task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | 7. If the set $A=\left\{2,4, x^{3}-2 x^{2}-x+7\right\}, B=\left\{-4, y+3, y^{2}-2 y+2, y^{3}+y^{2}+3 y+7\right\}$, and $A \cap B=\{2,5\}$, then $A \cup B=$ | {-4,2,4,5,25} | 91 | 12 |
math | 5. Find all natural numbers $n$ such that $n$ has as many digits as it has distinct prime divisors, and the sum of the distinct prime divisors is equal to the sum of the powers of the same divisors. | 48,72,4 | 49 | 7 |
math | 10.141. Determine the lateral sides of an isosceles trapezoid if its bases and area are respectively $8 \mathrm{~cm}, 14 \mathrm{~cm}$, and $44 \mathrm{~cm}^{2}$. | 5\mathrm{~} | 60 | 6 |
math | Let $\alpha$ be a positive real number. Determine for this $\alpha$ the largest real number $C$ such that for all positive real numbers $x, y,$ and $z$ with $xy + yz + zx = \alpha$, the inequality
$$
\left(1+\frac{\alpha}{x^{2}}\right)\left(1+\frac{\alpha}{y^{2}}\right)\left(1+\frac{\alpha}{z^{2}}\right) \geq C \cdot\... | 16 | 143 | 2 |
math | Example 3. Solve the inequality $x+\sqrt{x+1}>\frac{11}{4}$. | x>\frac{5}{4} | 24 | 8 |
math | Jimmy runs a successful pizza shop. In the middle of a busy day, he realizes that he is running low on ingredients. Each pizza must have 1 lb of dough, $\frac14$ lb of cheese, $\frac16$ lb of sauce, and $\frac13$ lb of toppings, which include pepperonis, mushrooms, olives, and sausages. Given that Jimmy currently ha... | 80 | 153 | 2 |
math | 3. If the positive real numbers $x \backslash y$ satisfy $y=2016 x$, and $x^{y}=y^{x}$, then $\log _{2016} x+\log _{2016} y=$ $\qquad$ | \frac{2017}{2015} | 61 | 13 |
math | 13. A homemade deck of cards consists of 52 cards (including 4 suits: Hearts, Diamonds, Spades, and Clubs. Each suit has one card for each point value from 1 to 13). After shuffling, the cards are placed face down. To ensure that at least 2 cards have the same point value and color, you must draw at least $\qquad$ card... | 27;37 | 124 | 5 |
math | 1751. Find the expression for the third-order central moment in terms of the initial moments. | v_{3}-3v_{1}v_{2}+2v_{1}^{3} | 21 | 22 |
math | ## Problem Statement
Find the derivative.
$y=\frac{1}{x} \sqrt{1-4 x^{2}}+\ln \frac{1+\sqrt{1+4 x^{2}}}{2 x}$ | -\frac{1}{x^{2}\sqrt{1-4x^{2}}}-\frac{1}{x\sqrt{1+4x^{2}}} | 47 | 35 |
math | Let $s(a)$ denote the sum of digits of a given positive integer a. The sequence $a_1, a_2,..., a_n, ...$ of positive integers is such that $a_{n+1} = a_n+s(a_n)$ for each positive integer $n$. Find the greatest possible n for which it is possible to have $a_n = 2008$. | 6 | 84 | 1 |
math | ## 296. Math Puzzle $1 / 90$
Sabine values physical exercise a lot, and so her daily commute to school by bike means more to her than just fulfilling a necessary "evil."
On her route, she has to cross a $100 \mathrm{~m}$ long bridge. One day, she meets a classmate on the bridge after $40 \mathrm{~m}$, who was cycling... | 14\mathrm{~}/\mathrm{} | 151 | 10 |
math | Find all $10$-digit whole numbers $N$, such that first $10$ digits of $N^2$ coincide with the digits of $N$ (in the same order). | 1000000000 | 41 | 10 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{1+4+7+\ldots+(3 n-2)}{\sqrt{5 n^{4}+n+1}}
$$ | \frac{3}{2\sqrt{5}} | 56 | 11 |
math | Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$? | 171 | 78 | 3 |
math | 1. (6 points) Calculate: $121 \times \frac{13}{25}+12 \times \frac{21}{25}$. | 73 | 38 | 2 |
math | 18. Given points $A(3 ; 2), B(-1 ; 5), C(0 ; 3)$. Find the coordinates of the vectors $\overrightarrow{A B}, \overrightarrow{B C}, \overrightarrow{A C}$. | \overrightarrow{AB}=(-4;3),\overrightarrow{BC}=(1;-2),\overrightarrow{AC}=(-3;1) | 56 | 33 |
math | Question 77: Given non-negative real numbers $a$, $b$, $c$ satisfying $a+b+c=1$, try to find the maximum value of $a^{2}(b-c)+b^{2}(c-a)+$ $c^{2}(a-b)$.
