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 | 5,6,7 |
| :---: | :---: | :---: |
| | [ Divisibility of numbers. General properties ] | |
| | [ Decimal number system ] | |
| | Case analysis | |
Which digits can stand in place of the letters in the example $A B \cdot C=D E$, if different letters represent different digits and the digits are written from left t... | 13\cdot6=78 | 94 | 8 |
math | Dedalo buys a finite number of binary strings, each of finite length and made up of the binary digits 0 and 1. For each string, he pays $(\frac{1}{2})^L$ drachmas, where $L$ is the length of the string. The Minotaur is able to escape the labyrinth if he can find an infinite sequence of binary digits that does not conta... | c = 75 | 249 | 6 |
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 | 357. Find the dimensions of a cylindrical closed tank with a given volume $v$ and the smallest total surface area. | r=\sqrt[3]{\frac{v}{2\pi}},=2\sqrt[3]{\frac{v}{2\pi}} | 26 | 31 |
math | 3. If the thought number is multiplied by 6, and then 382 is added to the product, the result will be the largest three-digit number written with two identical even numbers and one odd number. Find the thought number. | 101 | 49 | 3 |
math | 6.46 A number of students from a certain district participate in a mathematics competition. Each student's score is an integer, and the total score is 8250. The scores of the top three students are $88, 85, 80$, and the lowest score is 30. No more than 3 students can have the same score. How many students at minimum ha... | 60 | 98 | 2 |
math | 82. Andrey and Grandpa Grisha went mushroom picking sometime between six and seven in the morning, at the moment when the clock hands were aligned. They returned home between twelve and one in the afternoon, at the moment when the clock hands were pointing in exactly opposite directions. How long did their mushroom "hu... | 6 | 67 | 1 |
math | Find all integer solutions $(a, b)$ of the equation $3 a^{2} b^{2}+b^{2}=517+30 a^{2}$. | (2,7),(-2,7),(2,-7),(-2,-7) | 38 | 19 |
math | GS. 2 Given that $1^{3}+2^{3}+\ldots+k^{3}=\left(\frac{k(k+1)}{2}\right)^{2}$. Find the value of $11^{3}+12^{3}+\ldots+24^{3}$. | 86975 | 67 | 5 |
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 | 7.231. $\log _{4} \log _{2} x+\log _{2} \log _{4} x=2$. | 16 | 35 | 2 |
math | ## Task A-1.5.
Azra thought of four real numbers and on the board wrote down the sums of all possible pairs of the thought-of numbers, and then erased one of those sums. The numbers left on the board are $-2,1,2,3$ and 6. What numbers did Azra think of? | -\frac{3}{2},-\frac{1}{2},\frac{5}{2},\frac{7}{2} | 70 | 28 |
math | Let $\triangle A_1B_1C_1$ be an equilateral triangle of area $60$. Chloe constructs a new triangle $\triangle A_2B_2C_2$ as follows. First, she flips a coin. If it comes up heads, she constructs point $A_2$ such that $B_1$ is the midpoint of $\overline{A_2C_1}$. If it comes up tails, she instead constructs $A_2$ su... | 195 | 169 | 3 |
math | Example 6 (9th China High School Mathematics Olympiad Winter Camp Problem) Find all functions $f$: $[1,+\infty) \rightarrow[1,+\infty)$ that satisfy the following conditions:
(1) $f(x) \leqslant 2(x+1)$;
(2) $f(x+1)=\frac{1}{x}\left[(f(x))^{2}-1\right]$. | f(x)=x+1 | 93 | 6 |
math | 17. Let $p(x)$ be a polynomial with integer coefficients such that $p(m)-p(n)$ divides $m^{2}-n^{2}$ for all integers $m$ and $n$. If $p(0)=1$ and $p(1)=2$, find the largest possible value of $p(100)$. | 10001 | 73 | 5 |
