problem stringlengths 10 5.15k | answer dict |
|---|---|
Oleg drew an empty $50 \times 50$ table and wrote a non-zero number above each column and to the left of each row. It turned out that all 100 numbers written were different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the product of the numbers correspondin... | {
"answer": "1250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The weight of grain in a sample of 256 grains is 18 grains, and the total weight of rice is 1536 dan. Calculate the amount of mixed grain in the total batch of rice. | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rearrange the digits of 124669 to form a different even number. | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pentagon is inscribed around a circle, with the lengths of its sides being whole numbers, and the lengths of the first and third sides equal to 1. Into what segments does the point of tangency divide the second side? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three couples dine at the same restaurant every Saturday at the same table. The table is round and the couples agreed that:
(a) under no circumstances should husband and wife sit next to each other; and
(b) the seating arrangement of the six people at the table must be different each Saturday.
Disregarding rotations o... | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line parallel to the base of a triangle divides it into parts whose areas are in the ratio $2:1$, counting from the vertex. In what ratio does this line divide the sides of the triangle? | {
"answer": "(\\sqrt{6} + 2) : 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Independent trials are conducted, in each of which event \( A \) can occur with a probability of 0.001. What is the probability that in 2000 trials, event \( A \) will occur at least two and at most four times? | {
"answer": "0.541",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property: for every $x,y\in A$ with $x\neq y$ , it holds that
\[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\]
Determine the largest possible number of elements that the set $A$ can have. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of triangle \(ABC\) are divided by points \(M, N\), and \(P\) such that \(AM : MB = BN : NC = CP : PA = 1 : 4\). Find the ratio of the area of the triangle bounded by lines \(AN, BP\), and \(CM\) to the area of triangle \(ABC\). | {
"answer": "3/7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the total number of triples of integers $(x,y,n)$ satisfying the equation $\tfrac 1x+\tfrac 1y=\tfrac1{n^2}$ , where $n$ is either $2012$ or $2013$ . | {
"answer": "338",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 5-primable positive integers are there that are less than 500? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the coordinate plane, a rectangle has vertices with coordinates $(34,0), (41,0), (34,9), (41,9)$. Find the smallest value of the parameter $a$ such that the line $y = ax$ divides this rectangle into two parts where the area of one part is twice the area of the other. If the answer is not an integer, write it as a d... | {
"answer": "0.08",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the Saturday of a weekend softball tournament, Team A plays Team D, Team B plays Team E, and Team C gets a bye (no match). The winner of Team A vs. Team D plays against Team C in the afternoon, while the winner of Team B vs. Team E has no further matches on Saturday. On Sunday, the winners of Saturday's afternoon ma... | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle, chords $A B$ and $C D$, which are not diameters, are drawn perpendicular to each other. Chord $C D$ divides chord $A B$ in the ratio $1:5$, and it divides the longer arc of $A B$ in the ratio $1:2$. In what ratio does chord $A B$ divide chord $C D$? | {
"answer": "1 : 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A car and a truck start traveling towards each other simultaneously from points $A$ and $B$, respectively. It is known that the car's speed is twice the speed of the truck. The car arrives at point $C$ at 8:30, and the truck arrives at point $C$ at 15:00 on the same day. Both vehicles continue moving without stopping a... | {
"answer": "10:40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We wrote the reciprocals of natural numbers from 2 to 2011 on a board. In one step, we erase two numbers, \( x \) and \( y \), and replace them with the number
$$
\frac{xy}{xy + (1 - x)(1 - y)}
$$
By repeating this process 2009 times, only one number remains. What could this number be? | {
"answer": "\\frac{1}{2010! + 1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Determine the volume, in cubic light years, of the set of all possible locations for a base such that the sum of squares of the distances from the base to each of the stations does not exceed 15 square lig... | {
"answer": "\\frac{27 \\sqrt{6} \\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set \( S \) is given by \( S = \{1, 2, 3, 4, 5, 6\} \). A non-empty subset \( T \) of \( S \) has the property that it contains no pair of integers that share a common factor other than 1. How many distinct possibilities are there for \( T \)? | {
"answer": "27",