保留源文本的换行和格式，翻译结果如下：
Question 77: Given non-negative real numbers $a$, $b$, $c$ satisfying $a+b+c=1$, try to find the maximum value of $a^{2}(b-c)+b^{2}(c-a)+$ $c^{2}... | \frac{\sqrt{3}}{18} | 131 | 11 |
math | Find the last two digits of $\tbinom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g. if the last two digits are $05$, just write $5$.) | 20 | 55 | 2 |
math | Adam is playing Minesweeper on a $9\times9$ grid of squares, where exactly $\frac13$ (or 27) of the squares are mines (generated uniformly at random over all such boards). Every time he clicks on a square, it is either a mine, in which case he loses, or it shows a number saying how many of the (up to eight) adjacent sq... | \frac{88}{379} | 147 | 10 |
math | Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer. | 11 | 37 | 2 |
math | 20. There is a sum of money, used to buy a notebook for each student in Class 4 (1). If each notebook costs 3 yuan, then 6 more can be bought; if each notebook costs 5 yuan, then 30 yuan is short. If the money is used up, just enough to buy a notebook for each student, then a total of $\qquad$ notebooks are bought, amo... | 24,15 | 101 | 5 |
math | Let $a, b, c$ be positive real numbers such that: $$ab - c = 3$$ $$abc = 18$$ Calculate the numerical value of $\frac{ab}{c}$ | 2 | 42 | 1 |
math |
Problem 4. Given a positive integer $n$, denote by $\tau(n)$ the number of positive divisors of $n$, and by $\sigma(n)$ the sum of all positive divisors of $n$. Find all positive integers $n$ satisfying
$$
\sigma(n)=\tau(n) \cdot\lceil\sqrt{n}\rceil
$$
(Here, $\lceil x\rceil$ denotes the smallest integer not less th... | n\in{1,3,5,6} | 106 | 12 |
math | 13.436 On the river section from $A$ to $B$, the current is so weak that it can be neglected; on the section from $B$ to $C$, the current affects the boat's movement. The boat covers the distance downstream from $A$ to $C$ in 6 hours, and upstream from $C$ to $\boldsymbol{A}$ in 7 hours. If the current on the section f... | 7.7 | 173 | 3 |
math | ## Problem Statement
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\ln \sqrt{\frac{1-t}{1+t}} \\
y=\sqrt{1-t^{2}}
\end{array}\right.
$$ | \cdot\sqrt{1-^{2}} | 61 | 10 |
math | 4・82 Solve the equation $\log _{3} x+\log _{x} 3-2 \log _{3} x \log _{x} 3=\frac{1}{2}$. | x=\sqrt{3}, x=9 | 47 | 9 |
math | 4. A robot is located in one of the cells of an infinite grid paper, to which the following commands can be given:
- up (the robot moves to the adjacent cell above);
- down (the robot moves to the adjacent cell below);
- left (the robot moves to the adjacent cell to the left);
- right (the robot moves to the adjacent ... | 4900 | 142 | 4 |
math | 4. Let the base $A B C D$ of the right square prism $A B C D-A_{1} B_{1} C_{1} D_{1}$ be a unit square. If the dihedral angle $A_{1}-B D-C_{1}$ is $\frac{\pi}{3}$, then $A A_{1}=$ $\qquad$ | \frac{\sqrt{6}}{2} | 80 | 10 |
math | Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime.
