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 | 4. Given that $\overline{73 a b c 6}$ is divisible by 56 $(b<4)$, and $a$ leaves the same remainder when divided by 40, 61, and 810, find all six-digit numbers that satisfy the requirements. | 731136,737016,737296 | 63 | 20 |
math | 一、(20 points) Find all real numbers $k$ such that the quadratic equation in $x$
$$
k x^{2}-2(3 k-1) x+9 k-1=0
$$
has two integer roots. | k=-\frac{1}{4}, \frac{1}{9}, \frac{1}{5} | 53 | 23 |
math | Determine the value of $2^{4}\left(1+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\frac{1}{2^{4}}\right)$. | 31 | 51 | 2 |
math | N51 (43-3, Romania) Find all pairs of positive integers $m, n \geqslant 3$, such that there exist infinitely many positive integers $a$, for which
$$
\frac{a^{m}+a-1}{a^{n}+a^{2}-1}
$$
is an integer. | (,n)=(5,3) | 74 | 8 |
math | II. (40 points) Let $m, n$ be positive integers, and $p$ be a prime. Find all triples $(m, n, p)$ that satisfy $\mathrm{C}_{m}^{3}-4=p^{n}$. | (,n,p)=(7,1,31),(6,4,2) | 53 | 18 |
math | Find the least positive integer $n$ such that the decimal representation of the binomial coefficient $\dbinom{2n}{n}$ ends in four zero digits. | 313 | 34 | 3 |
math | 5. From $26=1^{2}+5^{2}=1^{2}+3^{2}+4^{2}$, it can be concluded that 26 can be represented as the sum of squares of at most 3 distinct non-zero natural numbers. Please determine the maximum number of distinct non-zero natural numbers whose squares can sum up to 360? | 360=1^{2}+2^{2}+3^{2}+4^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2} | 80 | 49 |
math | 3. (HUN) Let $x$ be an angle and let the real numbers $a, b, c, \cos x$ satisfy the following equation:
$$
a \cos ^{2} x+b \cos x+c=0 .
$$
Write the analogous quadratic equation for $a, b, c, \cos 2 x$. Compare the given and the obtained equality for $a=4, b=2, c=-1$. | 4\cos^{2}2x+2\cos2x-1=0 | 94 | 18 |
math | A8. In my desk, the number of pencils and pens was in the ratio $4: 5$. I took out a pen and replaced it with a pencil and now the ratio is $7: 8$. What is the total number of pencils and pens in my desk? | 45 | 58 | 2 |
math | Problem 12.1. The sequence $\left\{x_{n}\right\}_{n=1}^{\infty}$ is defined by $x_{1}=2$ and $x_{n+1}=1+a x_{n}$, $n \geq 1$, where $a$ is a real number. Find all values of $a$ for which the sequence is:
a) an arithmetic progression;
b) convergent and find its limit.
Oleg Mushkarov | 1thelimitis\frac{1}{1-}for\in(-1,1) | 104 | 21 |
math | 10.3. What is the maximum number of digits that a natural number can have, where all digits are different, and it is divisible by each of its digits? | 7 | 35 | 1 |
math | Example 2 A class participated in a math competition, with a total of $a$, $b$, and $c$ three questions. Each question either scores full marks or 0 points, where question $a$ is worth 20 points, and questions $b$ and $c$ are worth 25 points each. After the competition, every student answered at least one question corr... | 42 | 220 | 2 |
math | 1. The positive integer $n=$ $\qquad$ that makes $2^{n}+256$ a perfect square. | 11 | 28 | 2 |
math | Thirteen, for $\{1,2,, 3 \cdots, n\}$ and each of its non-empty subsets, we define the alternating sum as follows: arrange the numbers in the subset in descending order, then alternately add and subtract the numbers starting from the largest (for example, the alternating sum of $\{1,2,4,6,9\}$ is $9-6+4-2+1=6$, and the... | 448 | 121 | 3 |
math | Problem 10.1. Consider the inequality $\sqrt{x}+\sqrt{2-x} \geq \sqrt{a}$, where $a$ is a real number.
a) Solve the inequality for $a=3$.
б) Find all $a$, for which the set of solutions of the inequality is a segment (possibly, a point) of length less than or equal to $\sqrt{3}$.
