task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | ## Task 4 - 140734
In a VEB, a certain type of workpiece was first processed in department A1 and then in department A2. Initially, the same number of workpieces could be processed daily in one department as in the other.
With the help of rationalization measures in both departments, the 53 workers in department A1 i... | 7 | 206 | 1 |
math | It is known that ЖЖ + Ж = МЁД. What is the last digit of the product: В $\cdot И \cdot H \cdot H \cdot U \cdot \Pi \cdot У \cdot X$ (different letters represent different digits, the same letters represent the same digits)?
# | 0 | 62 | 1 |
math | 8. (40 points) On one of the planets in the Alpha Centauri system, elderly women love to paint the cells of $2016 \times 2016$ boards with gold and silver paints. One day, it turned out that in all the boards painted that day, each $3 \times 3$ square had exactly $A$ gold cells, and each $2 \times 4$ or $4 \times 2$ re... | A=Z=0orA=9,Z=8 | 121 | 12 |
math | 7. Let $t=\left(\frac{1}{2}\right)^{x}+\left(\frac{2}{3}\right)^{x}+\left(\frac{5}{6}\right)^{x}$, then the sum of all real solutions of the equation $(t-1)(t-2)(t-3)=0$ with respect to $x$ is $\qquad$ . | 4 | 85 | 1 |
math | 4. $\sin 7.5^{\circ}+\cos 7.5^{\circ}=$ | \frac{\sqrt{4+\sqrt{6}-\sqrt{2}}}{2} | 23 | 19 |
math | 13. The function $f(x)$ defined on $\mathbf{R}$ satisfies: $f(x+2)=2-f(x), f(x+3) \geqslant f(x)$, try to find $f(x)$. | f(x)=1 | 51 | 4 |
math | Problem 1. Let $a, b, c$ be numbers different from zero, such that $b(c+a)$ is the arithmetic mean of the numbers $a(b+c)$ and $c(a+b)$. If $b=\frac{2019}{2020}$, calculate the arithmetic mean of the numbers $\frac{1}{a}, \frac{1}{b}$ and $\frac{1}{c}$. | \frac{2020}{2019} | 91 | 13 |
math | 4. Let $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ be two points on the ellipse $\frac{y^{2}}{a^{2}}+$ $\frac{x^{2}}{b^{2}}=1(a>b>0)$, $m=\left(\frac{x_{1}}{b}, \frac{y_{1}}{a}\right)$, $n$ $=\left(\frac{x_{2}}{b}, \frac{y_{2}}{a}\right)$, and $\boldsymbol{m} \cdot \boldsymbol{n}=0$. The eccentricity ... | 1 | 192 | 1 |
math | ## problem statement
Approximately calculate using the differential.
$y=\sqrt{x^{2}+5}, x=1.97$ | 2.98 | 29 | 4 |
math | 1. 2. A car with a load traveled from one city to another at a speed of 60 km/h, and returned empty at a speed of 90 km/h. Find the average speed of the car for the entire route. Give your answer in kilometers per hour, rounding to the nearest whole number if necessary.
$\{72\}$ | 72 | 74 | 2 |
math | (14 Given real numbers $a$, $b$, $c$, $d$ satisfy $ab=c^2+d^2=1$, then $(a-c)^2+$ $(b-d)^2$ the minimum value is $\qquad$ . | 3-2\sqrt{2} | 52 | 8 |
math | Three. (25 points) Given $x_{1}, x_{2}, \cdots, x_{2021}$ take values of 1 or $-\mathrm{i}$, let
$$
\begin{aligned}
S= & x_{1} x_{2} x_{3}+x_{2} x_{3} x_{4}+\cdots+x_{2019} x_{2020} x_{2021}+ \\
& x_{2020} x_{2021} x_{1}+x_{2021} x_{1} x_{2} .