[i]Lucian Petrescu[/i] | n = 2, 3, 6 | 54 | 11 |
math | Four. (50 points) The International Mathematical Olympiad Chief Examination Committee has $n$ countries participating, with each country being represented by a team leader and a deputy team leader. Before the meeting, the participants shake hands with each other, but the team leader does not shake hands with their own ... | m=0, n=50 | 141 | 8 |
math | Let $ P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that
\[ P(0) \equal{} 2007, P(1) \equal{} 2006, P(2) \equal{} 2005, \dots, P(2007) \equal{} 0.
\]Determine the value of $ P(2008)$. You may use factorials in your answer. | 2008! - 1 | 105 | 8 |
math | 5. [20] The curves $y=x^{2}(x-3)^{2}$ and $y=\left(x^{2}-1\right)(x-2)$ intersect at a number of points in the real plane. Determine the sum of the $x$-coordinates of these points of intersection. | 7 | 65 | 1 |
math | 5. (2002 Japan Mathematical Olympiad) 14 people participate in a Japanese chess round-robin tournament, where each person plays against the other 13 people. There are no ties in the matches. Find the maximum number of "triangular ties" (here, a "triangular tie" refers to a situation where 3 people each have one win and... | 112 | 84 | 3 |
math | If $x$ is a real number and $k$ is a nonnegative integer, recall that the binomial coefficient $\binom{x}{k}$ is defined by the formula
\[
\binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, .
\]
Compute the value of
\[
\frac{\binom{1/2}{2014} \cdot 4^{2014}}{\binom{4028}{2014}} \, .
\] | -\frac{1}{4027} | 132 | 12 |
math | Let $n$ be a fixed positive integer. How many sequences $1 \leq a_{1}<a_{2}<\cdots<a_{k} \leq n$ are there, in which the terms with odd indices are odd, and the terms with even indices are even integers? | A_{n}=\frac{(\frac{1+\sqrt{5}}{2})^{n+2}-(\frac{1-\sqrt{5}}{2})^{n+2}}{\sqrt{5}}-1 | 61 | 49 |
math | How many two-digit numbers are there where the digit in the tens place is greater than the digit in the units place?
# | 45 | 25 | 2 |
math | 13.249. It was planned to divide the bonus equally among the most distinguished employees of the enterprise. However, it turned out that there were three more employees deserving the bonus than was initially expected. In this case, each would receive 400 rubles less. The union and administration found a way to increase... | 18 | 104 | 2 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 4} \frac{\sqrt{1+2 x}-3}{\sqrt{x}-2}$ | \frac{4}{3} | 39 | 7 |
math | Petya is playing a shooting game. If he scores less than 1000 points, the computer will add $20 \%$ of his score. If he scores from 1000 to 2000 points, the computer will add $20 \%$ of the first thousand points and $30 \%$ of the remaining points. If Petya scores more than 2000 points, the computer will add $20 \%$ of... | 470 | 152 | 3 |
math | 1. Determine all distinct pairs of natural numbers $(x, y)$ such that by swapping the last two digits of the number $x^{2}$, the number $y^{2}$ is obtained. | (13,14)(14,13) | 41 | 13 |
math | 4. In how many different ways can $n$ books be arranged on a shelf so that $k$ of those books, which are numbered in advance, are in ascending or descending order. | 2\frac{n!}{k!} | 39 | 9 |
math | 3. In the set of complex numbers, consider the equations $x^{19}=-i$ and $y^{53}=i$
a) Calculate $(a \cdot b)^{2014}$ where $a$ is a solution to one of the equations and $b$ is a solution to the other equation.
b) Find the common solutions of the two equations.
Prof. Voiculeț Septimius, Videle | i | 92 | 1 |
math | 4. Let $X Y Z$ be a triangle with $\angle X=60^{\circ}$ and $\angle Y=45^{\circ}$. A circle with center $P$ passes through points $A$ and $B$ on side $X Y, C$ and $D$ on side $Y Z$, and $E$ and $F$ on side $Z X$. Suppose $A B=C D=E F$. Find $\angle X P Y$ in degrees. | \frac{255}{2} | 102 | 9 |
math | For a positive integer $K$, de fine a sequence, $\{a_n\}$, as following: $a_1 = K$ and
$a_{n+1} =a_n -1$ if $a_n$ is even
$a_{n+1} =\frac{a_n - 1}{2}$ if $a_n$ is odd , for all $n \ge 1$.
Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to $0$.
| 2^{1003} - 2 | 118 | 10 |
math | 4. (ROM) Solve the equation
$$
\cos ^{2} x+\cos ^{2} 2 x+\cos ^{2} 3 x=1 .
$$ | x\in{\pi/2+\pi,\pi/4+\pi/2,\pi/6+\pi/3\mid\inZ} | 40 | 31 |
math | 5】 Given $x_{i}$ are non-negative real numbers, $i=1,2,3,4 . x_{1}+x_{2}+x_{3}+x_{4}=1$. Let $S=1-$ $\sum_{i=1}^{4} x_{i}^{3}-6 \sum_{1 \leqslant i<j<k \leqslant 4} x_{i} x_{j} x_{k}$, find the range of $S$. | S \in \left[0, \frac{3}{4}\right] | 110 | 17 |
math | 8.9 (1) Two people, A and B, are playing a game: A writes two rows, each containing 10 numbers, such that they satisfy the following rule: if $b$ is below $a$, and $d$ is below $c$, then $a+d=b+c$. Knowing this rule, B wants to determine all the numbers written. B can ask A questions like, "What is the number in the th... | 11 | 226 | 2 |
math | What are the $(x, y, z)$ integers such that $x^{3}+2 y^{3}=4 z^{3}$? | (0,0,0) | 30 | 7 |
math | ## Task 2 - 160612
Knut is a very trained cyclist. On a trip, he covered on average $320 \mathrm{~m}$ per minute on his bicycle.
He set off on his bike at 7:00 AM and reached his destination at 11:00 AM. From 9:00 AM to 9:20 AM, he rested, and during the rest of the time, he cycled continuously.
How long (in $\mathr... | 70.4\mathrm{~} | 121 | 9 |
math | 13. (20 points) Two circles on the same side of the $x$-axis: a moving circle $C_{1}$ and the circle $4 a^{2} x^{2}+4 a^{2} y^{2}-4 a b x-2 a y+b^{2}=0$ are externally tangent $(a, b \in \mathbf{N}, a \neq 0)$, and the moving circle $C_{1}$ is tangent to the $x$-axis. Find
(1) the equation of the locus $\Gamma$ of the ... | a=686, b=784 | 196 | 11 |
math | 1. [5] Determine all pairs $(a, b)$ of real numbers such that $10, a, b, a b$ is an arithmetic progression. | (4,-2),(\frac{5}{2},-5) | 34 | 15 |
math | 3. Given real numbers $x, y$ satisfy $x y+6=x+9 y$, and $y \in(-\infty, 1)$. Then the maximum value of $(x+3)(y+1)$ is
$\qquad$ . | 27-12\sqrt{2} | 56 | 10 |
math | A bronze lion was carved, and on it the following inscription was written: "I can pour water from my eyes, throat, and right leg. If I open my right eye, I fill my basin in 2 days, if I open my left eye, in 3 days. The water flowing from my leg fills the basin in 4 days, and the water flowing from my throat fills it in... | 4\frac{44}{61} | 112 | 10 |
math | 3.063. $1-\sin \left(\frac{\alpha}{2}-3 \pi\right)-\cos ^{2} \frac{\alpha}{4}+\sin ^{2} \frac{\alpha}{4}$. | 2\sqrt{2}\sin\frac{\alpha}{4}\sin(\frac{\alpha+\pi}{4}) | 52 | 24 |
math | 247. System of two quadratic equations
$$
\left\{\begin{array}{r}
x^{2}-y^{2}=0 \\
(x-a)^{2}+y^{2}=1
\end{array}\right.
$$
generally has four solutions. For what values of \( a \) does the number of solutions of this system decrease to three or to two? | \1,\\sqrt{2} | 83 | 8 |
math | Example 9 Let $n(n \geqslant 2)$ be a positive integer, try to find all real-coefficient polynomials $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+$ $\cdots+a_{1} x+a_{0}$, such that $P(x)$ has exactly $n$ real roots no greater than -1, and $a_{0}^{2}+a_{1} a_{n}=a_{n}^{2}+a_{0} a_{n-1}$. | P(x)=a_{n}(x+1)^{n-1}(x+\beta)(a_{n}\neq0,\beta\geqslant1) | 124 | 36 |
math | Three, (50 points) Solve the equation $x^{2} \equiv 1\left(\bmod 2^{t}\right)$.
