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 | 448. The sides $a$ and $b$ of a rectangle change according to the laws $a=(2 t+1)$ cm, $b=(3 t+2)$ cm. At what rate is its area $S$ changing at the moment $t=4$ s? | 55 | 61 | 2 |
math | 2. a) Determine the integers $a, b, c$ that satisfy the relation: $a^{2}-a+|b-3|+|c-9|=0$.
b) Determine the rational numbers $x, y, z$ that satisfy the relations: $|x-y|=2|y-z|=3|z-x|$ and $x+y+z=2014$. | \in{0,1},b=3,=9,\frac{2014}{3} | 84 | 23 |
math | 7. Let the set $T=\left\{(a, b, c) \mid a, b, c \in \mathbf{N}^{*}\right.$, and $a, b, c$ can form the side lengths of some triangle $\}$. Then the value of $\sum_{(a, b, c) \in T} \frac{2^{a}}{3^{b} \cdot 5^{c}}$ is $\qquad$. | \frac{17}{21} | 100 | 9 |
math | 5. Given real numbers $a, b, c$, and $b \neq 0$. If real numbers $x_{1},$ $x_{2}, y_{1}, y_{2}$ satisfy $x_{1}^{2}+a x_{2}^{2}=b, x_{2} y_{1}-x_{1} y_{2}=a$, $x_{1} y_{1}+a x_{2} y_{2}=c$, then the value of $y_{1}^{2}+a y_{2}^{2}$ is $\qquad$ | \frac{c^{2}+a^{3}}{b} | 128 | 15 |
math | Problem 2. On a shelf, there are three books. The first has 90, the second 110, and the third 150 pages. The covers of the books are of equal thickness, and each of them is 2 mm thick. How many millimeters thick are the books together if it is known that 10 pages have a thickness of $1 \mathrm{~mm}$? | 47\mathrm{~} | 87 | 7 |
math | 3. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=1, a_{2}=2$, $a_{n} a_{n+1} a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$, and $a_{n+1} a_{n+2} \neq$ 1. Then $a_{1}+a_{2}+\cdots+a_{2004}=$ $\qquad$ | 4008 | 114 | 4 |
math | ## Task A-1.1.
Determine the natural number $n$ such that the sum of its two smallest divisors is 6, and the sum of its two largest divisors is 1122. | 935 | 46 | 3 |
math | Consider numbers of the form $1a1$, where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?
[i]Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$, $91719$.[/i] | 55 | 76 | 2 |
math | Ana has $22$ coins. She can take from her friends either $6$ coins or $18$ coins, or she can give $12$ coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have. | 4 | 59 | 1 |
math | ## 192. Math Puzzle $5 / 81$
The students of class 7 b collected a total of $336 \mathrm{~kg}$ of waste paper. From $1 \mathrm{~kg}$ of waste paper, a paper factory produces $700 \mathrm{~g}$ of pure white paper, and from every $30 \mathrm{~g}$ of this, a writing pad is made.
What is the maximum number of pads that c... | 7840 | 110 | 4 |
math | 20. Find the value of $\frac{1}{\sin 10^{\circ}}-4 \sin 70^{\circ}$. | 2 | 33 | 1 |
math | 1. Given real numbers $a>0, b>0$, satisfying $a+\sqrt{a}=2008, b^{2}+b=2008$. Then the value of $a+b$ is $\qquad$ | 2008 | 52 | 4 |
math | Let $\Phi$.
On the board, $N \geq 9$ different non-negative numbers, each less than one, are written. It turns out that for any eight different numbers on the board, there is a ninth, different from them, such that the sum of these nine numbers is an integer. For which $N$ is this possible? | 9 | 73 | 1 |
math | . Determine the units digit of $1789^{1789}$.
. | 9 | 19 | 1 |
math | Yasinsky V.
