problem stringlengths 10 5.15k | answer dict |
|---|---|
Given a geometric sequence $\{a_n\}$ with a common ratio $q=-5$, and $S_n$ denotes the sum of the first $n$ terms of the sequence, the value of $\frac{S_{n+1}}{S_n}=\boxed{?}$. | {
"answer": "-4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A ship travels from Port A to Port B against the current at a speed of 24 km/h. After arriving at Port B, it returns to Port A with the current. It is known that the journey with the current takes 5 hours less than the journey against the current. The speed of the current is 3 km/h. Find the distance between Port A and Port B. | {
"answer": "350",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two watermelons and one banana together weigh 8100 grams, and two watermelons and three bananas together weigh 8300 grams, calculate the weight of one watermelon and one banana. | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The solid \( T \) consists of all points \((x,y,z)\) such that \(|x| + |y| \leq 2\), \(|x| + |z| \leq 2\), and \(|y| + |z| \leq 2\). Find the volume of \( T \). | {
"answer": "\\frac{1664}{81}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $M$ be the greatest integer multiple of 9, no two of whose digits are the same. What is the remainder when $M$ is divided by 1000? | {
"answer": "963",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $a$, $b$, $c$, and $d$ are integers satisfying the equations: $a - b + c = 7$, $b - c + d = 8$, $c - d + a = 5$, and $d - a + b = 4$. What is the value of $a + b + c + d$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ in a plane, satisfying $|\overrightarrow {a}|=1$, $|\overrightarrow {b}|=2$, and the dot product $(\overrightarrow {a}+ \overrightarrow {b})\cdot (\overrightarrow {a}-2\overrightarrow {b})=-7$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Sia and Kira count sequentially, where Sia skips every fifth number, find the 45th number said in this modified counting game. | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $XYZ$, $XY = 12$, $XZ = 15$, and $YZ = 23$. The medians $XM$, $YN$, and $ZO$ of triangle $XYZ$ intersect at the centroid $G$. Let $Q$ be the foot of the altitude from $G$ to $YZ$. Find $GQ$. | {
"answer": "\\frac{40}{23}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of integers $a$ with $1\le a\le 2012$ for which there exist nonnegative integers $x,y,z$ satisfying the equation
\[x^2(x^2+2z) - y^2(y^2+2z)=a.\]
*Ray Li.*
<details><summary>Clarifications</summary>[list=1][*] $x,y,z$ are not necessarily distinct.[/list]</details> | {
"answer": "1257",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that we are given 52 points equally spaced around the perimeter of a rectangle, such that four of them are located at the vertices, 13 points on two opposite sides, and 12 points on the other two opposite sides. If \( P \), \( Q \), and \( R \) are chosen to be any three of these points which are not collinear, how many different possible positions are there for the centroid of triangle \( PQR \)? | {
"answer": "805",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One hundred bricks, each measuring $3''\times12''\times20''$, are to be stacked to form a tower 100 bricks tall. Each brick can contribute $3''$, $12''$, or $20''$ to the total height of the tower. However, each orientation must be used at least once. How many different tower heights can be achieved? | {
"answer": "187",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the arithmetic sequence $\{a_n\}$, $d=-2$, $a_1+a_4+a_7+\ldots+a_{31}=50$. Calculate the value of $a_2+a_6+a_{10}+\ldots+a_{42}$. | {
"answer": "82",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Professor Lewis has twelve different language books lined up on a bookshelf: three Arabic, four German, three Spanish, and two French, calculate the number of ways to arrange the twelve books on the shelf keeping the Arabic books together, the Spanish books together, and the French books together. | {
"answer": "362,880",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence \(\{a_n\}\) is a geometric sequence with a common ratio of \(q\), where \(|q| > 1\). Let \(b_n = a_n + 1 (n \in \mathbb{N}^*)\), if \(\{b_n\}\) has four consecutive terms in the set \(\{-53, -23, 19, 37, 82\}\), find the value of \(q\). | {
"answer": "-\\dfrac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 11 seats, and now we need to arrange for 2 people to sit down. It is stipulated that the middle seat (the 6th seat) cannot be occupied, and the two people must not sit next to each other. How many different seating arrangements are there? | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid $PQRS$, leg $\overline{QR}$ is perpendicular to bases $\overline{PQ}$ and $\overline{RS}$, and diagonals $\overline{PR}$ and $\overline{QS}$ are perpendicular. Given that $PQ=\sqrt{23}$ and $PS=\sqrt{2023}$, find $QR^2$. | {
"answer": "100\\sqrt{46}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the total number of positive four-digit integers \( N \) satisfying both of the following properties:
(i) \( N \) is divisible by 7, and
(ii) when the first and last digits of \( N \) are interchanged, the resulting positive integer is also divisible by 7. (Note that the resulting integer need not be a four-digit number.) | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For what value of the parameter \( p \) will the sum of the squares of the roots of the equation
\[ p x^{2}+(p^{2}+p) x-3 p^{2}+2 p=0 \]
be the smallest? What is this smallest value? | {
"answer": "1.10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Find the domain of the function $y= \sqrt {\sin x}+ \sqrt { \frac{1}{2}-\cos x}$.
