problem stringlengths 10 5.15k | answer dict |
|---|---|
Let \( n \) be an integer between 1 and 1990, and let \( x^2 + x - 3n \) be factored into the product of two linear factors with integer coefficients. Determine the number of integers \( n \) that satisfy this condition. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $11^{-1} \equiv 3 \pmod{31}$, find $20^{-1} \pmod{31}$. Provide the result as a residue modulo 31 (a number between 0 and 30, inclusive). | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a $4 \times 4$ matrix filled with each number from $1$ to $16$. Each number must be placed such that the entries in each row and in each column are in increasing order. Find the number of such matrices that exist. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product of several distinct positive integers is divisible by ${2006}^{2}$ . Determine the minimum value the sum of such numbers can take. | {
"answer": "228",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given set \( A = \{0, 1, 2, 3, 4, 5, 9\} \), and \( a, b \in A \) where \( a \neq b \). The number of functions of the form \( y = -a x^2 + (4 - b)x \) whose vertex lies in the first quadrant is ___.
| {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The segments of two lines, enclosed between two parallel planes, are in the ratio of \( 5:9 \), and the acute angles between these lines and one of the planes are in the ratio of \( 2:1 \), respectively. Find the cosine of the smaller angle. | {
"answer": "0.9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the last two digits of \(\left[(\sqrt{29}+\sqrt{21})^{1984}\right]\). | {
"answer": "71",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can one arrange the natural numbers from 1 to 9 in a $3 \times 3$ square table so that the sum of the numbers in each row and each column is odd? (Numbers can repeat) | {
"answer": "6 * 4^6 * 5^3 + 9 * 4^4 * 5^5 + 5^9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of four-digit numbers, composed of the digits 1, 2, 3, 4, 5, 6, 7 (each digit can be used no more than once), that are divisible by 15. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a six-digit phone number, how many seven-digit numbers can be obtained by removing one digit? | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the isosceles triangle \(ABC\) (\(AB = BC\)), a point \(D\) is taken on the side \(BC\) such that \(BD : DC = 1 : 4\). In what ratio does the line \(AD\) divide the altitude \(BE\) of triangle \(ABC\), counting from vertex \(B\)? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$ s that may occur among the $100$ numbers. | {
"answer": "95",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{\pi / 4}^{\arcsin \sqrt{2 / 3}} \frac{8 \tan x \, dx}{3 \cos ^{2} x+8 \sin 2 x-7}
$$ | {
"answer": "\\frac{4}{21} \\ln \\left| \\frac{7\\sqrt{2} - 2}{5} \\right| - \\frac{4}{3} \\ln |2 - \\sqrt{2}|",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A $7 \times 7$ board is either empty or contains an invisible $2 \times 2$ ship placed "on the cells." Detectors can be placed on certain cells of the board and then activated simultaneously. An activated detector signals if its cell is occupied by the ship. What is the minimum number of detectors needed to guarantee determining whether the ship is on the board and, if so, identifying the cells it occupies? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$ . How many quadratics are there of the form $ax^2+2bx+c$ , with equal roots, and such that $a,b,c$ are distinct elements of $X$ ? | {
"answer": "9900",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of ordered integer pairs \((a, b)\) such that the equation \(x^{2} + a x + b = 167 y\) has integer solutions \((x, y)\), where \(1 \leq a, b \leq 2004\). | {
"answer": "2020032",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the integer root of the polynomial \[2x^3 + ax^2 + bx + c = 0,\] where $a, b$, and $c$ are rational numbers. The equation has $4-2\sqrt{3}$ as a root and another root whose sum with $4-2\sqrt{3}$ is 8. | {
"answer": "-8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parabola \(C\) with the center of ellipse \(E\) as its focus, the parabola \(C\) passes through the two foci of the ellipse \(E\), and intersects the ellipse \(E\) at exactly three points. Find the eccentricity of the ellipse \(E\). | {
"answer": "\\frac{2 \\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Through two vertices of an equilateral triangle \(ABC\) with an area of \(21 \sqrt{3} \ \text{cm}^2\), a circle is drawn such that two sides of the triangle are tangent to the circle. Find the radius of this circle. | {
"answer": "2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Winnie wrote all the integers from 1 to 2017 inclusive on a board. She then erased all the integers that are a multiple of 3. Next, she reinstated all those integers that are a multiple of 6. Finally, she erased all integers then on the board which are a multiple of 27. Of the 2017 integers that began in the list, how many are now missing? | {
"answer": "373",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In each cell of a $5 \times 5$ table, a natural number is written with invisible ink. It is known that the sum of all the numbers is 200, and the sum of the three numbers inside any $1 \times 3$ rectangle is 23. What is the value of the central number in the table? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all values of \( a \) for which the system
\[
\left\{
\begin{array}{l}
x^{2} + 4y^{2} = 1 \\
x + 2y = a
\end{array}
\right.