\end{aligned}
$$
Find the smallest non-negative value that $S$ can take. | 1 | 159 | 1 |
math | 5. The cells of a $9 \times 9$ board are painted in black and white in a checkerboard pattern. How many ways are there to place 9 rooks on the same-colored cells of the board so that they do not attack each other? (A rook attacks any cell that is in the same row or column as itself.) | 2880 | 72 | 4 |
math | 13 If a positive integer $n$ can be written in the form $a^{b}$ (where $a, b \in \mathbf{N}, a \geqslant 2, b \geqslant 2$), then $n$ is called a "good number". Among the positive integers adjacent to the positive integer powers of 2, find all the "good numbers". | 9 | 85 | 1 |
math | 8.2 Let $s(n)$ denote the sum of the digits of a natural number $n$. Solve the equation $n+s(n)=2018$. | 2008 | 34 | 4 |
math | Let's determine the pairs of positive integers $(x, y)$ for which
$$
x^{y}-y^{x}=1
$$ | x_{1}=2,y_{1}=1x_{2}=3,y_{2}=2 | 29 | 20 |
math | 27. Suppose $a$ and $b$ are the roots of $x^{2}+x \sin \alpha+1=0$ while $c$ and $d$ are the roots of the equation $x^{2}+x \cos \alpha-1=0$. Find the value of $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}+\frac{1}{d^{2}}$. | 1 | 102 | 1 |
math | ## Task B-1.3.
Each member of Marića's family drank 4 deciliters of a coffee and milk mixture. The amount of coffee and milk is different in each cup, but it is never zero. Marića drank one quarter of the total amount of milk and one sixth of the total amount of coffee. How many members are there in Marića's family? | 5 | 81 | 1 |
math | 2. In the interval $[0, \pi]$, the number of solutions to the trigonometric equation $\cos 7 x=\cos 5 x$ is $\qquad$ | 7 | 39 | 1 |
math | 1. Add parentheses to the numerical expression $36+144: 9-3 \cdot 2$ so that its value is:
a) 84 ;
b) 14 . | 14 | 43 | 2 |
math | A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled and n cards are dealt to $A$ and $B$. $A$ and $B$ alternately discard a card face up, starting with $A$. The game when the sum of the discards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins if neither ... | 0 | 97 | 1 |
math | 25.15. Calculate $\lim _{n \rightarrow \infty}(\sqrt{n+1}-\sqrt{n})$. | 0 | 29 | 1 |
math | Raskina I.V.
Every day the sheep learns the same number of languages. By the evening of his birthday, he knew 1000 languages. By the evening of the first day of the same month, he knew 820 languages, and by the last day of the month - 1100 languages. When is the sheep's birthday? | February19 | 76 | 3 |
math | 1. A student did not notice the multiplication sign between two seven-digit numbers and wrote a single fourteen-digit number, which turned out to be three times their product. Find these numbers. | x=1666667,y=3333334 | 37 | 18 |
math | Kazitsyna T.v.
Four mice: White, Gray, Fat, and Thin were dividing a piece of cheese. They cut it into 4 visually identical slices. Some slices had more holes, so Thin's slice weighed 20 grams less than Fat's, and White's slice weighed 8 grams less than Gray's. However, White was not upset because his slice weighed ex... | 14 | 139 | 2 |
math | \section*{Problem 1 - 021041}
Determine all pairs \((x ; y)\) of positive integers \(x\) and \(y\) for which \(\sqrt{x}+\sqrt{y}=\sqrt{50}\) holds! | (32,2),(18,8),(8,18),(2,32) | 58 | 21 |
math | 15. In the sequence $20,18,2,20,-18, \ldots$ the first two terms $a_{1}$ and $a_{2}$ are 20 and 18 respectively. The third term is found by subtracting the second from the first, $a_{3}=a_{1}-a_{2}$. The fourth is the sum of the two preceding elements, $a_{4}=a_{2}+a_{3}$. Then $a_{5}=a_{3}-a_{4}$, $a_{6}=a_{4}+a_{5}$,... | 38 | 155 | 2 |
math | Let $A, B$ and $C$ be three sets such that:
- $|A|=100,|B|=50$ and $|C|=48$,
- the number of elements belonging to exactly one of the three sets is equal to twice the number of elements belonging to exactly two of the sets,
- the number of elements belonging to exactly one of the three sets is equal to three times th... | 22 | 106 | 2 |
math | 373. A discrete random variable $X$ is given by the distribution law:
$$
\begin{array}{cccc}
X & 1 & 3 & 5 \\
p & 0.4 & 0.1 & 0.5
\end{array}
$$
Find the distribution law of the random variable $Y=3X$. | \begin{pmatrix}Y&3&9&15\\p&0.4&0.1&0.5\end{pmatrix} | 76 | 34 |
math | 1. Let integer $n \geqslant 2$, set
$$
A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbf{Z}, 0 \leqslant k<n\right\} \text {. }
$$
Find the largest positive integer such that it cannot be written as the sum of one or more numbers (repetition allowed) from the set $A_{n}$. | 2^{n}(n-2)+1 | 97 | 9 |
math | 11. C4 (POL) Consider a matrix of size $n \times n$ whose entries are real numbers of absolute value not exceeding 1 , and the sum of all entries is 0 . Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not excee... | n/2 | 86 | 3 |
math | 1. (7 points) Percival's castle had a square shape. One day, Percival decided to expand his domain and added a square extension to the castle. As a result, the perimeter of the castle increased by $10 \%$. By what percentage did the area of the castle increase? | 4 | 61 | 1 |
math | 5. For any integer $n(n \geqslant 2)$, the positive numbers $a$ and $b$ that satisfy $a^{n}=a+1, b^{2 n}=b+3 a$ have the following relationship ( ).