Kerope Chakaryan | \in[3,4] | 95 | 7 |
math | 5. If the function $f(x)=\lg \left(\sin ^{6} x+\cos ^{6} x+a \sin x \cos x\right)$ has the domain $\mathbf{R}$, then the range of the real number $a$ is $\qquad$ . | (-\frac{1}{2},\frac{1}{2}) | 63 | 15 |
math | ### 3.19. Calculate
$$
\int_{0}^{i} z \sin z d z
$$ | -\frac{i}{e} | 27 | 6 |
math | If $x_{1}, x_{2},\ldots ,x_{n}$ are positive real numbers with $x_{1}^2+x_2^{2}+\ldots +x_{n}^{2}=1$, find the minimum value of $\sum_{i=1}^{n}\frac{x_{i}^{5}}{x_{1}+x_{2}+\ldots +x_{n}-x_{i}}$. | \frac{1}{n(n-1)} | 97 | 11 |
math | The set $M$ consists of all 7-digit positive integers that (in decimal notation) contain each of the digits 1, 3, 4, 6, 7, 8, and 9 exactly once.
(a) Determine the smallest positive difference $d$ between two numbers in $M$.
(b) How many pairs $(x, y)$ with $x$ and $y$ from $M$ are there such that $x-y=d$?
Answer. a... | 480 | 122 | 3 |
math | Task B-2.4. Let $f(x)$ be a quadratic function such that
$$
x^{2}-2 x+2 \leq f(x) \leq 2 x^{2}-4 x+3
$$
for every $x \in \mathbb{R}$. If $f(11)=181$, what is $f(16)$? | 406 | 84 | 3 |
math | 4. If acute angles $\alpha, \beta$ satisfy
$$
\sin \alpha=\cos (\alpha+\beta) \cdot \sin \beta,
$$
then the maximum value of $\tan \alpha$ is $\qquad$ | \frac{\sqrt{2}}{4} | 50 | 10 |
math | 2. Find the solution to the system
$$
\left\{\begin{array}{l}
5 x^{7}+3 y^{2}+5 u+4 v^{4}=-2 \\
2 x^{7}+8 y^{2}+7 u+4 v^{4}=\frac{6^{5}}{3^{4} \cdot 4^{2}} \\
8 x^{7}+2 y^{2}+3 u+6 v^{4}=-6 \\
5 x^{7}+7 y^{2}+7 u+8 v^{4}=\frac{8^{3}}{2^{6} \cdot 4}
\end{array}\right.
$$ | {-1,\1,0,0} | 152 | 9 |
math | Consider a circle centered at $O$. Parallel chords $AB$ of length $8$ and $CD$ of length $10$ are of distance $2$ apart such that $AC < AD$. We can write $\tan \angle BOD =\frac{a}{b}$ , where $a, b$ are positive integers such that gcd $(a, b) = 1$. Compute $a + b$. | 113 | 87 | 3 |
math | 15. Let $x>1, y>1, S=\min \left\{\log _{x} 2, \log _{2} y\right.$ , $\left.\log _{y}\left(8 x^{2}\right)\right\}$. Then the maximum value of $S$ is $\qquad$ . | 2 | 74 | 1 |
math | (French-Slovak Competition 1996) Find all strictly positive integers $x, y, p$ such that $p^{x}-y^{p}=1$ with $p$ prime. | 2 | 42 | 1 |
math | 7.14. On a plane, two points $A$ and $B$ are given. Find the locus of points $M$ such that $A M: B M=k$ (Apollonian circle). | 圆心为(-\frac{1+k^{2}} | 45 | 12 |
math | 2. (8 points) Shuaishuai finished memorizing English words in five days. It is known that in the first three days, he memorized $\frac{1}{2}$ of all the words, and in the last three days, he memorized $\frac{2}{3}$ of all the words, and he memorized 120 fewer words in the first three days than in the last three days. T... | 120 | 107 | 3 |
math | 3. Given that the three non-zero real roots of the equation $x^{3}+a x^{2}+b x+c=0$ form a geometric sequence, then the value of $a^{3} c-b^{3}$ is
$\qquad$ . | 0 | 57 | 1 |
math | 13. Given 10 points on a plane, no three of which are collinear, draw 4 line segments, each connecting two of the points. These line segments are chosen arbitrarily, and each has an equal chance of being selected. The probability that three of these line segments form a triangle with three of the given 10 points as ver... | 489 | 99 | 3 |
