problem stringlengths 10 5.15k | answer dict |
|---|---|
Given triangle \( \triangle ABC \) with \( AB < AC \), the altitude \( AD \), angle bisector \( AE \), and median \( AF \) are drawn from \( A \), with \( D, E, F \) all lying on \(\overline{BC}\). If \( \angle BAD = 2 \angle DAE = 2 \angle EAF = \angle FAC \), what are all possible values of \( \angle ACB \)? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two motorcyclists departed simultaneously from points \( A \) and \( B \) towards each other and met 50 km from point \( B \). After arriving at points \( A \) and \( B \), the motorcyclists immediately turned back and met for the second time 25 km from point \( A \). How many kilometers are there from \( A \) to \( B \)? | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the arc lengths of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\frac{1}{2} \cos t-\frac{1}{4} \cos 2 t \\
y=\frac{1}{2} \sin t-\frac{1}{4} \sin 2 t
\end{array}\right. \\
& \frac{\pi}{2} \leq t \leq \frac{2 \pi}{3}
\end{aligned}
$$ | {
"answer": "\\sqrt{2} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Andrey found the product of all the numbers from 1 to 11 inclusive and wrote the result on the board. During a break, someone accidentally erased three digits, leaving the number $399 * 68 * *$. Help restore the missing digits without recalculating the product. | {
"answer": "39916800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( p(x) \) be a polynomial with integer coefficients such that \( p(m) - p(n) \) divides \( m^2 - n^2 \) for all integers \( m \) and \( n \). If \( p(0) = 1 \) and \( p(1) = 2 \), find the largest possible value of \( p(100) \). | {
"answer": "10001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute values of the numbers in the sequence are nonincreasing? | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a four-digit number \( A \). By rearranging the digits (none of which are 0), the largest possible number is 7668 greater than \( A \), and the smallest possible number is 594 less than \( A \). What is \( A \)? | {
"answer": "1963",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). A chord \( OB \) of the 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"
} |
The numbers \( a, b, c, d \) belong to the interval \([-7.5, 7.5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school program will randomly start between 8:30AM and 9:30AM and will randomly end between 7:00PM and 9:00PM. What is the probability that the program lasts for at least 11 hours and starts before 9:00AM? | {
"answer": "5/16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a positive integer \( N \) that has exactly nine positive divisors, with three of these divisors \( a, b, \) and \( c \) satisfying
\[ a + b + c = 2017 \]
and
\[ ac = b^2. \]
Find the value of \( N \). | {
"answer": "82369",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A courier set out on a moped from city $A$ to city $B$, which are 120 km apart. One hour later, a second courier on a motorcycle left from $A$. The second courier caught up to the first, delivered a message, and immediately returned to $A$ at the same speed, arriving back in $A$ at the same time the first courier reached $B$. What is the speed of the first courier, if the speed of the second courier is 50 km/h? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three circles are drawn around vertices \( A, B, \) and \( C \) of a regular hexagon \( ABCDEF \) with side length 2 units, such that the circles touch each other externally. What is the radius of the smallest circle? | {
"answer": "2 - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Zhang departs from point A to point B at 8:00 AM, traveling at a speed of 60 km/h. At 9:00 AM, Xiao Wang departs from point B to point A. After arriving at point B, Xiao Zhang immediately returns along the same route and arrives at point A at 12:00 PM, at the same time as Xiao Wang. How many kilometers from point A do they meet each other? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular tetrahedron is inscribed in a cylinder such that two opposite edges of the tetrahedron are the diameters of the cylinder's bases. Find the ratio of the volume of the cylinder to the volume of the tetrahedron. | {
"answer": "\\frac{3 \\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right quadrilateral pyramid \( V-ABCD \) with the height equal to half the length of \( AB \), \( M \) is the midpoint of the lateral edge \( VB \), and \( N \) is a point on the lateral edge \( VD \) such that \( DN=2VN \). Find the cosine of the angle between the skew lines \( AM \) and \( BN \). | {
"answer": "\\frac{\\sqrt{11}}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the numbers \(x, y, z \) form an arithmetic progression in that order with a common difference \(\alpha = \arcsin \frac{\sqrt{7}}{4}\), and the numbers \(\frac{1}{\sin x}, \frac{4}{\sin y}, \frac{1}{\sin z}\) also form an arithmetic progression in that order. Find \(\sin ^{2} y\). | {