(A) $a>b>1$.
(B) $b>a>1$
(C) $a>1,01$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result d... | A | 106 | 1 |
math | (10) Given positive real numbers $a$ and $b$ satisfying $a^{2}+b^{2}=1$, and $a^{3}+b^{3}+1=m(a+b+1)^{3}$, find the range of values for $m$. | [\frac{3\sqrt{2}-4}{2},\frac{1}{4}) | 60 | 20 |
math | Example 1 In $\triangle ABC$, if $AB=5, BC=6$, $CA=7, H$ is the orthocenter, then the length of $AH$ is $\qquad$
$(2000$, Shanghai Junior High School Mathematics Competition) | \frac{19 \sqrt{6}}{12} | 56 | 14 |
math | 4. Let $k$ be a positive integer, find all polynomials $P(x)=a_{n}+a_{n-1} x+\cdots+a_{0} x^{n}$, where $a_{i}$ are real numbers, that satisfy the equation $P(P(x))=(P(x))^{k}$.
| P(x)=x^k | 70 | 6 |
math | I2.2 In the trapezium $A B C D, A B / / D C . A C$ and $B D$ intersect at $O$. The areas of triangles $A O B$ and $C O D$ are $P$ and 25 respectively. Given that the area of the trapezium is $Q$, find the value of $Q$. | 81 | 82 | 2 |
math | Example. Find the derivative of the function
$$
u=x^{2}-\operatorname{arctg}(y+z)
$$
at the point $A(2,1,1)$ in the direction of the point $B(2,4,-3)$. | \frac{1}{25} | 57 | 8 |
math | Example 4 Find the largest positive integer $n$, such that the system of equations
$$
(x+1)^{2}+y_{1}^{2}=(x+2)^{2}+y_{2}^{2}=\cdots=(x+k)^{2}+y_{k}^{2}=\cdots=(x+n)^{2}+y_{n}^{2}
$$
has integer solutions $\left(x, y_{1}, y_{2}, \cdots, y_{n}\right)$. | 3 | 115 | 1 |
math | Exercise 6. Find all strictly positive integers $p, q$ such that
$$
\mathrm{p} 2^{\mathrm{q}}=\mathrm{q} 2^{\mathrm{p}}
$$ | (p,p)forp\in\mathbb{N}^{*}(2,1),(1,2) | 46 | 24 |
math | 2. (5 points) Xiao Fou buys 3 erasers and 5 pencils for 10.6 yuan. If he buys 4 erasers and 4 pencils of the same type, he needs to pay 12 yuan. Then the price of one eraser is $\qquad$ yuan. | 2.2 | 64 | 3 |
math | 1. Determine all polynomials $P$ such that for all real numbers $x$,
$$
(P(x))^{2}+P(-x)=P\left(x^{2}\right)+P(x) .