math | 4.1. The number $3^{2009}$ is represented as the sum of $k$ consecutive natural numbers. What is the greatest possible value of $k$? | 2\cdot3^{1004} | 38 | 10 |
math | 2. $\frac{\sqrt{2} \cos 55^{\circ}-\sin 20^{\circ}}{\sqrt{2} \cos 5^{\circ}+\sin 20^{\circ}}=$ | 2-\sqrt{3} | 50 | 6 |
math | 4.018. The denominator of the geometric progression is $1 / 3$, the fourth term of this progression is $1 / 54$, and the sum of all its terms is 121/162. Find the number of terms in the progression. | 5 | 59 | 1 |
math | \section*{Exercise 2 - 171012}
Determine the set of all real numbers \(x\) for which the term \(\frac{1}{\sqrt{33-8 x-x^{2}}}\) is defined! | -11<x<3 | 54 | 6 |
math | 13.132. In a laboratory setup, a certain liquid flows into a vessel through three inlet valves. If all valves are opened simultaneously, the vessel will be filled in 6 minutes. If the vessel is filled only through the second valve, it will take 0.75 of the time it takes to fill the vessel only through the first valve. ... | \frac{56}{3},14,24 | 112 | 13 |
math | Task 1. There are 3 types of installations, no more than 200 in total. The number of installations of type 2 is 4 times the number of type 1, and the number of installations of type 3 is a multiple of the number of installations of type 1. If the number of installations of type 3 were 5 times more, then they would be 9... | 9,36,27 | 105 | 7 |
math | Find all differentiable functions $f:(0,\infty) \to \mathbb{R}$ such that
$$f(b)-f(a)=(b-a)f’(\sqrt{ab}) \qquad \text{for all}\qquad a,b>0.$$
[i]Proposed by Orif Ibrogimov, National University of Uzbekistan[/i]
| f(x) = a + bx + c/x | 77 | 11 |
math | # Problem 3.
The number $b$ is the arithmetic mean of the numbers $a$ and $c$. Find all ordered triples $(a, b, c)$ of such numbers for which at least one of the numbers $1 / a, 1 / b, 1 / c$ is the arithmetic mean of the other two. | (x,x,x)wherex\neq0,or(-4x,-x,2x)wherex\neq0 | 70 | 29 |
math | 1*. Into how many regions do $n$ planes divide space if every three of them have exactly one common point, and no four of them have a common point? | F_{3}(n)=C_{n}^{3}+C_{n}^{2}+C_{n}^{1}+C_{n}^{0} | 34 | 37 |
math | Ilya Muromets meets the three-headed Zmei Gorynych. Every minute, Ilya cuts off one head of the Zmei. Let $x$ be the resilience of the Zmei ($x>0$). The probability $p_{s}$ that $s$ new heads will grow in place of the cut head ($s=0,1$, 2) is $\frac{x^{s}}{1+x+x^{2}}$. During the first 10 minutes of the battle, Ilya re... | \frac{\sqrt{97}+1}{8}\approx1.36 | 174 | 18 |
math | 1. Find all integers $a$ for which the equation
$$
a\left(x^{2}+x\right)=3\left(x^{2}+1\right)
$$
has at least one integer root.
(Jaromír Šimša) | =3or=5 | 59 | 5 |
math | Problem 1. In one class, there are 24 students. In this class, the number of boys is equal to $\frac{3}{5}$ of the number of girls. How many boys and how many girls are there in the class? | 9 | 52 | 1 |
math | 2. Given an integer $n>1$, let $a_{1}, a_{2}, \cdots, a_{n}$ be distinct non-negative real numbers, and define the sets
$$
A=\left\{a_{i}+a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\}, B=\left\{a_{i} a_{j} \mid 1 \leqslant i \leqslant j \leqslant n\right\} .
