problem stringlengths 10 5.15k | answer dict |
|---|---|
Given vectors $\overrightarrow{a} = (x, -3)$, $\overrightarrow{b} = (-2, 1)$, $\overrightarrow{c} = (1, y)$ on a plane. If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b} - \overrightarrow{c}$, and $\overrightarrow{b}$ is parallel to $\overrightarrow{a} + \overrightarrow{c}$, find the projection of $\overrightarrow{a}$ onto the direction of $\overrightarrow{b}$. | {
"answer": "-\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair six-sided die is rolled twice. Let $a$ and $b$ be the numbers obtained from the first and second roll respectively. Determine the probability that three line segments of lengths $a$, $b$, and $5$ can form an isosceles triangle. | {
"answer": "\\frac{7}{18}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$ and $B$ be two opposite vertices of a cube with side length 1. What is the radius of the sphere centered inside the cube, tangent to the three faces that meet at $A$ and to the three edges that meet at $B$? | {
"answer": "2 - \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many different positive three-digit integers can be formed using only the digits in the set $\{4, 4, 5, 6, 6, 7, 7\}$, with no digit used more times than it appears in the set? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Add together all natural numbers less than 1980 for which the sum of their digits is even! | {
"answer": "979605",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle, there are two mutually perpendicular chords $AB$ and $CD$. Determine the distance between the midpoint of segment $AD$ and the line $BC$, given that $AC=6$, $BC=5$, and $BD=3$. If necessary, round the answer to two decimal places. | {
"answer": "4.24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$ . We have such strange rules:
(1) Define $P(n)$ : the number of prime numbers that are smaller than $n$ . Only when $P(i)\equiv j\pmod7$ , wolf $i$ may eat sheep $j$ (he can also choose not to eat the sheep).
(2) If wolf $i$ eat sheep $j$ , he will immediately turn into sheep $j$ .
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end? | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The center of a semicircle, inscribed in a right triangle such that its diameter lies on the hypotenuse, divides the hypotenuse into segments of 30 and 40. Find the length of the arc of the semicircle that is enclosed between the points where it touches the legs. | {
"answer": "12\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Is it possible to append two digits to the right of the number 277 so that the resulting number is divisible by any number from 2 to 12? | {
"answer": "27720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the share of the Japanese yen in the currency structure of the National Wealth Fund (NWF) as of 01.12.2022 using one of the following methods:
First method:
a) Find the total amount of NWF funds placed in Japanese yen as of 01.12.2022:
\[ J P Y_{22} = 1388.01 - 41.89 - 2.77 - 309.72 - 554.91 - 0.24 = 478.48 \text{ (billion rubles)} \]
b) Determine the share of the Japanese yen in the currency structure of NWF funds as of 01.12.2022:
\[ \alpha_{22}^{J P Y} = \frac{478.48}{1388.01} \approx 34.47\% \]
c) Calculate by how many percentage points and in what direction the share of the Japanese yen in the currency structure of NWF funds changed during the period considered in the table:
\[ \Delta \alpha^{J P Y} = \alpha_{22}^{J P Y} - \alpha_{21}^{J P Y} = 34.47 - 47.06 = -12.59 \approx -12.6 \text{ (percentage points)} \]
Second method:
a) Determine the share of the euro in the currency structure of NWF funds as of 01.12.2022:
\[ \alpha_{22}^{E U R} = \frac{41.89}{1388.01} \approx 3.02\% \]
b) Determine the share of the Japanese yen in the currency structure of NWF funds as of 01.12.2022:
\[ \alpha_{22}^{J P Y} = 100 - 3.02 - 0.2 - 22.31 - 39.98 - 0.02 = 34.47\% \]
c) Calculate by how many percentage points and in what direction the share of the Japanese yen in the currency structure of NWF funds changed during the period considered in the table:
\[ \Delta \alpha^{J P Y} = \alpha_{22}^{J P Y} - \alpha_{21}^{J P Y} = 34.47 - 47.06 = -12.59 \approx -12.6 \text{ (percentage points)} \] | {
"answer": "-12.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ f\left( x \right) \equal{} \frac {x^5}{5x^4 \minus{} 10x^3 \plus{} 10x^2 \minus{} 5x \plus{} 1}$ . $ \sum_{i \equal{} 1}^{2009} f\left( \frac {i}{2009} \right) \equal{} ?$ | {
"answer": "1005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a race, four cars each independently run timed laps around a circuit. Each car's lap time is discretely measured in seconds and can be any integer value between 150 and 155 seconds, uniformly distributed. The winner is the car with the shortest lap time. In case of a tie, the involved cars re-run the lap until a single winner is determined. What is the probability that Maria's first lap time was 152 seconds, given that she won the race?