On the plane, there are $n(n>2)$ points, no three of which lie on the same line. In how many different ways can this set of points be divided into two non-empty subsets such that the convex hulls of these subsets do not intersect? | \frac{1}{2}n(n-1) | 59 | 12 |
math | 7.1. (GDR, 74). What is greater: $\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}-\sqrt{2}$ or 0? | 0 | 43 | 1 |
math | 9. Let $[x]$ denote the greatest integer not exceeding the real number $x$, and $\{x\}=x-[x]$. Then the solution to the equation
$$
[x]^{4}+\{x\}^{4}+x^{4}=2048
$$
is | -3-\sqrt{5} | 65 | 7 |
math | 13.337. At a plant that manufactures instant coffee, a batch of coffee beans for processing was delivered in late May. One mechanism, which grinds the beans, was put into operation on Monday, June 1st, and ground $m$ kg daily. Starting from June 6th, a second mechanism was added, which ground $n$ kg daily. By the end o... | June17 | 139 | 3 |
math | Five. (20 points) Given the sequence of positive integers $\left\{T_{m, n}\right\}$ satisfies:
(1) $T_{m, m}=2^{m}$;
(2) $T_{m, n}=T_{n, m}$;
(3) $\left(T_{m, n}\right)^{m+n}=\left(T_{m, m+n}\right)^{n}$.
Here $m, n$ are any positive integers. Try to find the general term formula for the sequence $\left\{T_{m, n}\righ... | 2^{(m \cdot n)} | 127 | 8 |
math | Convex quadrilateral $ABCD$ satisfies $\angle{CAB} = \angle{ADB} = 30^{\circ}, \angle{ABD} = 77^{\circ}, BC = CD$ and $\angle{BCD} =n^{\circ}$ for some positive integer $n$. Compute $n$. | 68^\circ | 71 | 4 |
math | 10.64 On the plane, what is the minimum number of points needed so that the distances between each pair of points can take on each of the values $1,2,4,8,16,32,64$? | 8 | 52 | 1 |
math | 2. In $\triangle A B C$, if $\sin A+\cos A=-\frac{1}{3}$, then $\cos 2 A=$ . $\qquad$ | \frac{\sqrt{17}}{9} | 37 | 11 |
math | * Find all positive integers $n$, such that $n^{4}-4 n^{3}+22 n^{2}-36 n+18$ is a perfect square. | 1or3 | 39 | 3 |
math | Let $n\geq 2$ an integer. Find the least value of $\gamma$ such that for any positive real numbers $x_1,x_2,...,x_n$ with $x_1+x_2+...+x_n=1$ and any real $y_1+y_2+...+y_n=1$ and $0\leq y_1,y_2,...,y_n\leq \frac{1}{2}$ the following inequality holds:
$$x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right)$$ | \frac{1}{2(n-1)^{n-1}} | 144 | 15 |
math | 1.49 Find the values of $a$ such that the roots of the equation $x^{2}-a x+9 a=0$ are integers.
(Recommended by the Soviet Ministry of Education, 1990) | 100,-64,48,-12,36,0 | 49 | 17 |
math | 7. (5 points) There are several stamps with denominations of 0.5 yuan, 0.8 yuan, and 1.2 yuan, with a total denomination of 60 yuan. The number of 0.8 yuan stamps is 4 times the number of 0.5 yuan stamps. Therefore, the number of 1.2 yuan stamps is $\qquad$.
| 13 | 83 | 2 |
math | 6.30 There is a sports competition consisting of $M$ events, with athletes $A, B, C$ participating. In each event, the first, second, and third places receive $p_{1}, p_{2}, p_{3}$ points, respectively, where $p_{1}, p_{2}, p_{3}$ are positive integers, and $p_{1}>p_{2}>p_{3}$. In the end, $A$ scores 22 points, $B$ and... | C | 150 | 1 |
math | Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$,$$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$$
[i](Macedonia)[/i] | f(x) = x + 1 | 83 | 9 |
math | Example 5 Given $x y z=1, x+y+z=2, x^{2}+$ $y^{2}+z^{2}=16$. Then $\frac{1}{x y+2 z}+\frac{1}{y z+2 x}+\frac{1}{z x+2 y}=$ | -\frac{4}{13} | 70 | 8 |
math | 21.3. The numbers $2^{n}$ and $5^{n}$ start with the digit $a$. What is $a$? | 3 | 31 | 1 |
math | 4.6.15 ** Let $x, y, z \geqslant 0$, and $x+y+z=1$. Find the minimum and maximum values of $x y+y z+z x-3 x y z$.
