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 | Find all pairs $(m, n)$ of positive integers, for which number $2^n - 13^m$ is a cube of a positive integer.
[i]Proposed by Oleksiy Masalitin[/i] | (2, 9) | 47 | 7 |
math | ## Task $9 / 85$
Given are two similar right-angled triangles, one inscribed and the other circumscribed to the same circle. Determine the minimal similarity ratio $k>1$! | 1+\sqrt{2} | 43 | 6 |
math | 3. Find the area of the region defined by the inequality: $|y-| x-2|+| x \mid \leq 4$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 32 | 59 | 2 |
math | 15th Chinese 2000 Problem B2 Find all positive integers n for which we can find one or more integers m 1 , ... , m k each at least 4 such that (1) n = m 1 m 2 ... m k , (2) n = 2 M - 1, where M = (m 1 - 1)(m 2 - 1) ... (m k - 1)/2 k . Solution | 7 | 98 | 1 |
math | 4. A monkey is climbing an 8-rung ladder, each time it can climb one rung or jump two rungs, and at most jump three rungs. From the ground to the top rung, there are $\qquad$ different ways to climb and jump. | 81 | 57 | 2 |
math | 2. At 8 PM, two equally long candles were lit. After some time, the candles were extinguished, and it was observed that the remaining part of the first candle was 4 times larger than the remaining part of the second candle. Calculate at what time the candles were extinguished, given that the first candle burns out in 5... | 23 | 80 | 2 |
math | 4.3. [6] Find all finite arithmetic progressions with sum 1, each term of which is of the form $\frac{1}{k}$, where $k$ is a natural number. | \frac{1}{6},\frac{1}{3},\frac{1}{2} | 43 | 21 |
math | Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends w... | 1007 | 148 | 4 |
math | Find all functions $f: (1,\infty)\text{to R}$ satisfying
$f(x)-f(y)=(y-x)f(xy)$ for all $x,y>1$.
[hide="hint"]you may try to find $f(x^5)$ by two ways and then continue the solution.
I have also solved by using this method.By finding $f(x^5)$ in two ways
I found that $f(x)=xf(x^2)$ for all $x>1$.[/hide] | f(x) = \frac{k}{x} | 107 | 11 |
math | Task 1. (5 points) Find $\frac{a^{8}+256}{16 a^{4}}$, if $\frac{a}{2}+\frac{2}{a}=3$. | 47 | 45 | 2 |
math | 1 Hugo plays a game: he places a chess piece on the top left square of a $20 \times 20$ chessboard and makes 10 moves with it. On each of these 10 moves, he moves the piece either one square horizontally (left or right) or one square vertically (up or down). After the last move, he draws an $\mathrm{X}$ on the square t... | 36 | 129 | 2 |
math | 8,9
Given a regular tetrahedron \( PABC \) with edge length \( a \). A sphere is drawn through the points \( C, E, M, P \), where \( E \) is the midpoint of \( AB \), and \( M \) is the midpoint of \( AC \). Find the radius of the sphere. | \frac{\sqrt{22}}{8} | 73 | 11 |
math | 5.1. Calculate: $\frac{1 \cdot 2+2 \cdot 3+3 \cdot 4+\ldots+2013 \cdot 2014}{(1+2+3+\ldots+2014) \cdot \frac{1}{5}}$. If necessary, round the answer to the nearest hundredths. | 6710 | 79 | 4 |
math | 11.1. At first, there were natural numbers, from 1 to 2021. And they were all white. Then the underachiever Borya painted every third number blue. Then the top student Vova came and painted every fifth number red. How many numbers remained white? (7 points)
# | 1078 | 68 | 4 |
math | Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$? | 11 | 48 | 4 |
math | 18. [10] Let $x, y, z$ be real numbers satisfying
$$
\frac{1}{x}+y+z=x+\frac{1}{y}+z=x+y+\frac{1}{z}=3 .
$$
The sum of all possible values of $x+y+z$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $\operatorname{gcd}(m, n)=1$. Find $100 m+n$. | 6106 | 109 | 4 |
math | 10. Given that the real number $a$ satisfies $a^{2}-a-1=0$. Then the value of $a^{8}+7 a^{-4}$ is - $\qquad$ | 48 | 44 | 2 |
math | \section*{Problem 4 - 310914}
a) A school has a total of 825 students. It has been calculated that during a school year, the number of participants in an interest group increased by \(4\%\) of its initial value, and, equivalently, the number of non-participants decreased by \(7\%\) of its initial value.