A) $\frac{1}{8}$
B) $\frac{1}{6}$
C) $\frac{1}{4}$
D) $\frac{1}{3}$
E) $\frac{1}{2}$ | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $L$ be the intersection point of the diagonals $C E$ and $D F$ of a regular hexagon $A B C D E F$ with side length 5. Point $K$ is such that $\overrightarrow{L K}=\overrightarrow{F B}-3 \overrightarrow{A B}$. Determine whether point $K$ lies inside, on the boundary, or outside of $A B C D E F$, and also find the length of the segment $K F$. | {
"answer": "\\frac{5 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence of 0s and 1s of length 23 that begins with a 0, ends with a 0, contains no two consecutive 0s, and contains no four consecutive 1s, determine the number of such sequences. | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Little Pang, Little Dingding, Little Ya, and Little Qiao's four families, totaling 8 parents and 4 children, went to the amusement park together. The ticket prices are as follows: adult tickets are 100 yuan per person; children's tickets are 50 yuan per person; if there are 10 or more people, they can buy group tickets, which are 70 yuan per person. What is the minimum amount they need to spend to buy the tickets? | {
"answer": "800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bottle of cola costs 2 yuan, and two empty bottles can be exchanged for one more bottle of cola. With 30 yuan, what is the maximum number of bottles of cola that you can drink? | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the ellipse $C_1$ and the parabola $C_2$ have a common focus $F(1,0)$. The center of $C_1$ and the vertex of $C_2$ are both at the origin. A line $l$ passes through point $M(4,0)$ and intersects the parabola $C_2$ at points $A$ and $B$ (with point $A$ in the fourth quadrant).
1. If $|MB| = 4|AM|$, find the equation of line $l$.
2. If the symmetric point $P$ of the origin $O$ with respect to line $l$ lies on the parabola $C_2$, and line $l$ intersects the ellipse $C_1$ at common points, find the minimum length of the major axis of the ellipse $C_1$. | {
"answer": "\\sqrt{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The focus of a vertically oriented, rotational paraboloid-shaped tall vessel is at a distance of 0.05 meters above the vertex. If a small amount of water is poured into the vessel, what angular velocity $\omega$ is needed to rotate the vessel around its axis so that the water overflows from the top of the vessel? | {
"answer": "9.9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's call an integer "extraordinary" if it has exactly one even divisor other than 2. How many extraordinary numbers exist in the interval $[1 ; 75]$? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A semicircular sponge with a diameter of $20 \text{ cm}$ is used to wipe a corner of a room's floor such that the ends of the diameter continuously touch the two walls forming a right angle. What area does the sponge wipe? | {
"answer": "100\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area in the plane contained by the graph of
\[
|x + 2y| + |2x - y| \le 6.
\] | {
"answer": "5.76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The edges meeting at one vertex of a rectangular parallelepiped are in the ratio of $1: 2: 3$. What is the ratio of the lateral surface areas of the cylinders that can be circumscribed around the parallelepiped? | {
"answer": "\\sqrt{13} : 2\\sqrt{10} : 3\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles \(\omega_{1}\) and \(\omega_{2}\) intersect at points \(A\) and \(B\). Segment \(PQ\) is tangent to \(\omega_{1}\) at \(P\) and to \(\omega_{2}\) at \(Q\), and \(A\) is closer to \(PQ\) than \(B\). Point \(X\) is on \(\omega_{1}\) such that \(PX \parallel QB\), and point \(Y\) is on \(\omega_{2}\) such that \(QY \parallel PB\). Given that \(\angle APQ=30^{\circ}\) and \(\angle PQA=15^{\circ}\), find the ratio \(AX / AY\). | {
"answer": "2 - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \(ABCD\) is a right trapezoid with \(AD = 2\) as the upper base, \(BC = 6\) as the lower base. Point \(E\) is on \(DC\). The area of triangle \(ABE\) is 15.6 and the area of triangle \(AED\) is 4.8. Find the area of trapezoid \(ABCD\). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$ .) If $M$ is the maximum possible final value remaining on the board, and if m is the minimum, compute $M - m$ . | {
"answer": "2014",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=2\sqrt{3}\sin ^{2}x+2\sin x\cos x-\sqrt{3}$, where $x\in\left[ \frac{\pi}{3}, \frac{11\pi}{24}\right]$.
