\qquad$ . | {
"answer": "2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The center of sphere $\alpha$ lies on the surface of sphere $\beta$. The ratio of the surface area of sphere $\beta$ that is inside sphere $\alpha$ to the entire surface area of sphere $\alpha$ is $1 / 5$. Find the ratio of the radii of spheres $\alpha$ and $\beta$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the following expression (accurate to 8 decimal places):
$$
16\left(\frac{1}{5}-\frac{1}{3} \times \frac{1}{5^{3}}+\frac{1}{5} \times \frac{1}{5^{5}}-\frac{1}{7} \times \frac{1}{5^{7}}+\frac{1}{9} \times \frac{1}{5^{9}}-\frac{1}{11} \times \frac{1}{5^{11}}\right)-4\left(\frac{1}{239}-\frac{1}{3} \times \frac{1}{239^{3}}\right)
$$ | {
"answer": "3.14159265",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the regular quadrangular pyramid \(P-ABCD\), \(M\) and \(N\) are the midpoints of \(PA\) and \(PB\) respectively. If the tangent of the dihedral angle between a side face and the base is \(\sqrt{2}\), find the cosine of the angle between skew lines \(DM\) and \(AN\). | {
"answer": "1/6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A palindrome is a number, word, or text that reads the same backward as forward. How much time in a 24-hour day display palindromes on a clock, showing time from 00:00:00 to 23:59:59? | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board, there are \( n \) different integers, each pair of which differs by at least 10. The sum of the squares of the three largest among them is less than three million. The sum of the squares of the three smallest among them is also less than three million. What is the greatest possible \( n \)? | {
"answer": "202",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two spheres touch the plane of triangle \(ABC\) at points \(B\) and \(C\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 11, and the distance between their centers is \(5 \sqrt{17}\). The center of a third sphere with a radius of 8 is at point \(A\), and it is externally tangent to each of the first two spheres. Find the radius of the circumcircle of triangle \(ABC\). | {
"answer": "2\\sqrt{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AB=10$, $AC=8$, and $BC=6$. Circle $P$ passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection of circle $P$ with sides $AC$ and $BC$ (excluding $C$). The length of segment $QR$ is | {
"answer": "4.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $H$ defined by the points $(x,y)$ with integer coordinates, $2\le|x|\le8$, $2\le|y|\le8$, calculate the number of squares of side at least $5$ that have their four vertices in $H$. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( A B C \) with side \( A C = 8 \), a bisector \( B L \) is drawn. It is known that the areas of triangles \( A B L \) and \( B L C \) are in the ratio \( 3: 1 \). Find the bisector \( B L \), for which the height dropped from vertex \( B \) to the base \( A C \) will be the greatest. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given numbers \(a, b, c\) satisfy \(a b c+a+c-b\). Then the maximum value of the algebraic expression \(\frac{1}{1+a^{2}}-\frac{1}{1+b^{2}}+\frac{1}{1+c^{2}}\) is | {
"answer": "\\frac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $w,x,y,z$ satisfy \begin{align*}w+x+y+z&=25,wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1,w_2>0$ . Find $w_1+w_2$ . | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $| \overrightarrow{a}|=12$, $| \overrightarrow{b}|=9$, and $\overrightarrow{a} \cdot \overrightarrow{b}=-54 \sqrt {2}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If
\[(1 + \tan 0^\circ)(1 + \tan 1^\circ)(1 + \tan 2^\circ) \dotsm (1 + \tan 30^\circ) = 2^m,\]
find the value of $m$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We say that two natural numbers form a perfect pair when the sum and the product of these two numbers are perfect squares. For example, 5 and 20 form a perfect pair because $5+20=25=5^{2}$ and $5 \times 20=100=10^{2}$. Does 122 form a perfect pair with any other natural number? | {
"answer": "122 \\times 121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the circle $C$: $x^{2}+y^{2}-2x=0$, find the coordinates of the circle center $C$ and the length of the chord intercepted by the line $y=x$ on the circle $C$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.
All the students in this class sat at a round table.
- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."