math | A1. At a bazaar, you can win a prize by guessing the exact number of ping pong balls in a glass jar. Arie guesses there are 90, Bea guesses there are 97, Cor guesses there are 99, and Dirk guesses there are 101. None of the four win the prize. It turns out that one of the four is off by 7, one is off by 4, and one is o... | 94 | 109 | 2 |
math | 4. In the number $2016 * * * * 02 * *$, each of the 6 asterisks needs to be replaced with any of the digits $0,2,4,5,7,9$ (digits can be repeated) so that the resulting 12-digit number is divisible by 15. In how many ways can this be done? | 5184 | 80 | 4 |
math | Let $S$ be the set of positive integer divisors of $20^9.$ Three numbers are chosen independently and at random with replacement from the set $S$ and labeled $a_1,a_2,$ and $a_3$ in the order they are chosen. The probability that both $a_1$ divides $a_2$ and $a_2$ divides $a_3$ is $\tfrac{m}{n},$ where $m$ and $n$ are ... | 77 | 113 | 2 |
math | II. (50 points) Given the sequence $\left\{a_{n}\right\}$ defined by
$$
a_{1}=\frac{2}{3}, a_{n+1}=a_{n}^{2}+a_{n-1}^{2}+\cdots+a_{1}^{2}\left(n \in \mathbf{N}_{+}\right)
$$
Determine. If for any $n\left(n \in \mathbf{N}_{+}\right)$,
$$
\frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{n}+1}<M
$$
holds true. Fin... | \frac{57}{20} | 166 | 9 |
math | 3. A covered football field of rectangular shape with a length of $90 \mathrm{m}$ and a width of 60 m is being designed, which should be illuminated by four spotlights, each hanging at some point on the ceiling. Each spotlight illuminates a circle, the radius of which is equal to the height at which the spotlight hangs... | 27.1 | 142 | 4 |
math | 2.96 Let $a_{1}, a_{2}, a_{3}, \cdots$ be a non-decreasing sequence of positive integers. For $m \geqslant 1$, define
$$b_{m}=\min \left\{n \mid a_{n} \geqslant m\right\}$$
i.e., $b_{m}$ is the smallest value of $n$ such that $a_{n} \geqslant m$.
If $a_{19}=85$, find the maximum value of $a_{1}+a_{2}+\cdots+a_{19}+b_{... | 1700 | 158 | 4 |
math | Find a four digit number $M$ such that the number $N=4\times M$ has the following properties.
(a) $N$ is also a four digit number
(b) $N$ has the same digits as in $M$ but in reverse order. | 2178 | 56 | 4 |
math | 3. If three integers $a, b, c (a \neq 0)$ make the equation $a x^{2}$ $+b x+c=0$ have two roots $a$ and $b$, then $a+b+c$ equals. $\qquad$ | 18 | 58 | 2 |
math | Let's determine the digits $a, b, c, d$ if the number written in the decimal system with these digits satisfies:
$$
\overline{a b c d}=16\left(a^{2}+b^{2}+c^{2}+d^{2}\right)+a+b
$$
and additionally, $b^{2}-a^{2}=2\left(c^{2}+d^{2}\right)$. | 1962 | 95 | 4 |
math | Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$
$$
f(f(x)+y)=2 x+f(-f(f(x))+f(y))
$$ | f(x)=x | 51 | 4 |
math | Four. (20 points) Given points $A$ and $B$ are the upper and lower vertices of the ellipse $\frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1(a>b>0)$, and $P$ is a point on the hyperbola $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$ in the first quadrant. The lines $PA$ and $PB$ intersect the ellipse at points $C$ and $D$, respe... | \frac{\sqrt{7}}{2} | 174 | 10 |
math | 8. Given that $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ are both arithmetic sequences, and $a_{11}=32, b_{21}=43$. Let $c_{n}=(-1)^{n} \cdot\left(a_{n}-b_{n}\right)$, the sum of the first 10 terms of the sequence $\left\{c_{n}\right\}$ is 5, and the sum of the first 13 terms is -5, then the sum of the first 100... | 10200 | 144 | 5 |
math | $6 \cdot 162$ Find all functions $f:[1, \infty) \rightarrow[1, \infty)$ that satisfy the following conditions:
(1) $f(x) \leqslant 2(x+1)$;
(2) $f(x+1)=\frac{1}{x}\left[(f(x))^{2}-1\right]$. | f(x)=x+1 | 84 | 6 |
math | M1. The positive integer $N$ has five digits.