(1) Find the range of the function $f(x)$.
(2) Suppose that the lengths of two sides of an acute-angled triangle $ABC$ are the maximum and minimum values of the function $f(x)$, respectively, and the radius of the circumcircle of $\triangle ABC$ is $\frac{3\sqrt{2}}{4}$. Find the area of $\triangle ABC$. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fill the numbers $1,2,\cdots,36$ into a $6 \times 6$ grid, placing one number in each cell, such that the numbers in each row are in increasing order from left to right. What is the minimum possible sum of the six numbers in the third column? | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We have 10 springs, each originally $0.5 \mathrm{~m}$ long with a spring constant of $200 \mathrm{~N}/\mathrm{m}$. A mass of $2 \mathrm{~kg}$ is hung on each spring, and the springs, along with the masses, are hung in a series. What is the length of the resulting chain? (Neglect the mass of the springs.) | {
"answer": "10.39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A teacher drew a rectangle $ABCD$ on the board. A student named Petya divided this rectangle into two rectangles with a line parallel to side $AB$. It turned out that the areas of these parts are in the ratio 1:2, and their perimeters are in the ratio 3:5 (in the same order). Another student named Vasya divided this rectangle into two parts with a line parallel to side $BC$. The areas of the new parts are also in the ratio 1:2. What is the ratio of their perimeters? | {
"answer": "20/19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the family of curves
$$
2(2 \sin \theta - \cos \theta + 3) x^{2} - (8 \sin \theta + \cos \theta + 1) y = 0,
$$
where $\theta$ is a parameter. Find the maximum length of the chord that these curves cut on the line $y = 2 x$. | {
"answer": "8\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points \( A(3,1) \) and \( B\left(\frac{5}{3}, 2\right) \), and the four vertices of quadrilateral \( \square ABCD \) are on the graph of the function \( f(x)=\log _{2} \frac{a x+b}{x-1} \), find the area of \( \square ABCD \). | {
"answer": "\\frac{26}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \), \( f(1) = 1 \), and \( |f'(x)| \leq 2 \) for all real numbers \( x \). If \( a \) and \( b \) are real numbers such that the set of possible values of \( \int_{0}^{1} f(x) \, dx \) is the open interval \( (a, b) \), determine \( b - a \). | {
"answer": "3/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the sum of $1001101_2$ and $111000_2$, and then add the decimal equivalent of $1010_2$. Write your final answer in base $10$. | {
"answer": "143",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many pairs of positive integer solutions \((x, y)\) satisfy \(\frac{1}{x+1} + \frac{1}{y} + \frac{1}{(x+1) y} = \frac{1}{1991}\)? | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a \) be a nonzero real number. In the Cartesian coordinate system \( xOy \), the quadratic curve \( x^2 + ay^2 + a^2 = 0 \) has a focal distance of 4. Determine the value of \( a \). | {
"answer": "\\frac{1 - \\sqrt{17}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([4, 6]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{9 - \sqrt{19}}{2}\right)) \ldots) \). If necessary, round the answer to two decimal places. | {
"answer": "6.68",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum (identical) number of pencils that need to be placed in each of the 6 boxes so that in any 4 boxes there are pencils of any of the 26 pre-specified colors (assuming there are enough pencils available)? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest square in which 5 circles, each with a radius of 1, can be arranged so that no two circles share any interior points. | {
"answer": "2\\sqrt{2} + 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a defect rate of 3%, products are drawn from the batch without replacement until a non-defective product is found or a maximum of three draws have been made. Let $X$ represent the number of products drawn, and calculate $P(X=3)$. | {
"answer": "(0.03)^2 \\times 0.97 + (0.03)^3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The last 5 digits of $99 \times 10101 \times 111 \times 1001001$ are _____. | {
"answer": "88889",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During a long voyage of a passenger ship, it was observed that at each dock, a quarter of the passenger composition is renewed, that among the passengers leaving the ship, only one out of ten boarded at the previous dock, and finally, that the ship is always fully loaded.