\]
has a unique solution. If necessary, round your answer to two decimal places. If there are no solutions, answer with 0. | {
"answer": "-1.41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{1, 2, \cdots, 2005\} \). If every subset of \( S \) with \( n \) pairwise coprime numbers always contains at least one prime number, find the minimum value of \( n \). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the lateral side \(CD\) of trapezoid \(ABCD (AD \parallel BC)\), point \(M\) is marked. A perpendicular \(AH\) is dropped from vertex \(A\) to segment \(BM\). It turns out that \(AD = HD\). Find the length of segment \(AD\), given that \(BC = 16\), \(CM = 8\), and \(MD = 9\). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain class, with any distribution of 200 candies, there will be at least two students who receive the same number of candies (possibly none). What is the minimum number of students in such a class? | {
"answer": "21",
"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 $1:3$ as measured from the vertex of its acute angle. What fraction of the rhombus's area is the area of the circle inscribed in it? | {
"answer": "\\frac{\\pi \\sqrt{15}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A quadrilateral \(ABCD\) is inscribed in a circle with a diameter of 1, where \(\angle D\) is a right angle and \(AB = BC\).
Find the area of quadrilateral \(ABCD\) if its perimeter is \(\frac{9\sqrt{2}}{5}\). | {
"answer": "\\frac{8}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that points $ D$ and $ E$ lie on $ \overline{BC}$ and $ \overline{AC}$ respectively, if $ \overline{AD}$ and $ \overline{BE}$ intersect at $ T$ so that $ AT/DT \equal{} 3$ and $ BT/ET \equal{} 4$, calculate the value of $ CD/BD$. | {
"answer": "\\frac{4}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest divisor of 360 that is smaller than 60 and also a factor of 90? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the arc lengths of the curves given by the equations in the rectangular coordinate system.
$$
y=2-e^{x}, \ln \sqrt{3} \leq x \leq \ln \sqrt{8}
$$ | {
"answer": "1 + \\frac{1}{2} \\ln \\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \) and \( b \) be real numbers such that \( a + b = 1 \). Then, the minimum value of
\[
f(a, b) = 3 \sqrt{1 + 2a^2} + 2 \sqrt{40 + 9b^2}
\]
is ______. | {
"answer": "5 \\sqrt{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rodney uses the following clues to try to guess a secret number:
- It is a two-digit integer.
- The tens digit is odd.
- The units digit is even.
- The number is greater than 75.
What is the probability that Rodney will guess the correct number if he guesses a number that meets all these criteria? Express your answer as a common fraction. | {
"answer": "\\frac{1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A football association stipulates that in the league, a team earns $a$ points for a win, $b$ points for a draw, and 0 points for a loss, where $a$ and $b$ are real numbers such that $a > b > 0$. If a team has 2015 possible total scores after $n$ games, find the minimum value of $n$. | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A five-character license plate is composed of English letters and digits. The first four positions must contain exactly two English letters (letters $I$ and $O$ cannot be used). The last position must be a digit. Xiao Li likes the number 18, so he hopes that his license plate contains adjacent digits 1 and 8, with 1 preceding 8. How many different choices does Xiao Li have for his license plate? (There are 26 English letters in total.) | {
"answer": "23040",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shopping mall's main staircase from the 1st floor to the 2nd floor consists of 15 steps. Each step has a height of 16 centimeters and a depth of 26 centimeters. The width of the staircase is 3 meters. If the cost of carpeting is 80 yuan per square meter, how much will it cost to buy the carpet needed for the staircase from the 1st floor to the 2nd floor? | {
"answer": "1512",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fill the five numbers $2015, 2016, 2017, 2018, 2019$ into the five boxes labeled " $D, O, G, C, W$ " such that $D+O+G=C+O+W$. How many different ways can this be done? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The electronic clock on the International Space Station displayed time in the format HH:MM. Due to an electromagnetic storm, the device malfunctioned, and each digit on the display either increased by 1 or decreased by 1. What was the actual time of the storm if the clock displayed 20:09 immediately after it? | {