$$
(P. Calábek) | Thesoughtpolynomialstheconstants01,themonomialsx^{2},x^{4},x^{6},\ldotsthebinomialsx+1,x^{3}+1,x^{5}+1,\ldots | 53 | 53 |
math | 4. For a positive integer $n$, if $(x y-5 x+3 y-15)^{n}$ is expanded and like terms are combined, $x^{i} y^{j}(i, j=0,1$, $\cdots, n)$ after combining have at least 2021 terms, then the minimum value of $n$ is $\qquad$. | 44 | 82 | 2 |
math | 3. Given that $x, y, z$ are positive numbers and satisfy $\left\{\begin{array}{l}x+y+x y=8, \\ y+z+y z=15, \\ z+x+z x=35 .\end{array}\right.$ Then $x+y+z+x y=$ | 15 | 66 | 2 |
math | Two numbers $a$ and $b(a>b)$ are written on the board. They are erased and replaced by the numbers ${ }^{a+b} / 2$ and ${ }^{a-b} / 2$. The same procedure is applied to the newly written numbers. Is it true that after several erasures, the difference between the numbers written on the board will become less than $\frac... | 2^{-k}-2^{-k}b<1/2002 | 93 | 16 |
math | Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have:
\[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\]
Show that this sequence has a finite limit. Determine this limit. | 3 | 128 | 1 |
math | 3. Solve the rebus UHA = LCM(UX, UA, HA). Here U, X, A are three different digits. Two-digit and three-digit numbers cannot start with zero. Recall that the LCM of several natural numbers is the smallest natural number that is divisible by each of them. | 150 | 62 | 3 |
math | Find all positive integers $n$ such that the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{n}$ has exactly $2011$ positive integer solutions $(x,y)$ where $x \leq y$. | n = p^{2010} | 57 | 10 |
math | 10. Find all four-digit numbers $m$, such that $m<2006$, and there exists a positive integer $n$, such that $m-n$ is a prime number, and $m n$ is a perfect square. | 1156, 1296, 1369, 1600, 1764 | 51 | 28 |
math | 6. Find all pairs of natural numbers $m$ and $n$, such that $m+1$ is divisible by $n$ and $n^{2}-n+1$ is divisible by $m$. | (,n)=(1,1),(1,2),(3,2) | 44 | 16 |
math | Find all positive integer $N$ which has not less than $4$ positive divisors, such that the sum of squares of the $4$ smallest positive divisors of $N$ is equal to $N$. | 130 | 44 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow-1} \frac{x^{3}-3 x-2}{x^{2}-x-2}$ | 0 | 40 | 1 |
math | ## Task A-3.6.
Calculate the product
$$
\left(1-\frac{\cos 61^{\circ}}{\cos 1^{\circ}}\right)\left(1-\frac{\cos 62^{\circ}}{\cos 2^{\circ}}\right) \ldots\left(1-\frac{\cos 119^{\circ}}{\cos 59^{\circ}}\right)
$$ | 1 | 97 | 1 |
math | 779. Find the mutual correlation function of two random functions: $X(t)=t^{2} U$ and $Y(t)=t^{3} U$, where $U$ is a random variable, and $D(U)=5$. | 5t_{1}^{2}t_{2}^{3} | 51 | 15 |
math | Consider a function $ f(x)\equal{}xe^{\minus{}x^3}$ defined on any real numbers.
(1) Examine the variation and convexity of $ f(x)$ to draw the garph of $ f(x)$.
(2) For a positive number $ C$, let $ D_1$ be the region bounded by $ y\equal{}f(x)$, the $ x$-axis and $ x\equal{}C$. Denote $ V_1(C)$ the volume obtai... | \pi \left( e^{-\frac{1}{3}} - \frac{2}{3} \right) | 230 | 25 |
math | Given an integer $n \ge 2$, determine the number of ordered $n$-tuples of integers $(a_1, a_2,...,a_n)$ such that
(a) $a_1 + a_2 + .. + a_n \ge n^2$ and
(b) $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$ | a_i = n \text{ for all } 1 \le i \le n | 93 | 18 |
math | A company has five directors. The regulations of the company require that any majority (three or more) of the directors should be able to open its strongroom, but any minority (two or less) should not be able to do so. The strongroom is equipped with ten locks, so that it can only be opened when keys to all ten locks a... | n = 6 | 112 | 5 |
