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 | Problem 5.2. During a physical education class, 25 students from 5B class lined up. Each of the students is either an excellent student who always tells the truth, or a troublemaker who always lies.
Excellent student Vlad stood in the 13th place. Everyone except Vlad stated: "There are exactly 6 troublemakers between ... | 12 | 87 | 2 |
math | 3 . If $\mathrm{iog}_{2}\left(\log _{8} x\right)=\log _{8}\left(\log _{2} x\right)$, find $\left(\log _{2} x\right)^{2}$. | 27 | 56 | 2 |
math | 15. (6 points) There are three numbers $a, b, c$, $a \times b=24, a \times c=36, b \times c=54$, then $a+b+c=$ | 19 | 49 | 2 |
math | 24. Find the least possible value of $f(x)=\frac{9}{1+\cos 2 x}+\frac{25}{1-\cos 2 x}$, where $x$ ranges over all real numbers for which $f(x)$ is defined. | 32 | 57 | 2 |
math | # Problem No. 8 (10 points)
A water heater with a power of \( P = 500 \mathrm{W} \) is used to heat a certain amount of water. When the heater is turned on for \( t_{1} = 1 \) minute, the temperature of the water increases by \( \Delta T = 2^{\circ} \mathrm{C} \), and after the heater is turned off, the temperature de... | 2.38 | 167 | 4 |
math | 5. Let $p$ be a prime number. How many colors are needed at minimum to place tokens on a square $p \times p$ board, each token being colored with one of these colors, so that each field of the board has exactly one token and there are no two tokens of the same color that attack each other? Two tokens attack each other ... | p | 101 | 1 |
math | B1. Find all real numbers $x$ for which $\left(x^{2}-7 x+11\right)^{x^{2}-13 x+42}=1$. | 2,3,4,5,6,7 | 40 | 11 |
math | For real numbers $a$ and $b$, define
$$f(a,b) = \sqrt{a^2+b^2+26a+86b+2018}.$$
Find the smallest possible value of the expression $$f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b).$$ | 4 \sqrt{2018} | 80 | 9 |
math | Example 4 Let $D=\{1,2, \cdots, 10\}, f(x)$ be a one-to-one mapping from $D$ to $D$, and let $f_{n+1}(x)=$ $f\left(f_{n}(x)\right), n \in \mathbf{N}_{+} ; f_{1}(x)=f(x)$. Try to find a permutation $x_{1}, x_{2}, \cdots, x_{10}$ of $D$, such that $\sum_{i=1}^{10} x_{i} f_{2520}(i)=220$. | 10,9,8,7,6,5,4,3,2,1 | 141 | 20 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{(2 n+1)!+(2 n+2)!}{(2 n+3)!}
$$ | 0 | 48 | 1 |
math | 3. The father is now 35 years old, and the son is 7 years old. In how many years will the father be twice as old as the son? | 21 | 36 | 2 |
math | 2. Find all quadratic trinomials $a x^{2}+b x+c$ such that if any of the coefficients $a, b, c$ is increased by 1, the resulting new quadratic trinomial will have a double root. | \frac{1}{8}x^{2}-\frac{3}{4}x+\frac{1}{8} | 53 | 26 |
math | 52. Find the particular solution of the equation $2 y d x=(1+x) d y$, if $y=4$ when $x=1$. | (1+x)^2 | 34 | 5 |
math | Example 6 Arrange the terms of the arithmetic sequence $2,6,10,14, \cdots, 2006$ tightly together to form a “large number”: $A=261014 \cdots 2006$. Find the remainder when $A$ is divided by 9. | 8 | 71 | 1 |
math | $\mathrm{Na}$ our planned cottage, we brought the cat Vilda. On Monday, she caught $\frac{1}{2}$ of all the mice, on Tuesday $\frac{1}{3}$ of the remaining, on Wednesday $\frac{1}{4}$ of those left after Tuesday's hunt, and on Thursday only $\frac{1}{5}$ of the remainder. On Friday, the remaining mice preferred to move... | 60 | 144 | 2 |
math | 8.2. Find the largest natural number with all distinct digits such that the sum of any two of its digits is a prime number. | 520 | 28 | 3 |
math | 6. For the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, the left and right foci are $F_{1}, F_{2}$, and $P$ is any point on the ellipse that does not coincide with the left or right vertices. Points $I, G$ are the incenter and centroid of $\triangle P F_{1} F_{2}$, respectively. Given that for any point ... | \frac{1}{3} | 148 | 7 |
math | Described quadrilaterals Area of a trapezoid
The bases of an isosceles trapezoid circumscribed around a circle are 2 and 18. Find the area of the trapezoid.