What is the minimum number of troublemakers that can be in the class? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Mr. Smith's class, the ratio of boys to girls is 3 boys for every 4 girls and there are 42 students in his class, calculate the percentage of students that are boys. | {
"answer": "42.857\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a_{n} = 1 + 2 + \cdots + n \), where \( n \in \mathbf{N}_{+} \), and \( S_{m} = a_{1} + a_{2} + \cdots + a_{m} \), \( m = 1, 2, \cdots, m \). Find the number of values among \( S_{1}, S_{2}, \cdots, S_{2017} \) that are divisible by 2 but not by 4. | {
"answer": "252",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bus traveling a 100 km route is equipped with a computer that forecasts the remaining time to arrival at the final destination. This time is calculated based on the assumption that the average speed of the bus on the remaining part of the route will be the same as it was on the part already traveled. Forty minutes after departure, the expected time to arrival was 1 hour and remained the same for five more hours. Could this be possible? If yes, how many kilometers did the bus travel by the end of these five hours? | {
"answer": "85",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\pi / 4} \frac{5 \operatorname{tg} x+2}{2 \sin 2 x+5} d x
$$ | {
"answer": "\\frac{1}{2} \\ln \\left(\\frac{14}{5}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The secant \( ABC \) intercepts an arc \( BC \), which contains \( 112^\circ \); the tangent \( AD \) at point \( D \) divides this arc in the ratio \( 7:9 \). Find \(\angle BAD\). | {
"answer": "31.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain operation is performed on a positive integer: if it is even, divide it by 2; if it is odd, add 1. This process continues until the number becomes 1. How many integers become 1 after exactly 10 operations? | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane, the number of integer points (points where both the x-coordinate and y-coordinate are integers) that satisfy the system of inequalities
\[
\begin{cases}
y \leq 3x, \\
y \geq \frac{1}{3}x, \\
x + y \leq 100
\end{cases}
\]
is ___. | {
"answer": "2551",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three lathes \( A, B, C \) each process the same type of standard parts at a certain work efficiency. Lathe \( A \) starts 10 minutes earlier than lathe \( C \), and lathe \( C \) starts 5 minutes earlier than lathe \( B \). After lathe \( B \) has been working for 10 minutes, the number of standard parts processed by lathes \( B \) and \( C \) is the same. After lathe \( C \) has been working for 30 minutes, the number of standard parts processed by lathes \( A \) and \( C \) is the same. How many minutes after lathe \( B \) starts will it have processed the same number of standard parts as lathe \( A \)? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the quadrilateral pyramid \( S A B C D \):
- The lateral faces \( S A B \), \( S B C \), \( S C D \), and \( S D A \) have areas 9, 9, 27, 27 respectively;
- The dihedral angles at the edges \( A B \), \( B C \), \( C D \), \( D A \) are equal;
- The quadrilateral \( A B C D \) is inscribed in a circle, and its area is 36.
Find the volume of the pyramid \( S A B C D \). | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation
$$
\sin ^{4} x + 5(x - 2 \pi)^{2} \cos x + 5 x^{2} + 20 \pi^{2} = 20 \pi x
$$
Find the sum of its roots that belong to the interval $[-\pi ; 6 \pi]$, and provide the answer, rounding to two decimal places if necessary. | {
"answer": "31.42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A transgalactic ship encountered an astonishing meteor stream. Some meteors fly along a straight line at the same speed, equally spaced from each other. Another group of meteors flies in the exact same manner along another straight line, parallel to the first, but in the opposite direction, also equally spaced. The ship flies parallel to these lines. Astronaut Gavril observed that every 7 seconds the ship meets meteors flying towards it, and every 13 seconds it meets meteors flying in the same direction as the ship. Gavril wondered how often the meteors would pass by if the ship were stationary. He thought that he should take the arithmetic mean of the two given times. Is Gavril correct? If yes, write down this arithmetic mean as the answer. If not, specify the correct time in seconds, rounded to the nearest tenth. | {