math | Let $P_1, P_2, \ldots, P_6$ be points in the complex plane, which are also roots of the equation $x^6+6x^3-216=0$. Given that $P_1P_2P_3P_4P_5P_6$ is a convex hexagon, determine the area of this hexagon. | 9\sqrt{3} | 84 | 6 |
math | 5. Rewrite the equation as
$$
5^{(x-1)^{2}} \log _{7}\left((x-1)^{2}+2\right)=5^{2|x-a|} \log _{7}(2|x-a|+2)
$$
Let $f(t)=5^{t} \log _{7}(t+2)$. This function is increasing. Therefore,
$$
f\left((x-1)^{2}\right)=f(2|x-a|) \Longleftrightarrow(x-1)^{2}=2|x-a|
$$
There will be three solutions in the case of tangency f... | 1/2;1;3/2 | 191 | 9 |
math | A palindrome is a positive integer that reads the same forward and backward, like 2552 or 1991. Find a positive integer greater than 1 that divides all four-digit palindromes. | 11 | 45 | 2 |
math | Example 4 Two seventh-grade students are allowed to participate in the chess competition for eighth-grade students. Each pair of players competes once, with a win earning 1 point, a draw earning half a point, and a loss earning zero points. The two seventh-grade students scored a total of 8 points, and each eighth-grad... | 7 \text{ and } 14 | 100 | 9 |
math | 12.206. A section is made through the vertex of a regular triangular pyramid and the midpoints of two sides of the base. Find the area of the section and the volume of the pyramid, given the side $a$ of the base and the angle $\alpha$ between the section and the base. | \frac{^{2}\sqrt{3}}{48\cos\alpha};\frac{^{3}\operatorname{tg}\alpha}{48} | 65 | 34 |
math | Let $E$ be a finite (non-empty) set. Find all functions $f: \mathcal{P} \rightarrow \mathbb{R}$ such that for all subsets $A$ and $B$ of $E, f(A \cap B)+f(A \cup B)=f(A)+f(B)$, and for any bijection $\sigma: E \rightarrow E$ and any subset $A$ of $E, f(\sigma(A))=f(A)$. | f(A)=\cdot\operatorname{Card}(A)+b | 101 | 14 |
math | (15) (50 points) Find all positive integers $x, y$ that satisfy the following conditions: (1) $x$ and $y-1$ are coprime; (2) $x^{2}-x+1=y^{3}$. | (x,y)=(1,1),(19,7) | 57 | 12 |
math | Alex and Bobette are playing on a $20 \times 20$ grid where the cells are square and have a side length of 1. The distance between two cells is the distance between their centers. They take turns playing as follows: Alex places a red stone on a cell, ensuring that the distance between any two cells with red stones is n... | 100 | 133 | 3 |
math | Jeffrey rolls fair three six-sided dice and records their results. The probability that the mean of these three numbers is greater than the median of these three numbers can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i] | 101 | 72 | 3 |
math | 7.1. Find the smallest natural solution of the inequality $2^{x}+2^{x+1}+\ldots+2^{x+2000}>2^{2017}$. Answer. 17. | 17 | 51 | 2 |
math | 6. Determine the minimum value of $\sum_{k=1}^{50} x_{k}$, where the summation is done over all possible positive numbers $x_{1}, \ldots, x_{50}$ satisfying $\sum_{k=1}^{50} \frac{1}{x_{k}}=1$. | 2500 | 72 | 4 |
math | 4. Is the number $\operatorname{tg} \sqrt{5 \pi}-1$ positive or negative?
# | positive | 26 | 1 |
math | A group of cows and horses are randomly divided into two equal rows. (The animals are welltrained and stand very still.) Each animal in one row is directly opposite an animal in the other row. If 75 of the animals are horses and the number of cows opposite cows is 10 more than the number of horses opposite horses, dete... | 170 | 79 | 3 |
math | The class received the following task. Determine the smallest natural number $H$ such that its half is equal to a perfect cube, its third is equal to a perfect fifth power, and its fifth is equal to a perfect square.