# | 60 | 50 | 2 |
math | Zhendarov R.G.
On the lateral sides $AB$ and $BC$ of an isosceles triangle $ABC$, points $K$ and $L$ are taken respectively, such that $AK + LC = KL$. From the midpoint $M$ of segment $KL$, a line parallel to $BC$ is drawn, and this line intersects side $AC$ at point $N$. Find the measure of angle $KNL$. | 90 | 91 | 2 |
math | 1. Find the number of points in the $x O y$ plane having natural coordinates $(x, y)$ and lying on the parabola $y=-\frac{x^{2}}{9}+33$. | 5 | 46 | 1 |
math | 9. (12 points) Shuaishuai memorized words for 7 days. Starting from the 2nd day, he memorized 1 more word each day than the previous day, and the sum of the number of words memorized in the first 4 days is equal to the sum of the number of words memorized in the last 3 days. How many words did Shuaishuai memorize in to... | 84 | 98 | 2 |
math | II. Fill-in-the-blank Questions (9 points each, total 54 points)
1. Remove all perfect squares and cubes from the natural numbers, and arrange the remaining numbers in ascending order to form a sequence $\left\{a_{n}\right\}$. Then $a_{2008}=$ $\qquad$ | 2062 | 71 | 4 |
math | ## Task Condition
Approximately calculate using the differential.
$y=\sqrt{x^{3}}, x=0.98$ | 0.97 | 26 | 4 |
math | 2. In a sports store, over two days, thirteen pairs of sneakers, two sports suits, and one T-shirt were sold, with the same amount of money earned on the first day as on the second day (from the sale of the aforementioned items). One pair of sneakers is cheaper than a sports suit and more expensive than a T-shirt by th... | 8 | 91 | 1 |
math | $1 \cdot 53$ A student did not notice the multiplication sign written between two 7-digit numbers, mistaking it for a 14-digit number. This 14-digit number happens to be 3 times the original product. Try to find these 3 numbers. | 1666667,3333334,16666673333334 | 59 | 30 |
math | 21. (Problem from the 54th Belarusian Mathematical Olympiad) Let the positive integer $A=\overline{a_{n} a_{n-1} \cdots a_{1} a_{0}}, a_{n}, a_{n-1}, \cdots$, $a_{0}$ are all non-zero and not all equal ($n$ is a positive integer). The numbers
$$
\begin{array}{l}
A_{1}=\overline{a_{n-1} \cdots a_{1} a_{0} a_{n}}, \\
A_{... | \underbrace{142857}\underbrace{142857}\cdots\underbrace{142857},k\in{N}_{+} | 287 | 42 |
math | Example 3.15 (2007 Girls' Mathematical Olympiad) Let $n>3$ be an integer, and let $a_{1}, a_{2}, \cdots, a_{n}$ be non-negative real numbers satisfying $a_{1}+a_{2}+\cdots+a_{n}=2$. Find the minimum value of
$$\frac{a_{1}}{a_{2}^{2}+1}+\frac{a_{2}}{a_{3}^{2}+1}+\cdots+\frac{a_{n}}{a_{1}^{2}+1}.$$ | \frac{3}{2} | 136 | 7 |
math | 3. Let integer $n(n>3)$, non-negative real numbers $a_{1}, a_{2}$, $\cdots, a_{n}$ satisfy $a_{1}+a_{2}+\cdots+a_{n}=2$.
Find the minimum value of $\frac{a_{1}}{a_{2}^{2}+1}+\frac{a_{2}}{a_{3}^{2}+1}+\cdots+\frac{a_{n}}{a_{1}^{2}+1}$.