"answer": "11:18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A boy wrote the first twenty natural numbers on a piece of paper. He didn't like how one of them was written, so he crossed it out. Among the remaining 19 numbers, there is one number that equals the arithmetic mean of these 19 numbers. Which number did he cross out? If the problem has more than one solution, write down the sum of these numbers. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In tetrahedron \(ABCD\), the dihedral angle between face \(ABC\) and face \(BCD\) is \(60^\circ\). The orthogonal projection of vertex \(A\) onto face \(BCD\) is \(H\), which is the orthocenter of \(\triangle BCD\). Point \(G\) is the centroid of \(\triangle ABC\). Given that \(AH = 4\) and \(AB = AC\), find the length of \(GH\). | {
"answer": "\\frac{4 \\sqrt{21}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a\in \mathbb{R}$, $i$ is the imaginary unit. If $z=1+ai$ and $z\cdot \overline{z}=4$, solve for the value of $a$. | {
"answer": "-\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid \(ABCD\), \(AD\) is parallel to \(BC\) and \(BC : AD = 5 : 7\). Point \(F\) lies on \(AD\) and point \(E\) lies on \(DC\) such that \(AF : FD = 4 : 3\) and \(CE : ED = 2 : 3\). If the area of quadrilateral \(ABEF\) is 123, determine the area of trapezoid \(ABCD\). | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value for \(a, b > 0\) of the expression
$$
\frac{|6a - 4b| + |3(a + b\sqrt{3}) + 2(a\sqrt{3} - b)|}{\sqrt{a^2 + b^2}}
$$ | {
"answer": "\\sqrt{39}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular octagon has a side length of 8 cm. What is the number of square centimeters in the area of the shaded region formed by diagonals connecting alternate vertices (forming a square in the center)? | {
"answer": "192 + 128\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Foma and Erema were traveling along a straight road to Moscow in a cart at a constant speed.
- At 12:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "82."
- At 13:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "71."
- At 15:00, Foma asked: "How many versts to Moscow?"
- Erema replied: "46."
It is known that Erema rounded the distance to the nearest whole number each time, and if there were two nearest whole numbers, he could choose either.
At 16:00, Foma asked again: "How many versts to Moscow?" This time Erema gave an exact answer, without rounding. What was Erema's answer? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pedestrian and a cyclist set off from point $A$ to point $B$ simultaneously. At point $B$, the cyclist turns back and meets the pedestrian 20 minutes after starting. Without stopping, the cyclist continues to point $A$, turns around, and catches up with the pedestrian 10 minutes after their first meeting. How long will it take for the pedestrian to travel from $A$ to $B$? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The three-digit even numbers \( A \, , B \, , C \, , D \, , E \) satisfy \( A < B < C < D < E \). Given that \( A + B + C + D + E = 4306 \), find the smallest value of \( A \). | {
"answer": "326",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to 1:00 pm of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer. | {
"answer": "132",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer \(1 \leqq k \leqq 100\), let \(a_{k}\) denote the sum \(\frac{1}{k}+\frac{1}{k+1}+\ldots+\frac{1}{100}\). Calculate the value of
\[ a_{1} + a_{1}^{2} + a_{2}^{2} + \ldots + a_{100}^{2}. \] | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hyperbola $C: x^{2}-y^{2}=2$ has its right focus at $F$. Let $P$ be any point on the left branch of the hyperbola, and point $A$ has coordinates $(-1,1)$. Find the minimum perimeter of $\triangle A P F$. | {
"answer": "3\\sqrt{2} + \\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three clients are at the hairdresser, each paying their bill at the cash register.
- The first client pays the same amount that is in the register and takes 10 reais as change.
- The second client performs the same operation as the first.
- The third client performs the same operation as the first two.