Determine the proportion of passengers at any time, while the ship is en route, who did not board at either of the two previous docks? | {
"answer": "21/40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For real numbers \( x \), \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). Find the largest positive integer \( n \) such that the following equation holds:
\[
\lfloor \log_2 1 \rfloor + \lfloor \log_2 2 \rfloor + \lfloor \log_2 3 \rfloor + \cdots + \lfloor \log_2 n \rfloor = 1994
\]
(12th Annual American Invitational Mathematics Examination, 1994) | {
"answer": "312",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the number of birch trees in a certain mixed forest plot ranges from $13\%$ to $14\%$ of the total number of trees. Find the minimum possible total number of trees in this plot. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are $$\begin{cases} x=2 \sqrt {3}+at \\ y=4+ \sqrt {3}t\end{cases}$$ (where $t$ is the parameter), and in the polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinates of point $A$ are $(2, \frac {π}{6})$. Line $l$ passes through point $A$, and curve $C$ has the polar equation $ρ\sin^2θ=4\cosθ$.
1. Find the general equation of line $l$ and the rectangular equation of curve $C$.
2. Draw a perpendicular line to line $l$ through point $P( \sqrt {3},0)$, intersecting curve $C$ at points $D$ and $E$ (with $D$ above the $x$-axis). Find the value of $\frac {1}{|PD|}- \frac {1}{|PE|}$. | {
"answer": "\\frac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Oleg drew an empty 50×50 table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, 50 of which are rational and the remaining 50 are irrational. Then, in each cell of the table, he recorded the product of the numbers written near its row and its column (a "multiplication table"). What is the maximum number of products in this table that could turn out to be rational numbers? | {
"answer": "1250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The orthogonal projections of the triangle \(ABC\) onto two mutually perpendicular planes are equilateral triangles with sides of length 1. Find the perimeter of triangle \(ABC\), given that \(AB = \frac{\sqrt{5}}{2}\). | {
"answer": "\\sqrt{2} + \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \[\mathbf{N} = \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\] be a matrix with real entries such that $\mathbf{N}^3 = \mathbf{I}.$ If $xyz = -1$, find the possible values of $x^3 + y^3 + z^3.$ | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given three composite numbers \( A, B, \) and \( C \) which are pairwise coprime, and \( A \times B \times C = 11011 \times 28 \). What is the maximum value of \( A + B + C \)? | {
"answer": "1626",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A tangent and a secant drawn from the same point to a circle are mutually perpendicular. The length of the tangent is 12, and the internal segment of the secant is 10. Find the radius of the circle. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number is written on the board. If its last digit (in the units place) is erased, the remaining non-zero number is divisible by 20. If the first digit is erased, the remaining number is divisible by 21. What is the smallest number that could be on the board if its second digit is not 0? | {
"answer": "1609",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $\triangle ABC$, $A=120^{\circ}$, $D$ is a point on side $BC$, $AD\bot AC$, and $AD=2$. Calculate the possible area of $\triangle ABC$. | {
"answer": "\\frac{8\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cylinder with height $OO_1 = 12$ and a base radius $r = 5$. There are points $A$ and $B$ on the circumferences of the top and bottom bases respectively, with $AB = 13$. Find the distance between the axis $OO_1$ and line segment $AB$. | {
"answer": "\\frac{5}{2} \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are a batch of wooden strips with lengths of \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10,\) and 11 centimeters, with an adequate quantity of each length. If you select 3 strips appropriately to form a triangle with the requirement that the base is 11 centimeters long, how many different triangles can be formed? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\[\left(\frac{a}{b} + \frac{b}{a} + 2\right) \cdot \left(\frac{a+b}{2a} - \frac{b}{a+b}\right) \div \left(\left(a + 2b + \frac{b^2}{a}\right) \cdot \left(\frac{a}{a+b} + \frac{b}{a-b}\right)\right);\ a = 0.75,\ b = \frac{4}{3}.\] | {
"answer": "-\\frac{7}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest real number \(\lambda\) such that for the real-coefficient polynomial \(f(x) = x^3 + ax^2 + bx + c\) with all roots non-negative real numbers, we have \(f(x) \geqslant \lambda(x - a)^3\) for all \(x \geqslant 0\). Also, determine when the equality holds. | {
"answer": "-\\frac{1}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( p \) and \( q \) are positive integers, \(\max (p, q)\) is the maximum of \( p \) and \( q \) and \(\min (p, q)\) is the minimum of \( p \) and \( q \). For example, \(\max (30,40)=40\) and \(\min (30,40)=30\). Also, \(\max (30,30)=30\) and \(\min (30,30)=30\).