(Provided by Zhu Huawei) | \frac{3}{2} | 124 | 7 |
math | Let's construct a $12.75^{\circ}$ angle as simply as possible. | 12.75=\frac{36+15}{2^{2}} | 20 | 18 |
math | The class teacher calculated the class average grades for each subject, and Kati helped her by recalculating the grades based on how many students received each consecutive grade. When comparing the results of the first subject, it turned out that Kati had used the data for consecutive fives in reverse order, taking th... | 6 | 107 | 1 |
math | 7.3. Yesterday, Sasha cooked soup and put in too little salt, so he had to add more salt. Today, he put in twice as much salt, but still had to add more salt, though only half the amount he added yesterday. By what factor does Sasha need to increase today's portion of salt so that he doesn't have to add more salt tomor... | 1.5 | 87 | 3 |
math | For a given rational number $r$, find all integers $z$ such that \[2^z + 2 = r^2\mbox{.}\]
[i](Swiss Mathematical Olympiad 2011, Final round, problem 7)[/i] | (\pm 2, 1) | 58 | 9 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{1-2+3-4+\ldots+(2 n-1)-2 n}{\sqrt[3]{n^{3}+2 n+2}}
$$ | -1 | 63 | 2 |
math | Let $ABC$ be a triangle with $AB = 9$, $BC = 10$, and $CA = 17$. Let $B'$ be the reflection of the point $B$ over the line $CA$. Let $G$ be the centroid of triangle $ABC$, and let $G'$ be the centroid of triangle $AB'C$. Determine the length of segment $GG'$. | \frac{48}{17} | 83 | 9 |
math | Example 2 Determine the smallest natural number $k$, such that for any $a \in [0,1]$ and any $n \in \mathbf{N}$ we have
$$
a^{k}(1-a)^{n}<\frac{1}{(n+1)^{3}} .
$$ | 4 | 66 | 1 |
math | 13. Let $x \geqslant y \geqslant z \geqslant \frac{\pi}{12}$, and $x+y+z=\frac{\pi}{2}$. Find the maximum and minimum values of the product $\cos x \sin y \cos z$. | \frac{2+\sqrt{3}}{8} | 64 | 12 |
math | The length of every edge of a rectangular cuboid is an integer. By adding the volume of the cuboid, half of its surface area, and the lengths of the edges meeting at one vertex, we get 2000. What are the lengths of the cuboid's edges? | 28,22,2 | 59 | 7 |
math | ## Task A-3.6.
Let $a, b$ and $c$ be the lengths of the sides of a triangle opposite to angles of size $\alpha, \beta$ and $\gamma$.
If $9 a^{2}+9 b^{2}=19 c^{2}$, determine $\frac{\operatorname{ctg} \gamma}{\operatorname{ctg} \alpha+\operatorname{ctg} \beta}$. | \frac{5}{9} | 96 | 7 |
math | Example 3. Find the integral $\int \frac{x}{x^{3}+1} d x$. | -\frac{1}{3}\ln|x+1|+\frac{1}{6}\ln(x^{2}-x+1)+\frac{1}{\sqrt{3}}\operatorname{arctg}\frac{2x-1}{\sqrt{3}}+C | 23 | 60 |
math | I5.4 Given that $\left(R x^{2}-x+1\right)^{1999} \equiv a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{3998} x^{3998}$. If $S=a_{0}+a_{1}+a_{2}+\ldots+a_{3997}$, find the value of $S$. | 0 | 98 | 1 |
math | 6. Given $f(x)=x^{3}+b x^{2}+c x+d$ is an increasing function on $(-\infty, 0)$, and a decreasing function on $[0,2]$, and the equation $f(x)=0$ has three roots, which are $\alpha, 2, \beta$, then the range of $|\alpha-\beta|$ is | [3,+\infty) | 84 | 7 |
math | 6-8 Given $\sin \alpha+\sin \beta=p, \cos \alpha+\cos \beta=q$, try to find the values of $\sin (\alpha+\beta)$ and $\cos (\alpha+\beta)$. (Beijing Mathematical Competition, 1963) | \sin(\alpha+\beta)=\frac{2pq}{p^{2}+q^{2}};\cos(\alpha+\beta)=\frac{q^{2}-p^{2}}{q^{2}+p^{2}} | 58 | 50 |