(2) Find the range of the function $y=\cos ^{2}x-\sin x$, where $x\in\left[-\frac {\pi}{4}, \frac {\pi}{4}\right]$. | {
"answer": "\\frac{1-\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A lateral face of a regular triangular pyramid $SABC$ is inclined to the base plane $ABC$ at an angle $\alpha = \operatorname{arctg} \frac{3}{4}$. Points $M, N, K$ are midpoints of the sides of the base $ABC$. The triangle $MNK$ serves as the lower base of a rectangular prism. The edges of the upper base of the prism intersect the lateral edges of the pyramid $SABC$ at points $F, P,$ and $R$ respectively. The total surface area of the polyhedron with vertices at points $M, N, K, F, P, R$ is $53 \sqrt{3}$. Find the side length of the triangle $ABC$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangles $\triangle DEF$ and $\triangle D'E'F'$ are positioned in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(14,0)$, $D'(20,20)$, $E'(30,20)$, $F'(20,8)$. Determine the angle of rotation $n$ degrees clockwise around the point $(p,q)$ where $0<n<180$, that transforms $\triangle DEF$ to $\triangle D'E'F'$. Find $n+p+q$. | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute \(\frac{x^{10} + 15x^5 + 125}{x^5 + 5}\) when \( x=3 \). | {
"answer": "248.4032",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \(PQ\) is a diameter of a larger circle, point \(R\) is on \(PQ\), and smaller semi-circles with diameters \(PR\) and \(QR\) are drawn. If \(PR = 6\) and \(QR = 4\), what is the ratio of the area of the shaded region to the area of the unshaded region? | {
"answer": "2: 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the volume of a regular tetrahedron with the side of its base equal to $\sqrt{3}$ and the angle between its lateral face and the base equal to $60^{\circ}$. | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain commodity has a cost price of 200 yuan and a marked price of 400 yuan. What is the maximum discount that can be offered to ensure that the profit margin is not less than 40%? | {
"answer": "30\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $ xyz$ space, find the volume of the solid expressed by the sytem of inequality:
$ 0\leqq x\leqq 1,\ 0\leqq y\leqq 1,\ 0\leqq z\leqq 1$
$ x^2 \plus{} y^2 \plus{} z^2 \minus{} 2xy \minus{} 1\geqq 0$ | {
"answer": "\\frac{\\pi}{3} - \\left(1 + \\frac{\\sqrt{3}}{4}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \(a\) and \(b\) are positive numbers such that \(a^b=b^a\) and \(b=27a\), then find the value of \(a\). | {
"answer": "\\sqrt[26]{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $2 - 5\sqrt{3}$ is a root of the equation \[x^3 + ax^2 + bx - 48 = 0\] and that $a$ and $b$ are rational numbers, compute $a.$ | {
"answer": "-\\frac{332}{71}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence ${a_n}$ that satisfies: $a_1=3$, $a_{n+1}=9\cdot 3a_{n} (n\geqslant 1)$, find $\lim\limits_{n\to∞}a_{n}=$ \_\_\_\_\_\_. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, a road network between the homes of five friends is shown. The shortest distance by road from Asya to Galia is 12 km, from Galia to Borya is 10 km, from Asya to Borya is 8 km, from Dasha to Galia is 15 km, and from Vasya to Galia is 17 km. What is the shortest distance by road from Dasha to Vasya? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the set of integers $\{1,2,3,\dots,3009\}$, choose $k$ pairs $\{a_i,b_i\}$ such that $a_i < b_i$ and no two pairs have a common element. Assume all the sums $a_i+b_i$ are distinct and less than or equal to 3009. Determine the maximum possible value of $k$. | {
"answer": "1203",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Bulls are playing the Knicks in the NBA playoffs. To win this playoff series, a team must secure 4 victories before the other team. If the Knicks win each game with a probability of $\dfrac{3}{5}$ and there are no ties, what is the probability that the Bulls will win the playoff series and that the contest will require all seven games to be decided? Express your answer as a fraction. | {