If this were ... | 294,295,\ldots,306 | 213 | 14 |
math | 7. For $n \in \mathbf{Z}_{+}, n \geqslant 2$, let
$$
S_{n}=\sum_{k=1}^{n} \frac{k}{1+k^{2}+k^{4}}, T_{n}=\prod_{k=2}^{n} \frac{k^{3}-1}{k^{3}+1} \text {. }
$$
Then $S_{n} T_{n}=$ . $\qquad$ | \frac{1}{3} | 107 | 7 |
math | ## Task 3B - 321233B
For each natural number $n \geq 2$, determine all $n$-tuples $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ of positive integers $x_{1}, x_{2}, \ldots, x_{n}$ that satisfy the following system of equations:
$$
\begin{aligned}
& x_{1} \cdot x_{2}=3 \cdot\left(x_{1}+x_{2}\right) \\
& x_{2} \cdot x_{3}=... | (x_{1},x_{2},x_{3},\ldots,x_{n})=(6,6,6,\ldots,6) | 216 | 31 |
math | 10. Non-negative numbers $a$ and $d$, and positive numbers $b$ and $c$ satisfy the condition $b+c \geqslant a+d$. What is the minimum value that the expression $\frac{b}{c+d} + \frac{c}{a+b}$ can take? (1988 22nd All-Soviet Union Mathematical Olympiad) | \sqrt{2}-\frac{1}{2} | 83 | 12 |
math | 4. (CAN) Let $\left\{f_{n}\right\}$ be the Fibonacci sequence $\{1,1,2,3,5, \ldots\}$.
(a) Find all pairs $(a, b)$ of real numbers such that for each $n, a f_{n}+b f_{n+1}$ is a member of the sequence.
(b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $u f_{n}^{2}+v f_{n+1}^{2}$ is a member o... | =f_{k-1},b=f_{k} | 127 | 11 |
math | Let's determine those real numbers $x, y, z$ for which
$$
(x-y-3)^{2}+(y-z)^{2}+(x-z)^{2}=3
$$ | z+1,z-1 | 43 | 6 |
math | $\left[\begin{array}{l}{[\text { Integer and fractional parts. Archimedes' principle ] }} \\ {[\quad \underline{\text { equations in integers }} \underline{\text { ] }}}\end{array}\right]$
How many solutions in natural numbers does the equation $\left[{ }^{x} / 10\right]=\left[{ }^{x} / 11\right]+1$ have? | 110 | 93 | 3 |
math | 4. If three angles $x, y, z$ form an arithmetic sequence with a common difference of $\frac{\pi}{3}$, then $\tan x \tan y+\tan y \tan z+\tan z \tan x=$ $\qquad$ | -3 | 53 | 2 |
math | 4. Given that the three sides of $\triangle A B C$ are exactly three consecutive positive integers, and its perimeter and area are $p_{1}$ and $S_{1}$, respectively. If the three sides of $\triangle A B C$ are each increased by 10, the new $\triangle A^{\prime} B^{\prime} C^{\prime}$ has a perimeter and area of $p_{2}$... | \frac{3}{5} | 136 | 7 |
math | Grandpa is twice as strong as Grandma, Grandma is three times stronger than Granddaughter, Granddaughter is four times stronger than Bug, Bug is five times stronger than Cat, and Cat is six times stronger than Mouse. Grandpa, Grandma, Granddaughter, Bug, and Cat together with Mouse can pull out the Turnip, but without ... | 1237 | 91 | 4 |
math | $\begin{array}{l}\text { 9. } \frac{1-2 \log _{6} 5 \cdot \log _{10} 3 \cdot \log _{15} 2}{\log _{6} 5 \cdot \log _{10} 3+\log _{10} 3 \cdot \log _{15} 2+\log _{15} 2 \cdot \log _{6} 5} \\ =\end{array}$ | 1 | 116 | 1 |