math | 85. Bacteria have the following development rule: each one lives for 1 hour and every half hour it produces one new one (a total of two during its life). What will be the offspring of one bacterium 6 hours after its birth? | 377 | 52 | 3 |
math | 1. I 1 (USA 4) ${ }^{\mathrm{IMO}}$ Alice, Betty, and Carol took the same series of examinations. There was one grade of $A$, one grade of $B$, and one grade of $C$ for each examination, where $A, B, C$ are different positive integers. The final test scores were | Alice | Betty | Carol | | :---: | :---: | :---: | | 20... | Carol | 126 | 1 |
math | 1. Let $A=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$. Determine the set of matrices:
$$
M=\left\{X \in M_{2}(\mathbb{Z}) \mid A X=X^{2} A\right\}
$$
Prof.Burtea Marius, Alexandria | {O_{2},I_{2},(\begin{pmatrix}0&1\\-1&-1\end{pmatrix}),(\begin{pmatrix}-1&1\\-1&0\end{pmatrix}),(\begin{pmatrix}-1&-1\\1&0\end{pmatrix}),(\begin{array} | 81 | 75 |
math | Example 1. Calculate the definite integral
$$
\int_{0}^{\pi / 2} \frac{\sin x}{2+\sin x} d x
$$ | \frac{\pi}{2}-\frac{2\pi}{3\sqrt{3}} | 38 | 20 |
math | Let's determine the value of the product
$$
z=a \sqrt{a} \sqrt[4]{a} \sqrt[8]{a} \ldots \sqrt[2^{n}]{a} \ldots
$$
when $n$ is infinitely large. | ^2 | 59 | 2 |
math | Calculate the remainder of the Euclidean division of $2022^{2023^{2024}}$ by 19. | 8 | 31 | 1 |
math | 4. (1) Find all positive integers $n$, such that there exist $a, b \in \mathbf{N}^{*}$, satisfying: $[a, b]=n!$, $(a, b)=1998 ;$
(2) Under the condition that (1) holds, to ensure that the number of pairs of positive integers $(a, b)$ with $a \leqslant b$ does not exceed 1998, what condition should $n$ satisfy? | 37 \leqslant n \leqslant 40 | 108 | 16 |
math | One, (20 points) Real numbers $x, y, z, w$ satisfy $x \geqslant y \geqslant z \geqslant w \geqslant 0$, and $5 x+4 y+3 z+6 w=100$. Find the maximum and minimum values of $x+y+z+w$. | 20 | 78 | 2 |
math | Let's calculate $x$, if
$$
x^{2}+(2 m p+2 n q)^{2}+(2 m q-2 n p)^{2}=\left(m^{2}+n^{2}+p^{2}+q^{2}\right)^{2}
$$ | x_{1}=^{2}+n^{2}-p^{2}-q^{2},\quadx_{2}=-^{2}-n^{2}+p^{2}+q^{2} | 65 | 44 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$:
$$
f(x) f(y)=f(x-y)
$$ | f(x)=0orf(x)=1 | 47 | 8 |
math | Zhendarov R.G.
Find all natural numbers that have exactly six divisors, the sum of which is equal to 3500. | 1996 | 30 | 4 |
math | 3.362. $\frac{6 \sin \alpha-7 \cos \alpha+1}{8 \sin \alpha+9 \cos \alpha-1}$, if $\operatorname{tg} \frac{\alpha}{2}=4$. | -\frac{85}{44} | 54 | 9 |
math | Let $a, b, c, d$ be the roots of the quartic polynomial $f(x) = x^4 + 2x + 4$. Find the value of
$$\frac{a^2}{a^3 + 2} + \frac{b^2}{b^3 + 2} + \frac{c^2}{c^3 + 2} + \frac{d^2}{d^3 + 2}.$$ | \frac{3}{2} | 100 | 7 |
math | 4. Given real numbers $x, y, z$ satisfy
$$
x^{2}+2 y^{2}+3 z^{2}=24 \text {. }
$$
Then the minimum value of $x+2 y+3 z$ is $\qquad$ | -12 | 59 | 3 |
math | In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the giv... | 11 | 125 | 2 |
math | 10. (15 points) 30 students are lined up in ascending order of height, with the height difference between any two adjacent students being the same. The sum of the heights of the first 10 students is 1450 cm, and the sum of the heights of the first 20 students is 3030 cm. What is the sum of the heights of all 30 student... | 4740 | 95 | 4 |