$$
Find the minimum value of $\frac{|A|}{|B|}$. H... | \frac{2(2n-1)}{n(n+1)} | 156 | 16 |
math | Lord Moneybag said to his grandson: "Bill, pay attention! Christmas is coming soon. I have taken a sum between 300 and 500 pounds, which is a multiple of 6. You will get 5 pounds in 1-pound coins. When I give you one pound at a time, the remaining amount will first be divisible by 5, then by 4, then by 3, then by 2, an... | 426 | 130 | 3 |
math | 113. For what least natural $n$ is each of the fractions
$$
\frac{7}{n+9}, \frac{8}{n+10}, \ldots, \frac{31}{n+33}
$$
irreducible
146 | 35 | 62 | 2 |
math | 3. Given $x, y, z \in \mathbf{R}, x y+y z+z x=-1$. Then the minimum value of $x^{2}+5 y^{2}+8 z^{2}$ is
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 4 | 75 | 1 |
math | Task B-2.2. One solution of the equation $a x^{2}+b x+5=0$ is five times larger than the other. The coefficients $a$ and $b$ are natural numbers less than 20. Determine all such equations. | x^{2}+6x+5=0,4x^{2}+12x+5=0,9x^{2}+18x+5=0 | 57 | 39 |
math | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$f(x)=\left\{\begin{array}{c}x^{2} \cos \left(\frac{4}{3 x}\right)+\frac{x^{2}}{2}, x \neq 0 \\ 0, x=0\end{array}\right.$ | 0 | 82 | 1 |
math | Example 3. Find the maximum and minimum values of the function $y=\sqrt[4]{7-2 \sin x}$ $+\sqrt[4]{2 \sin x+1}$. | y_{\text{minimum}} = \sqrt[4]{8}, \quad y_{\text{maximum}} = \sqrt[4]{5} + \sqrt[4]{3} | 41 | 40 |
math | 2.084. $\left(\frac{2-b}{b-1}+2 \cdot \frac{a-1}{a-2}\right):\left(b \cdot \frac{a-1}{b-1}+a \cdot \frac{2-b}{a-2}\right)$;
$a=\sqrt{2}+0.8 ; b=\sqrt{2}-0.2$. | 1 | 88 | 1 |
math | 10. (5 points) There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm, then starting from the right end, a line is drawn every 6 cm, and the stick is cut at the lines. Among the small sticks obtained, the number of 3 cm long sticks is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻... | 12 | 103 | 2 |
math | 6. Viewers rate a movie with an integer number of points from 0 to 10. At any given time, the movie's rating is calculated as the sum of all the given ratings divided by their number. At some point in time $T$, the rating was an integer, and then with each new voting viewer, it decreased by one. What is the maximum num... | 5 | 103 | 1 |
math | Yan and Jacob play the following game. Yan shows Jacob a weighted 4-sided die labelled 1, 2, 3, 4, with weights $\frac{1}{2}, \frac{1}{3}, \frac{1}{7}, \frac{1}{42}$, respectively. Then, Jacob specifies 4 positive real numbers $x_{1}, x_{2}, x_{3}, x_{4}$ such that $x_{1}+\cdots+x_{4}=1$. Finally, Yan rolls the dice, a... | (x_{1},x_{2},x_{3},x_{4})=(\frac{1}{2},\frac{1}{3},\frac{1}{7},\frac{1}{42}) | 184 | 46 |
math | Problem 21. In a triangle with a perimeter of $2 \sqrt{3}$, the product of its three angle bisectors is 1, and the radius of the inscribed circle is $\frac{1}{3}$. Find the angles of the triangle. | 60;60;60 | 56 | 8 |
math | 96 Let the sequence $\left\{a_{1}, a_{2}, \cdots,\right\}=\left\{\frac{1}{1}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{2}{2}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2}, \frac{2}{3}, \frac{1}{4}, \cdots\right\}$, then the 1988th term of this sequence $a_{1988}=$ $\qquad$ . | \frac{29}{35} | 131 | 9 |
math | ## Task A-1.2.
On the playground, there are 2014 athletes who have numbers from 1 to 2014 on their jerseys (each number is on exactly one jersey). At the beginning, all of them are standing. At certain time intervals, the coach shouts out all natural numbers from 1 to 2014 in sequence. Athletes whose jersey numbers ar... | 44 | 120 | 2 |
math | Task 1. Find all pairs of prime numbers $(p, q)$ for which there exist positive integers $(m, n)$ such that
$$
(p+q)^{m}=(p-q)^{n}
$$ | (p,q)=(3,5)(p,q)=(5,3) | 46 | 14 |
math | Calculate $1+3+5+\ldots+2 n+1$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | (n+1)^{2} | 42 | 7 |
math | A bucket full of milk weighed $35 \mathrm{~kg}$. The same bucket with half the amount of milk weighed $18 \mathrm{~kg}$.
How much does the empty bucket weigh?
