problem stringlengths 10 5.15k | answer dict |
|---|---|
The teacher asked the students to calculate \(\overline{AB} . C + D . E\). Xiao Hu accidentally missed the decimal point in \(D . E\), getting an incorrect result of 39.6; while Da Hu mistakenly saw the addition sign as a multiplication sign, getting an incorrect result of 36.9. What should the correct calculation result be? | {
"answer": "26.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The scent of blooming lily of the valley bushes spreads within a radius of 20 meters around them. How many blooming lily of the valley bushes need to be planted along a straight 400-meter-long alley so that every point along the alley can smell the lily of the valley? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\left\{a_{n}\right\}$ is defined such that $a_{n}$ is the last digit of the sum $1 + 2 + \cdots + n$. Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Calculate $S_{2016}$. | {
"answer": "7066",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane Cartesian coordinate system, the area of the region corresponding to the set of points $\{(x, y) \mid(|x|+|3 y|-6)(|3 x|+|y|-6) \leq 0\}$ is ________. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A quadrilateral is divided into 1000 triangles. What is the maximum number of distinct points that can be the vertices of these triangles? | {
"answer": "1002",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At 1:00 PM, two identical recreational boats set off in opposite directions from a pier on a river. At the same time, a raft also departed from the pier. An hour later, one of the boats turned around and started moving back. The other boat did the same at 3:00 PM. What is the speed of the current if, at the moment the boats met, the raft had drifted 7.5 km from the pier? | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCD$ is a square of side length 1. $X$ and $Y$ are points on $BC$ and $CD$ respectively such that $CX = CY = m$. When extended, $AB$ meets $DX$ at $P$, $AD$ meets $BY$ at $Q$, $AX$ meets $DC$ at $R$, and $AY$ meets $BC$ at $S$. If $P, Q, R, S$ are collinear, find $m$. | {
"answer": "\\frac{3 - \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle \(ABC\) is to be constructed so that \(A\) is at \((3,2)\), \(B\) is on the line \(y=x\), and \(C\) is on the \(x\)-axis. Find the minimum possible perimeter of \(\triangle ABC\). | {
"answer": "\\sqrt{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Karlsson eats three jars of jam and one jar of honey in 25 minutes, while Little Brother does it in 55 minutes. Karlsson eats one jar of jam and three jars of honey in 35 minutes, while Little Brother does it in 1 hour 25 minutes. How long will it take them to eat six jars of jam together? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \( S = \{1,2,3, \cdots, 10\} \). Given that subset \( A \) of \( S \) satisfies \( A \cap \{1,2,3\} \neq \varnothing \) and \( A \cup \{4,5,6\} \neq S \), determine the number of such subsets \( A \).
(Note: The original problem includes determining the count of subset \( A \) that meets the given conditions.) | {
"answer": "888",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Is the following number rational or irrational?
$$
\sqrt[3]{2016^{2} + 2016 \cdot 2017 + 2017^{2} + 2016^{3}} ?
$$ | {
"answer": "2017",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten numbers are written around a circle with their sum equal to 100. It is known that the sum of each triplet of consecutive numbers is at least 29. Identify the smallest number \( A \) such that, in any such set of numbers, each number does not exceed \( A \). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( p(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a, b, c, d \) are constants, and given that \( p(1) = 1993 \), \( p(2) = 3986 \), and \( p(3) = 5979 \). Calculate \( \frac{1}{4} [p(11) + p(-7)] \). | {
"answer": "5233",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular chessboard of size \( m \times n \) is composed of unit squares (where \( m \) and \( n \) are positive integers not exceeding 10). A piece is placed on the unit square in the lower-left corner. Players A and B take turns moving the piece. The rules are as follows: either move the piece any number of squares upward, or any number of squares to the right, but you cannot move off the board or stay in the same position. The player who cannot make a move loses (i.e., the player who first moves the piece to the upper-right corner wins). How many pairs of integers \( (m, n) \) are there such that the first player A has a winning strategy? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parallelepiped \( A B C D A_{1} B_{1} C_{1} D_{1} \). On edge \( A_{1} D_{1} \), point \( X \) is selected, and on edge \( B C \), point \( Y \) is selected. It is known that \( A_{1} X = 5 \), \( B Y = 3 \), and \( B_{1} C_{1} = 14 \). The plane \( C_{1} X Y \) intersects the ray \( D A \) at point \( Z \). Find \( D Z \). | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are five distinct nonzero natural numbers; the smallest one is 7. If one of them is decreased by 20, and the other four numbers are each increased by 5, the resulting set of numbers remains the same. What is the sum of these five numbers? | {