"answer": "\\frac{864}{15625}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cube with an edge length of 1, find the surface area of the smaller sphere that is tangent to the larger sphere and the three faces of the cube. | {
"answer": "7\\pi - 4\\sqrt{3}\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(ABCD\) is a parallelogram with \(AB = 7\), \(BC = 2\), and \(\angle DAB = 120^\circ\). Parallelogram \(ECFA\) is contained within \(ABCD\) and is similar to it. Find the ratio of the area of \(ECFA\) to the area of \(ABCD\). | {
"answer": "39/67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \( g(x) \) is a rational function such that \( 4g\left(\dfrac{1}{x}\right) + \dfrac{3g(x)}{x} = 2x^2 \) for \( x \neq 0 \). Find \( g(-3) \). | {
"answer": "\\frac{98}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( M \) lies on the edge \( AB \) of cube \( ABCD A_1 B_1 C_1 D_1 \). Rectangle \( MNLK \) is inscribed in square \( ABCD \) in such a way that one of its vertices is at point \( M \), and the other three vertices are located on different sides of the base square. Rectangle \( M_1N_1L_1K_1 \) is the orthogonal projection of rectangle \( MNLK \) onto the plane of the upper face \( A_1B_1C_1D_1 \). The ratio of the side lengths \( MK_1 \) and \( MN \) of quadrilateral \( MK_1L_1N \) is \( \sqrt{54}:8 \). Find the ratio \( AM:MB \). | {
"answer": "1:4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the probability that each of 5 different boxes contains exactly 2 fruits when 4 identical pears and 6 different apples are distributed into the boxes? | {
"answer": "0.0074",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Randomly select a number $x$ in the interval $[0,4]$, the probability of the event "$-1 \leqslant \log_{\frac{1}{3}}(x+ \frac{1}{2}) \leqslant 1$" occurring is ______. | {
"answer": "\\frac{3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions? | {
"answer": "1691",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the store "Everything for School," three types of chalk packs are sold: regular, unusual, and excellent. Initially, the quantitative ratio of the types was 3:4:6. As a result of sales and deliveries from the warehouse, this ratio changed to 2:5:8. It is known that the number of packs of excellent chalk increased by 80%, and the number of regular chalk packs decreased by no more than 10 packs. How many total packs of chalk were in the store initially? | {
"answer": "390",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the surface integral
$$
\iint_{\Sigma}(-x+3 y+4 z) d \sigma
$$
where $\Sigma$ is the part of the plane
$$
x + 2y + 3z = 1
$$
located in the first octant (i.e., $x \geq 0, y \geq 0, z \geq 0$). | {
"answer": "\\frac{\\sqrt{14}}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle is inscribed in a circle. The vertices of the triangle divide the circle into three arcs of lengths 5, 6, and 7. What is the area of the triangle? | {
"answer": "\\frac{202.2192}{\\pi^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $n$-gon $S-A_{1} A_{2} \cdots A_{n}$ has its vertices colored such that each vertex is colored with one color, and the endpoints of each edge are colored differently. Given $n+1$ colors available, find the number of different ways to color the vertices. (For $n=4$, this was a problem in the 1995 National High School Competition) | {
"answer": "420",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A small ball is released from a height \( h = 45 \) m without an initial velocity. The collision with the horizontal surface of the Earth is perfectly elastic. Determine the moment in time after the ball starts falling when its average speed equals its instantaneous speed. The acceleration due to gravity is \( g = 10 \ \text{m}/\text{s}^2 \). | {
"answer": "4.24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation $\frac{20}{x^2 - 9} - \frac{3}{x + 3} = 2$, determine the root(s). | {
"answer": "\\frac{-3 - \\sqrt{385}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular octagon is inscribed in a circle of radius 2 units. What is the area of the octagon? Express your answer in simplest radical form. | {