The six-digit integer $P$ is formed by appending the digit 2 to the front of $N$. The six-digit integer $Q$ is formed by appending the digit 2 to the end of $N$. Given that $Q=3 P$, what values of $N$ are possible? | 85714 | 74 | 5 |
math | Example 3: There are two red balls, one black ball, and one white ball. Questions: (1) How many different ways are there to select the balls? Try to enumerate them respectively; (2) If 3 balls are randomly selected each time, how many different ways are there to select them? | 12 | 64 | 2 |
math | 3. From a point on the Earth's surface, a large number of small balls are launched in all directions at the same speed of 10 m/s. Among all the balls that land at a distance from the starting point not closer than $96 \%$ of the distance at which the farthest flying ball lands, find the one that will spend the most tim... | 1.6 | 120 | 3 |
math | A natural number $N$ greater than 10 is called a "super-square" if the number formed by each pair of consecutive digits of the number $N$ (considered in the same order) is always a perfect square. For example, 8164 is a "super-square" because the numbers 81, 16, and 64 are perfect squares. Other examples of super-squar... | 14 | 120 | 2 |
math | 【Example 5】How many positive integer solutions does the equation $x+y+z+w=23$ have? | 1540 | 24 | 4 |
math | 1. In a class, there are boys and girls, such that the number of boys divides the number of girls. Florin distributes 1000 candies equally among the students in the class. Knowing that there are at least 10 boys in the class, and if 10 more students were to join, Florin could, again, distribute the 1000 candies equally... | 25 | 102 | 2 |
math | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-4.5 ; 4.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 90 | 58 | 2 |
math | 36. Given $x=\sqrt{\frac{a-\sqrt{a^{2}-4}}{2 a}}(\mathrm{a}>0)$, then $\frac{x}{\sqrt{1-x^{2}}}+\frac{\sqrt{1-x^{2}}}{x}=$ $\qquad$ | a | 64 | 1 |
math | 9. Find the maximum value of the function $f(x)=9 \sin x+12 \cos x$. | 15 | 24 | 2 |
math | 11. (China Mathematical Olympiad) Let $S=\{1,2,3, \cdots, 98\}$, find the smallest natural number $n$, such that in any $n$-element subset of $S$, one can always select 10 numbers, and no matter how these 10 numbers are divided into two groups, there is always a number in one group that is coprime with the other four n... | 50 | 114 | 2 |
math | 3. It is known that the numbers $x, y, z$ form an arithmetic progression in the given order with a common difference $\alpha=\arccos \frac{1}{9}$, and the numbers $5+\cos x, 5+\cos y, 5+\cos z$ form a non-constant geometric progression in the given order. Find $\cos y$. | -\frac{1}{9} | 79 | 7 |
math | 2. Let the set $M=\{1,99,-1,0,25,-36,-91,19,-2,11\}$, and denote all non-empty subsets of $M$ as $M_{i}, i=1,2, \cdots$, 2013. The product of all elements in each $M_{i}$ is $m_{i}$. Then $\sum_{i=1}^{2013} m_{i}=$ $\qquad$ . | -1 | 112 | 2 |
math | Determine all positive integers $k$ for which there exist positive integers $r$ and $s$ that satisfy the equation $$(k^2-6k+11)^{r-1}=(2k-7)^{s}.$$ | k \in \{2, 3, 4, 8\} | 52 | 18 |
math | 4. Find all polynomials $P(x)$ with real coefficients such that
$$
P(a) \in \mathbb{Z} \quad \text { implies that } \quad a \in \mathbb{Z} \text {. }
$$ | P(x)=\frac{x}{p}+\frac{q}{p}wherep,q\in\mathbb{Z}p\neq0 | 53 | 32 |
math |
Problem 2. Find all pairs of integers $(a, b)$ so that each of the two cubic polynomials
$$
x^{3}+a x+b \text { and } x^{3}+b x+a
$$
has all the roots to be integers.