"answer": "85",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex quadrilateral $ABCD$, $M$ and $N$ are the midpoints of sides $AD$ and $BC$, respectively. Given that $|\overrightarrow{AB}|=2$, $|\overrightarrow{MN}|=\frac{3}{2}$, and $\overrightarrow{MN} \cdot (\overrightarrow{AD} - \overrightarrow{BC}) = \frac{3}{2}$, find $\overrightarrow{AB} \cdot \overrightarrow{CD}$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 4 people in the family. If Masha's scholarship is doubled, the total family income will increase by 5%. If the mother's salary is doubled instead, the total family income will increase by 15%. If the father's salary is doubled, the total family income will increase by 25%. By how many percent will the family's income increase if the grandfather's pension is doubled? | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. If a plane is tangent to these two spheres and intersects the surface of the cylinder forming an ellipse, then the sum of the lengths of the major axis and the minor axis of this ellipse is ___. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Every day, from Monday to Friday, an old man went to the blue sea and cast his net into the sea. Each day, the number of fish caught in the net was no greater than the number caught on the previous day. In total, over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could have caught over three specific days: Monday, Wednesday, and Friday? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence \(\{a_n\}\) with the first term 2, and it satisfies
\[ 6 S_n = 3 a_{n+1} + 4^n - 1. \]
Find the maximum value of \(S_n\). | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cyclic quadrilateral \(ABCD\) has side lengths \(AB = 1\), \(BC = 2\), \(CD = 3\), and \(AD = 4\). Determine \(\frac{AC}{BD}\). | {
"answer": "5/7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fast train with a weight of $P=150$ tons travels at a maximum speed of $v=72 \frac{\text{km}}{\text{hour}}$ on a horizontal track with a friction coefficient of $\rho=0,005$. What speed can it reach on a track with the same friction conditions but with an incline having $e=0.030$?
(Note: In this problem, $\rho$ is the friction coefficient and $e$ is the sine of the inclination angle.) | {
"answer": "10.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The quadrilateral \(P Q R S\) is inscribed in a circle. Diagonals \(P R\) and \(Q S\) are perpendicular and intersect at point \(M\). It is known that \(P S = 13\), \(Q M = 10\), and \(Q R = 26\). Find the area of the quadrilateral \(P Q R S\). | {
"answer": "319",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We use \( S_{k} \) to represent an arithmetic sequence with the first term \( k \) and common difference \( k^{2} \). For example, \( S_{3} \) is \( 3, 12, 21, \cdots \). If 306 is a term in \( S_{k} \), the sum of all possible \( k \) that satisfy this condition is ____. | {
"answer": "326",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles trapezoid with bases \(a = 21\), \(b = 9\) and height \(h = 8\), find the radius of the circumscribed circle. | {
"answer": "\\frac{85}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A four-digit number $\overline{abcd}$ has the properties that $a + b + c + d = 26$, the tens digit of $b \cdot d$ equals $a + c$, and $bd - c^2$ is a multiple of 2. Find this four-digit number (provide justification). | {
"answer": "1979",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given 2017 lines separated into three sets such that lines in the same set are parallel to each other, what is the largest possible number of triangles that can be formed with vertices on these lines? | {
"answer": "673 * 672^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular dodecagon $A B C D E F G H I J K L$ inscribed in a circle with a radius of $6 \mathrm{~cm}$, determine the perimeter of the pentagon $A C F H K$. | {
"answer": "18 + 12\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{2} x \, dx}{(1+\cos x+\sin x)^{2}}
$$ | {
"answer": "\\frac{1}{2} - \\frac{1}{2} \\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A black and white chocolate bar consists of individual pieces arranged in $n$ horizontal rows and $m$ vertical columns, painted in a checkerboard pattern. Ian ate all the black pieces, and Max ate all the white pieces. What is the sum of $m + n$ if it is known that Ian ate $8 \frac{1}{3} \%$ more pieces than Max? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers from 1 to 9 are placed in the cells of a \(3 \times 3\) grid such that the sum of the numbers on one diagonal is 7 and on the other diagonal is 21. What is the sum of the numbers in the five shaded cells? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The square $A B C D$ is enlarged from vertex $A$ resulting in the square $A B^{\prime} C^{\prime} D^{\prime}$. The intersection point of the diagonals of the enlarged square is $M$. It is given that $M C = B B^{\prime}$. What is the scale factor of the enlargement? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An assembly line produces, on average, 85% first grade products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the proportion of first grade products from 0.85 in absolute value does not exceed 0.01? | {
"answer": "11475",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