math | 9. Suppose $f$ is a function satisfying $f\left(x+x^{-1}\right)=x^{6}+x^{-6}$, for all $x \neq 0$. Determine $f(3)$. | 322 | 48 | 3 |
math | 1. In a cup, there is 3 dl of white coffee. The ratio of milk to coffee in this mixture is $3: 2$. Marko drank a sixth of the white coffee and then topped up the cup to the brim with milk. After that, he drank a fifth of the white coffee and refilled the cup to the brim, adding five times more milk than coffee. After d... | 2.7\mathrm{dl} | 127 | 8 |
math | 7、Remove the big and small jokers from a deck of cards, and randomly draw 5 cards from the remaining 52 cards. The probability that at least two of the cards have the same number (or letters $\mathrm{J}, \mathrm{Q}, \mathrm{K}, \mathrm{A}$) is $\underline{0.49}$ (require calculating this probability value, accurate to ... | 0.49 | 90 | 4 |
math | 3. The set
$$
A=\{\sqrt[n]{n} \mid n \in \mathbf{N} \text { and } 1 \leqslant n \leqslant 2020\}
$$
has the maximum element as $\qquad$ . | \sqrt[3]{3} | 63 | 7 |
math | 4. In grandmother's garden, apples have ripened: Antonovka, Grushovka, and White Naliv. If there were three times as many Antonovka apples, the total number of apples would increase by $70 \%$. If there were three times as many Grushovka apples, it would increase by $50 \%$. By what percentage would the total number of... | 80 | 97 | 2 |
math | 4. The record of a natural number x in a base-23 numeral system consists of $2 m$ identical digits. It turned out that in the 23-ary representation of the number $x^{2}$, the extreme digits are the same, and the other $4 m-2$ digits are zero. Find all such numbers x. Provide the answer in the 23-ary numeral system. (Di... | DD_{23} | 107 | 5 |
math | In the rectangle $ABCD$, the sides are $AB=3, BC=2$. $P$ is a point on the side $AB$ such that the line $PD$ touches, at point $E$, the circle with diameter $BC$. The line passing through the center of the circle and $E$ intersects the side $AB$ at $Q$. What is the area of the triangle $PQE$? | \frac{1}{24} | 87 | 8 |
math | 2. On a line, several points were marked, including points $A$ and $B$. All possible segments with endpoints at the marked points are considered. Vasya calculated that point $A$ is inside 40 of these segments, and point $B$ is inside 42 segments. How many points were marked? (The endpoints of a segment are not consider... | 14 | 81 | 2 |
math | 2. From the 99 natural numbers $1,2,3, \cdots, 99$, the number of ways to choose two different numbers such that their sum is less than 99 is $\qquad$ ways. | 2352 | 50 | 4 |
math | Find all positive integers $n$ such that there exists a sequence $a_{1}$, $a_{2}, \cdots, a_{n}$ consisting of $-1,1$ satisfying $\sum_{i=1}^{n} a_{i} i^{2}=0$. | {\begin{pmatrix}4k+3,&k\geqslant1,\\4k,\quadk\geqslant20\end{pmatrix}.} | 61 | 39 |
math | 2. (2 points) In a class, 10 people gathered, each of whom is either a knight, who always tells the truth, or a liar, who always lies. Each of them was asked to first name the number of knights in the room, and then the number of liars. It turned out that each number from 0 to 9 was named exactly twice. How many knight... | from0to2 | 101 | 4 |