| (0,0) | 59 | 5 |
math | 3. Given the real-coefficient equation $x^{3}+a x^{2}+b x+c=0$ whose three roots can serve as the eccentricities of an ellipse, a hyperbola, and a parabola, the range of $\frac{b}{a}$ is $\qquad$ | (-2,-\frac{1}{2}) | 65 | 10 |
math | 14.15. Let $P(x)$ be a polynomial of degree $n$, and $P(x) = 2^x$ for $x = 1, 2, \ldots, n+1$. Compute $P(n+2)$. | 2^{n+2}-2 | 56 | 7 |
math | ## Task 1 - 060521
In each of five boxes, there is exactly the same number of apples. If you remove 60 apples from each box, the total number of apples left in the boxes is the same as the number of apples that were originally in two boxes. Determine the total number of apples that were initially in the boxes! | 500 | 76 | 3 |
math | 3.4.1. Find the equations of the tangents to the curve $y=\frac{3 x+2}{3 x-2}$ that are parallel to the line $3 x+y+3=0$. | 3x+y+1=0\text{}3x+y-7=0 | 46 | 17 |
math | Find all integers $ x$ such that $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square
It's a nice problem ...hope you enjoy it!
Daniel | -9, -8, -7, -4, -1, 0, 1 | 46 | 20 |
math | A certain function $f$ has the properties that $f(3x)=3f(x)$ for all positive real values of $x$, and that $f(x)=1-\mid x-2 \mid$ for $1\leq x \leq 3$. Find the smallest $x$ for which $f(x)=f(2001)$. | 429 | 77 | 3 |
math | Let $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether
$$\exp \left( \sum_{n=1}^{\infty} \frac{ B(n)}{n(n+1)} \right)$$
is a rational number. | 4 | 65 | 3 |
math | 1. Among the positive integers not greater than 2017, there are $\qquad$ numbers that are divisible by 12 but not by 20. | 135 | 36 | 3 |
math | How many distinct sums can be made from adding together exactly 8 numbers that are chosen from the set $\{ 1,4,7,10 \}$, where each number in the set is chosen at least once? (For example, one possible sum is $1+1+1+4+7+7+10+10=41$.) | 13 | 77 | 2 |
math | 33. Two players $A$ and $B$ play rock-paper-scissors continuously until player $A$ wins 2 consecutive games. Suppose each player is equally likely to use each hand-sign in every game. What is the expected number of games they will play? | 12 | 55 | 2 |
math | Mumchov D:
On a circle of length 2013, 2013 points are marked, dividing it into equal arcs. A chip is placed at each marked point. We define the distance between two points as the length of the shorter arc between them. For what largest $n$ can the chips be rearranged so that there is again one chip at each marked poi... | 670 | 102 | 3 |
math | Example 3. Calculate the mass of the surface $z=x y$, located inside the cylinder $x^{2}+\frac{y^{2}}{4}=1$, if the density is $\rho=\frac{|z|}{\sqrt{1+x^{2}+y^{2}}}$. | 2 | 62 | 1 |
math | The number 61 is written on the board. Every minute, the number is erased from the board and the product of its digits, increased by 13, is written in its place. After the first minute, the number 19 is written on the board ( $6 \cdot 1+13=19$). What number will be on the board after an hour? | 16 | 81 | 2 |
math | 12.55*. In a right triangle $ABC$ with a right angle at $A$, a circle is constructed on the altitude $AD$ as a diameter, intersecting side $AB$ at point $K$ and side $AC$ at point $M$. Segments $AD$ and $KM$ intersect at point $L$. Find the acute angles of triangle $ABC$, given that $AK: AL = AL: AM$. | 15 | 91 | 2 |