100 people participated in a quick calculation test consisting of 10 questions. The number of people who answered each question correctly is given in the table below:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Problem Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Number of Correct Answers & 93 & 90 & 86 & 91 & 80 & 83 & 72 & 75 & 78 & 59 \\
\hline
\end{tabular}
Criteria: To pass, one must answer at least 6 questions correctly. Based on the table, calculate the minimum number of people who passed. | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given natural numbers \( m \) and \( n \) where \( n > m > 1 \), the last three digits of the decimal representation of \( 1978^m \) and \( 1978^n \) are the same. Find \( m \) and \( n \) such that \( m+n \) is minimized. | {
"answer": "106",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In "The Three-Body Problem," the operation involves intercepting the "Judgment Day" ship with a "nano-blade" material at the Panama Canal locks. As long as the "Judgment Day" ship passes through the "nano-blade" material completely undetected, the operation is a success. If the entire length of the "Judgment Day" ship is 400 meters, and it takes 50 seconds to pass through a 100-meter long tunnel at a constant speed, how many seconds will it take for the ship to pass through the "nano-blade" material at the same speed? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(\triangle ABC\) is equilateral with side length 4. \(D\) is a point on \(BC\) such that \(BD = 1\). If \(r\) and \(s\) are the radii of the inscribed circles of \(\triangle ADB\) and \(\triangle ADC\) respectively, find \(rs\). | {
"answer": "4 - \\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parallelepiped \(A B C D A_{1} B_{1} C_{1} D_{1}\), a point \(X\) is chosen on edge \(A_{1} D_{1}\), and a point \(Y\) is chosen on edge \(B C\). It is known that \(A_{1} X = 5\), \(B Y = 3\), and \(B_{1} C_{1} = 14\). The plane \(C_{1} X Y\) intersects the ray \(D A\) at point \(Z\). Find \(D Z\). | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three balls are lying on a table, each pair touching each other. The radii of the balls form a geometric progression with a common ratio \( q \neq 1 \). The radius of the middle ball is 2012. Find the ratio of the sum of the squares of the sides of the triangle formed by the points of contact of the balls with the table to the sum of the sides of the triangle formed by the centers of the balls. | {
"answer": "4024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many times does the digit 0 appear in the integer equal to \( 20^{10} \)? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the process of making a steel cable, it was determined that the length of the cable is the same as the curve given by the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=10 \\
x y+y z+x z=18
\end{array}\right.
$$
Find the length of the cable. | {
"answer": "4 \\pi \\sqrt{\\frac{23}{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Masha looked at the drawing and said: "There are seven rectangles here: one big one and six small ones." "There are also various middle-sized rectangles here," said her mother. How many rectangles are there in total in this drawing? Explain your answer. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-102)^{2}
$$
If you obtain a non-integer number, round the result to the nearest whole number. | {
"answer": "46852",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sasha wrote the numbers $7, 8, 9, \ldots, 17$ on the board and then erased one or more of them. It turned out that the remaining numbers on the board cannot be divided into several groups such that the sums of the numbers in the groups are equal. What is the maximum value that the sum of the remaining numbers on the board can have? | {
"answer": "121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the area enclosed by the curves \( y = \sin x \) and \( y = \left(\frac{4}{\pi}\right)^{2} \sin \left(\frac{\pi}{4}\right) x^{2} \) (the latter is a quadratic function). | {
"answer": "1 - \\frac{\\sqrt{2}}{2}\\left(1 + \\frac{\\pi}{12}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( AC = 1 \ \text{cm} \) and \( AD = 4 \ \text{cm} \), what is the relationship between the areas of triangles \( \triangle ABC \) and \( \triangle CBD \)? | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A two-digit number, when multiplied by 109, yields a four-digit number. It is divisible by 23, and the quotient is a one-digit number. The maximum value of this two-digit number is $\qquad$. | {
"answer": "69",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\) and \(\sin \beta = 2 \cos (\alpha + \beta) \cdot \sin \alpha \left(\alpha + \beta \neq \frac{\pi}{2}\right)\), find the maximum value of \(\tan \beta\). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number \( n \) for which \( (n+1)(n+2)(n+3)(n+4) \) is divisible by 1000. | {
"answer": "121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangular prism \(P-ABC\), \(\triangle ABC\) is an equilateral triangle with side length \(2\sqrt{3}\), \(PB = PC = \sqrt{5}\), and the dihedral angle \(P-BC-A\) is \(45^\circ\). Find the surface area of the circumscribed sphere around the triangular prism \(P-ABC\). | {
"answer": "25\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of a right triangle is 1, and its hypotenuse is \(\sqrt{5}\). Find the cosine of the acute angle between the medians of the triangle that are drawn to its legs. | {