Determine the number of ordered pairs \((x, y)\) that satisfy the equation
$$
\max (60, \min (x, y))=\min (\max (60, x), y)
$$
where \(x\) and \(y\) are positive integers with \(x \leq 100\) and \(y \leq 100\). | {
"answer": "4100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of distinct numbers from 1 to 1000 that can be selected so that the difference between any two selected numbers is not equal to 4, 5, or 6? | {
"answer": "400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \( f(x) \) is defined on the set of real numbers, and satisfies the equations \( f(2+x) = f(2-x) \) and \( f(7+x) = f(7-x) \) for all real numbers \( x \). Let \( x = 0 \) be a root of \( f(x) = 0 \). Denote the number of roots of \( f(x) = 0 \) in the interval \(-1000 \leq x \leq 1000 \) by \( N \). Find the minimum value of \( N \). | {
"answer": "401",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One material particle entered the opening of a pipe, and after 6.8 minutes, a second particle entered the same opening. Upon entering the pipe, each particle immediately began linear motion along the pipe: the first particle moved uniformly at a speed of 5 meters per minute, while the second particle covered 3 meters in the first minute and in each subsequent minute covered 0.5 meters more than in the previous minute. How many minutes will it take for the second particle to catch up with the first? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A battery of three guns fired a volley, and two shells hit the target. Find the probability that the first gun hit the target, given that the probabilities of hitting the target by the first, second, and third guns are $p_{1}=0,4$, $p_{2}=0,3$, and $p_{3}=0,5$, respectively. | {
"answer": "20/29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the numbers \( x, y, z \) form an arithmetic progression in the given order with a common difference \( \alpha = \arccos \left(-\frac{3}{7}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{7}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^{2} y \). | {
"answer": "\\frac{10}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular triangular prism \(A B C A_{1} B_{1} C_{1}\) with base \(A B C\) and lateral edges \(A A_{1}, B B_{1}, C C_{1}\) is inscribed in a sphere. The segment \(C D\) is the diameter of this sphere, and point \(K\) is the midpoint of edge \(A A_{1}\). Find the volume of the prism if \(C K = 2 \sqrt{3}\) and \(D K = 2 \sqrt{2}\). | {
"answer": "9\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the cells of an $80 \times 80$ table, pairwise distinct natural numbers are placed. Each number is either prime or the product of two prime numbers (possibly the same). It is known that for any number $a$ in the table, there is a number $b$ in the same row or column such that $a$ and $b$ are not coprime. What is the largest possible number of prime numbers that can be in the table? | {
"answer": "4266",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle is tangent to two adjacent sides \(AB\) and \(AD\) of square \(ABCD\) and cuts off a segment of length 8 cm from vertices \(B\) and \(D\) at the points of tangency. On the other two sides, the circle cuts off segments of 4 cm and 2 cm respectively from the vertices at the points of intersection. Find the radius of the circle. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with its center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of circle \( \Omega \) intersects the second circle at point \( A \). Find the length of the segment \( AB \) if \( BD \cdot BC = 5 \). | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( a, b, c, d \) are four distinct positive integers such that \( a \times b \times c \times d = 2277 \), what is the maximum value of \( a + b + c + d \)? | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let a three-digit number \( n = \overline{abc} \), where \( a \), \( b \), and \( c \) can form an isosceles (including equilateral) triangle as the lengths of its sides. How