Pista and Sanyi were absent that day, and one of their friends told them the problem from memory, so P... | P | 209 | 1 |
math | 287. $y=x^{2}+\frac{2}{x^{4}}-\sqrt[3]{x}$
287. $y=x^{2}+\frac{2}{x^{4}}-\sqrt[3]{x}$ | 2x-\frac{8}{x^{5}}-\frac{1}{3\sqrt[3]{x^{2}}} | 53 | 26 |
math | Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$? | 589 | 42 | 3 |
math | 7・10 A fishing boat is fishing in the territorial sea of a foreign country without permission. Each time it casts a net, it causes the same value of loss in the fishing catch of the country. The probability of the boat being detained by the foreign coast patrol during each net casting is $1 / k$, where $k$ is a natural... | k-1 | 194 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\sin b x - \sin a x}{\ln \left(\tan\left(\frac{\pi}{4} + a x\right)\right)}
$$ | \frac{b-}{2a} | 59 | 9 |
math | 1. (5 points) Find the degree measure of the angle
$$
\delta=\arccos \left(\left(\sin 3271^{\circ}+\sin 3272^{\circ}+\cdots+\sin 6871^{\circ}\right)^{\cos } 3240^{\circ}+\cos 3241^{\circ}+\cdots+\cos 6840^{\circ}\right)
$$ | 59 | 104 | 2 |
math | # Problem 6.
B-1
Find the minimum value of the expression $4 x+9 y+\frac{1}{x-4}+\frac{1}{y-5}$ given that $x>4$ and $y>5$. | 71 | 53 | 2 |
math | 5.1. From one point on a circular track, a pedestrian and a cyclist started simultaneously in the same direction. The cyclist's speed is $55\%$ greater than the pedestrian's speed, and therefore the cyclist overtakes the pedestrian from time to time. At how many different points on the track will the overtakes occur? | 11 | 69 | 2 |
math | 25.2.4 ** Find the sum $S=\sum_{k=1}^{n} k^{2} \cdot \mathrm{C}_{n}^{k}$. | n(n+1)\cdot2^{n-2} | 39 | 12 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(1 ; 0 ; 2)$
$M_{2}(1 ; 2 ;-1)$
$M_{3}(2 ;-2 ; 1)$
$M_{0}(-5 ;-9 ; 1)$ | \sqrt{77} | 88 | 6 |
math | 3.100. The base of the pyramid is a rectangle, where the angle between the diagonals is $\alpha$. A sphere of radius $R$ is circumscribed around this pyramid. Find the volume of the pyramid if all its lateral edges form an angle $\beta$ with the base. | \frac{4}{3}R^{3}\sin^{2}2\beta\sin^{2}\beta\sin\alpha | 62 | 28 |
math | 5. For any set $S$, let $|S|$ denote the number of elements in the set, and let $n(S)$ denote the number of subsets of set $S$. If $A, B, C$ are three sets that satisfy the following conditions:
(1) $n(A)+n(B)+n(C)=n(A \cup B \cup C)$;
(2) $|A|=|B|=100$.
Find the minimum value of $|A \cap B \cap C|$. | 97 | 109 | 2 |
math | Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$? | 81 | 42 | 2 |
math | 2. Let $\left(a_{n}\right)_{n \geq 1}$ be a sequence defined by: $a_{1}=\frac{1}{2}$ and $a_{n+1}+a_{n}=\frac{2}{n^{2}+2 n}, \forall n \geq 1$.
a. Find the general term of the sequence.
b. Calculate the sum $S=\sum_{k=1}^{m}(2 k+1) a_{k}^{2}$ and show that $S<1$.
Prof. Traian Tămâian | 1-\frac{1}{(n+1)^2} | 129 | 13 |
math | Given an equilateral triangle $ABC$ of side $a$ in a plane, let $M$ be a point on the circumcircle of the triangle. Prove that the sum $s = MA^4 +MB^4 +MC^4$ is independent of the position of the point $M$ on the circle, and determine that constant value as a function of $a$. | 2a^4 | 79 | 6 |
math | Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions. | n \in \{1344, 1728, 1792, 1920, 1984\} | 38 | 36 |
math | 23rd Swedish 1983 Problem 1 The positive integers are grouped as follows: 1, 2+3, 4+5+6, 7+8+9+10, ... Find the value of the nth sum. | \frac{1}{2}n(n^2+1) | 54 | 14 |
math | In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively, and $G$ is the centroid of $\triangle A B C$. If
$$