"answer": "4.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Perpendiculars \( B E \) and \( D F \), dropped from the vertices \( B \) and \( D \) of parallelogram \( A B C D \) onto sides \( A D \) and \( B C \) respectively, divide the parallelogram into three parts of equal area. On the extension of diagonal \( B D \) past vertex \( D \), a segment \( D G \) is laid off equal to segment \( B D \). Line \( B E \) intersects segment \( A G \) at point \( H \). Find the ratio \( A H: H G \). | {
"answer": "1:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron on the plane containing the given edge. (12 points) | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $PQR$ has $PQ = 28.$ The incircle of the triangle evenly trisects the median $PS.$ If the area of the triangle is $p \sqrt{q}$ where $p$ and $q$ are integers, and $q$ is prime, find $p+q.$ | {
"answer": "199",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle \(ABC\), the legs \(AB\) and \(AC\) measure 4 and 3 respectively. Point \(D\) bisects the hypotenuse \(BC\). Find the distance between the centers of the incircles of triangles \(ADC\) and \(ABD\). | {
"answer": "\\frac{5 \\sqrt{13}}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
I bought a lottery ticket with a five-digit number such that the sum of its digits equals the age of my neighbor. Determine the number of this ticket, given that my neighbor easily solved this problem. | {
"answer": "99999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, \cdots, 98\} \). Find the smallest positive integer \( n \) such that, in any subset of \( S \) with \( n \) elements, it is always possible to select 10 numbers, and no matter how these 10 numbers are evenly divided into two groups, there will always be one number in one group that is relatively prime to the other 4 numbers in the same group, and one number in the other group that is not relatively prime to the other 4 numbers in that group. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hotel has 5 distinct rooms, each with single beds for up to 2 people. The hotel has no other guests, and 5 friends want to stay there for the night. In how many ways can the 5 friends choose their rooms? | {
"answer": "2220",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a $2\times 3$ grid where each entry is either $0$ , $1$ , or $2$ . For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$ ? One valid grid is shown below: $$ \begin{bmatrix} 1 & 2 & 0 2 & 1 & 0 \end{bmatrix} $$ | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sets
$$
\begin{array}{l}
A=\{(x, y) \mid |x| + |y| = a, a > 0\}, \\
B=\{(x, y) \mid |xy| + 1 = |x| + |y|\}
\end{array}
$$
If $A \cap B$ forms the vertices of a regular octagon in the plane, find the value of $a$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 13 students in a class (one of them being the monitor) and 13 seats in the classroom. Every day, the 13 students line up in random order and then enter the classroom one by one. Except for the monitor, each student will randomly choose an unoccupied seat and sit down. The monitor, however, prefers the seat next to the door and chooses it if possible. What is the probability that the monitor can choose his favourite seat? | {
"answer": "7/13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the complex numbers \( z_{1} \) and \( z_{2} \) such that \( \left| z_{2} \right| = 4 \) and \( 4z_{1}^{2} - 2z_{1}z_{2} + z_{2}^{2} = 0 \), find the maximum value of \( \left| \left( z_{1} + 1 \right)^{2} \left( z_{1} - 2 \right) \right| \). | {
"answer": "6\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.
\[ y = \ln \frac{5}{2 x}, \quad \sqrt{3} \leq x \leq \sqrt{8} \] | {
"answer": "1 + \\frac{1}{2} \\ln \\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain school holds a men's table tennis team competition. The final match adopts a points system. The two teams in the final play three matches in sequence, with the first two matches being men's singles matches and the third match being a men's doubles match. Each participating player can only play in one match in the final. A team that has entered the final has a total of five team members. Now, the team needs to submit the lineup for the final, that is, the list of players for the three matches.
$(I)$ How many different lineups are there in total?
$(II)$ If player $A$ cannot participate in the men's doubles match due to technical reasons, how many different lineups are there in total? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three points \( A \), \( B \), and \( C \) are randomly selected on the unit circle. Find the probability that the side lengths of triangle \( \triangle ABC \) do not exceed \( \sqrt{3} \). | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the mass of the plate $D$ with surface density $\mu = \frac{x^2}{x^2 + y^2}$, bounded by the curves
$$
y^2 - 4y + x^2 = 0, \quad y^2 - 8y + x^2 = 0, \quad y = \frac{x}{\sqrt{3}}, \quad x = 0.