(L. Hozová) | 1\mathrm{~} | 51 | 6 |
math | I1.1 Find $a$ if $a=\log _{5} \frac{(125)(625)}{25}$.
I1.2 If $\left(r+\frac{1}{r}\right)^{2}=a-2$ and $r^{3}+\frac{1}{r^{3}}=b$, find $b$.
I1.3 If one root of the equation $x^{3}+c x+10=b$ is 2 , find $c$.
I1.4 Find $d$ if $9^{d+2}=(6489+c)+9^{d}$. (Reference: 1986 FG7.4) | 5,0,-9,2 | 153 | 7 |
math | Solve the equation
\[
\sin 9^\circ \sin 21^\circ \sin(102^\circ + x^\circ) = \sin 30^\circ \sin 42^\circ \sin x^\circ
\]
for $x$ where $0 < x < 90$. | x = 9^\circ | 71 | 7 |
math | 6.251. For what positive $p$ are the roots of the equation
$5 x^{2}-4(p+3) x+4=p^{2}$
of opposite signs? Find these roots. | p>2;x_{1}=\frac{-p+2}{5},x_{2}=p+2 | 46 | 23 |
math | Question 217, Determine the smallest possible value of the largest term in an arithmetic sequence composed of seven distinct prime numbers.
Translate the above text into English, keep the original text's line breaks and format, and output the translation result directly. | 907 | 50 | 3 |
math | 14. Given that for all $x \in \mathbf{R}$,
$$
3 \sin ^{2} x-\cos ^{2} x+4 a \cos x+a^{2} \leqslant 31 \text{. }
$$
Find the range of real numbers $a$. | [-4,4] | 69 | 5 |
math | A natural number $n$ has the following property:
For arbitrary real numbers $a_{1}, a_{2}, \ldots, a_{d}$, which both satisfy $a_{1}+a_{2}+\ldots+a_{d}=2013$ and $0 \leq a_{i} \leq 1$ for $i=1,2, \ldots, d$, there exists a partition of the set of these real numbers into $n$ pairwise disjoint subsets (some of which may ... | 4025 | 142 | 4 |
math | [ Classical combinatorics (miscellaneous).] $[$ Inclusion-exclusion principle $]$
From the sequence of natural numbers, all numbers that are squares or cubes of integers have been erased. Which of the remaining numbers is in the hundredth place? | 112 | 52 | 3 |
math | 3. Find the smallest $n>2016$, such that $1^{n}+2^{n}+3^{n}+4^{n}$ is not divisible by 10. | 2020 | 43 | 4 |
math | 1. Person A and Person B are standing by the railway waiting for a train. It is known that the train is moving at a constant speed. At a certain moment, when the front of the train passes them, A starts walking in the same direction as the train at a constant speed, while B walks in the opposite direction at the same s... | 180 | 114 | 3 |
math | 111. Reduce the equations of the lines to normal form:
1) $2 x-3 y-10=0$
2) $3 x+4 y=0$ | \frac{2}{\sqrt{13}}x-\frac{3}{\sqrt{13}}y-\frac{10}{\sqrt{13}}=0\frac{3}{5}x+\frac{4}{5}0 | 39 | 54 |
math | 26. Investigate the possibility of equality:
$$
k y^{3}=x^{3}+z^{3}
$$
where $k$ is an integer, if $x, y, z$ are integers forming an increasing arithmetic progression. | (1-),,(1+) | 52 | 7 |
math | ## Task 12/68
Calculate the sum
$$
\sum_{k=1}^{n} k \cdot\binom{n}{k}
$$ | n\cdot2^{n-1} | 36 | 9 |
math | A quadruplet of distinct positive integers $(a, b, c, d)$ is called $k$-good if the following conditions hold:
1. Among $a, b, c, d$, no three form an arithmetic progression.
2. Among $a+b, a+c, a+d, b+c, b+d, c+d$, there are $k$ of them, forming an arithmetic progression.
$a)$ Find a $4$-good quadruplet.