math | Let $ABCD$ be a square of side length $13$. Let $E$ and $F$ be points on rays $AB$ and $AD$ respectively, so that the area of square $ABCD$ equals the area of triangle $AEF$. If $EF$ intersects $BC$ at $X$ and $BX = 6$, determine $DF$. | \sqrt{13} | 78 | 6 |
math | Let $E$ and $F$ be the respective midpoints of $BC,CD$ of a convex quadrilateral $ABCD$. Segments $AE,AF,EF$ cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$. | 6 | 61 | 1 |
math | One. (20 points) Given the equations about $x$: $2 x^{2}-4 n x$ $-2 n=1$ and $x^{2}-(3 n-1) x+2 n^{2}-3 n=2$. Does there exist a value of $n$ such that the sum of the squares of the two real roots of the first equation equals an integer root of the second equation? If it exists, find such $n$ values; if not, explain th... | n=0 | 107 | 3 |
math | 3. Find the positive integer solutions $(p, n, k)$ of the equation $3^{p}+4^{p}=n^{k}$, where $p$ is a prime number, $k>1$. | (p,n,k)=(2,5,2) | 46 | 10 |
math | 1A. Factorize the polynomial $x^{5}+x+1$. | (x^{2}+x+1)(x^{3}-x^{2}+1) | 17 | 20 |
math | ## Task Condition
Find the $n$-th order derivative.
$$
y=\lg (5 x+2)
$$ | y^{(n)}=\frac{(-1)^{n-1}\cdot(n-1)!\cdot5^{n}}{\ln10\cdot(5x+2)^{n}} | 26 | 42 |
math | The highest known volcano on Earth is Mauna Kea in the Hawaiian Islands. Its height from base to summit is even 358 meters greater than the elevation of the world's highest mountain, Mount Everest. However, it does not rise from land but from the bottom of the Pacific Ocean, from a depth of 5000 meters. If the sea leve... | 4206 | 192 | 4 |
math | $\left[\begin{array}{l}[\text { Combinatorics (other) })] \\ [\text { Extreme Principle (other) })]\end{array}\right.$
In how many ways can the numbers from 1 to 100 be permuted so that adjacent numbers differ by no more than 1? | 2 | 69 | 1 |
math | \section*{Problem 3 - V01203}
A clock, equipped with a synchronous motor, has ideally uniformly moving hands.
Determine exactly the times at which the hands are positioned such that one hour later, the angle between the hands is of the same size!
(Hint: The relevant angles are less than \(180^{\circ}\).) | \mu_{0}=\sigma_{0}-165\quad\text{or}\quad\mu_{0}=\sigma_{0}+15 | 77 | 34 |
math | 3. Philip's grandfather was born in this century. Philip noticed that the sum of the digits of the birth year of his grandfather is equal to the sum of the digits of the number of years his grandfather had in 1985. In which year was Philip's grandfather born? | 1906 | 58 | 4 |
math | Question 81, In quadrilateral $ABCD$ on the plane, points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively, and $AB=1, EF=\sqrt{2}$, $\mathrm{CD}=3$. If $\overrightarrow{\mathrm{AD}} \cdot \overrightarrow{\mathrm{BC}}=15$, find $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{BD}}$. | 16 | 102 | 2 |
math | 103 Using the digits $1,2,3,4$ to form a 6-digit number, each digit must be used at least once, the number of 6-digit numbers that can be formed is $\qquad$.