"answer": "16 \\sqrt{2} - 8(2)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, it is known that $P$ is a moving point on the graph of the function $f(x) = \ln x$ ($x > 0$). The tangent line $l$ at point $P$ intersects the $x$-axis at point $E$. A perpendicular line to $l$ through point $P$ intersects the $x$-axis at point $F$. Suppose the midpoint of the line segment $EF$ is $T$ with the $x$-coordinate $t$, then the maximum value of $t$ is __________. | {
"answer": "\\dfrac{1}{2}\\left(e - \\dfrac{1}{e}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The development of new energy vehicles worldwide is advancing rapidly. Electric vehicles are mainly divided into three categories: pure electric vehicles, hybrid electric vehicles, and fuel cell electric vehicles. These three types of electric vehicles are currently at different stages of development and each has its own development strategy. China's electric vehicle revolution has long been underway, replacing diesel vehicles with new energy vehicles. China is vigorously implementing a plan that will reshape the global automotive industry. In 2022, a certain company plans to introduce new energy vehicle production equipment. Through market analysis, it is determined that a fixed cost of 20 million yuan needs to be invested throughout the year. For each production of x (hundreds of vehicles), an additional cost C(x) (in million yuan) needs to be invested, and C(x) is defined as follows: $C(x)=\left\{\begin{array}{l}{10{x^2}+100x,0<x<40,}\\{501x+\frac{{10000}}{x}-4500,x\geq40}\end{array}\right.$. It is known that the selling price of each vehicle is 50,000 yuan. According to market research, all vehicles produced within the year can be sold.
$(1)$ Find the function relationship of the profit $L(x)$ in 2022 with respect to the annual output $x$ (in hundreds of vehicles).
$(2)$ For how many hundreds of vehicles should be produced in 2022 to maximize the profit of the company? What is the maximum profit? | {
"answer": "2300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $9\times 1$ board is completely covered by $m\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $9\times 1$ board in which all three colors are used at least once. Find the remainder when $N$ is divided by $1000$. | {
"answer": "838",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1,2, \cdots, 15\} \). From \( S \), extract \( n \) subsets \( A_{1}, A_{2}, \cdots, A_{n} \), satisfying the following conditions:
(i) \(\left|A_{i}\right|=7, i=1,2, \cdots, n\);
(ii) \(\left|A_{i} \cap A_{j}\right| \leqslant 3,1 \leqslant i<j \leqslant n\);
(iii) For any 3-element subset \( M \) of \( S \), there exists some \( A_{K} \) such that \( M \subset A_{K} \).
Find the minimum value of \( n \). | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let triangle $PQR$ be a right triangle in the xy-plane with a right angle at $R$. Given that the length of the hypotenuse $PQ$ is $50$, and that the medians through $P$ and $Q$ lie along the lines $y=x+2$ and $y=3x+5$ respectively, find the area of triangle $PQR$. | {
"answer": "\\frac{125}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\left\{a_{n}\right\}$ defined by
$$
a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad (n \in \mathbf{Z}_{+}),
$$
where $[x]$ denotes the greatest integer not exceeding the real number $x$, determine the minimum value of the constant $C$ such that for any positive integer $n$, the following inequality holds:
$$
\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C.
$$ | {
"answer": "1/288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The following grid represents a mountain range; the number in each cell represents the height of the mountain located there. Moving from a mountain of height \( a \) to a mountain of height \( b \) takes \( (b-a)^{2} \) time. Suppose that you start on the mountain of height 1 and that you can move up, down, left, or right to get from one mountain to the next. What is the minimum amount of time you need to get to the mountain of height 49? | {
"answer": "212",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, forming a pyramid $DABC$ with all triangular faces.