math | [Examples and counterexamples. Constructions] Evaluation + example $\quad]$
On an island, there are 100 knights and 100 liars. Each of them has at least one friend. One day, exactly 100 people said: "All my friends are knights," and exactly 100 people said: "All my friends are liars." What is the smallest possible num... | 50 | 102 | 2 |
math | 3. Let $\alpha$, $\beta$, and $\gamma$ be the angles formed by the diagonal of a rectangular prism with the three faces sharing a common vertex. Then the range of $\alpha+\beta+\gamma$ is $\qquad$ . | \left(\frac{\pi}{2}, 3 \arcsin \frac{\sqrt{3}}{3}\right] | 50 | 27 |
math | The following equation needs to be solved:
$$
\sin x + \cos x = \frac{\cos 2x}{1 - 2 \sin x}
$$ | 0,\frac{3\pi}{4},\frac{3\pi}{2},\frac{7\pi}{4} | 35 | 28 |
math | 3.21. Find all solutions of the system of equations
$$
x\left(1-\frac{1}{2^{n}}\right)+y\left(1-\frac{1}{2^{n+1}}\right)+z\left(1-\frac{1}{2^{n+2}}\right)=0
$$
where $n=1,2,3,4, \ldots$ | -3x,2x | 91 | 6 |
math | 6. In the quadrilateral pyramid $P-ABCD$, it is known that $AB // CD$, $AB \perp AD$, $AB=4$, $AD=2\sqrt{2}$, $CD=2$, $PA \perp$ plane $ABCD$, $PA=4$. Let $Q$ be a point on the line segment $PB$, and the sine of the angle formed by line $QC$ and plane $PAC$ is $\frac{\sqrt{3}}{3}$. Then
$\frac{PQ}{PB}$ is $\qquad$ | \frac{7}{12} | 124 | 8 |
math | ## Task 18/79
The measures of the inradius and the sides of a triangle are to be integral members of an arithmetic sequence of the first order. How large are they if they are as small as possible? | 4,15,26,37 | 47 | 10 |
math | Solve the equation, where $x$ and $y$ are positive integers: $$ x^3-y^3=999$$ | (x, y) = (12, 9) | 29 | 14 |
math | 2. Given that $A$ is a two-digit number, the remainder when $A^{2}$ is divided by 15 is 1, then the number of $A$ that satisfies the condition is $($ )
The translation preserves the original text's line breaks and format. | 24 | 58 | 2 |
math | Example 3. Find the value of $\cos \frac{\pi}{7}+\cos \frac{3 \pi}{7}+\cos \frac{5 \pi}{7}$. | \frac{1}{2} | 40 | 7 |
math | I place $n$ pairs of socks (thus $2 n$ socks) in a line in such a way that the left sock is to the right of the right sock for each pair. How many different ways can I place my socks like this? | \frac{(2n)!}{2^n} | 51 | 10 |
math | $4 \cdot 36$ Try to find the $n$ $n$-th roots of 1, and find the sum of their $n$-th powers.
untranslated part:
$4 \cdot 36$ 试求 1 的 $n$ 个 $n$ 次方根, 并求它们 $n$ 次幂的和. | n | 84 | 1 |
math | 315*. Find all roots of the equation
$$
8 x\left(2 x^{2}-1\right)\left(8 x^{4}-8 x^{2}+1\right)=1
$$
satisfying the condition $0<x<1$. | \cos\pi/9,\cos\pi/3,\cos2\pi/7 | 59 | 19 |
math | 7. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $E$ is the midpoint of $A B$, $F$ is the midpoint of $C C_{1}$, the cosine value of the angle formed by the skew lines $E F$ and $A C_{1}$ is . $\qquad$ | \frac{2\sqrt{2}}{3} | 80 | 12 |
math | 53. Point $K$ lies on the base $AD$ of trapezoid $ABCD$, such that $|AK|=\lambda|AD|$. Find the ratio $|AM|:|AD|$, where $M$ is the point of intersection with $AD$ of the line passing through the points of intersection of the lines $AB$ and $CD$ and the lines $BK$ and $AC$.