"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 \( \alpha = \arccos \left(-\frac{1}{3}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{3}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^2 y \). | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Express the number $15.7$ billion in scientific notation. | {
"answer": "1.57\\times 10^{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a trapezoid \( MNPQ \) with bases \( MQ \) and \( NP \). A line parallel to the bases intersects the lateral side \( MN \) at point \( A \), and the lateral side \( PQ \) at point \( B \). The ratio of the areas of the trapezoids \( ANPB \) and \( MABQ \) is \( \frac{2}{7} \). Find \( AB \) if \( NP = 4 \) and \(... | {
"answer": "\\frac{2\\sqrt{46}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$. | {
"answer": "306",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From May 1st to May 3rd, the provincial hospital plans to schedule 6 doctors to be on duty, with each person working 1 day and 2 people scheduled per day. Given that doctor A cannot work on the 2nd and doctor B cannot work on the 3rd, how many different scheduling arrangements are possible? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Once, Carlson and Winnie the Pooh competed in the speed of eating honey and jam. Carlson, an expert in jam, eats a jar of jam in 2 minutes, while Winnie the Pooh takes a full 7 minutes to finish a jar of jam. Meanwhile, Winnie the Pooh can finish a pot of honey in 3 minutes, but Carlson requires 5 minutes to do the sam... | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A \) and \( B \) be the endpoints of a semicircular arc of radius \( 3 \). The arc is divided into five congruent arcs by four equally spaced points \( C_1, C_2, C_3, C_4 \). All chords of the form \( \overline{AC_i} \) or \( \overline{BC_i} \) are drawn. Find the product of the lengths of these eight chords. | {
"answer": "32805",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence ${a_n}$ satisfying $a_1=1$, $a_2=2$, $a_3=3$, $a_{n+3}=a_n$ ($n\in\mathbb{N}^*$). If $a_n=A\sin(\omega n+\varphi)+c$ $(ω>0,|\varphi|<\frac{\pi}{2})$, find the real number $A$. | {
"answer": "-\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height of a cone and its slant height are 4 cm and 5 cm, respectively. Find the volume of a hemisphere inscribed in the cone, whose base lies on the base of the cone. | {
"answer": "\\frac{1152}{125} \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there are \( c \) prime numbers less than 100 such that their unit digits are not square numbers, find the values of \( c \). | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of a prism is an equilateral triangle $ABC$. The lateral edges of the prism $AA_1$, $BB_1$, and $CC_1$ are perpendicular to the base. A sphere, whose radius is equal to the edge of the base of the prism, touches the plane $A_1B_1C_1$ and the extensions of the segments $AB_1$, $BC_1$, and $CA_1$ beyond the poin... | {
"answer": "\\sqrt{44} - 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square board with three rows and three columns contains nine cells. In how many different ways can we write the three letters A, B, and C in three different cells, so that exactly one of these three letters is written in each row? | {
"answer": "162",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A package of seeds was passed around a table. The first person took 1 seed, the second person took 2 seeds, the third took 3 seeds, and so forth, with each subsequent person taking one more seed than the previous one. It is known that during the second round a total of 100 more seeds were taken than during the first ro... | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the prism \( A B C - A_{1} B_{1} C_{1} \), \( A B \) is perpendicular to the lateral face \( B B_{1} C_{1} C \). Point \( E \) lies on edge \( C C_{1} \) such that \( E \ne C \) and \( E \ne C_{1} \). Given that \( E A \perp E B_1 \), \( A B = \sqrt{2} \), \( B B_1 = 2 \), \( B C = 1 \), and \( \angle B C C_1 = \fr... | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
By permuting the digits of 20130518, how many different eight-digit positive odd numbers can be formed? | {
"answer": "3600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all odd natural numbers greater than 500 but less than 1000, each of which has the property that the sum of the last digits of all its divisors (including 1 and the number itself) is equal to 33. | {
"answer": "729",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural decimal number \(n\) whose square starts with the digits 19 and ends with the digits 89. | {
"answer": "1383",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For the four-digit number \(\overline{abcd}\) where \(1 \leqslant a \leqslant 9\) and \(0 \leqslant b, c, d \leqslant 9\), if \(a > b, b < c, c > d\), then \(\overline{abcd}\) is called a \(P\)-type number. If \(a < b, b > c, c < d\), then \(\overline{abcd}\) is called a \(Q\)-type number. Let \(N(P)\) and \(N(Q)\) rep... | {
"answer": "285",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three positive integers are each greater than $1$, have a product of $1728$, and are pairwise relatively prime. What is their sum? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya wrote a note on a piece of paper, folded it into quarters, and labeled the top with "MAME". Then he unfolded the note, wrote something else on it, folded the note along the creases randomly (not necessarily as before), and left it on the table with a random side facing up. Find the probability that the inscriptio... | {