Suppose every edge of $DABC$ has a length either $25$ or $60$, and no face of $DABC$ is equilateral. Determine the total surface area of $DABC$. | {
"answer": "3600\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twenty kindergarten children are arranged in a line at random, consisting of 11 girls and 9 boys. Find the probability that there are no more than five girls between the first and the last boys in the line. | {
"answer": "0.058",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given 95 numbers \( a_{1}, a_{2}, \cdots, a_{95} \) where each number can only be +1 or -1, find the minimum value of the sum of the products of each pair of these numbers, \( \sum_{1 \leq i<j \leq 95} a_{i} a_{j} \). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A(2,-5,1)$, $B(2,-2,4)$, $C(1,-4,1)$, the angle between vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is equal to ______. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( g(x) = 10x + 5 \). Find the sum of all \( x \) that satisfy the equation \( g^{-1}(x) = g((3x)^{-2}) \). | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A store has equal amounts of candies priced at 2 rubles per kilogram and candies priced at 3 rubles per kilogram. At what price should the mixture of these candies be sold? | {
"answer": "2.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $ABCD$ is circumscribed around a circle. Another square $IJKL$ is inscribed inside a smaller concentric circle. The side length of square $ABCD$ is $4$ units, and the radius of the smaller circle is half the radius of the larger circle. Find the ratio of the area of square $IJKL$ to the area of square $ABCD$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bus ticket costs 1 yuan each. Xiaoming and 6 other children are lining up to buy tickets. Each of the 6 children has only 1 yuan, while Xiaoming has a 5-yuan note. The seller has no change. In how many ways can they line up so that the seller can give Xiaoming change when he buys a ticket? | {
"answer": "10800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a memorable $9$-digit telephone number defined as $d_1d_2d_3d_4-d_5d_6d_7d_8d_9$. A number is memorable if the prefix sequence $d_1d_2d_3d_4$ is exactly the same as either of the sequences $d_5d_6d_7d_8$ or $d_6d_7d_8d_9$. Each digit $d_i$ can be any of the ten decimal digits $0$ through $9$. Find the number of different memorable telephone numbers.
A) 199980
B) 199990
C) 200000
D) 200010
E) 200020 | {
"answer": "199990",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four math teams in a region consist of 6, 8, 9, and 11 students respectively. Each team has three co-captains. If you randomly select a team, and then randomly select two members from that team to gift a copy of *Introduction to Geometry*, what is the probability that both recipients are co-captains? | {
"answer": "\\frac{1115}{18480}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( 5^{\log 30} \times \left(\frac{1}{3}\right)^{\log 0.5} = d \), find the value of \( d \). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parametric graph is defined by:
\[
x = \cos t + \frac{t}{3}, \quad y = \sin t.
\]
Determine the number of times the graph intersects itself between \(x = 3\) and \(x = 45\). | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 6 rectangular prisms with edge lengths of \(3 \text{ cm}\), \(4 \text{ cm}\), and \(5 \text{ cm}\). The faces of these prisms are painted red in such a way that one prism has only one face painted, another has exactly two faces painted, a third prism has exactly three faces painted, a fourth prism has exactly four faces painted, a fifth prism has exactly five faces painted, and the sixth prism has all six faces painted. After painting, each rectangular prism is divided into small cubes with an edge length of \(1 \text{ cm}\). What is the maximum number of small cubes that have exactly one red face? | {
"answer": "177",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $z$ be a complex number such that
\[ |z - 2i| + |z - 5| = 7. \]
Find the minimum value of $|z|$. | {
"answer": "\\sqrt{\\frac{100}{29}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x \in (0,1) \) and \( \frac{1}{x} \notin \mathbf{Z} \), define
\[
a_{n}=\frac{n x}{(1-x)(1-2 x) \cdots(1-n x)} \quad \text{for} \quad n=1,2, \cdots
\]
A number \( x \) is called a "good number" if and only if it makes the sequence \(\{a_{n}\}\) satisfy
\[
a_{1}+a_{2}+\cdots+a_{10} > -1 \quad \text{and} \quad a_{1} a_{2} \cdots a_{10} > 0
\]
Find the sum of the lengths of all intervals on the number line that represent all such "good numbers". | {
"answer": "61/210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right triangle has legs of lengths 126 and 168 units. What is the perimeter of the triangle formed by the points where the angle bisectors intersect the opposite sides? | {
"answer": "230.61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose four solid iron balls are placed in a cylinder with the radius of 1 cm, such that every two of the four balls are tangent to each other, and the two balls in the lower layer are tangent to the cylinder base. Now put water into the cylinder. Find, in $\text{cm}^2$ , the volume of water needed to submerge all the balls. | {
"answer": "\\frac{4 \\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Here is a fairly simple puzzle: EH is four times greater than OY. AY is four times greater than OH. Find the sum of all four. | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Yvan and Zoé play the following game. Let \( n \in \mathbb{N} \). The integers from 1 to \( n \) are written on \( n \) cards arranged in order. Yvan removes one card. Then, Zoé removes 2 consecutive cards. Next, Yvan removes 3 consecutive cards. Finally, Zoé removes 4 consecutive cards.