| 0 | 51 | 1 |
math | II. (50 points)
$n$ is a positive integer, let the integer closest to $\sqrt{n}$ be $a_{n}$, let $b_{n}=n+a_{n}$, remove $b_{n}$ from the set of all positive integers, and arrange the remaining positive integers in ascending order to form the sequence $\left\{c_{n}\right\}$, try to express $c_{n}$ in terms of $n$. | c_{n}=n^{2} | 95 | 8 |
math | (7) The four real roots of the quartic polynomial $f(x)$ form an arithmetic sequence with a common difference of 2. Then, the difference between the largest and smallest roots of $f^{\prime}(x)$ is | 2\sqrt{5} | 48 | 6 |
math | $A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$. | |A + B| \ge |A| > 0 | 38 | 13 |
math | 2. Let real numbers $x, y, z, w$ satisfy $x \geqslant y \geqslant z \geqslant w \geqslant 0$, and $5 x+4 y+3 z+6 w=100$. Denote the maximum value of $x+y+z+w$ as $a$, and the minimum value as $b$. Then $a+b=$ $\qquad$ | 45 | 94 | 2 |
math | Example 1 Find all functions $f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+}$ such that for all $m, n \in \mathbf{Z}_{+}$, we have
$$
\begin{array}{l}
f(m n)=f(m) f(n), \\
(m+n) \mid(f(m)+f(n)) .{ }^{[1]}
\end{array}
$$
(2016, Turkey National Team Selection Exam) | f(n)=n^{k}(n\in{Z}_{+},k.ispositiveodd) | 106 | 20 |
math | 8.397. $\left\{\begin{array}{l}\sin x \cos y=0.25, \\ \sin y \cos x=0.75\end{array}\right.$ | x_{1}=\frac{\pi}{6}+\pi(k_{1}-k_{2}),y_{1}=\frac{\pi}{3}+\pi(k_{1}+k_{2});x_{2}=-\frac{\pi}{6}+\pi(k_{1}-k_{2}),y_{2}=\frac{2}{3}\pi+\pi(k_{1} | 46 | 83 |
math | \section*{Problem 13 - V01013}
How many diagonals does a 4775-gon have? | 11,393,150 | 31 | 10 |
math | 10. Given the standard equation of ellipse $C$ as $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>1)$, point $P$ is a moving point on the unit circle, points $A, F$ are the right vertex and right focus of ellipse $C$ respectively, and when $\angle P O A=\frac{\pi}{3}$, $|P F|=1,|P A|=\sqrt{3}$.
(1) Find the standard equ... | [3,4] | 207 | 5 |
math | 6.357 Solve the equation $3 x^{3}+2 \sqrt{3} x^{2}-21 x+6 \sqrt{3}=0$, given that the product of two of its roots is 1. | x_{1}=\sqrt{3},x_{2}=\frac{\sqrt{3}}{3},x_{3}=-2\sqrt{3} | 50 | 34 |
math | Problem 1. The watermelon and the cantaloupe have a total mass of $30 \mathrm{~kg}$. The cantaloupe and a 3 $\mathrm{kg}$ weight have half the mass of the watermelon. What is the difference in kilograms between the mass of the watermelon and the cantaloupe? | 14\mathrm{~} | 72 | 7 |
math | 4. Add three digits after 764 to make the resulting six-digit number divisible by 8, 9, and 11. | 764280 | 30 | 6 |
math | 13.306. Two hours after departure, the train stopped for 30 minutes. On the remaining part of the route to the station, repair work was being carried out, and the train was allowed a speed that was $1 / 3$ of its initial speed, as a result of which the train arrived at the station 1 hour and 10 minutes late. The next d... | 196 | 126 | 3 |
math | Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 3$ and $\frac{\cos x}{\cos y} = \frac12$. The value of $\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}$ can be expressed in the form $\frac pq$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 107 | 106 | 3 |
math | ## Task 24/70