Translating the text into English while preserving the original formatting and line breaks, the result is as follows:
103 Using the digits $1,2,3,4$ to form a 6... | 1560 | 119 | 4 |
math | \section*{Problem 22}
What is the smallest \(\mathrm{n}\) for which there is a solution to \(\sin \mathrm{x}_{1}+\sin \mathrm{x}_{2}+\ldots+\sin \mathrm{x}_{\mathrm{n}}=0, \sin \mathrm{x}_{1}+\) \(2 \sin \mathrm{x}_{2}+\ldots+\mathrm{n} \sin \mathrm{x}_{\mathrm{n}}=100 ?\)
| 20 | 103 | 2 |
math | On the board we write a series of $n$ numbers, where $n \geq 40$, and each one of them is equal to either $1$ or $-1$, such that the following conditions both hold:
(i) The sum of every $40$ consecutive numbers is equal to $0$.
(ii) The sum of every $42$ consecutive numbers is not equal to $0$.
We denote by $S_n$ the... | 20 | 124 | 2 |
math | 35. Let $f(x)=x^{2}+a x+b \cos x$, find all pairs of real numbers $(a, b)$, such that the equation $f(x)=0$ and $f(f(x))=0$ have the same and non-empty set of real solutions. | (,b)\mid0\leqslant<4,b=0 | 62 | 16 |
math | ## Task 4 - 040824
Peter is at a summer camp. He wants to buy soda for his group at 21 Pf per bottle and takes empty bottles with him.
With the refund money (30 Pf for each of the empty bottles), he wants to buy as many bottles of soda as possible. For each bottle, 30 Pf deposit must be paid again.
It turns out that... | 14or13 | 124 | 5 |
math | 1. let $S$ be the set of all $n$-tuples $\left(X_{1}, \ldots, X_{n}\right)$, where $X_{1}, \ldots, X_{n}$ are subsets of $\{1,2, \ldots, 1000\}$, which do not all have to be different, and which can also be empty. For $a=\left(X_{1}, \ldots, X_{n}\right) \in S$ denote
$$
E(a)=\text { number of elements of } X_{1} \cup... | 1000(2^{n}-1)2^{999n} | 162 | 18 |
math | 4. (a) Determine all pairs $(x, y)$ of (real) numbers with $0<x<1$ and $0<y<1$ for which $x+3 y$ and $3 x+y$ are both integers. An example of such a pair is $(x, y)=\left(\frac{3}{8}, \frac{7}{8}\right)$, because $x+3 y=\frac{3}{8}+\frac{21}{8}=\frac{24}{8}=3$ and $3 x+y=\frac{9}{8}+\frac{7}{8}=\frac{16}{8}=2$.
(b) Fi... | 11 | 229 | 2 |
math | A non-negative integer $n$ is said to be [i]squaredigital[/i] if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital. | 0, 1, 81 | 42 | 8 |
math | 12.224. In a regular triangular pyramid, the dihedral angle at the base is equal to $\alpha$, the lateral surface area is $S$. Find the distance from the center of the base to the lateral face. | \frac{\sin\alpha}{3}\sqrt{S\sqrt{3}\cos\alpha} | 49 | 21 |
math | Question 23: If real numbers $a$, $b$, $c$ satisfy $2^{a}+4^{b}=2^{c}, 4^{a}+2^{b}=4^{c}$, find the minimum value of $c$.
| \log_{2}3-\frac{5}{3} | 56 | 13 |
math | Vojta started writing the number of the current school year in his notebook: 2019202020192020... and continued doing so indefinitely. When he wrote 2020 digits, he got bored and stopped.
How many twos did he write?