"answer": "1/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that two congruent 30°-60°-90° triangles with hypotenuses of 12 are overlapped such that their hypotenuses exactly coincide, calculate the area of the overlapping region. | {
"answer": "9 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the cosine of the angle between the non-intersecting diagonals of two adjacent lateral faces of a regular triangular prism, where the lateral edge is equal to the side of the base. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) with \( a > 0 \) and \( b > 0 \), the eccentricity is \( \frac{\sqrt{17}}{3} \). Let \( F \) be the right focus, and points \( A \) and \( B \) lie on the right branch of the hyperbola. Let \( D \) be the point symmetric to \( A \) with respect ... | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a connected simple graph \( G \) with \( e \) edges and pieces placed on each vertex of \( G \) (where each piece can only be placed on a single vertex of \( G \)), you are allowed to perform the following operation: if the number of pieces on a vertex \( v \) is at least the number of vertices adjacent to \( v \... | {
"answer": "e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Majka examined multi-digit numbers in which odd and even digits alternate regularly. Those that start with an odd digit, she called "funny," and those that start with an even digit, she called "cheerful" (for example, the number 32387 is funny, the number 4529 is cheerful).
Majka created one three-digit funny number a... | {
"answer": "635040",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In tetrahedron \(ABCD\), \(AB = 1\), \(BC = 5\), \(CD = 7\), \(DA = 5\), \(AC = 5\), \(BD = 2\sqrt{6}\). Find the distance between skew lines \(AC\) and \(BD\). | {
"answer": "\\frac{3\\sqrt{11}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $(128)^{\frac{1}{3}}(729)^{\frac{1}{2}}$. | {
"answer": "108 \\cdot 2^{\\frac{1}{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(\mathbb{N}\) be the set of all positive integers. A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfies \( f(m + n) = f(f(m) + n) \) for all \( m, n \in \mathbb{N} \), and \( f(6) = 2 \). Also, no two of the values \( f(6), f(9), f(12) \), and \( f(15) \) coincide. How many three-digit positive integers... | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During the first eleven days, 700 people responded to a survey question. Each respondent chose exactly one of the three offered options. The ratio of the frequencies of each response was \(4: 7: 14\). On the twelfth day, more people participated in the survey, which changed the ratio of the response frequencies to \(6:... | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find a positive integer \( n \) with 1000 digits, none of which are 0, such that we can group the digits into 500 pairs so that the sum of the products of the numbers in each pair divides \( n \). | {
"answer": "111...111211221122112211221122112211221122112211221122112211221122112211221122112",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two teams of 20 people each participated in a relay race from Moscow to Petushki. Each team divided the distance into 20 segments, not necessarily of equal length, and assigned them to participants such that each member runs exactly one segment (each participant maintains a constant speed, but the speeds of different p... | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A group of adventurers displays their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Additionally, it is known that:
- If an adventurer has diamonds, they either have rubies or sapphires (but not both simultaneously);
- If an adven... | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the quadratic function \( f(x) = a x^{2} + b x + c \) where \( a, b, c \in \mathbf{R}_{+} \), if the function has real roots, determine the maximum value of \( \min \left\{\frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c}\right\} \). | {
"answer": "5/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of the pyramid is a parallelogram with adjacent sides of 9 cm and 10 cm, and one of the diagonals measuring 11 cm. The opposite lateral edges are equal, and each of the longer edges is 10.5 cm. Calculate the volume of the pyramid. | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 3,$ and $\|\mathbf{c}\| = 6,$ and
\[2\mathbf{a} + 3\mathbf{b} + 4\mathbf{c} = \mathbf{0}.\]Compute $2(\mathbf{a} \cdot \mathbf{b}) + 3(\mathbf{a} \cdot \mathbf{c}) + 4(\mathbf{b} \cdot \mathbf{c}).$ | {
"answer": "-673/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sasha wrote down numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 contain the digit one, 19 contain the digit two, and 30 contain neither one nor two. How many numbers did Misha erase? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two water particles fall freely in succession from a $300 \mathrm{~m}$ high cliff. The first one has already fallen $\frac{1}{1000} \mathrm{~mm}$ when the second one starts to fall.