| x\equiv1,1+2^{-1},-1,-1+2^{-1}(\bmod2^{}) | 32 | 26 |
math | 3. There are 10 positive integers arranged from smallest to largest: $1, 4, 8$, $10, 16, 19, 21, 25, 30, 43$. How many groups of consecutive numbers have a sum that is divisible by 11? | 7 | 70 | 1 |
math | 30th Swedish 1990 Problem 3 Find all a, b such that sin x + sin a ≥ b cos x for all x. | (4n+1)\frac{\pi}{2},0 | 32 | 13 |
math | On a circular running track, two people are running in the same direction at constant speeds. At a certain moment, runner $A$ is 10 meters ahead of runner $B$, but after $A$ runs 22 meters, $B$ catches up.
How many points on the track can $B$ later overtake $A$? | 5 | 72 | 1 |
math | Consider the set
$\mathbb A=\{f\in C^1([-1,1]):f(-1)=-1,f(1)=1\}$.
Prove that there is no function in this function space that gives us the minimum of $S=\int_{-1}^1x^2f'(x)^2dx$. What is the infimum of $S$ for the functions of this space? | \inf_{f \in \mathbb{A}} S(f) = 0 | 88 | 19 |
math | 7. The line $x-2 y-1=0$ intersects the parabola $y^{2}=4 x$ at points $A$ and $B$, and $C$ is a point on the parabola such that $\angle A C B$ $=90^{\circ}$. Then the coordinates of point $C$ are $\qquad$ . | (1,-2) \text{ or } (9,-6) | 79 | 15 |
math | 10. The function $y=f(x)$ defined on $\mathbf{R}$ has the following properties:
（1）For any $x \in \mathbf{R}$, $f\left(x^{3}\right)=f^{3}(x)$;
（2）For any $x_{1} 、 x_{2} \in \mathbf{R}, x_{1} \neq x_{2}$, $f\left(x_{1}\right) \neq f\left(x_{2}\right)$. Then the value of $f(0)+f(1)+f(-1)$ is $\qquad$. | 0 | 137 | 1 |
math | 3. Find the value of the expression $4 \sin 40^{\circ}-\operatorname{tg} 40^{\circ}$. | \sqrt{3} | 33 | 5 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{\sqrt[3]{1+\ln ^{2} x}-1}{1+\cos \pi x}$ | \frac{2}{3\pi^{2}} | 46 | 11 |
math | # Task 4.
In modern conditions, digitalization - the conversion of all information into digital code - is considered relevant. Each letter of the alphabet can be assigned a non-negative integer, called the code of the letter. Then, the weight of a word can be defined as the sum of the codes of all the letters in that ... | 100 | 169 | 3 |
math | 43. For each of the numbers from 1 to 1000000000, the sum of its digits is calculated, and for each of the resulting billion numbers, the sum of its digits is calculated again, and so on, until a billion single-digit numbers are obtained. Which number will there be more of: 1 or 2? | 1 | 77 | 1 |
math | 6. Two vectors $\overrightarrow{O A}, \overrightarrow{O B}$ on a plane satisfy $|\overrightarrow{O A}|=a,|\overrightarrow{O B}|=b$, and $a^{2}+b^{2}=4, \overrightarrow{O A} \cdot \overrightarrow{O B}=0$. If the vector $\overrightarrow{O C}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}(\lambda, \mu \in \mathbf{R... | 2 | 179 | 1 |
math | 1. Among the natural numbers from $1 \sim 10000$, the integers that are neither perfect squares nor perfect cubes are $\qquad$ in number. | 9883 | 36 | 4 |
math | XXXIII OM - I - Problem 9
In a chessboard created by dividing a square of side length $ n $ into unit squares using lines parallel to the sides of the square, we consider all squares whose sides are contained in the lines forming the chessboard. Let $ 1 \leq k \leq n $ and $ P(k,n) $ denote the number of these squares... | \frac{\sqrt[3]{4}}{2} | 149 | 12 |
math | 16. (15 points) Two cars, A and B, start from points $A$ and $B$ respectively, heading towards each other. The two cars meet after 5 hours, at which point car A has passed the midpoint by 25 kilometers. After meeting, the two cars continue to travel, and 3 hours later, car A reaches point $B$. How many kilometers does ... | 15 | 89 | 2 |
math | 7.085. $10^{1+x^{2}}-10^{1-x^{2}}=99$. | -1;1 | 28 | 4 |
math | Mr. Celer had two identical tanks in the shape of a quadrilateral prism with a square base in his garden. Together, he had 300 liters of water in both. In the first tank, the water formed a perfect cube and filled $62.5\%$ of the tank, while in the second tank, there was 50 liters more water. What were the dimensions o... | 5\mathrm{}\times5\mathrm{}\times8\mathrm{} | 97 | 14 |
math | 2. 22 Let $S=\{1,2, \cdots, 1990\}$. If the sum of the elements of a 31-element subset of $S$ is divisible by 5, it is called a good subset of $S$. Find the number of good subsets of $S$. | \frac{1}{5}C_{1990}^{31} | 69 | 18 |
math |
Problem 8.2. Given a $\triangle A B C$. Let $M$ be the midpoint of $A B$, $\angle C A B=15^{\circ}$ and $\angle A B C=30^{\circ}$.