many such three-digit numbers \( n \) are there? | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \(C\) lies on \(AE\) and \(AB=BC=CD\). If \(\angle CDE=t^{\circ}, \angle DEC=(2t)^{\circ}\), and \(\angle BCA=\angle BCD=x^{\circ}\), determine the measure of \(\angle ABC\). | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a rectangular coordinate system, a circle centered at the point $(1,0)$ with radius $r$ intersects the parabola $y^2 = x$ at four points $A$, $B$, $C$, and $D$. If the intersection point $F$ of diagonals $AC$ and $BD$ is exactly the focus of the parabola, determine $r$. | {
"answer": "\\frac{\\sqrt{15}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A necklace consists of 100 beads of red, blue, and green colors. It is known that among any five consecutive beads, there is at least one blue bead, and among any seven consecutive beads, there is at least one red bead. What is the maximum number of green beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last one is adjacent to the first one.) | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radius of the circle inscribed in triangle \(ABC\) is 4, with \(AC = BC\). On the line \(AB\), point \(D\) is chosen such that the distances from \(D\) to the lines \(AC\) and \(BC\) are 11 and 3 respectively. Find the cosine of the angle \(DBC\). | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the numbers $1, 2, 3, \cdots, 50$, if 10 consecutive numbers are selected, what is the probability that exactly 3 of them are prime numbers? | {
"answer": "22/41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest possible value of the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\), where \(a, b, c\) are distinct non-zero digits? | {
"answer": "1/203",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|= \sqrt {2}$, and $\overrightarrow{a}\perp (\overrightarrow{a}- \overrightarrow{b})$, then the angle between vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$ is ______. | {
"answer": "\\dfrac {\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The three different points \(A(x_1, y_1)\), \(B\left(4, \frac{9}{5}\right)\), and \(C(x_2, y_2)\) on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), along with the focus \(F(4,0)\) have distances that form an arithmetic sequence. If the perpendicular bisector of line segment \(AC\) intersects the x-axis at point \(T\), find the slope \(k\) of the line \(BT\). | {
"answer": "5/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many triangles exist such that the lengths of the sides are integers not greater than 10? | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Although I am certain that my clock is 5 minutes fast, it is actually 10 minutes slow. On the other hand, my friend's clock is really 5 minutes fast, even though he thinks it is correct. We scheduled a meeting for 10 o'clock and plan to arrive on time. Who will arrive first? After how much time will the other arrive? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S = \{ 1, 2, \cdots, 2005 \} \). Find the smallest number \( n \) such that in any subset of \( n \) pairwise coprime numbers from \( S \), there is at least one prime number. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Al and Bert must arrive at a town 22.5 km away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle, and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle, and rides at 10 km/h. For how many minutes was the bicycle not in motion? | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A barcode is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either color) has a width of 1 or 2. The total width of the barcode is 12. The barcodes are always read from left to right. How many distinct barcodes are possible? | {
"answer": "116",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the expanded hexagonal lattice shown below, where each point is one unit from its nearest neighbor. Determine the number of equilateral triangles whose vertices lie on this lattice.