a \overrightarrow{G A}+b \overrightarrow{G B}+\frac{\sqrt{3}}{3} c \overrightarrow{G C}=0 \text {, }
$$
then $\angle A=$ . $\qquad$ | 30^{\circ} | 104 | 6 |
math | 4. The incident took place in 1968. A high school graduate returned from a written university entrance exam and told his family that he couldn't solve the following problem:
Several identical books and identical albums were bought. The books cost 10 rubles 56 kopecks, and the albums cost 56 kopecks. The number of book... | 8 | 231 | 1 |
math | 2. Let $D(k)$ denote the number of positive divisors of $k$. For a quadruple of positive integers $(a, b, c, d)$, if
$$
b=a^{2}+1, c=b^{2}+1, d=c^{2}+1 \text {, }
$$
and $D(a)+D(b)+D(c)+D(d)$ is odd, then it is called "green". How many green quadruples $(a, b, c, d)$ are there with $a, b, c, d$ less than 1000000? | 2 | 130 | 1 |
math | One. (20 points) Equations
$$
x^{2}+a x+b=0 \text { and } x^{2}+b x+a=0
$$
have a common root, and let the other two roots be $x_{1} 、 x_{2}$; Equations
$$
x^{2}-c x+d=0 \text { and } x^{2}-d x+c=0
$$
have a common root, and let the other two roots be $x_{3} 、 x_{4}$. Find the range of $x_{1} x_{2} x_{3} x_{4}$ $(a, ... | 0 < x_{1} x_{2} x_{3} x_{4} < \frac{1}{16} | 166 | 27 |
math | Example 4 Evaluate:
(1) $\cos 40^{\circ}+\cos 80^{\circ}+\cos 160^{\circ}$;
(2) $\cos 40^{\circ} \cdot \cos 80^{\circ}+\cos 80^{\circ} \cdot \cos 160^{\circ}+$ $\cos 160^{\circ} \cdot \cos 40^{\circ}$;
(3) $\cos 40^{\circ} \cdot \cos 80^{\circ} \cdot \cos 160^{\circ}$. | 0, -\frac{3}{4}, -\frac{1}{8} | 140 | 18 |
math | A triangle has two sides of length 12 and 20 units, and the angle bisector of the angle between them is 15 units. What is the length of the third side of the triangle? | 8 | 44 | 1 |
math | 4. Find all functions $f: \mathscr{E} \rightarrow \mathscr{B}$, for any $x, y \in \mathscr{F}, f$ satisfies
$$
f(x y)(f(x)-f(y))=(x-y) f(x) f(y) \text {. }
$$
where $\mathscr{B}$ is the set of real numbers. | f(x)=\left\{\begin{array}{cc} C x, & x \in G, \\ 0, & x \notin G . \end{array}\right.} | 83 | 40 |
math | # Problem 5.
Solve the equation with three unknowns
$$
X^{Y}+Y^{Z}=X Y Z
$$
in natural numbers.
# | (1;1;2),(2;2;2),(2;2;3),(4;2;3),(4;2;4) | 36 | 31 |
math | $\left[\begin{array}{l}{[\text { Ratio of areas of triangles with a common angle }} \\ {\left[\begin{array}{l}\text { Law of Cosines }\end{array}\right]}\end{array}\right.$
In triangle $A B C$, angle $A$ is $60^{\circ} ; A B: A C=3: 2$. Points $M$ and $N$ are located on sides $A B$ and $A C$ respectively, such that $B... | \frac{4}{25} | 140 | 8 |
math | 10. (20 points) Given the ellipse $C: \frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, and the moving circle $\Gamma: x^{2}+y^{2}=r^{2}(3<r<5)$. If $M$ is a point on the ellipse $C$, and $N$ is a point on the moving circle $\Gamma$, and the line $M N$ is tangent to both the ellipse $C$ and the moving circle $\Gamma$, find the maxi... | 2 | 127 | 1 |
math | ## Task A-4.3.
How many complex numbers $z$ satisfy the following two conditions:
$$
|z|=1, \quad \operatorname{Re}\left(z^{100}\right)=\operatorname{Im}\left(z^{200}\right) \quad ?
$$ | 400 | 64 | 3 |
math | LX OM - I - Task 5
For each integer $ n \geqslant 1 $, determine the largest possible number of different subsets of the set $ \{1,2,3, \cdots,n\} $ with the following property: Any two of these subsets are either disjoint or one is contained in the other. | 2n | 72 | 2 |
math | ## Task A-1.1.
Determine all ordered pairs of integers $(x, y)$ for which
$$
x^{2} y+4 x^{2}-3 y=51
$$ | (x,y)=(0,-17),(2,35),(-2,35),(4,-1),(-4,-1) | 43 | 28 |
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