What is the probability of rolling a sum of 12 with three dice? | \frac{25}{216} | 25 | 10 |
math | 10. Let $[x]$ denote the greatest integer not exceeding $x$, then the value of $\sum_{k=1}^{2008}\left[\frac{2008 k}{2009}\right]$ is | 2015028 | 52 | 7 |
math | 8. Robots A and B simultaneously conduct a $100 \mathrm{~m}$ track test at a uniform speed, and the automatic recorder shows: when A is $1 \mathrm{~m}$ away from the finish line, B is $2 \mathrm{~m}$ away from the finish line; when A reaches the finish line, B is $1.01 \mathrm{~m}$ away from the finish line. After calc... | 1 | 122 | 1 |
math | 12. (16 points) On the Cartesian plane, a point whose both coordinates are rational numbers is called a rational point. Find the smallest positive integer $k$ such that: for every circle that contains $k$ rational points on its circumference, the circle must contain infinitely many rational points on its circumference. | 3 | 64 | 1 |
math | 98 For $i=1,2, \cdots, n$, we have $\left|x_{i}\right|<1$, and assume $\left|x_{1}\right|+\left|x_{2}\right|+\cdots+\left|x_{n}\right|=19+$ $\left|x_{1}+x_{2}+\cdots+x_{n}\right|$, then the minimum value of the integer $n$ is $\qquad$ | 20 | 97 | 2 |
math | 1. Find the last three digits of $9^{100}-1$. | 0 | 17 | 1 |
math | 3. The arithmetic mean of the numbers $x, y, z, p$ and $q$ is equal to $a$. What is the arithmetic mean of the numbers $x+2y-3, y+2z-1, z+2p, p+2q+1$ and $q+2x+3$? | 3a | 72 | 2 |
math | Example 5 Let $x, y, z, w$ be four real numbers, not all zero. Find:
$S=\frac{x y+2 y z+z w}{x^{2}+y^{2}+z^{2}+w^{2}}$'s maximum value. | \frac{1}{2}(1+\sqrt{2}) | 62 | 13 |
math | ## Task A-3.1. (4 points)
Solve the equation $\quad \sin x \cdot \cos 2 x \cdot \cos 4 x=1$. | \frac{3\pi}{2}+2k\pi | 38 | 14 |
math |
Problem 1. Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of 0 's and 1's of length $2 n+2$ with the following property: any sequence of 0 's and 1 's of length $2 n+2$ coincides in at least $n+2$ positions with some of these $k$ sequences.
| 4 | 92 | 1 |
math | (2) Consider a tangent line to the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{3^{2}}=1$, which intersects the two symmetry axes of the ellipse at points $A$ and $B$, respectively. Then the minimum length of the line segment $AB$ is | 8 | 68 | 1 |
math | 1. Find the number of all different solutions to the equation
$$
x_{1}+x_{2}+\ldots+x_{n}=k
$$
( $k$ and $n$ are given numbers) in the set $\mathbb{N}$. | \binom{k-1}{n-1} | 57 | 11 |
math | 3. Solve the inequality $\log _{5} 250+\left(4-\log _{5}^{2} 2\right) \log _{50} 5 \leqslant 125^{\log _{5}^{2} x}-24 \cdot x^{\log _{5} x}$. Answer. $x \in\left(0 ; \frac{1}{5}\right] \cup[5 ;+\infty)$. | x\in(0;\frac{1}{5}]\cup[5;+\infty) | 108 | 21 |
math | 4. The numbers $x$ and $y$ are such that the equalities $\operatorname{tg} x - \operatorname{tg} y = 7$ and $2 \sin (2 x - 2 y) = \sin 2 x \sin 2 y$ hold. Find $\operatorname{tg} x \operatorname{tg} y$. | -\frac{7}{6} | 79 | 7 |
math | [ Layouts and partitions ] [ Product rule $\quad]$
How many four-digit numbers (from 0001 to 9999) exist such that the sum of the first two digits equals the sum of the last two digits
# | 669 | 52 | 3 |
math | 7. Let $a=\sqrt{3 x+1}+\sqrt{3 y+1}+\sqrt{3 z+1}$, where $x+y+z=1, x, y, z \geqslant 0$. Then $[a]=$ $\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$). | 4 | 78 | 1 |