"answer": "\\frac{5\\sqrt{34}}{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A loaf of sausage is marked with thin transverse rings. If you cut along the red rings, you get 5 pieces; if along the yellow rings, you get 7 pieces; and if along the green rings, you get 11 pieces. How many pieces of sausage will you get if you cut along the rings of all three colors? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function
$$
f(x) = \left|8x^3 - 12x - a\right| + a
$$
The maximum value of this function on the interval \([0, 1]\) is 0. Find the maximum value of the real number \(a\). | {
"answer": "-2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, where $AB = BC > AC$, let $AH$ and $AM$ be the altitude and median to side $BC$, respectively. Given $\frac{S_{\triangle AMH}}{S_{\triangle ABC}} = \frac{3}{8}$, determine the value of $\cos \angle BAC$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chocolate bar weighed 250 g and cost 50 rubles. Recently, for cost-saving purposes, the manufacturer reduced the weight of the bar to 200 g and increased its price to 52 rubles. By what percentage did the manufacturer's income increase? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The angle $A$ at the vertex of the isosceles triangle $ABC$ is $100^{\circ}$. On the ray $AB$, a segment $AM$ is laid off, equal to the base $BC$. Find the measure of the angle $BCM$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all real numbers \( r, s, t \) satisfying \( 1 \leq r \leq s \leq t \leq 4 \), find the minimum value of \( (r-1)^{2}+\left(\frac{s}{r}-1\right)^{2} +\left(\frac{t}{s}-1\right)^{2}+\left(\frac{4}{t}-1\right)^{2} \). | {
"answer": "4(\\sqrt{2} - 1)^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube with a side of 10 is divided into 1000 smaller cubes each with an edge of 1. A number is written in each small cube such that the sum of the numbers in each column of 10 cubes (along any of the three directions) equals 0. In one of the cubes (denoted as A), the number 1 is written. There are three layers passing through cube A, and these layers are parallel to the faces of the cube (each layer has a thickness of 1). Find the sum of all the numbers in the cubes that do not lie in these layers. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\left\{a_{n}\right\}$ defined by $a_{0} = \frac{1}{2}$ and $a_{n+1} = a_{n} + \frac{a_{n}^{2}}{2023}$ for $n=0,1,2,\ldots$, find the integer $k$ such that $a_{k} < 1 < a_{k+1}$. | {
"answer": "2023",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya is creating a 4-digit password for a combination lock. He dislikes the digit 2, so he does not use it. Additionally, he dislikes when two identical digits stand next to each other. Furthermore, he wants the first digit to match the last. How many possible combinations must be tried to guarantee guessing Vasya's password? | {
"answer": "504",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Unlucky Emelya was given several metal balls. He broke the 3 largest ones (their mass was 35% of the total mass of all the balls), then lost the 3 smallest ones, and brought home the remaining balls (their mass was \( \frac{8}{13} \) of the unbroken ones). How many balls was Emelya given? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At an old estate, the house is surrounded in a circle by tall trees: spruces, pines, and birches. There are a total of 96 trees. These trees have a strange property: from any coniferous tree, if you take two trees skipping one tree in between, one of them is coniferous and the other is deciduous; and from any coniferous tree, if you take two trees skipping two trees in between, one of them is coniferous and the other is deciduous. How many birches are planted around the house? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A workshop produces items of types $A$ and $B$. One item of type $A$ requires 10 kg of steel and 23 kg of non-ferrous metals, while an item of type $B$ requires 70 kg of steel and 40 kg of non-ferrous metals. The profit from selling an item of type $A$ is 80 thousand rubles, and for type $B$ it is 100 thousand rubles. The daily supply of steel is 700 kg, and non-ferrous metals is 642 kg. How many items of types $A$ and $B$ should be produced per shift to maximize profit from sales, given that resource consumption should not exceed the allocated supplies for the shift? State the maximum profit (in thousand rubles) that can be obtained under these conditions as a single number without specifying the unit. | {
"answer": "2180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integer solutions does the inequality
$$
|x| + |y| < 1000
$$
have, where \( x \) and \( y \) are integers? | {
"answer": "1998001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a positive integer \( n \), consider the function
\[
f(n)=\frac{4n+\sqrt{4n^2-1}}{\sqrt{2n+1}+\sqrt{2n-1}}
\]
Calculate the value of
\[
f(1)+f(2)+f(3)+\cdots+f(40)
\] | {
"answer": "364",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 20 chairs arranged in a circle. There are \(n\) people sitting in \(n\) different chairs. These \(n\) people stand, move \(k\) chairs clockwise, and then sit again. After this happens, exactly the same set of chairs is occupied. For how many pairs \((n, k)\) with \(1 \leq n \leq 20\) and \(1 \leq k \leq 20\) is this possible? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \( x_{1}, x_{2}, \ldots, x_{49} \) are real numbers such that