"answer": "\\frac{7}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five identical balls roll on a smooth horizontal surface towards each other. The velocities of the first and second are $v_{1}=v_{2}=0.5$ m/s, and the velocities of the others are $v_{3}=v_{4}=v_{5}=0.1$ m/s. The initial distances between the balls are the same, $l=2$ m. All collisions are perfectly elastic. How much time will pass between the first and last collisions in this system? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Captain Billy the Pirate looted 1010 gold doubloons and set sail on his ship to a deserted island to bury his treasure. Each evening of their voyage, he paid each of his pirates one doubloon. On the eighth day of sailing, the pirates plundered a Spanish caravel, doubling Billy's treasure and halving the number of pirates. On the 48th day of sailing, the pirates reached the deserted island, and Billy buried all his treasure at the marked spot—exactly 1000 doubloons. How many pirates set off with Billy to the deserted island? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x+1)$ is an odd function, $f(x-1)$ is an even function, and $f(0)=2$, find $f(4)=$ | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Liang went to the science museum. The science museum is located 400 meters northeast of Xiao Liang's home. When Xiao Liang left home, he walked 600 meters in the northwest direction by mistake. At this time, he did not see the science museum. He asked a lady, and she didn't know where the science museum was, so she told him, "Walk 400 meters in the northeast direction, and there is a supermarket. You can ask the uncle in the supermarket!" After arriving at the supermarket, Xiao Liang needs to walk how many meters in which direction to reach the science museum? | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a moving point \( P \) on the \( x \)-axis, \( M \) and \( N \) lie on the circles \((x-1)^{2}+(y-2)^{2}=1\) and \((x-3)^{2}+(y-4)^{2}=3\) respectively. Find the minimum value of \( |PM| + |PN| \). | {
"answer": "2\\sqrt{10} - \\sqrt{3} - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( 1991 = 2^{\alpha_{1}} + 2^{\alpha_{2}} + \cdots + 2^{\alpha_{n}} \), where \( \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} \) are distinct non-negative integers, find the sum \( \alpha_{1} + \alpha_{2} + \cdots + \alpha_{n} \). | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The different ways to obtain the number of combinations of dice, as discussed in Example 4-15 of Section 4.6, can also be understood using the generating function form of Pólya’s enumeration theorem as follows:
$$
\begin{aligned}
P= & \frac{1}{24} \times\left[\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)^{6}\right. \\
& +6\left(x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}\right)^{2}\left(x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}+x_{5}^{4}+x_{6}^{4}\right) \\
& +3\left(x_{1}+x+x_{3}+x_{4}+x_{5}+x_{6}\right)^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}\right)^{2} \\
& \left.+6\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}\right)^{3}+8\left(x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}+x_{5}^{3}+x_{6}^{3}\right)^{2}\right],
\end{aligned}
$$
where $x_{i}$ represents the $i^{th}$ color for $i=1,2,\cdots,6$. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The analysis of bank accounts revealed that the balances on each of them are more than 10 rubles. Additionally, there was a group of clients, each having the same amount of money on their account. This amount is a number consisting solely of ones. If the total amount of money on the accounts of this group of clients is calculated, this sum will also be a number consisting solely of ones. Find the minimum number of clients in the group for which this is possible, given that there are more than one person in the group. | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a positive number \( r \), let the set \( T = \left\{(x, y) \mid x, y \in \mathbb{R}, \text{ and } x^{2} + (y-7)^{2} \leq r^{2} \right\} \). This set \( T \) is a subset of the set \( S = \{(x, y) \mid x, y \in \mathbb{R}, \text{ and for any } \theta \in \mathbb{R}, \ \cos 2\theta + x \cos \theta + y \geq 0\} \). Determine the maximum value of \( r \). | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a cubic planet, cubic mice live only on the faces of the cube and not on the edges or vertices. The number of mice on different faces is different and on any two neighboring faces, this number differs by at least 2. What is the minimum number of cubic mice that can live on this planet if there is at least one mouse on each face? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The school organized a picnic with several participants. The school prepared many empty plates. Each attendee counts the empty plates and takes one empty plate to get food (each person can only take one plate, no more). The first attendee counts all the empty plates, the second attendee counts one less plate than the first attendee, and so on. The last attendee finds that there are 4 empty plates left. It is known that the total number of plates prepared by the school plus the number of attendees equals 2015. How many people attended the picnic? | {
"answer": "1006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the extension of side $AD$ of rectangle $ABCD$ beyond point $D$, point $E$ is taken such that $DE = 0.5 AD$ and $\angle BEC = 30^\circ$.