Find the initial amount of money in the cash register, knowing that at the end of the three operations, the cash register is empty. | {
"answer": "8.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an $11 \times 11$ table, integers from 0 to 10 are placed (naturally, numbers can repeat, and not necessarily all listed numbers occur). It is known that in every $3 \times 2$ or $2 \times 3$ rectangle, the sum of the numbers is 10. Find the smallest possible value of the sum of the numbers in the entire table. | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\overline{AB}$ be a diameter in a circle with radius $6$. Let $\overline{CD}$ be a chord in the circle that intersects $\overline{AB}$ at point $E$ such that $BE = 3$ and $\angle AEC = 60^{\circ}$. Find the value of $CE^{2} + DE^{2}$. | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=2$, $PD=6$, and $DE=2\sqrt{10}$. Determine the area of quadrilateral $AEDC$. | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diameters of two pulleys with parallel axes are 80 mm and 200 mm, respectively, and they are connected by a belt that is 1500 mm long. What is the distance between the axes of the pulleys if the belt is tight (with millimeter precision)? | {
"answer": "527",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the trapezoid \(MPQF\), the bases are \(MF = 24\) and \(PQ = 4\). The height of the trapezoid is 5. Point \(N\) divides the side into segments \(MN\) and \(NP\) such that \(MN = 3NP\).
Find the area of triangle \(NQF\). | {
"answer": "22.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \( 62, 63, 64, 65, 66, 67, 68, 69, \) and \( 70 \) are divided by, in some order, the numbers \( 1, 2, 3, 4, 5, 6, 7, 8, \) and \( 9 \), resulting in nine integers. The sum of these nine integers is \( S \). What are the possible values of \( S \)? | {
"answer": "187",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose a regular tetrahedron \( P-ABCD \) has all edges equal in length. Using \(ABCD\) as one face, construct a cube \(ABCD-EFGH\) on the other side of the regular tetrahedron. Determine the cosine of the angle between the skew lines \( PA \) and \( CF \). | {
"answer": "\\frac{2 + \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange all positive integers whose digits sum to 8 in ascending order to form a sequence $\{a_n\}$, called the $P$ sequence. Then identify the position of 2015 within this sequence. | {
"answer": "83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an equilateral triangle with side length \( n \), divided into unit triangles as illustrated, let \( f(n) \) be the number of paths from the top-row triangle to the triangle in the center of the bottom row. The path must move through adjacent triangles sharing a common edge, never revisiting any triangle, and never moving upwards (from a lower row to an upper row). An example path is illustrated for \( n = 5 \). Determine the value of \( f(2012) \). | {
"answer": "2011!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that (1+ex)<sup>2019</sup>=a<sub>0</sub>+a<sub>1</sub>x+a<sub>2</sub>x<sup>2</sup>+……+a<sub>2019</sub>x<sup>2019</sup>, find the value of:
- $$\frac {a_{1}}{e}$$+ $$\frac {a_{2}}{e^{2}}$$\- $$\frac {a_{3}}{e^{3}}$$+ $$\frac {a_{4}}{e^{4}}$$\-……- $$\frac {a_{2019}}{e^{2019}}$$ | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya has seven cards with the digits 2, 2, 3, 4, 5, 6, 8. He wants to use all the cards to form the largest natural number that is divisible by 12. What number should he get? | {
"answer": "8654232",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From an external point \(A\), a tangent \(AB\) and a secant \(ACD\) are drawn to a circle. Find the area of triangle \(CBD\), given that the ratio \(AC : AB = 2 : 3\) and the area of triangle \(ABC\) is 20. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagonals of a trapezoid are mutually perpendicular, and one of them is 13. Find the area of the trapezoid if its height is 12. | {
"answer": "1014/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many days have passed from March 19, 1990, to March 23, 1996, inclusive? | {
"answer": "2197",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), angle \(C\) is \(60^\circ\) and the radius of the circumcircle of this triangle is \(2\sqrt{3}\).