$b)$ What ... | k = 4 | 120 | 5 |
math | Example 1 Let $a, b, c > 0$, and $abc + a + c = b$. Find the maximum value of
$$
p=\frac{2}{a^{2}+1}-\frac{2}{b^{2}+1}+\frac{3}{c^{2}+1}
$$
(1999, Vietnam Mathematical Olympiad) | \frac{10}{3} | 82 | 8 |
math | Consider the set $S=\{1,2,...,n\}$. For every $k\in S$, define $S_{k}=\{X \subseteq S, \ k \notin X, X\neq \emptyset\}$. Determine the value of the sum \[S_{k}^{*}=\sum_{\{i_{1},i_{2},...,i_{r}\}\in S_{k}}\frac{1}{i_{1}\cdot i_{2}\cdot...\cdot i_{r}}\] [hide]in fact, this problem was taken from an austrian-polish[/hide... | \frac{nk - 1}{k + 1} | 134 | 13 |
math | 1. A set of integers $T$ is orensan if there exist integers $a<b<c$ such that $a$ and $c$ belong to $T$ and $b$ does not belong to $T$. Find the number of subsets $T$ of $\{1,2, \ldots, 2019\}$ that are orensan. | 2^{2019}-2039191 | 79 | 14 |
math | 2. Find the sum $a_{1}+a_{2}+\ldots+a_{1985}$, if
$$
a_{k-1}=\frac{3 k^{2}-3 k+1}{\left(k^{2}-k\right)^{3}} \text { for } k \geq 2
$$ | \frac{1986^{3}-1}{1986^{3}} | 75 | 19 |
math | 0.35 \cdot 160+0.1 x=0.2 \cdot 160+0.2 x, 0.15 \cdot 160=0.1 x, x=240 \text {. }
$$
In total, it results in $160+240=400$ g of solution. | 400 | 82 | 3 |
math | If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$. | 97 | 57 | 2 |
math | 8.2. In the bus, there are single and double seats. In the morning, 13 people were sitting in the bus, and there were 9 completely free seats. In the evening, 10 people were sitting in the bus, and 6 seats were completely free. How many seats are there in the bus? | 16 | 69 | 2 |
math | 2. Given the complex number $z=\frac{-1+i \sqrt{3}}{2}$. Calculate the product:
$$
\left(z+\frac{1}{z}\right)\left(z^{2}+\frac{1}{z^{2}}\right) \ldots\left(z^{2012}+\frac{1}{z^{2012}}\right)
$$ | 2^{670} | 85 | 6 |
math | 7. Let $f(x)=a x+b$, where $a, b$ are real numbers, $f_{1}(x)=f(x), f_{n+1}(x)=f\left(f_{n}(x)\right)$, $n=1,2, \cdots$, If $f_{7}(x)=128 x+381$, then $a+b=$ $\qquad$ . | 5 | 89 | 1 |
math | 6. Find the sum of the integers that belong to the set of values of the function $f(x)=\log _{3}(40 \cos 2 x+41)$ for $x \in[(5 / 3)(\operatorname{arctg}(1 / 5)) \cos (\pi-\arcsin (-0.8)) ; \operatorname{arctg} 3]$ (10 points) | 9 | 93 | 1 |
math | 2. let $M$ be a finite set of real numbers with the following property: From every three different elements of $M$, two can always be selected whose sum lies in $M$. What is the maximum number of elements $M$ can have?
## Solution | 7 | 54 | 1 |
math | I1 (2-2, Hungary) For which values of $x$ is the inequality $\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+$ 9 satisfied. | x\in[-\frac{1}{2},0)\cup(0,\frac{45}{8}) | 48 | 24 |
math | $4.76 \sin 20^{\circ} \sin 40^{\circ} \sin 60^{\circ} \sin 80^{\circ}=\frac{3}{16}$. | \frac{3}{16} | 50 | 8 |
math | 6. Let $[x]$ denote the greatest integer not exceeding the real number $x$. If
$$
A=\left[\frac{7}{8}\right]+\left[\frac{7^{2}}{8}\right]+\cdots+\left[\frac{7^{2019}}{8}\right]+\left[\frac{7^{2020}}{8}\right],
$$
then the remainder when $A$ is divided by 50 is | 40 | 99 | 2 |
math | 2. Polynomial
$$
p(x)=x^{3}-224 x^{2}+2016 x-d
$$
has three roots that form a geometric progression. Then the value of $d$ is $\qquad$ | 729 | 51 | 3 |
math | XXVIII - II - Task 3
In a hat, there are 7 slips of paper. On the $ n $-th slip, the number $ 2^n-1 $ is written ($ n = 1, 2, \ldots, 7 $). We draw slips randomly until the sum exceeds 124. What is the most likely value of this sum? | 127 | 82 | 3 |
math | With a brief calculation, determine the value of the following product:
$$
\frac{6 \cdot 27^{12}+2 \cdot 81^{9}}{8000000^{2}} \cdot \frac{80 \cdot 32^{3} \cdot 125^{4}}{9^{19}-729^{6}}
$$ | 10 | 86 | 2 |
math | Four. (This question is worth 20 points) In the cyclic hexagon $A B C D E F$, $A B=B C=C D=3 \text{~cm}, D E=E F=$ $F A=5 \text{~cm}$. Find the area of this hexagon.