Find the ratio of the sides of rectangle $ABCD$. | {
"answer": "\\sqrt{3}/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The medians \( A M \) and \( B E \) of triangle \( A B C \) intersect at point \( O \). Points \( O, M, E, C \) lie on the same circle. Find \( A B \) if \( B E = A M = 3 \). | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the arc length of the curve defined by the equation in the rectangular coordinate system.
\[ y = \ln 7 - \ln x, \sqrt{3} \leq x \leq \sqrt{8} \] | {
"answer": "1 + \\frac{1}{2} \\ln \\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
This month, I spent 26 days exercising for 20 minutes or more, 24 days exercising 40 minutes or more, and 4 days of exercising 2 hours exactly. I never exercise for less than 20 minutes or for more than 2 hours. What is the minimum number of hours I could have exercised this month? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a container filled to the brim with $100\%$ juice, fifth-grader Masha drank 1 liter of juice in a day and in the evening refilled the container with 1 liter of water. The next day, after thoroughly mixing the contents, she drank 1 liter of the mixture and in the evening refilled the container with 1 liter of water. On the third day, after mixing the contents again, she drank 1 liter of the mixture and in the evening refilled the container with 1 liter of water. The next morning, her parents discovered that the volume of water in the container was 1.5 liters more than the volume of the remaining juice. How many liters of juice did Masha ultimately drink? If the answer is ambiguous, provide the sum of all possible values of the desired quantity. | {
"answer": "1.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( x_{1} \) satisfies \( 2x + 2^{x} = 5 \) and \( x_{2} \) satisfies \( 2x + 2 \log_{2}(x - 1) = 5 \), then \( x_{1} + x_{2} = \) ? | {
"answer": "\\frac{7}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a round table with 9 chairs, and 4 people are seated randomly. What is the probability that no two people are sitting next to each other? | {
"answer": "1/14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the minimum of the expression
$$
\frac{2}{|a-b|}+\frac{2}{|b-c|}+\frac{2}{|c-a|}+\frac{5}{\sqrt{ab+bc+ca}}
$$
under the conditions that \(ab + bc + ca > 0\), \(a + b + c = 1\), and \(a, b, c\) are distinct. | {
"answer": "10\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mat is digging a hole. Pat asks him how deep the hole will be. Mat responds with a riddle: "I am $90 \mathrm{~cm}$ tall and I have currently dug half the hole. When I finish digging the entire hole, the top of my head will be as far below the ground as it is above the ground now." How deep will the hole be when finished? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Person A departs from location A to location B, while persons B and C depart from location B to location A. After A has traveled 50 kilometers, B and C start simultaneously from location B. A and B meet at location C, and A and C meet at location D. It is known that A's speed is three times that of C and 1.5 times that of B. The distance between locations C and D is 12 kilometers. Determine the distance between locations A and B in kilometers. | {
"answer": "130",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), \(E\) and \(F\) are the midpoints of \(AC\) and \(AB\) respectively, and \( AB = \frac{2}{3} AC \). If \( \frac{BE}{CF} < t \) always holds, then the minimum value of \( t \) is ______. | {
"answer": "\\frac{7}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All the edges of the regular tetrahedron \(P-ABC\) have length \(1\). Let \( L, M, N \) be the midpoints of the edges \( PA, PB, PC \) respectively. Find the area of the cross-section of the circumsphere of the tetrahedron when cut by the plane \( LMN \). | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of the pyramid \( SABC \) is a triangle \( ABC \) such that \( AB = AC = 10 \) cm and \( BC = 12 \) cm. The face \( SBC \) is perpendicular to the base and \( SB = SC \). Calculate the radius of the sphere inscribed in the pyramid if the height of the pyramid is 1.4 cm. | {