How far apart will the two particles be at the moment when the first particle reaches the base of the cliff? (The result should be calcul... | {
"answer": "34.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\sum_{k=1}^{36}\sin 4k=\tan \frac{p}{q},$ where angles are measured in degrees, and $p$ and $q$ are relatively prime positive integers that satisfy $\frac{p}{q}<90,$ find $p+q.$ | {
"answer": "73",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fill in the appropriate numbers in the parentheses:
(1) 7÷9= $$\frac {( )}{( )}$$
(2) $$\frac {12}{7}$$=\_\_\_\_÷\_\_\_\_\_
(3) 3 $$\frac {5}{8}$$= $$\frac {( )}{( )}$$
(4) 6= $$\frac {()}{11}$$ | {
"answer": "\\frac {66}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the coefficient of \(x^{29}\) in the expansion of \(\left(1 + x^{5} + x^{7} + x^{9}\right)^{16}\). | {
"answer": "65520",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose two equally strong tennis players play against each other until one player wins three games in a row. The results of each game are independent, and each player will win with probability $\frac{1}{2}$ . What is the expected value of the number of games they will play? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular triangular prism \(ABC-A_1B_1C_1\), all 9 edges are equal in length. Point \(P\) is the midpoint of \(CC_1\). The dihedral angle \(B-A_1P-B_1 = \alpha\). Find \(\sin \alpha\). | {
"answer": "\\frac{\\sqrt{10}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a function \( f \), whose domain is positive integers, such that:
$$
f(n)=\begin{cases}
n-3 & \text{if } n \geq 1000 \\
f(f(n+7)) & \text{if } n < 1000
\end{cases}
$$
Find \( f(90) \). | {
"answer": "999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of eight-digit integers comprising the eight digits from 1 to 8 such that \( (i+1) \) does not immediately follow \( i \) for all \( i \) that runs from 1 to 7. | {
"answer": "16687",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which integers from 1 to 60,000 (inclusive) are more numerous and by how much: those containing only even digits in their representation, or those containing only odd digits in their representation? | {
"answer": "780",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a board game played with dice, our piece is four spaces away from the finish line. If we roll at least a four, we reach the finish line. If we roll a three, we are guaranteed to finish in the next roll.
What is the probability that we will reach the finish line in more than two rolls? | {
"answer": "1/12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pedestrian traffic light allows pedestrians to cross the street for one minute and prohibits crossing for two minutes. Find the average waiting time for a pedestrian who approaches the intersection. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A polynomial with integer coefficients is of the form
\[12x^3 - 4x^2 + a_1x + 18 = 0.\]
Determine the number of different possible rational roots of this polynomial. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We draw 6 circles of equal radius on the surface of a unit sphere such that the circles do not intersect. What is the maximum possible radius of these circles?