math | 4. Given $41^{x}=2009,7^{y}=2009$. Then the value of $\frac{1}{x}+\frac{2}{y}$ is . $\qquad$ | 1 | 47 | 1 |
math | Biot and Gay-Lussac, French natural scientists, in 1804 ascended in a balloon and reached a height of $6825 \mathrm{~m}$. Question: How far did the horizon extend on this occasion and how much $\mathrm{km}^{2}$ of land area could be seen from the balloon? (The radius of the Earth is $6377.4 \mathrm{~km}$. ) | =295\mathrm{~},\quadf=273188\mathrm{~}^{2} | 94 | 27 |
math | 5.3 Find the range of the function $y=\sqrt{a+x}+\sqrt{b-x}\left(a, b \in \mathbf{R}^{+}\right)$. | \sqrt{+b}\leqslanty\leqslant\sqrt{2(+b)} | 41 | 23 |
math | Problem 6. Find the smallest positive root of the equation
$$
2 \sin (6 x)+9 \cos (6 x)=6 \sin (2 x)+7 \cos (2 x)
$$ | \frac{\alpha+\beta}{8}\approx0.1159 | 44 | 16 |
math | A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$
Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$ | c = 5 | 83 | 5 |
math | 10. (20 points) Given the function
$$
f(x)=x^{4}+a x^{3}+b x^{2}+a x+1(a, b \in \mathbf{R})
$$
has at least one root. Find the minimum value of $a^{2}-b$. | 1 | 69 | 1 |
math | 1. Find all roots of the equation $\frac{1}{\cos ^{3} x}-\frac{1}{\sin ^{3} x}=4 \sqrt{2}$, lying in the interval $\left(-\frac{\pi}{2}, 0\right)$. Write the answer in degrees. | -45 | 67 | 3 |
math | Example 11 (1994 Sichuan Province High School Competition Question) Given that point $P$ moves on the circle $x^{2}+(y-4)^{2}=1$, and point $Q$ moves on the ellipse $\frac{x^{2}}{9}+y^{2}=1$, find the maximum value of $|P Q|$.
Translate the above text into English, please retain the original text's line breaks and for... | 3\sqrt{3}+1 | 104 | 8 |
math | An express train overtakes a freight train. The speed of the express is as many times greater than that of the freight train as the time it takes for them to pass each other side by side is greater than the time it would take if they were passing each other head-on. What is this ratio? | 1+\sqrt{2} | 61 | 6 |
math | Problem 5. On 6 trees there are 129 birds. At one moment, 6 birds flew away from the first tree, 11 from the second, 8 from the third, 10 from the fourth, 7 from the fifth, and 9 from the sixth. Then on all the trees, the same number of birds remained. How many birds were there on each tree at the beginning? | 19,24,21,23,20,22 | 88 | 17 |
math | Solve the following equation:
$$
\sqrt[3]{(a+x)^{2}}+20 \sqrt[3]{(a-x)^{2}}=9 \sqrt[3]{a^{2}-x^{2}}
$$ | x_{1}=\frac{63}{65},x_{2}=\frac{62}{63} | 51 | 26 |
math | 7. Solve the equation $x^{6}=y^{3}+217$ in integers.
ANSWER: $(-1,-6)(1,-6)(-3,8),(3,8)$. | (-1,-6),(1,-6),(-3,8),(3,8) | 45 | 18 |
math | 106. Calculate the sum:
$$
S=\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{99 \cdot 100}
$$ | 0.99 | 62 | 4 |
math | 5. (5 points) From the 100 natural numbers $1 \sim 100$, remove two consecutive even numbers. The average of the remaining numbers is 50. Then the product of the two removed numbers is $\qquad$.
| 5624 | 54 | 4 |
math | Exercise 6. The strictly positive integers $x, y$, and $z$ satisfy the following two equations: $x+2 y=z$ and $x^{2}-4 y^{2}+z^{2}=310$. Find all possible values of the product $x y z$.