"answer": "12/19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When triangle $EFG$ is rotated by an angle $\arccos(_{1/3})$ around point $O$, which lies on side $EG$, vertex $F$ moves to vertex $E$, and vertex $G$ moves to point $H$, which lies on side $FG$. Find the ratio in which point $O$ divides side $EG$. | {
"answer": "3:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Piercarlo chooses \( n \) integers from 1 to 1000 inclusive. None of his integers is prime, and no two of them share a factor greater than 1. What is the greatest possible value of \( n \)? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Come up with at least one three-digit number PAU (all digits are different), such that \((P + A + U) \times P \times A \times U = 300\). (Providing one example is sufficient) | {
"answer": "235",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many divisors of \(88^{10}\) leave a remainder of 4 when divided by 6? | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 91. Find the volume of the original cone. | {
"answer": "94.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the triangular pyramid \( P-ABC \) with edge lengths \( PA=1 \), \( PB=2 \), and \( PC=3 \), and \( PA \perp PB \), \( PB \perp PC \), \( PC \perp PA \), determine the maximum distance from a point \( Q \) on the circumsphere of this pyramid to the face \( ABC \). | {
"answer": "\\frac{3}{7} + \\frac{\\sqrt{14}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the angle bisector \(BL\) is drawn. Find the area of the triangle, given that \(AL = 2\), \(BL = \sqrt{30}\), and \(CL = 5\). | {
"answer": "\\frac{7\\sqrt{39}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a moving line $l$ that tangentially touches the circle $O: x^{2}+y^{2}=1$ and intersects the ellipse $\frac{x^{2}}{9}+y^{2}=1$ at two distinct points $A$ and $B$, find the maximum distance from the origin to the perpendicular bisector of line segment $AB$. | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A train leaves station K for station L at 09:30, while another train leaves station L for station K at 10:00. The first train arrives at station L 40 minutes after the trains pass each other. The second train arrives at station K 1 hour and 40 minutes after the trains pass each other. Each train travels at a constant speed. At what time did the trains pass each other? | {
"answer": "10:50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right triangle \(ABC\) is inscribed in a circle. A chord \(CM\) is drawn from the vertex \(C\) of the right angle, intersecting the hypotenuse at point \(K\). Find the area of triangle \(ABM\) if \(AK : AB = 1 : 4\), \(BC = \sqrt{2}\), and \(AC = 2\). | {
"answer": "\\frac{9}{19} \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 10 boys, each with different weights and heights. For any two boys $\mathbf{A}$ and $\mathbf{B}$, if $\mathbf{A}$ is heavier than $\mathbf{B}$, or if $\mathbf{A}$ is taller than $\mathbf{B}$, we say that " $\mathrm{A}$ is not inferior to B". If a boy is not inferior to the other 9 boys, he is called a "strong boy". What is the maximum number of "strong boys" among the 10 boys? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many Pythagorean triangles are there in which one of the legs is equal to 2013? (A Pythagorean triangle is a right triangle with integer sides. Identical triangles count as one.). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The apex of a regular pyramid with a square base $ABCD$ of unit side length is $E$. Point $P$ lies on the base edge $AB$ and point $Q$ lies on the lateral edge $EC$ such that $PQ$ is perpendicular to both $AB$ and $EC$. Additionally, we know that $AP : PB = 6 : 1$. What are the lengths of the lateral edges? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an island, there are only knights, who always tell the truth, and liars, who always lie. There are at least two knights and at least two liars. One day, each islander pointed to each of the others in turn and said either "You are a knight!" or "You are a liar!". The phrase "You are a liar!" was said exactly 230 times. How many times was the phrase "You are a knight!" said? | {