| {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins. If Nathaniel goes first, determine ... | {
"answer": "\\frac{5}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Draw a tangent line MN to the circle $(x-2)^2+(y-2)^2=1$ at point N, where N is the point of tangency. If $|MN|=|MO|$ (where O is the origin), then the minimum value of $|MN|$ is \_\_\_\_\_\_. | {
"answer": "\\frac{7\\sqrt{2}}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Today our cat gave birth to kittens! It is known that the two lightest kittens together weigh 80 g, the four heaviest kittens together weigh 200 g, and the total weight of all the kittens is 500 g. How many kittens did the cat give birth to? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a rectangular coordinate system with origin point $O$, vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ satisfy $|\overrightarrow {a}|=|\overrightarrow {b}|=1$ and $\overrightarrow {a}\cdot \overrightarrow {b}=\frac {1}{2}$. Let $\overrightarrow {c} = (m, 1-m)$ and $\overrightarrow {d} = (n, 1-n)$. For any... | {
"answer": "\\frac{\\sqrt{6} - \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer that is divisible by 111 and has the last four digits as 2004? | {
"answer": "662004",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person is waiting at the $A$ HÉV station. They get bored of waiting and start moving towards the next $B$ HÉV station. When they have traveled $1 / 3$ of the distance between $A$ and $B$, they see a train approaching $A$ station at a speed of $30 \mathrm{~km/h}$. If they run at full speed either towards $A$ or $B$ st... | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In every acyclic graph with 2022 vertices we can choose $k$ of the vertices such that every chosen vertex has at most 2 edges to chosen vertices. Find the maximum possible value of $k$ . | {
"answer": "1517",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the largest integer $x$ for which $4^{27} + 4^{1010} + 4^{x}$ is a perfect square. | {
"answer": "1992",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the median \(BK\), the angle bisector \(BE\), and the altitude \(AD\) are given.
Find the side \(AC\), if it is known that the lines \(BK\) and \(BE\) divide the segment \(AD\) into three equal parts, and \(AB=4\). | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Equilateral triangles $ABC$ and $A_{1}B_{1}C_{1}$ with a side length of 12 are inscribed in a circle $S$ such that point $A$ lies on the arc $B_{1}C_{1}$, and point $B$ lies on the arc $A_{1}B_{1}$. Find $AA_{1}^{2} + BB_{1}^{2} + CC_{1}^{2}$. | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. If $\cos A= \frac {3}{4}$, $\cos C= \frac {1}{8}$,
(I) find the ratio $a:b:c$;
(II) if $| \overrightarrow{AC}+ \overrightarrow{BC}|= \sqrt {46}$, find the area of $\triangle ABC$. | {
"answer": "\\frac {15 \\sqrt {7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer is *happy* if:
1. All its digits are different and not $0$ ,
2. One of its digits is equal to the sum of the other digits.
For example, 253 is a *happy* number. How many *happy* numbers are there? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a cube \( C \) with side length 1, an inscribed sphere \( O_{1} \) is placed. Another sphere \( O_{2} \) is also placed inside \( C \) such that it is tangent to the inscribed sphere \( O_{1} \) and also tangent to three faces of the cube. What is the surface area of sphere \( O_{2} \)? | {
"answer": "(7 - 4\\sqrt{3})\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
My friend Ana likes numbers that are divisible by 8. How many different pairs of last two digits are possible in numbers that Ana likes? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a corridor that is 100 meters long, there are 20 rugs with a total length of 1 kilometer. Each rug is as wide as the corridor. What is the maximum possible total length of the sections of the corridor that are not covered by the rugs? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Acute angles \( A \) and \( B \) of a triangle satisfy the equation \( \tan A - \frac{1}{\sin 2A} = \tan B \) and \( \cos^2 \frac{B}{2} = \frac{\sqrt{6}}{3} \). Determine the value of \( \sin 2A \). | {
"answer": "\\frac{2\\sqrt{6} - 3}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle on a plane divides the plane into 2 parts. A circle and a line can divide the plane into a maximum of 4 parts. A circle and 2 lines can divide the plane into a maximum of 8 parts. A circle and 5 lines can divide the plane into a maximum of how many parts? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( ABC \) be a triangle with \(\angle A = 60^\circ\). Line \(\ell\) intersects segments \( AB \) and \( AC \) and splits triangle \( ABC \) into an equilateral triangle and a quadrilateral. Let \( X \) and \( Y \) be on \(\ell\) such that lines \( BX \) and \( CY \) are perpendicular to \(\ell\). Given that \( AB =... | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all values of \( x \) for which the smaller of the numbers \( \frac{1}{x} \) and \( \sin x \) is greater than \( \frac{1}{2} \). In the answer, provide the total length of the resulting intervals on the number line, rounding to the nearest hundredth if necessary. | {
"answer": "2.09",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average lifespan of a motor is 4 years. Estimate from below the probability that this motor will not last more than 20 years. | {
"answer": "0.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For what is the largest natural number \( m \) such that \( m! \cdot 2022! \) is a factorial of a natural number? | {
"answer": "2022! - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a quadrilateral pyramid \(S A B C D\):
- The lateral faces \(S A B, S B C, S C D, S D A\) have areas of 9, 9, 27, and 27 respectively;
- The dihedral angles at the edges \(A B, B C, C D, D A\) are equal;
- The quadrilateral \(A B C D\) is inscribed in a circle, and its area is 36.