math | 6. Let $\xi_{1}, \ldots, \xi_{N}$ be independent Bernoulli random variables, $\mathrm{P}\left\{\xi_{i}=1\right\}=p, \mathrm{P}\left\{\xi_{i}=-1\right\}=1-p, S_{i}=\xi_{1}+\ldots+\xi_{i}, 1 \leqslant i \leqslant N$, $S_{0}=0$. Let $\mathscr{R}_{N}$ be the range, i.e., the number of distinct points visited by the random ... | \frac{\mathscr{R}_{N}}{N}arrow\mathrm{P}{S_{n}\neq0,n\geqslant0} | 250 | 34 |
math | Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that
$$
f(f(x)+x f(y))=x+f(x) y
$$
where $\mathbb{Q}$ is the set of rational numbers. | f(x)=x | 56 | 4 |
math | Determine all the functions $f:\mathbb R\mapsto\mathbb R$ satisfies the equation
$f(a^2 +ab+ f(b^2))=af(b)+b^2+ f(a^2)\,\forall a,b\in\mathbb R $
| f(x) \equiv x \text{ or } f(x) \equiv -x | 59 | 18 |
math | 2. (7 points) It is known that $a+b+c<0$ and that the equation $a x^{2}+b x+c=0$ has no real roots. Determine the sign of the coefficient $c$.
| <0 | 49 | 2 |
math | 1. Given
$$
\frac{1}{x}+\frac{1}{y+z}=\frac{1}{2}, \frac{1}{y}+\frac{1}{z+x}=\frac{1}{3}, \frac{1}{z}+\frac{1}{x+y}=\frac{1}{4} \text {. }
$$
Find the value of $\frac{2}{x}+\frac{3}{y}+\frac{4}{z}$. | 2 | 104 | 1 |
math | Solve the following equation:
$$
2^{x+1}+2^{x+2}+2^{x+3}=3^{x+1}-3^{x+2}+3^{x+3}
$$ | -1 | 48 | 2 |
math | 8,9
On the plane, points $A(1 ; 2), B(2 ; 1), C(3 ;-3), D(0 ; 0)$ are given. They are the vertices of a convex quadrilateral $A B C D$. In what ratio does the point of intersection of its diagonals divide the diagonal $A C$? | 1:3 | 75 | 3 |
math | 4. If three angles $\alpha, \beta, \gamma$ form an arithmetic sequence with a common difference of $\frac{\pi}{3}$, then $\tan \alpha \cdot \tan \beta + \tan \beta \cdot \tan \gamma + \tan \gamma \cdot \tan \alpha$ $=$ . $\qquad$ | -3 | 72 | 2 |
math | Let $A$ be a set with $2n$ elements, and let $A_1, A_2...,A_m$ be subsets of $A$e ach one with n elements. Find the greatest possible m, such that it is possible to select these $m$ subsets in such a way that the intersection of any 3 of them has at most one element. | m \leq 4 \frac{n-1}{n-4} | 78 | 17 |
math | 6. In $\{1000,1001, \cdots, 2000\}$, how many pairs of consecutive integers can be added without carrying over a digit? | 156 | 42 | 3 |
math | Exercise 6. Find all prime numbers $p, q$ satisfying $p^{5}+p^{3}+2=q^{2}-q$. | (2,7)(3,17) | 32 | 10 |
math | Problem 1. Find the last digit of the number $n=2^{0}+2^{1}+2^{2}+\ldots+2^{2014}$. | 7 | 40 | 1 |
math | 8. Let $A B C$ be a triangle with circumradius $R=17$ and inradius $r=7$. Find the maximum possible value of $\sin \frac{A}{2}$. | \frac{17+\sqrt{51}}{34} | 44 | 15 |
math | 30. Find the largest positive number $x$ such that
$$
\left(2 x^{3}-x^{2}-x+1\right)^{1+\frac{1}{2 x+1}}=1
$$ | 1 | 50 | 1 |
math | 1. How many natural numbers from 1 to 2017 have exactly three distinct natural divisors? | 14 | 23 | 2 |
math | ## 5. Soba
Ana can paint the room in 15 hours, Barbara in 10 hours, and Cvijeta twice as fast as Ana. Ana starts painting and paints alone for one and a half hours, then Barbara joins her and they paint together until half the room is painted. After that, Cvijeta joins them and all three paint until the entire room is... | 334 | 104 | 3 |
math | 125. There are 60 squares, each with a side length of 1 cm. From these squares, all possible rectangles are formed, each using all 60 squares. Find the sum of the areas of all these rectangles.