a) Find $\angle A C M$.
b) Prove that $C M=\frac{A B \cdot B C}{2 A C}$.
Chavdar Lozanov
| MC=\frac{AB\cdotBC}{2AC} | 93 | 12 |
math | 1. Triangle $G R T$ has $G R=5, R T=12$, and $G T=13$. The perpendicular bisector of $G T$ intersects the extension of $G R$ at $O$. Find $T O$. | \frac{169}{10} | 55 | 10 |
math | 3. Let $\lambda>0$, find the largest constant $c=c(\lambda)$, such that for all non-negative real numbers $x, y$, we have $x^{2}+y^{2}+\lambda x y \geqslant c(x+y)^{2}$. | (\lambda)={\begin{pmatrix}1,\lambda\geqslant2,\\\frac{2+\lambda}{4},0<\lambda<20\end{pmatrix}.} | 61 | 43 |
math | 4. Given $P(1,4,5)$ is a fixed point in the rectangular coordinate system $O-x y z$, a plane is drawn through $P$ intersecting the positive half-axes of the three coordinate axes at points $A$, $B$, and $C$ respectively. Then the minimum value of the volume $V$ of all such tetrahedrons $O-A B C$ is $\qquad$ | 90 | 89 | 2 |
math | ## Task Condition
Find the derivative of the specified order.
$$
y=\frac{\log _{2} x}{x^{3}}, y^{\prime \prime \prime}=?
$$ | \frac{47-60\lnx}{\ln2\cdotx^{6}} | 40 | 21 |
math | Problem 7.3. Lёsha gathered friends to play hide and seek. He surveyed Andrey, Borya, Vasya, Gena, and Denis and found out the following.
- If Andrey goes to play hide and seek, then Borya will go too, but Vasya will not.
- If Borya goes to play hide and seek, then Gena or Denis will also go.
- If Vasya does not go to... | Borya,Vasya,Denis | 166 | 9 |
math | From vertex $A$ of an equilateral triangle $ABC$, a ray $Ax$ intersects $BC$ at point $D$. Let $E$ be a point on $Ax$ such that $BA =BE$. Calculate $\angle AEC$. | 30^\circ | 51 | 4 |
math | Find all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ for which $f(0)=0$ and
\[f(x^2-y^2)=f(x)f(y) \]
for all $x,y\in\mathbb{N}_0$ with $x>y$. | f(x) = 0 | 69 | 7 |
math | Example 2 If positive numbers $a, b, c$ satisfy
$$
\left(\frac{b^{2}+c^{2}-a^{2}}{2 b c}\right)^{2}+\left(\frac{c^{2}+a^{2}-b^{2}}{2 c a}\right)^{2}+\left(\frac{a^{2}+b^{2}-c^{2}}{2 a b}\right)^{2}=3 \text {, }
$$
find the value of the algebraic expression
$$
\frac{b^{2}+c^{2}-a^{2}}{2 b c}+\frac{c^{2}+a^{2}-b^{2}}{2 ... | 1 | 184 | 1 |
math | ## Task A-1.5.