$\underline{\text { | 2015 | 68 | 4 |
math | 4. Alana, Beatrix, Celine, and Deanna played 6 games of tennis together. In each game, the four of them split into two teams of two and one of the teams won the game. If Alana was on the winning team for 5 games, Beatrix for 2 games, and Celine for 1 game, for how many games was Deanna on the winning team? | 4 | 85 | 1 |
math | N3 (4-1, Poland) Find the smallest positive integer $n$ with the following properties:
(1) The unit digit of $n$ is 6;
(2) If the unit digit 6 of $n$ is moved to the front of the other digits, the resulting new number is 4 times $n$.
| 153846 | 71 | 6 |
math | 4. Given that $a$ is an integer, the two real roots of the equation $x^{2}+(2 a-1) x+$ $a^{2}=0$ are $x_{1}$ and $x_{2}$. Then $\left|\sqrt{x_{1}}-\sqrt{x_{2}}\right|=$ | 1 | 70 | 1 |
math | 4. Let $a$, $b$, $c$ be positive integers, and satisfy
$$
a^{2}+b^{2}+c^{2}-a b-b c-c a=19 \text {. }
$$
Then the minimum value of $a+b+c$ is $\qquad$ | 10 | 65 | 2 |
math | 1. Divide the set of positive even numbers $\{2,4, \cdots\}$ into groups in ascending order, with the $n$-th group containing $3 n-2$ numbers:
$$
\{2\},\{4,6,8,10\},\{12,14, \cdots, 24\}, \cdots \text {. }
$$
Then 2018 is in the group. $\qquad$ | 27 | 103 | 2 |
math | 7. (1976 Yugoslav Mathematical Olympiad) Let $a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}\left(n \in \mathbf{N}^{*}\right)$, then
$$
\sum_{k=1}^{99} a_{k}=
$$ | \frac{9}{10} | 79 | 8 |
math | Putnam 1992 Problem B1 Let R be the reals. Let S ⊆ R have n ≥ 2 elements. Let A S = { x ∈ R : x = (s + t)/2 for some s, t ∈ S with s ≠ t}. What is the smallest possible |A S |? Solution | 2n-3 | 69 | 4 |
math | ## Task Condition
Find the derivative.
$y=\ln \operatorname{tg}\left(\frac{\pi}{4}+\frac{x}{2}\right)$ | \frac{1}{\cosx} | 34 | 9 |
math | 4. Calculate the value of the expression
$$
\left(\frac{1+i}{\sqrt{2}}\right)^{n+4}+\left(\frac{1-i}{\sqrt{2}}\right)^{n+4}+\left(\frac{1+i}{\sqrt{2}}\right)^{n}+\left(\frac{1-i}{\sqrt{2}}\right)^{n},
$$
( i is the imaginary unit). | 0 | 100 | 1 |
math | 4. A rectangle with dimensions $m$ and $n$, where $m$ and $n$ are natural numbers and $n = m k$, is divided into $m \cdot n$ unit squares. Each path from point $A$ to point $C$ along the divided segments (sides of the small squares) where movement to the right and movement upwards is allowed, has a length of $m+n$. Fin... | k | 131 | 1 |
math | Example. Evaluate the definite integral
$$
\int_{0}^{2} \frac{4 \sqrt{2-x}-\sqrt{2+x}}{(\sqrt{x+2}+4 \sqrt{2-x})(x+2)^{2}} d x
$$ | \frac{\ln5}{16} | 59 | 9 |
math | [ Properties and characteristics of a parallelogram ] [ Isosceles, inscribed, and circumscribed trapezoids ]
A circle passing through the vertices $A, B$, and $C$ of parallelogram $A B C D$ intersects the lines $A D$ and $C D$ at points $M$ and $N$ respectively. Point $M$ is at distances 4, 3, and 2 from vertices $B, ... | \frac{8}{3} | 110 | 7 |
math | Analyzing the 4-digit natural numbers:
a) How many of them have all different digits?
b) How many have the digit 1 exactly once and all different digits?
c) How many have the digit 1? | 3168 | 46 | 4 |
math | 1. Let $a_{1}, a_{2}, \ldots, a_{n}$ be integers $(n>1)$ satisfying $a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=2005$. Find the smallest possible value of $n$.