Taking $\lambda=1 / n, n=1,2,3, \ldots$, obt... | \frac{|AM|}{|AD|}=\frac{\lambda}{\lambda+1} | 138 | 20 |
math | Find all triples of real numbers $(a, b, c)$ satisfying $$a+b+c=14, \quad a^2+b^2+c^2=84,\quad a^3+b^3+c^3=584.$$ | (4, 2, 8), (2, 4, 8), (8, 2, 4) | 52 | 28 |
math | 2. In an isosceles right $\triangle ABC$, $\angle C=90^{\circ}$, points $D$, $E$, and $F$ are on sides $AB$, $BC$, and $CA$ respectively, $FD=FE=\frac{AC}{2}$, $\angle DFE=90^{\circ}$. Then $FC: CE: EF=$ $\qquad$ . | 3: 4: 5 | 87 | 7 |
math | Let $p$, $q$, $r$, and $s$ be 4 distinct primes such that $p+q+r+s$ is prime, and the numbers $p^2+qr$ and $p^2+qs$ are both perfect squares. What is the value of $p+q+r+s$? | 23 | 67 | 2 |
math | 4. In square $A B C D$ with side 2, point $A_{1}$ lies on $A B$, point $B_{1}$ lies on $B C$, point $C_{1}$ lies on $C D$, point $D_{1}$ lies on $D A$. Points $A_{1}, B_{1}, C_{1}, D_{1}$ are the vertices of the square of the smallest possible area. Find the area of triangle $A A_{1} D_{1} .(\mathbf{1 1}$ points) | 0.5 | 119 | 3 |
math | 6. The number 25 is expressed as the sum of positive integers $x_{1}, x_{2}, \cdots, x_{k}$, where $k \leq 25$. What is the maximum value of the product of $x_{1}, x_{2}, x_{3}, \cdots$, and $x_{k}$ ? | 8748 | 76 | 4 |
math | Solve the following equation:
$$
x^{2}+5 x+4=5 \sqrt{x^{2}+5 x+28}
$$ | 4-9 | 33 | 3 |
math | Let $ABC$ be a triangle with sides $AB = 6$, $BC = 10$, and $CA = 8$. Let $M$ and $N$ be the midpoints of $BA$ and $BC$, respectively. Choose the point $Y$ on ray $CM$ so that the circumcircle of triangle $AMY$ is tangent to $AN$. Find the area of triangle $NAY$. | \frac{600}{73} | 87 | 10 |
math | [ Symmetry properties and center of symmetry ] [ Varignon parallelogram $]$
Point $O$, located inside a convex quadrilateral of area $S$, is reflected symmetrically with respect to the midpoints of its sides.
Find the area of the quadrilateral with vertices at the obtained points. | 2S | 61 | 2 |
math | Example 1. Calculate: $\sqrt{31 \cdot 30 \cdot 29 \cdot 28+1}$. (7th American Invitational Mathematics Examination) | 869 | 39 | 3 |
math | Example 1 (9th American Invitational Mathematics Examination AIME Problem) Let $r$ be a real number satisfying the condition:
$$
\left[r+\frac{19}{100}\right]+\left[r+\frac{20}{100}\right]+\left[r+\frac{21}{100}\right]+\cdots+\left[r+\frac{91}{100}\right]=546 \text {. }
$$
Find $[100 r]$. | 743 | 107 | 3 |
math | $$
\begin{array}{l}
\quad \mid\{(x, y) \mid x^{2}+y^{2} \equiv a(\bmod p), x, y \in\{0, \\
1, \cdots, p-1\}\} \mid
\end{array}
$$ | p+1 | 69 | 3 |
math | A positive integer $k$ when divided by the prime number $p$ leaves a remainder of 6. We get the same remainder when dividing $1000 - k$ by $p$; we also know that $10000 - k$ is divisible by $p$. Which is this prime number $p$? | 19 | 70 | 2 |
math | How many triangles are there in which the measures of the angles - measured in degrees - are integers? | 2700 | 20 | 4 |
math | Svyatlovsky M.
On a plane, grasshopper Kolya and 2020 of his friends are sitting. Kolya is going to jump over each of the other grasshoppers (in any order) such that the starting and ending points of each jump are symmetric relative to the grasshopper being jumped over. We will call a point a finish point if Kolya can... | C_{2020}^{1010} | 128 | 13 |
math | # Problem 5. (3 points)
In triangle $A B C$, the midpoints of sides $A B=40$ and $B C=26$ are marked as points $K$ and $L$ respectively. It turns out that the quadrilateral $A K L C$ is a tangential quadrilateral. Find the area of triangle $A B C$. | 264 | 79 | 3 |
math | Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(x) + y$ and $f(y) + x$ have the same number of $1$'s in their binary representations, for any $x,y \in \mathbb{N}$. | f(x) = x + c \ \ \forall x \in \mathbb{N} | 81 | 20 |
math | Let $P(n)$ be the number of permutations $\left(a_{1}, \ldots, a_{n}\right)$ of the numbers $(1,2, \ldots, n)$ for which $k a_{k}$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P(n)$ is a multiple of 2010 . | 4489 | 89 | 4 |
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