"answer": "526",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Once upon a time, a team of Knights and a team of Liars met in the park and decided to ride a circular carousel that can hold 40 people (the "Chain" carousel, where everyone sits one behind the other). When they took their seats, each person saw two others: one in front and one behind. Each person then said, "At least one of the people sitting in front of me or behind me belongs to my team." One spot turned out to be free, and they called one more Liar. This Liar said, "With me, we can arrange ourselves on the carousel so that this rule is met again." How many people were on the team of Knights? (A Knight always tells the truth, a Liar always lies). | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Losyash is walking to Sovunya's house along the river at a speed of 4 km/h. Every half hour, he launches paper boats that travel towards Sovunya at a speed of 10 km/h. At what time intervals do the boats arrive at Sovunya's house? (Provide a complete solution, not just the answer.) | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cone is inscribed in a sphere such that the slant height of the cone is equal to the diameter of the base. Find the ratio of the total surface area of the cone to the surface area of the sphere. | {
"answer": "9/16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Divide all coins into two parts of 20 coins each and weigh them. Since the number of fake coins is odd, one of the piles will be heavier. Thus, there is at most one fake coin in that pile. Divide it into two piles of 10 coins and weigh them. If the balance is even, then all 20 coins weighed are genuine. If one of the pans is heavier, it contains 10 genuine coins, and the other 10 coins have exactly one fake coin. Divide these 10 coins into three piles consisting of 4, 4, and 2 coins. The third weighing will compare the two piles of 4 coins. If they balance, all 8 coins are genuine, and we have found 18 genuine coins. If one pile is heavier, it has 4 genuine coins, the other pile contains the fake coin, and the 2 set aside coins are genuine, finding a total of 16 genuine coins. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the side \( BC \) of an equilateral triangle \( ABC \), points \( K \) and \( L \) are marked such that \( BK = KL = LC \). On the side \( AC \), point \( M \) is marked such that \( AM = \frac{1}{3} AC \). Find the sum of the angles \( \angle AKM \) and \( \angle ALM \). | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( F \) be the left focus of the ellipse \( E: \frac{x^{2}}{3}+y^{2}=1 \). A line \( l \) with a positive slope passes through point \( F \) and intersects \( E \) at points \( A \) and \( B \). Through points \( A \) and \( B \), lines \( AM \) and \( BN \) are drawn such that \( AM \perp l \) and \( BN \perp l \), each intersecting the x-axis at points \( M \) and \( N \), respectively. Find the minimum value of \( |MN| \). | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In quadrilateral \(ABCD\), \(\angle ABD = 70^\circ\), \(\angle CAD = 20^\circ\), \(\angle BAC = 48^\circ\), \(\angle CBD = 40^\circ\). Find \(\angle ACD\). | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An engraver makes plates with letters. He engraves the same letters in the same amount of time, but different letters may take different times. On two plates, "ДОМ МОДЫ" (DOM MODY) and "ВХОД" (VKHOD), together he spent 50 minutes, and one plate "В ДЫМОХОД" (V DYMOHOD) took him 35 minutes. How much time will it take him to make the plate "ВЫХОД" (VYKHOD)? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers from 1 to 8 are placed at the vertices of a cube so that the sum of the numbers at any three vertices, which lie on the same face, is at least 10. What is the minimum possible sum of the numbers on one face? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many solutions in natural numbers \(x, y\) does the system of equations have?
$$
\left\{\begin{array}{l}
\text{GCD}(x, y) = 20! \\
\text{LCM}(x, y) = 30!