What is the smallest value of \( n \) for which Zoé can ensure that she can play her two turns? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence of length 15 composed of zeros and ones, find the number of sequences where all zeros are consecutive, all ones are consecutive, or both. | {
"answer": "270",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In space, there are 3 planes and a sphere. How many distinct ways can a second sphere be placed in space so that it touches the three given planes and the first sphere? (In this problem, sphere touching is considered, i.e., it is not assumed that the spheres can only touch externally.) | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $8 \times 8 \times 8$ cube has three of its faces painted red and the other three faces painted blue (ensuring that any three faces sharing a common vertex are not painted the same color), and then it is cut into 512 $1 \times 1 \times 1$ smaller cubes. Among these 512 smaller cubes, how many have both a red face and a blue face? | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of the positive integer divisors of a positive integer \( n \) is 1024, and \( n \) is a perfect power of a prime. Find \( n \). | {
"answer": "1024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given complex numbers $\mathrm{z}_{\mathrm{i}} (\mathrm{i} = 1, 2, 3, 4, 5)$ satisfying:
$$\left\{\begin{array}{c}
\left|z_{1}\right| \leq 1, \quad \left|z_{2}\right| \leq 1 \\
\left|2 z_{3}-(\mathrm{z}_{1}+\mathrm{z}_{2})\right| \leq \left|\mathrm{z}_{1}-\mathrm{z}_{2}\right| \\
\left|2 \mathrm{z}_{4}-(\mathrm{z}_{1}+\mathrm{z}_{2})\right| \leq \left|\mathrm{z}_{1}-\mathrm{z}_{2}\right| \\
\left|2 z_{5}-(\mathrm{z}_{3}+\mathrm{z}_{4})\right| \leq \left|\mathrm{z}_{3}-\mathrm{z}_{4}\right|
\end{array} \right.$$
Find the maximum value of $\left|\mathrm{z}_{5}\right|$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A Senate committee consists of 10 Republicans and 8 Democrats. In how many ways can we form a subcommittee that has at most 5 members, including exactly 3 Republicans and at least 2 Democrats? | {
"answer": "10080",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chloe is baking muffins for a school event. If she divides the muffins equally among 8 of her friends, she'll have 3 muffins left over. If she divides the muffins equally among 5 of her friends, she'll have 2 muffins left over. Assuming Chloe made fewer than 60 muffins, what is the sum of the possible numbers of muffins that she could have made? | {
"answer": "118",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Hawkins, Dustin, and Lucas start playing a game where each begins with $\$2$. A bell rings every $10$ seconds, and each player with money independently chooses one of the other two players at random to give $\$1$. If a player has only $\$1$ left, there is a $\frac{1}{3}$ probability they will keep their money and not give it to anyone. What is the probability that after the bell has rung $2021$ times, each player will still have $\$1$?
A) $\frac{1}{7}$
B) $\frac{1}{4}$
C) $\frac{1}{3}$
D) $\frac{1}{2}$
E) $\frac{2}{3}$ | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are four pairs of siblings, each pair consisting of one boy and one girl. We need to divide them into three groups in such a way that each group has at least two members, and no siblings end up in the same group. In how many ways can this be done? | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the village of Znoynoe, there are exactly 1000 inhabitants, which exceeds the average population of the villages in the valley by 90 people.