(1 mark)
Let $a_{1}, a_{2}, \ldots, a_{n}$ be integers $(n>1)$, such that $a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=2005$. Find the smallest pos... | 5 | 153 | 1 |
math | Assume we are given a set of weights, $x_1$ of which have mass $d_1$, $x_2$ have mass $d_2$, etc, $x_k$ have mass $d_k$, where $x_i,d_i$ are positive integers and $1\le d_1<d_2<\ldots<d_k$. Let us denote their total sum by $n=x_1d_1+\ldots+x_kd_k$. We call such a set of weights [i]perfect[/i] if each mass $0,1,\ldots,n... | \{1, 1, 1, \ldots, 1\}, \{1, 2, 2, \ldots, 2\}, \{1, 1, 1, \ldots, 1, 997\} | 279 | 60 |
math | For a positive integer $n$, let $S(n)$ denote its digit sum. Find all positive integers $M$ such that for every positive integer $k$ not exceeding $M$, we have $S(M k)=S(M)$. | 10^{n}-1 | 49 | 6 |
math | Determine the largest constant $C$ such that for all real numbers $x_{1}, x_{2}, \ldots, x_{6}$ the inequality
$$
\left(x_{1}+x_{2}+\cdots+x_{6}\right)^{2} \geq C \cdot\left(x_{1}\left(x_{2}+x_{3}\right)+x_{2}\left(x_{3}+x_{4}\right)+\cdots+x_{6}\left(x_{1}+x_{2}\right)\right)
$$
holds.
Determine for this $C$ all $x... | C_{\max}=3 | 160 | 6 |
math | 3. The sum of three five-digit numbers of the form $\overline{\text { mat31 }}, \overline{\text { mat41 }}, \overline{\text { mat51 }}$ is 202023. What is $m+a+t$? | 16 | 62 | 2 |
math | Problem 1. Find the least positive integer $n$ with the following property: if $n$ distinct sums of the form $x_{p}+x_{q}+x_{r}, 1 \leq p<q<r \leq 5$, equal 0 , then $x_{1}=x_{2}=x_{3}=x_{4}=x_{5}=0$.
Sava Grozdev, Svetlozar Doychev | 7 | 99 | 1 |
math | 2.2. Find all possible values of
$$
\left\lfloor\frac{x-p}{p}\right\rfloor+\left\lfloor\frac{-x-1}{p}\right\rfloor,
$$
where $x$ is a real number and $p$ is a nonzero integer.
Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$. | -3,-2,-1,0 | 87 | 8 |
math | 175. $\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x$.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
175. $\int \frac{\sin \sqrt{x}}{\sqrt{x}} d x$. | -2\cos\sqrt{x}+C | 68 | 10 |
math | Task B-3.7. Solve the system of equations
$$
\begin{aligned}
& x=2^{\frac{4}{1+\log _{2} z}} \\
& y=2^{\frac{4}{1+\log _{2} x}} \\
& z=2^{\frac{9}{1+\log _{2} y}}
\end{aligned}
$$ | x_{1}=2,y_{1}=4,z_{1}=8,\quadx_{2}=\frac{1}{4},y_{2}=\frac{1}{16},z_{2}=\frac{1}{8} | 85 | 51 |
math | 5. Find all integer pairs $(x, y)$ such that $x^{3}=y^{3}+2 y^{2}+1$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | (x, y)=(-2, -3), (1, -2), (1, 0) | 57 | 22 |
math | Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying
(i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$.
(ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$. | f(x) = a^x e^{\frac{2ki\pi}{1987}} | 146 | 23 |
math | Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo coul... | 30 | 72 | 2 |
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