How many four-digit numbers are there that are composed of distinct digits from the set $\{0,1,2,3,4,5\}$ and are divisible by 5? | 108 | 46 | 3 |
math | $14 \cdot 49 k$ is a natural number, and $\frac{1001 \cdot 1002 \cdot \cdots \cdot 1985 \cdot 1986}{11^{k}}$ is an integer, what is the maximum value of $k$?
(China Beijing High School Grade 1 Mathematics Competition, 1986) | 99 | 87 | 2 |
math | 385*. Solve the system of equations:
$$
\left\{\begin{array}{l}
x=\frac{2 y^{2}}{1+z^{2}} \\
y=\frac{2 z^{2}}{1+x^{2}} \\
z=\frac{2 x^{2}}{1+y^{2}}
\end{array}\right.
$$ | (0;0;0),(1;1;1) | 77 | 13 |
math | Given a positive integer $n \ge 2$, determine the largest positive integer $N$ for which there exist $N+1$ real numbers $a_0, a_1, \dots, a_N$ such that
$(1) \ $ $a_0+a_1 = -\frac{1}{n},$ and
$(2) \ $ $(a_k+a_{k-1})(a_k+a_{k+1})=a_{k-1}-a_{k+1}$ for $1 \le k \le N-1$. | n | 121 | 1 |
math | 10. If $\sin \frac{\pi}{9}+\sin \frac{2 \pi}{9}+\cdots+\sin \frac{n \pi}{9}=\frac{1}{2} \tan \frac{4 \pi}{9}$, then the smallest positive integer $n$ is $\qquad$. | 4 | 69 | 1 |
math | Let $MATH$ be a square with $MA = 1$. Point $B$ lies on $AT$ such that $\angle MBT = 3.5 \angle BMT$. What is the area of $\vartriangle BMT$? | \frac{\sqrt{3}-1}{2} | 52 | 11 |
math | Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$. Find $3m + n$.
| 46 | 47 | 2 |
math | 5. Find the sum of the digits of the number $\underbrace{44 \ldots 4}_{2012 \text { times }} \cdot \underbrace{99 \ldots 9}_{2012 \text { times }}$ | 18108 | 56 | 5 |
math | A real number $ to $ is randomly and uniformly chosen from the $ [- 3,4] $ interval. What is the probability that all roots of the polynomial $ x ^ 3 + ax ^ 2 + ax + 1 $ are real? | \frac{3}{7} | 52 | 7 |
math | Determine all solutions of
\[ x + y^2 = p^m \]
\[ x^2 + y = p^n \]
For $x,y,m,n$ positive integers and $p$ being a prime. | (x, y, p, m, n) = (1, 1, 2, 1, 1), (5, 2, 3, 2, 3), (2, 5, 3, 3, 2) | 46 | 58 |
math | Let $\mathbb{R}_{\geq 0}$ denote the set of nonnegative real numbers. Find all functions $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ such that, for all $x, y \in \mathbb{R}_{\geq 0}$,
$$
f\left(\frac{x+f(x)}{2}+y\right)=2 x-f(x)+f(f(y))
$$
and
$$
(f(x)-f(y))(x-y) \geq 0
$$ | f(x)=x | 128 | 4 |
math | Example 6. Find a natural number $n$ such that $2^{8}+2^{11}+2^{n}$ is a perfect square.
(2nd All-Russian High School Mathematics Olympiad) | 12 | 45 | 2 |
math | Example 7. Calculate the central moments of a random variable distributed according to the normal law. | \mu_{2}=1\cdot3\ldots(2-1)\sigma^{2} | 19 | 21 |
math | Find all pairs of nonnegative integers $(x, y)$ for which $(xy + 2)^2 = x^2 + y^2 $. | (0, 2) \text{ and } (2, 0) | 30 | 17 |
math | 1. Compute the surface area
$$
x^{2}+y^{2}+z^{2}=4(x+y+z)
$$ | 48\pi | 29 | 4 |
math | Example 8 Solve the system of equations in the set of real numbers
$$\left\{\begin{array}{l}
x^{2}+y^{2}+z^{2}=\frac{9}{4} \\
-8 x+6 y-24 z=39 .
\end{array}\right.$$ | x=-\frac{6}{13}, y=\frac{9}{26}, z=-\frac{18}{13} | 69 | 30 |
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