```asy
size(100);
dot(origin);
dot(dir(30) + dir(90));
dot(dir(90));
dot(dir(90) + dir(150));
dot(dir(150));
dot(dir(150) + dir(210));
dot(dir(210));
dot(dir(210) + dir(270));
dot(dir(270));
dot(dir(270) + dir(330));
dot(dir(330));
dot(dir(330) + dir(30));
dot(dir(30));
dot(dir(30) + dir(90) + dir(150));
``` | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases: "My neighbor on the left is a liar" or "My neighbor on the right is a liar." What is the minimum number of knights that can be at the table? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( ABCD \) and \( WXYZ \) be two squares that share the same center such that \( WX \parallel AB \) and \( WX < AB \). Lines \( CX \) and \( AB \) intersect at \( P \), and lines \( CZ \) and \( AD \) intersect at \( Q \). If points \( P, W \), and \( Q \) are collinear, compute the ratio \( AB / WX \). | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation \(2 x^{3} + 24 x = 3 - 12 x^{2}\). | {
"answer": "\\sqrt[3]{\\frac{19}{2}} - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The point \( N \) is the center of the face \( ABCD \) of the cube \( ABCDEFGH \). Also, \( M \) is the midpoint of the edge \( AE \). If the area of \(\triangle MNH\) is \( 13 \sqrt{14} \), what is the edge length of the cube? | {
"answer": "2\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the first three terms of an arithmetic progression, as well as the sum of the first six terms, are natural numbers. Additionally, its first term \( d_{1} \) satisfies the inequality \( d_{1} \geqslant \frac{1}{2} \). What is the smallest possible value that \( d_{1} \) can take? | {
"answer": "5/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The density of a body is defined as the ratio of its mass to the volume it occupies. There is a homogeneous cube with a volume of $V=8 \, m^{3}$. As a result of heating, each of its edges increased by 4 mm. By what percentage did the density of this cube change? | {
"answer": "0.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A biased coin lands heads with a probability of $\frac{2}{3}$ and tails with $\frac{1}{3}$. A player can choose between Game C and Game D. In Game C, the player tosses the coin five times and wins if either the first three or the last three outcomes are all the same. In Game D, she tosses the coin five times and wins if at least one of the following conditions is met: the first two tosses are the same and the last two tosses are the same, or the middle three tosses are all the same.
A) Game C has a higher probability of $\frac{5}{81}$ over Game D.
B) Game D has a higher probability of $\frac{5}{81}$ over Game C.
C) Game C has a higher probability of $\frac{29}{81}$ over Game D.
D) Game D has a higher probability of $\frac{29}{81}$ over Game C. | {
"answer": "\\frac{29}{81}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let a constant $a$ make the equation $\sin x + \sqrt{3}\cos x = a$ have exactly three different solutions $x_{1}$, $x_{2}$, $x_{3}$ in the closed interval $\left[0,2\pi \right]$. The set of real numbers for $a$ is ____. | {
"answer": "\\{\\sqrt{3}\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alex wrote all natural divisors of a natural number \( n \) on the board in ascending order. Dima erased several of the first and several of the last numbers of the resulting sequence so that 151 numbers remained. What is the maximum number of these 151 divisors that could be fifth powers of natural numbers? | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of sides \( AB \) and \( BC \) of triangle \( ABC \) is 11, angle \( B \) is \( 60^\circ \), and the radius of the inscribed circle is \(\frac{2}{\sqrt{3}}\). It is also known that side \( AB \) is longer than side \( BC \). Find the height of the triangle dropped from vertex \( A \). | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From $A$ to $B$ it is 999 km. Along the road, there are kilometer markers with distances written to $A$ and to $B$:
$0|999,1|998, \ldots, 999|0$.
How many of these markers have only two different digits? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $k$ be a natural number. For which value of $k$ is $A_k = \frac{19^k + 66^k}{k!}$ maximized? | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given non-negative real numbers \( a, b, c, x, y, z \) that satisfy \( a + b + c = x + y + z = 1 \), find the minimum value of \( \left(a - x^{2}\right)\left(b - y^{2}\right)\left(c - z^{2}\right) \). | {
"answer": "-1/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find a three-digit number whose square is a six-digit number, such that each subsequent digit from left to right is greater than the previous one. | {
"answer": "367",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer \(n\) such that \(\frac{n}{n+75}\) is equal to a terminating decimal? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If three lines from the family of lines given by \( C: x \cos t + (y + 1) \sin t = 2 \) enclose an equilateral triangle \( D \), what is the area of the region \( D \)? | {
"answer": "12\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \(A\) lies on the line \(y = \frac{15}{8}x - 8\), and point \(B\) lies on the parabola \(y = x^2\). What is the minimum length of segment \(AB\)? | {
"answer": "1823/544",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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