Find the volume of the pyramid \(... | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many times must a die be rolled so that the probability of the inequality
\[ \left| \frac{m}{n} - \frac{1}{6} \right| \leq 0.01 \]
is at least as great as the probability of the opposite inequality, where \( m \) is the number of times a specific face appears in \( n \) rolls of the die? | {
"answer": "632",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Given non-negative real numbers \( x, y, z \) satisfying \( x^{2} + y^{2} + z^{2} + x + 2y + 3z = \frac{13}{4} \), determine the maximum value of \( x + y + z \).
2. Given \( f(x) \) is an odd function defined on \( \mathbb{R} \) with a period of 3, and when \( x \in \left(0, \frac{3}{2} \right) \), \( f(x) = \ln ... | {
"answer": "\\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The teacher of the summer math camp brought with him several shirts, several pairs of pants, several pairs of shoes, and two jackets for the entire summer. On each lesson, he wore pants, a shirt, and shoes, and wore a jacket for some lessons. On any two lessons, at least one element of his attire or shoes was different... | {
"answer": "216",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers \(a, b, c\) and a positive number \(\lambda\) such that \(f(x)=x^3 + ax^2 + bx + c\) has three real roots \(x_1, x_2, x_3\), satisfying
1. \(x_2 - x_1 = \lambda\);
2. \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\). | {
"answer": "\\frac{3 \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(-1,0)$ and $\overrightarrow{b}=\left( \frac{\sqrt{3}}{2}, \frac{1}{2}\right)$, calculate the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{5\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four points \( B, A, E, L \) are on a straight line. \( G \) is a point off the line such that \(\angle B A G = 120^\circ\) and \(\angle G E L = 80^\circ\). If the reflex angle at \( G \) is \( x^\circ \), then \( x \) equals: | {
"answer": "340",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equation $\frac{n!}{2}=k!+l!$ in natural numbers $n$, $k$, and $l$, where $n! = 1 \cdot 2 \cdots n$. In your answer, indicate 0 if there are no solutions, specify the value of $n$ if there is only one solution, or provide the sum of all $n$ values if there are multiple solutions. Recall that a solution is a t... | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A graph shows the number of books read in June by the top readers in a school library. The data points given are:
- 4 readers read 3 books each
- 5 readers read 5 books each
- 2 readers read 7 books each
- 1 reader read 10 books
Determine the mean (average) number of books read by these readers. | {
"answer": "5.0833",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $AB = 10$ and the height from $A$ to $BC$ is 3. When the product $AC \cdot BC$ is minimized, what is $AC + BC$? | {
"answer": "4\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Péter sent his son with a message to his brother, Károly, who in turn sent his son to Péter with a message. The cousins met $720 \, \text{m}$ away from Péter's house, had a 2-minute conversation, and then continued on their way. Both boys spent 10 minutes at the respective relative's house. On their way back, they met ... | {
"answer": "1760",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board, there are three two-digit numbers. One starts with 5, another starts with 6, and the third one starts with 7. The teacher asked three students to each choose any two of these numbers and add them together. The first student got 147, and the results of the second and third students were different three-dig... | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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