A point \(D\) is taken on the side \(AB\) such that \(AD = 2DB\) and \(CD = 2\sqrt{2}\). Find the area of triangle \(ABC\). | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of one hundred numbers is 1000. The largest of them was doubled, and another number was decreased by 10. It turned out that the sum did not change. Find the smallest of the original numbers. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the continuation of side \( BC \) of parallelogram \( ABCD \), a point \( F \) is taken beyond point \( C \). Segment \( AF \) intersects diagonal \( BD \) at point \( E \) and side \( CD \) at point \( G \), where \( GF=3 \) and \( AE \) is 1 more than \( EG \). What part of the area of parallelogram \( ABCD \) is the area of triangle \( ADE \)? | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a strip with a length of 100, and each cell of the strip contains a chip. You can swap any two adjacent chips for 1 ruble, or you can swap any two chips that have exactly three chips between them for free. What is the minimum number of rubles needed to rearrange the chips in reverse order? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equation \(x^{2}+5x+1=0\) has roots \(x_{1}\) and \(x_{2}\). Find the value of the expression
\[
\left(\frac{x_{1} \sqrt{6}}{1+x_{2}}\right)^{2}+\left(\frac{x_{2} \sqrt{6}}{1+x_{1}}\right)^{2}
\] | {
"answer": "220",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given fifty distinct natural numbers, twenty-five of which do not exceed 50, and the remaining are greater than 50 but do not exceed 100. Additionally, no two of these numbers differ by exactly 50. Find the sum of these numbers. | {
"answer": "2525",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all the roots of the equation
\[ 1 - \frac{x}{1} + \frac{x(x-1)}{2!} - \frac{x(x-1)(x-2)}{3!} + \frac{x(x-1)(x-2)(x-3)}{4!} - \frac{x(x-1)(x-2)(x-3)(x-4)}{5!} + \frac{x(x-1)(x-2)(x-3)(x-4)(x-5)}{6!} = 0 \]
(Where \( n! = 1 \cdot 2 \cdot 3 \cdots n \))
In the answer, specify the sum of the found roots. | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all positive real numbers \(c\) such that the graph of \(f: \mathbb{R} \rightarrow \mathbb{R}\) given by \(f(x) = x^3 - cx\) has the property that the circle of curvature at any local extremum is centered at a point on the \(x\)-axis. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Winnie the Pooh decided to give Piglet a birthday cake in the shape of a regular hexagon. On his way, he got hungry and cut off 6 pieces from the cake, each containing one vertex and one-third of a side of the hexagon (see the illustration). As a result, he gave Piglet a cake weighing 900 grams. How many grams of the cake did Winnie the Pooh eat on the way? | {
"answer": "112.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A large batch of tires contains $1.5\%$ defects. What should be the sample size for the probability of finding at least one defective tire in the sample to be more than $0.92 ?$ | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If Greg rolls five fair eight-sided dice, what is the probability that he rolls more 1's than 8's? | {
"answer": "\\frac{10246}{32768}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For which values of the parameter \(a\) does the equation \(x^{4} - 40 x^{2} + 144 = a(x^{2} + 4x - 12)\) have exactly three distinct solutions? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three children need to cover a distance of 84 kilometers using two bicycles. Walking, they cover 5 kilometers per hour, while bicycling they cover 20 kilometers per hour. How long will it take for all three to reach the destination if only one child can ride a bicycle at a time? | {
"answer": "8.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an island, there are two tribes: knights and liars. Knights always tell the truth, and liars always lie. One day, 80 people sat at a round table, and each of them declared: "Among the 11 people sitting immediately after me in a clockwise direction, there are at least 9 liars." How many knights are sitting at the round table? Indicate all possible options. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two vertices of a square with an area of \( 256 \, \text{cm}^2 \) lie on a circle, while the other two vertices lie on a tangent to this circle. Find the radius of the circle. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCD$ forms a rhombus. $E$ is the intersection of $AC$ and $BD$ . $F$ lie on $AD$ such that $EF$ is perpendicular to $FD$ . Given $EF=2$ and $FD=1$ . Find the area of the rhombus $ABCD$ | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the end of the school year, teachers of the third grade met with the parents of some of their students; exactly 31 people were present at this meeting. The Latin teacher was asked questions by 16 parents, the French teacher by 17 parents, the English teacher by 18 parents, and so on up to the Math teacher, who was asked questions by all the parents present at the meeting. How many parents were present at the meeting? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Robot Petya displays three three-digit numbers every minute, which sum up to 2019. Robot Vasya swaps the first and last digits of each of these numbers and then sums the resulting numbers. What is the maximum sum that Vasya can obtain? | {