\[ x_{1}^{2} + 2 x_{2}^{2} + \cdots + 49 x_{49}^{2} = 1. \]
Find the maximum value of \( x_{1} + 2 x_{2} + \cdots + 49 x_{49} \). | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of the pyramid \(SABC\) is an isosceles triangle \(ABC\) with \(AB = BC = 3\) and \(AC = 5\). The height of the pyramid \(SABC\) is the segment \(SO\), where \(O\) is the intersection point of the line passing through vertex \(B\) parallel to side \(AC\), and the line passing through \(C\) perpendicular to side \(AC\). Find the distance from the center of the inscribed circle of triangle \(ABC\) to the plane containing the lateral face \(BSC\), given that the height of the pyramid is \(7/12\). | {
"answer": "\\frac{35 \\sqrt{11}}{396}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle, the larger angle at the base is $45^{\circ}$, and the altitude divides the base into segments of 20 and 21. Find the length of the larger lateral side. | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Since the beginning of the school year, Andrey has been recording his math grades. Upon receiving a new grade (2, 3, 4, or 5), he called it unexpected if, up to that point, it had occurred less frequently than each of the other possible grades. (For example, if he received the grades 3, 4, 2, 5, 5, 5, 2, 3, 4, 3 in order, the first 5 and the second 4 would be unexpected.) Over the entire school year, Andrey received 40 grades - exactly 10 fives, 10 fours, 10 threes, and 10 twos (the order is unknown). Can you definitively determine the number of grades that were unexpected for him? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's define a number as complex if it has at least two different prime divisors. Find the greatest natural number that cannot be represented as the sum of two complex numbers. | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a rectangular sheet of paper, a picture in the shape of a "cross" made of two rectangles $ABCD$ and $EFGH$ was drawn, with sides parallel to the edges of the sheet. It is known that $AB = 9$, $BC = 5$, $EF = 3$, $FG = 10$. Find the area of the quadrilateral $AFCH$. | {
"answer": "52.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a plane with a Cartesian coordinate system, there are 16 grid points \((i, j)\), where \(0 \leq i \leq 3\) and \(0 \leq j \leq 3\). If \(n\) points are selected from these 16 points, there will always exist 4 points among the \(n\) points that are the vertices of a square. Find the minimum value of \(n\). | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All two-digit numbers divisible by 5, where the number of tens is greater than the number of units, were written on the board. There were \( A \) such numbers. Then, all two-digit numbers divisible by 5, where the number of tens is less than the number of units, were written on the board. There were \( B \) such numbers. What is the value of \( 100B + A \)? | {
"answer": "413",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of the expression \((\sqrt{8-4 \sqrt{3}} \sin x - 3 \sqrt{2(1+\cos 2x)} - 2) \cdot (3 + 2 \sqrt{11 - \sqrt{3}} \cos y - \cos 2y)\). If the answer is a non-integer, round it to the nearest whole number. | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the flux of the vector field
$$
\vec{a}=-x \vec{i}+2 y \vec{j}+z \vec{k}
$$
through the portion of the plane
$$
x+2 y+3 z=1
$$
located in the first octant (the normal forms an acute angle with the $OZ$ axis). | {
"answer": "\\frac{1}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number \( n \) is called "cubowat" if \( n^{3} + 13n - 273 \) is a cube of a natural number. Find the sum of all cubowat numbers. | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain group of men, 70% have brown eyes, 70% have dark hair, 85% are taller than 5 feet 8 inches, and 90% weigh more than 140 pounds. What percentage of men definitely possess all four of these characteristics? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 100 points on a coordinate plane. Let \( N \) be the number of triplets \((A, B, C)\) that satisfy the following conditions: the vertices are chosen from these 100 points, \( A \) and \( B \) have the same y-coordinate, and \( B \) and \( C \) have the same x-coordinate. Find the maximum value of \( N \). | {
"answer": "8100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One hundred number cards are laid out in a row in ascending order: \(00, 01, 02, 03, \ldots, 99\). Then, the cards are rearranged so that each subsequent card is obtained from the previous one by increasing or decreasing exactly one of the digits by 1 (for example, after 29 there can be 19, 39 or 28, but not 30 or 20). What is the maximum number of cards that could remain in their original positions? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \(A\) lies on the line \(y=\frac{12}{5} x-9\), and point \(B\) lies on the parabola \(y=x^{2}\). What is the minimum length of segment \(AB\)? | {
"answer": "189/65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The quadratic equations \(x^{2} + px + q\) and \(x^{2} + ax + b\) each have one root. Among the numbers \(p, q, a, b\) there are 16, 64, and 1024. What can the fourth number be?