How many inhabitants are there in the village of Raduzhny, which is also located in the Sunny Valley? | {
"answer": "900",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle \( ABC \) with \( AB = 12 \), \( BC = 10 \), and \( \angle ABC = 120^\circ \), find \( R^2 \), where \( R \) is the radius of the smallest circle that can contain this triangle. | {
"answer": "91",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rain drops fall vertically at a speed of $v = 2 \, \text{m/s}$. The rear window of a car is inclined at an angle of $\alpha = 60^{\circ}$ to the horizontal. At what speed $u$ must the car travel on a horizontal, flat road so that its rear window remains dry? | {
"answer": "1.15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers less than 100, which are also not divisible by 3, have an even number of positive divisors? | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x^{2}+y^{2}=1$, determine the maximum and minimum values of $x+y$. | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a fair 8-sided die is rolled until an odd number appears, determine the probability that the even numbers 2, 4, and 6 appear in strictly ascending order at least once before the first occurrence of any odd number. | {
"answer": "\\frac{1}{512}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parallelogram is defined by the vectors $\begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$. Determine the cosine of the angle $\theta$ between the diagonals of this parallelogram. | {
"answer": "\\frac{-11}{3\\sqrt{69}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square flag has a green cross of uniform width with a yellow square in the center on a white background. The cross is symmetric with respect to each of the diagonals of the square. If the entire cross (both the green arms and the yellow center) occupies 49% of the area of the flag, what percent of the area of the flag is yellow? | {
"answer": "25.14\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
I have a bag with $5$ marbles numbered from $1$ to $5.$ Mathew has a bag with $12$ marbles numbered from $1$ to $12.$ Mathew picks one marble from his bag and I pick two from mine, with the choice order being significant. In how many ways can we choose the marbles such that the sum of the numbers on my marbles exceeds the number on Mathew's marble by exactly one? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve for $y$: $\sqrt[4]{36y + \sqrt[3]{36y + 55}} = 11.$ | {
"answer": "\\frac{7315}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the digits \( a_{i} (i=1,2, \cdots, 9) \) satisfy
$$
a_{9} < a_{8} < \cdots < a_{5} \text{ and } a_{5} > a_{4} > \cdots > a_{1} \text{, }
$$
then the nine-digit positive integer \(\bar{a}_{9} a_{8} \cdots a_{1}\) is called a “nine-digit peak number”, for example, 134698752. How many nine-digit peak numbers are there? | {
"answer": "11875",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(a_{1}, a_{2}, a_{3}, \ldots\) is an increasing sequence of natural numbers. It is known that \(a_{a_{k}} = 3k\) for any \(k\).
Find
a) \(a_{100}\)
b) \(a_{1983}\). | {
"answer": "3762",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A quadrilateral pyramid \( S A B C D \) is given, with the base being the parallelogram \( A B C D \). A plane is drawn through the midpoint of edge \( A B \) that is parallel to the lines \( A C \) and \( S D \). In what ratio does this plane divide edge \( S B \)? | {
"answer": "1 : 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Complex numbers $z_1,$ $z_2,$ and $z_3$ are zeros of a polynomial $Q(z) = z^3 + pz + s,$ where $|z_1|^2 + |z_2|^2 + |z_3|^2 = 300$. The points corresponding to $z_1,$ $z_2,$ and $z_3$ in the complex plane are the vertices of a right triangle with the right angle at $z_3$. Find the square of the hypotenuse of this triangle. | {
"answer": "450",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all negative values of \( a \) for which the equation
$$
\frac{8 \pi a - \arcsin (\sin x) + 3 \arccos (\cos x) - a x}{3 + \operatorname{tg}^{2} x} = 0
$$
has exactly three solutions. Provide the sum of all found \( a \) (if no such \( a \) exist, indicate 0; if the sum of \( a \) is not an integer, round it to the nearest hundredth). | {
"answer": "-2.47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{r}$ and $\mathbf{s}$ be two three-dimensional unit vectors such that the angle between them is $45^\circ$. Find the area of the parallelogram whose diagonals correspond to $\mathbf{r} + 3\mathbf{s}$ and $3\mathbf{r} + \mathbf{s}$. | {
"answer": "\\frac{3\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=\overrightarrow{a}\cdot\overrightarrow{b}=1$, and $(\overrightarrow{a}-2\overrightarrow{c}) \cdot (\overrightarrow{b}-\overrightarrow{c})=0$, find the minimum value of $|\overrightarrow{a}-\overrightarrow{c}|$. | {
"answer": "\\frac{\\sqrt{7}-\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person receives an annuity at the end of each year for 15 years as follows: $1000 \mathrm{K}$ annually for the first five years, $1200 \mathrm{K}$ annually for the next five years, and $1400 \mathrm{K}$ annually for the last five years. If they received $1400 \mathrm{K}$ annually for the first five years and $1200 \mathrm{K}$ annually for the second five years, what would be the annual annuity for the last five years? | {
"answer": "807.95",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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