## Answers. Solutions | 360\mathrm{~}^{2} | 54 | 11 |
math | $a, b, c, d$ are integers with $ad \ne bc$. Show that $1/((ax+b)(cx+d))$ can be written in the form $ r/(ax+b) + s/(cx+d)$. Find the sum $1/1\cdot 4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001$. | \frac{1000}{3001} | 98 | 13 |
math | 2. A number divided by 20 has a quotient of 10 and a remainder of 10, this number is | 210 | 27 | 3 |
math | Problem 3. An inspector checks items for compliance with the standard. It is known that the probability of an item complying with the standard is 0.95. Find the probability that: a) both of the two inspected items will be standard, if the events of items complying with the standard are independent; b) only one of the t... | )0.9025;b)0.095 | 75 | 14 |
math | On the evening, more than $\frac 13$ of the students of a school are going to the cinema. On the same evening, More than $\frac {3}{10}$ are going to the theatre, and more than $\frac {4}{11}$ are going to the concert. At least how many students are there in this school? | 173 | 73 | 3 |
math | 19. (ROM 1) Consider the sequences $\left(a_{n}\right),\left(b_{n}\right)$ defined by
$$
a_{1}=3, \quad b_{1}=100, \quad a_{n+1}=3^{a_{n}}, \quad b_{n+1}=100^{b_{n}} .
$$
Find the smallest integer $m$ for which $b_{m}>a_{100}$. | 99 | 102 | 2 |
math | Example 1 Let $x, y \in \mathbf{R}$, find the minimum value of the function $f(x, y)=x^{2}+6 y^{2}-2 x y-14 x-6 y+72$ and the values of $x, y$ when the minimum value is achieved. | 3 | 70 | 1 |
math | Po writes down five consecutive integers and then erases one of them. The four remaining integers sum to 153. Compute the integer that Po erased.
[i]Proposed by Ankan Bhattacharya[/i] | 37 | 46 | 2 |
math | 9. (16 points) Let the moving point $P$ be on the right branch of the hyperbola
$$
\frac{x^{2}}{16}-\frac{y^{2}}{9}=1
$$
(excluding the vertex), $Q$ and $R$ are the left and right foci of the hyperbola, respectively, $S$ is the excenter of $\triangle PQR$ inside $\angle PQR$, and $D(0,1)$. Find the minimum value of $|... | \sqrt{25.9} | 116 | 8 |
math | [Help me] Determine the smallest value of the sum M =xy-yz-zx where x; y; z are real numbers satisfying the following condition $x^2+2y^2+5z^2 = 22$. | \frac{-55 - 11\sqrt{5}}{10} | 50 | 18 |
math | Example 2. Find the logarithmic residue of the function
$$
f(z)=\frac{\operatorname{ch} z}{e^{i z}-1}
$$
with respect to the contour $C:|z|=8$. | 3 | 50 | 1 |
math | 1. One worker in two hours makes 5 more parts than the other, and accordingly spends 2 hours less to manufacture 100 parts. How much time does each worker spend on manufacturing 100 parts?