. | \frac{47}{2} \sqrt{3} | 66 | 13 |
math | Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. What is the largest possible area of pentagon $ABCDE$?
$\text{(A) }9+3\sqrt{2}\qquad\text{(B) }13\qquad\text{(C) }12+\sqrt{2}\qquad\text{(D) }14\qquad\text{(E) }12+\sqrt{6}-\sqrt{3}$ | 9 + 3\sqrt{2} | 111 | 9 |
math | Three, (Full marks 30 points) In $\triangle A B C$, $A B=A C$, the altitude $A D=5$ on $B C$, $M$ is a point on $A D$, $M D=1$, and $\angle B M C=3 \angle B A C$. Try to find the perimeter of $\triangle A B C$. | \frac{10 \sqrt{7}}{7}(1+2 \sqrt{2}) | 79 | 21 |
math | 2. Compute $\frac{\tan ^{2}\left(20^{\circ}\right)-\sin ^{2}\left(20^{\circ}\right)}{\tan ^{2}\left(20^{\circ}\right) \sin ^{2}\left(20^{\circ}\right)}$. | 1 | 70 | 1 |
math | 1. Dima wrote a sequence of 0s and 1s in his notebook. Then he noticed that a 1 follows a 0 sixteen times, a 0 follows a 1 fifteen times, and a 0 follows 01 eight times. How many times does a 0 follow 11? | 7 | 66 | 1 |
math | # 3. Clone 1
On an island, there live knights who always tell the truth, and liars who always lie. Before a friendly match, 30 islanders gathered in T-shirts with numbers on them—arbitrary natural numbers. Each of them said: “I have a T-shirt with an odd number.” After that, they exchanged T-shirts, and each said: “I ... | 15 | 99 | 2 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{8 ;-7 ;-13\}$
$p=\{0 ; 1 ; 5\}$
$q=\{3 ;-1 ; 2\}$
$r=\{-1 ; 0 ; 1\}$ | -4p+3q+r | 73 | 7 |
math | 3. The sequence $\left(a_{n}\right)$ is defined by $a_{1}=\frac{1}{2}, a_{n}=\frac{a_{n-1}}{2 n a_{n-1}+1}$ for $n>1$. Determine the sum $a_{1}+\ldots+a_{k}$, for any natural number $k$. | \frac{k}{k+1} | 81 | 8 |
math | # Task 17-19. (2 points per Task)
Alena opened a multi-currency deposit at "Severny" Bank for 3 years. The deposit involves the deposit of funds in three currencies: euros, dollars, and rubles. At the beginning of the deposit agreement, Alena's account contained 3000 euros, 4000 dollars, and 240000 rubles. The interes... | 3280 | 425 | 4 |
math | ## Task 2 - 150722
The Winkler family has exactly three children.
On January 1, 1975, the oldest child was twice as old as the second, and this child in turn was twice as old as the youngest child. The mother was twice as old as her three children combined. The father was as old as the mother and the youngest child c... | 2,youngestchild:2,child:4,oldest:8,mother:28,father:30,grfather:72 | 151 | 33 |
math | 8. Let two strictly increasing sequences of positive integers $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ satisfy $a_{10}=b_{10}<2017$, for any positive integer $n$, there is $a_{n+2}=a_{n+1}+a_{n}, b_{n+1}=2 b_{n}$. Then all possible values of $a_{1}+b_{1}$ are . $\qquad$ | 13,20 | 110 | 5 |
math | 37 Let $a$, $b$, $c$ all be positive integers greater than 1. Find the minimum possible value of the algebraic expression $\frac{a+b+c}{2}-\frac{[a, b]+[b, c]+[c, a]}{a+b+c}$. | \frac{3}{2} | 64 | 7 |
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