\end{array}\right.
$$
(where \(n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n\)) | {
"answer": "1024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The points \( M \) and \( N \) are chosen on the angle bisector \( AL \) of a triangle \( ABC \) such that \( \angle ABM = \angle ACN = 23^\circ \). \( X \) is a point inside the triangle such that \( BX = CX \) and \( \angle BXC = 2\angle BML \). Find \( \angle MXN \). | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the mass of the body $\Omega$ with density $\mu = 20z$, bounded by the surfaces
$$
z = \sqrt{1 - x^{2} - y^{2}}, \quad z = \sqrt{\frac{x^{2} + y^{2}}{4}}
$$ | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the expression \(\frac{13 x^{2}+24 x y+13 y^{2}-14 x-16 y+61}{\left(4-16 x^{2}-8 x y-y^{2}\right)^{7 / 2}}\). If necessary, round your answer to the nearest hundredth. | {
"answer": "0.44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a tetrahedron \(A B C D\), a plane is drawn through the midpoint \(M\) of edge \(A D\), vertex \(C\), and point \(N\) on edge \(B D\) such that \(B N: N D = 2: 1\). In what ratio does this plane divide the segment \(K P\), where \(K\) and \(P\) are the midpoints of edges \(A B\) and \(C D\) respectively? | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The arithmetic mean of several consecutive natural numbers is 5 times greater than the smallest of them. How many times is the arithmetic mean smaller than the largest of these numbers? | {
"answer": "1.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number 2015 is split into 12 terms, and then all the numbers that can be obtained by adding some of these terms (from one to nine) are listed. What is the minimum number of numbers that could have been listed? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( ABC \), the angle bisector \( BL \) is drawn. Find the area of the triangle, given that \( AL=2 \), \( BL=3\sqrt{10} \), and \( CL=3 \). | {
"answer": "\\frac{15\\sqrt{15}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the center of a circular field, there is a geologist's cabin. From it extend 6 straight roads, dividing the field into 6 equal sectors. Two geologists start a journey from their cabin at a speed of 4 km/h each on a randomly chosen road. Determine the probability that the distance between them will be at least 6 km after one hour. | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's call two positive integers almost neighbors if each of them is divisible (without remainder) by their difference. In a math lesson, Vova was asked to write down in his notebook all the numbers that are almost neighbors with \(2^{10}\). How many numbers will he have to write down? | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right triangle \(ABC\), the leg \(AB = 3\), and the leg \(AC = 6\). The centers of circles with radii 1, 2, and 3 are located at points \(A\), \(B\), and \(C\) respectively. Find the radius of the circle that is externally tangent to each of these three given circles. | {
"answer": "\\frac{8 \\sqrt{11} - 19}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the expression \(\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\left(9-25 x^{2}+10 x y-y^{2}\right)^{5 / 2}}\). If necessary, round the answer to hundredths. | {
"answer": "0.19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All dwarves are either liars or knights. Liars always lie, while knights always tell the truth. Each cell of a $4 \times 4$ board contains one dwarf. It is known that among them there are both liars and knights. Each dwarf stated: "Among my neighbors (by edge), there are an equal number of liars and knights." How many liars are there in total? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \(D\) is the midpoint of the hypotenuse \(AB\) in the right triangle \(ABC\) with legs measuring 3 and 4.
Find the distance between the centers of the inscribed circles of triangles \(ACD\) and \(BCD\). | {
"answer": "\\frac{5 \\sqrt{13}}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \( a, b, c, d \) belong to the interval \([-7, 7]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Remove all perfect squares and perfect cubes from the set
$$
A=\left\{n \mid n \leqslant 10000, n \in \mathbf{Z}_{+}\right\}
$$
and arrange the remaining elements in ascending order. What is the 2014th element of this sequence? | {
"answer": "2068",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Box $A$ contains 1 red ball and 5 white balls, and box $B$ contains 3 white balls. Three balls are randomly taken from box $A$ and placed into box $B$. After mixing thoroughly, three balls are then randomly taken from box $B$ and placed back into box $A$. What is the probability that the red ball moves from box $A$ to box $B$ and then back to box $A$? | {
"answer": "1/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers \( x \), where \( x \neq 9 \), such that
\[
\log _{\frac{x}{9}}\left(\frac{x^{2}}{3}\right)<6+\log _{3}\left(\frac{9}{x}\right) .