# | 8 | 47 | 1 |
math | We need the approximate value of $\sqrt{17}$ rounded to eight decimal places. Can we obtain this from the following square roots rounded to seven decimal places?
$$
\begin{array}{ll}
\sqrt{17}=4.1231056, & \sqrt{425}=20.6155281 \\
\sqrt{68}=8.2462113, & \sqrt{612}=24.7386338 \\
\sqrt{153}=12.3693169, & \sqrt{833}=28.8... | 4.12310563 | 240 | 10 |
math | 5. Find all natural numbers $a$ and $b$ and prime numbers $p$ and $q$ such that at least one of the numbers $p$ and $q$ is greater than 12 and the following holds:
$$
p^{a}+q^{b}=19^{a}
$$ | (2,17,1,1)(17,2,1,1) | 66 | 19 |
math | Problem 8.2. Oleg bought a chocolate bar for $n$ rubles, and after some time, he sold it for 96 rubles. It turned out that he sold the chocolate bar for exactly $n \%$ more than he bought it for. For how many rubles did Oleg buy the chocolate bar? | 60 | 69 | 2 |
math | 【4】The smallest natural number that leaves a remainder of 2 when divided by 3, a remainder of 4 when divided by 5, and a remainder of 4 when divided by 7 is ( ). | 74 | 45 | 2 |
math | 2. Given that for any real number $x$ we have $a \cos x + b \cos 2x \geqslant -1$.
Then the maximum value of $a + b$ is $\qquad$ | 2 | 49 | 1 |
math | Example 5 Find the largest positive integer $x$, such that for every positive integer $y$, we have $x \mid\left(7^{y}+12 y-1\right)$.
| 18 | 43 | 2 |
math | 1. Given a parallelogram $A B C D$. It is known that the centers of the circles circumscribed around triangles $A B C$ and $C D A$ lie on the diagonal $B D$. Find the angle $D B C$, if $\angle A B D=40^{\circ}$. | 50 | 68 | 2 |
math | In one of three boxes there is a prize, the other two boxes are empty. You do not know which box contains the prize, but the host does. You must point to one of the boxes, which you think contains the prize. After this, the host opens one of the two remaining boxes. Since he does not want to give away the prize immedia... | \frac{2}{3} | 108 | 7 |
math | Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Find the smallest constant $c$ for which $d(n)\le c\sqrt n$ holds for all positive integers $n$. | \sqrt{3} | 47 | 5 |
math | 10. In the sequence $\left\{a_{n}\right\}$, all terms are positive, $S_{n}$ is the sum of the first $n$ terms, and $a_{n}+\frac{1}{a_{n}}=2 S_{n}$. The general term formula $a_{n}=$ $\qquad$. | a_{n}=\sqrt{n}-\sqrt{n-1} | 75 | 14 |
math | 3. Determine all pairs $(x, y)$ of real numbers for which
$$
\left(\sin ^{2} x+\frac{1}{\sin ^{2} x}\right)^{2}+\left(\cos ^{2} x+\frac{1}{\cos ^{2} x}\right)^{2}=12+\frac{1}{2} \sin y
$$ | \frac{\pi}{4}+\frac{k\pi}{2},\quad\frac{\pi}{2}+2\pi,\quadk,\in{Z} | 86 | 36 |
math | Cirlce $\Omega$ is inscribed in triangle $ABC$ with $\angle BAC=40$. Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$. Tangent form $D$ touches $\Omega$ in $E$. FInd $\angle BEC$. | 110^\circ | 81 | 5 |
math | 6. If the two quadratic equations $x^{2}+x+m=$ 0 and $m x^{2}+x+1=0$ each have two distinct real roots, but one of them is a common real root $\alpha$, then the range of the real root $\alpha$ is $\qquad$ . | \alpha=1 | 67 | 4 |
math | 1. (7p) Determine the number $\overline{a b}$ for which $\frac{\overline{a,(b)}+\overline{b,(a)}}{a+b}=\frac{a+b}{3 a}$.
GM12/2015 | 37 | 58 | 2 |
math | [ Arithmetic. Mental calculation, etc.]
From the ten-digit number 2946835107, five digits were erased. What is the largest number that could result from this?
# | 98517 | 42 | 5 |
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