$$ | {
"answer": "\\pi + \\frac{3\\sqrt{3}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the areas of all distinct rectangles that can be formed from 9 squares (not necessarily all), if the side of each square is $1 \text{ cm}$. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all real numbers $x$ and $y$, define the mathematical operation $\star$ such that the following conditions apply: $x\ \star\ 0 = x+1, x\ \star\ y = y\ \star\ x$, and $(x + 2)\ \star\ y = (x\ \star\ y) + y + 2$. What is the value of $7\ \star\ 3$? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three identical red balls, three identical yellow balls, and three identical green balls. In how many different ways can they be split into three groups of three balls each? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the positive numbers \( x \) and \( y \) satisfy \( x^{3} + y^{3} = x - y \). Find the maximum value of the real number \( \lambda \) such that \( x^{2} + \lambda y^{2} \leq 1 \) always holds. | {
"answer": "2 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The bases of a trapezoid are 2 cm and 3 cm long. A line passing through the intersection point of the diagonals and parallel to the bases intersects the legs at points X and Y. What is the distance between points X and Y? | {
"answer": "2.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chords \(AB\) and \(CD\) of a circle with center \(O\) both have a length of 5. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) intersect at point \(P\), where \(DP=13\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL:LC\). | {
"answer": "13/18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A block of iron solidifies from molten iron, and its volume reduces by $\frac{1}{34}$. Then, if this block of iron melts back into molten iron (with no loss in volume), by how much does its volume increase? | {
"answer": "\\frac{1}{33}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lateral edges of a triangular pyramid are mutually perpendicular, and the sides of the base are $\sqrt{85}$, $\sqrt{58}$, and $\sqrt{45}$. The center of the sphere, which touches all the lateral faces, lies on the base of the pyramid. Find the radius of this sphere. | {
"answer": "14/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer \( n \) with \( n \) digits is called an "auspicious number" if, when appended to the end of any two positive integers, the product of these two new numbers ends in \( x \). For example, 6 is an "auspicious number," but 16 is not, because \( 116 \times 216 = 25056 \), which does not end in 16. What is the sum of all "auspicious numbers" with up to 3 digits? | {
"answer": "1114",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron, each being equilateral triangles with side length 1, form a dihedral angle of 45 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this edge. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( E(n) \) denote the largest integer \( k \) such that \( 5^{k} \) divides the product \( 1^{1} \cdot 2^{2} \cdot 3^{3} \cdot 4^{4} \cdots \cdots n^{n} \). What is the value of \( E(150) \)? | {
"answer": "2975",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $S$, $P$ (not the origin) are two different points on the parabola $y=x^{2}$, the tangent line at point $P$ intersects the $x$ and $y$ axes at $Q$ and $R$, respectively.
(Ⅰ) If $\overrightarrow{PQ}=\lambda \overrightarrow{PR}$, find the value of $\lambda$;
(Ⅱ) If $\overrightarrow{SP} \perp \overrightarrow{PR}$, find the minimum value of the area of $\triangle PSR$. | {
"answer": "\\frac{4\\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four two-inch squares are placed with their bases on a line. The second square from the left is lifted out, rotated 45 degrees, then centered and lowered back until it touches its adjacent squares on both sides. Determine the distance, in inches, of point P, the top vertex of the rotated square, from the line on which the bases of the original squares were placed. | {
"answer": "1 + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An employee receives an average of two requests per hour. Assuming a simple flow of requests, what is the probability of receiving four requests in four hours? | {
"answer": "0.0572",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, points $X'$, $Y'$, and $Z'$ are on sides $YZ$, $XZ$, and $XY$ respectively. Given that $XX'$, $YY'$, and $ZZ'$ are concurrent at point $P$, and that $\frac{XP}{PX'} + \frac{YP}{PY'} + \frac{ZP}{PZ'} = 100$, find $\frac{XP}{PX'} \cdot \frac{YP}{PY'} \cdot \frac{ZP}{PZ'}$. | {
"answer": "98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When the number of repeated experiments is large enough, probability can be estimated using frequency. The mathematician Pearson once tossed a fair coin 24,000 times in an experiment. The number of times the coin landed heads up was 12,012 times, with a frequency of 0.5005. Therefore, the probability of a fair coin landing heads up is ____. | {
"answer": "0.5005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ordered triples \((x, y, z)\) satisfy the following conditions:
\[ x^2 + y^2 + z^2 = 9, \]
\[ x^4 + y^4 + z^4 = 33, \]
\[ xyz = -4? \] | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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