If there are multiple possible answers, input the larger one into the system, and specify all of them in the written solution. | {
"answer": "262144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$, measured from the vertex of its acute angle. What fraction of the area of the rhombus is occupied by the area of a circle inscribed in it? | {
"answer": "\\frac{\\pi \\sqrt{15}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Private Petrov took a bucket of unpeeled potatoes and peeled them in 1 hour. During this process, 25% of the potatoes went to peels. How long did it take him to collect half a bucket of peeled potatoes? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the definite integral:
$$
\int_{-\pi}^{0} 2^{8} \sin ^{6} x \cos ^{2} x \, dx
$$ | {
"answer": "10\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number \( N = 3^{16} - 1 \) has a divisor of 193. It also has some divisors between 75 and 85 inclusive. What is the sum of these divisors? | {
"answer": "247",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(a, b, c, d\) belong to the interval \([-5.5, 5.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | {
"answer": "132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find a number such that when it is multiplied by its reverse, the product is 78445. | {
"answer": "145",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Distribute 61 books to a class of students. If at least one person must receive at least 3 books, what is the maximum number of students in the class? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indefinitely. If Nathaniel goes first, determine the probability that he ends up winning. | {
"answer": "5/11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
By how much did the dollar exchange rate change over the course of 2014 (from January 1, 2014, to December 31, 2014)? Provide the answer in rubles, rounded to the nearest whole number (answer - whole number). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At each vertex of a cube with an edge length of 1, there is the center of a sphere. All the spheres are identical, and each touches three neighboring spheres. Find the length of the part of the space diagonal of the cube that lies outside the spheres. | {
"answer": "\\sqrt{3} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For which values of \(a\) does the equation \(|x-3| = a x - 1\) have two solutions? Enter the midpoint of the interval of parameter \(a\) in the provided field. Round the answer to three significant digits according to rounding rules and enter it in the provided field. | {
"answer": "0.667",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagonals of a trapezoid are mutually perpendicular, and one of them is equal to 17. Find the area of the trapezoid if its height is 15. | {
"answer": "4335/16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle is divided into 1000 smaller triangles. What is the minimum number of distinct points that can be the vertices of these triangles? | {
"answer": "503",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five integers are written on a board. By summing them in pairs, the following set of 10 numbers was obtained: $-1, 5, 8, 9, 11, 12, 14, 18, 20, 24$. Determine which numbers are written on the board. Provide their product as the answer. | {
"answer": "-2002",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \( S \) contain 2012 elements, where the ratio of any two elements is not an integer. An element \( x \) in \( S \) is called a "good element" if there exist distinct elements \( y \) and \( z \) in \( S \) such that \( x^2 \) divides \( y \cdot z \). Find the maximum possible number of good elements in \( S \). | {
"answer": "2010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board, two sums are written:
\[1+22+333+4444+55555+666666+7777777+88888888+999999999\]
\[9+98+987+9876+98765+987654+9876543+98765432+987654321\]
Determine which one is greater (or if they are equal). | {
"answer": "1097393685",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a linear function \( f(x) \). It is known that the distance between the points of intersection of the graphs \( y = x^2 - 1 \) and \( y = f(x) + 1 \) is \( 3\sqrt{10} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) + 3 \) is \( 3\sqrt{14} \). Find the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) \). | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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