"answer": "2118",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola \( y^{2} = 2 p x \) with focus \( F \) and directrix \( l \), a line passing through \( F \) intersects the parabola at points \( A \) and \( B \) such that \( |AB| = 3p \). Let \( A' \) and \( B' \) be the projections of \( A \) and \( B \) onto \( l \), respectively. If a point \( M \) is randomly chosen inside the quadrilateral \( AA'B'B \), what is the probability that \( M \) lies inside the triangle \( FA'B' \)? | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In square \(ABCD\) with a side length of 10, points \(P\) and \(Q\) lie on the segment joining the midpoints of sides \(AD\) and \(BC\). Connecting \(PA\), \(PC\), \(QA\), and \(QC\) divides the square into three regions of equal area. Find the length of segment \(PQ\). | {
"answer": "20/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive rational numbers less than \(\pi\) have denominator at most 7 when written in lowest terms? (Integers have denominator 1.) | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many points can be placed inside a circle of radius 2 such that one of the points coincides with the center of the circle and the distance between any two points is not less than 1? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mac is trying to fill 2012 barrels with apple cider. He starts with 0 energy. Every minute, he may rest, gaining 1 energy, or if he has \( n \) energy, he may expend \( k \) energy \((0 \leq k \leq n)\) to fill up to \( n(k+1) \) barrels with cider. What is the minimal number of minutes he needs to fill all the barrels? | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\( n \) is a positive integer that is not greater than 100 and not less than 10, and \( n \) is a multiple of the sum of its digits. How many such \( n \) are there? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the probability mass function of the random variable $X$ is $P(X=k)= \frac{k}{25}$ for $k=1, 2, 3, 4, 5$, find the value of $P(\frac{1}{2} < X < \frac{5}{2})$. | {
"answer": "\\frac{1}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school is hosting a Mathematics Culture Festival, and it was recorded that on that day, there were more than 980 (at least 980 and less than 990) students visiting. Each student visits the school for a period of time and then leaves, and once they leave, they do not return. Regardless of how these students schedule their visit, we can always find \( k \) students such that either all \( k \) students are present in the school at the same time, or at any time, no two of them are present in the school simultaneously. Find the maximum value of \( k \). | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( 169(157 - 77x)^2 + 100(201 - 100x)^2 = 26(77x - 157)(1000x - 2010) \), find the value of \( x \). | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 17 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). All of them said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( f(x) = x^{6} - 2 \sqrt{2006} x^{5} - x^{4} + x^{3} - 2 \sqrt{2007} x^{2} + 2 x - \sqrt{2006} \), then find \( f(\sqrt{2006} + \sqrt{2007}) \). | {
"answer": "\\sqrt{2007}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's call a natural number "remarkable" if all of its digits are different, it does not start with the digit 2, and by removing some of its digits, the number 2018 can be obtained. How many different seven-digit "remarkable" numbers exist? | {
"answer": "1800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The strengths of the two players are equal, meaning they have equal chances of winning each game. They agreed that the prize would go to the first player to win 6 games. They had to stop the game after the first player won 5 games and the second won 3. In what proportion should the prize be fairly divided? | {
"answer": "7:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $u_0 = \frac{1}{3}$, and for $k \ge 0$, let $u_{k+1} = \frac{3}{2}u_k - \frac{3}{2}u_k^2$. This sequence tends to a limit; call it $M$. Determine the least value of $k$ such that $|u_k - M| \le \frac{1}{2^{1000}}$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a plywood circle with a diameter of 30 cm, two smaller circles with diameters of 20 cm and 10 cm are cut out. What is the diameter of the largest circle that can be cut from the remaining piece of plywood? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the two regular tetrahedra \(A-OBC\) and \(D-OBC\) with coinciding bases, \(M\) and \(N\) are the centroids of \(\triangle ADC\) and \(\triangle BDC\) respectively. Let \(\overrightarrow{OA}=\boldsymbol{a}, \overrightarrow{OB}=\boldsymbol{b}, \overrightarrow{OC}=\boldsymbol{c}\). If point \(P\) satisfies \(\overrightarrow{OP}=x\boldsymbol{a}+y\boldsymbol{b}+z\boldsymbol{c}\) and \(\overrightarrow{MP}=2\overrightarrow{PN}\), then the real number \(9x+81y+729z\) equals \(\qquad\) | {
"answer": "439",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The journey from Petya's home to school takes him 20 minutes. One day, on his way to school, Petya remembered that he had forgotten a pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home for the pen and then goes to school at the same speed, he will be 7 minutes late for the start of the lesson. What fraction of the way to school had he covered when he remembered about the pen? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer n is called *primary divisor* if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$ , $2$ , $4$ , and $8$ each differ by $1$ from a prime number ( $2$ , $3$ , $5$ , and $7$ , respectively), while $9$ is not divisor primary, because the divisor $9$ does not differ by $1$ from a prime number (both $8$ and $10$ are composite). Determine the largest primary divisor number. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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