\] | {
"answer": "223",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(x\) and \(y\) are real numbers, and:
\[
z_1 = x + (y + 2)i,
\]
\[
z_2 = (x - 2) + yi,
\]
with the condition \(\left|z_1\right| + \left|z_2\right| = 4\),
Find the maximum value of \(|x + y|\). | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of two sides of a rectangle is 11, and the sum of three sides is 19.5. Find the product of all possible distinct values of the perimeter of such a rectangle. | {
"answer": "15400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given 8 planes in space, for each pair of planes, the line of their intersection is noted. For each pair of these noted lines, the point of their intersection is noted (if the lines intersect). What is the maximum number of noted points that could be obtained? | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right square pyramid $O-ABCD$, $\angle AOB=30^{\circ}$, the dihedral angle between plane $OAB$ and plane $OBC$ is $\theta$, and $\cos \theta = a \sqrt{b} - c$, where $a, b, c \in \mathbf{N}$, and $b$ is not divisible by the square of any prime number. Find $a+b+c=$ _______. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \( a, b, c, d \) belong to the interval \([-13.5, 13.5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | {
"answer": "756",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as its diameter, intersects side \( BC \) at point \( K \). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \( BOC \). | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum number of angles less than $150^{\circ}$ that a non-self-intersecting 2017-sided polygon can have if all its angles are strictly less than $180^{\circ}$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A function \( f(x) \) defined on the interval \([1,2017]\) satisfies \( f(1)=f(2017) \), and for any \( x, y \in [1,2017] \), \( |f(x) - f(y)| \leqslant 2|x - y| \). If the real number \( m \) satisfies \( |f(x) - f(y)| \leqslant m \) for any \( x, y \in [1,2017] \), find the minimum value of \( m \). | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the complex number \( z \) satisfies \( |z| = 2 \), then the maximum value of \( \frac{\left|z^{2}-z+1\right|}{|2z-1-\sqrt{3}i|} \) is _______. | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, dx}{(1+\cos x+\sin x)^{2}}
$$ | {
"answer": "\\ln 2 - \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( z \) be a complex number. If \( \frac{z-2}{z-\mathrm{i}} \) (where \( \mathrm{i} \) is the imaginary unit) is a real number, then the minimum value of \( |z+3| \) is \(\quad\). | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radius of the circumcircle of an acute triangle \( ABC \) is 1. It is known that the center of the circle passing through the vertices \( A \), \( C \), and the orthocenter of triangle \( ABC \) lies on this circumcircle. Find the length of side \( AC \). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sin ^{10} x+\cos ^{10} x=\frac{11}{36}$, find the value of $\sin ^{14} x+\cos ^{14} x$. | {
"answer": "\\frac{41}{216}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance between Luga and Volkhov is 194 km, between Volkhov and Lodeynoe Pole is 116 km, between Lodeynoe Pole and Pskov is 451 km, and between Pskov and Luga is 141 km. What is the distance between Pskov and Volkhov? | {
"answer": "335",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The length of the curve given by the parametric equations \(\left\{\begin{array}{l}x=2 \cos ^{2} \theta \\ y=3 \sin ^{2} \theta\end{array}\right.\) (where \(\theta\) is the parameter) is? | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively, and
$$
a=5, \quad b=4, \quad \cos(A-B)=\frac{31}{32}.
$$
Find the area of $\triangle ABC$. | {
"answer": "\\frac{15 \\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From Moscow to city \( N \), a passenger can travel by train, taking 20 hours. If the passenger waits for a flight (waiting will take more than 5 hours after the train departs), they will reach city \( N \) in 10 hours, including the waiting time. By how many times is the plane’s speed greater than the train’s speed, given that the plane will be above this train 8/9 hours after departure from the airport and will have traveled the same number of kilometers as the train by that time? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $19a, \frac{124b}{13}, \frac{c}{13}$ is an arithmetic progression. Find the common ratio of the geometric progression. | {
"answer": "247",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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