(L. Růžičková)
Hint. How many twos would Vojta write if he only wrote 20 digits? | 757 | 93 | 3 |
math | Example 5 A scientist stored the design blueprint of his time machine in a computer, setting the password to open the file as a permutation of $\{1,2, \cdots, 64\}$. He also designed a program that, when eight positive integers between $1 \sim 64$ are input each time, the computer will indicate the order (from left to ... | 45 | 112 | 2 |
math | Let $u_{n}$ be the sequence defined by $u_{1}=4, u_{2}=9$ and then $u_{n}=3 u_{n-1}-u_{n-2}-2 \cdot(-1)^{n}$. Find $u_{n}$. | u_{n}=F_{n+2}^{2} | 61 | 13 |
math | 4. A 1-meter long strip of paper, at 0.618 meters from one end there is a red dot. Fold the strip of paper in half, and mark a yellow dot where it aligns with the red dot, then unfold the strip and cut it at the red dot. Fold the segment with the yellow dot in half again, and cut it at the point where the yellow dot al... | 0.146 | 108 | 5 |
math | Determine the largest possible radius of a circle that is tangent to both the $x$-axis and $y$-axis, and passes through the point $(9,2)$.
(A circle in the $x y$-plane with centre $(a, b)$ and radius $r$ has equation $(x-a)^{2}+(y-b)^{2}=r^{2}$.) | 17 | 81 | 2 |
math | 16. Let $a, b, c$ be real numbers with $c \neq 1$. It is known that the two equations $x^{2}+a x+1=0$ and $x^{2}+b x+c=0$ have a common real root, and so do the two equations $x^{2}+x+a=0$ and $x^{2}+c x+b=0$. Find the value of $a+b+c$.
設 $a 、 b 、 c$ 為實數, 其中 $c \neq 1$ 。已知 $x^{2}+a x+1=0$ 和 $x^{2}+b x+c=0$ 這兩條方程有一個公共實根... | -3 | 217 | 2 |
math | Example 1. If in a Pythagorean triple, the difference between the hypotenuse and one of the legs is 1, then the form of the Pythagorean triple is
$$
2 a+1,2 a^{2}+2 a, 2 a^{2}+2 a+1 .
$$ | 2a+1, 2a^{2}+2a, 2a^{2}+2a+1 | 68 | 26 |
math | 32. On a $10 \times 10$ board for playing "Battleship," a four-cell "ship" $\square \square$ ( $\square$ is located. What is the minimum number of "shots" needed to hit the ship? (Indicate the method of delivering this number of shots and prove that with fewer shots, the ship can always be placed in such a way that it ... | 24 | 90 | 2 |
math | Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations:
$x_{1}+2x_{2}+...+nx_{n}=0$,
$x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$,
...
$x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$. | x_1 = x_2 = \cdots = x_n = 0 | 114 | 18 |
math | An iguana writes the number 1 on the blackboard. Every minute afterwards, if the number $x$ is written, the iguana erases it and either writes $\frac{1}{x}$ or $x+1$. Can the iguana eventually write the number $\frac{20}{17}$ ? | \frac{20}{17} | 69 | 9 |
math | Determine all non-constant monic polynomials $f(x)$ with integer coefficients for which there exists a natural number $M$ such that for all $n \geq M$, $f(n)$ divides $f(2^n) - 2^{f(n)}$
[i] Proposed by Anant Mudgal [/i] | f(x) = x | 69 | 6 |
math | Find the functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that $f(1)=2, \forall x, y \in \mathbb{Q}: f(x y)=f(x) f(y)-$ $f(x+y)+1$
## - Solution - | f(x)=x+1 | 64 | 6 |
math | Find the smallest prime $p$ such that the digits of $p$ (in base 10) add up to a prime number greater than $10$. | 29 | 34 | 2 |
math | Berlov S.L.
100 people came to a party. Then those who had no acquaintances among the attendees left. Then those who had exactly one acquaintance among the remaining people also left. Then, similarly, those who had exactly 2, 3, 4, ..., 99 acquaintances among the remaining people at the time of their departure also le... | 98 | 90 | 2 |
math | Ivan and Mirka were sharing pears from a plate. Ivan always took two pears, and Mirka took half of what was left on the plate. They proceeded in this manner: Ivan, Mirka, Ivan, Mirka, and finally Ivan, who took the last two pears.
Determine who ended up with more pears and by how many.
(M. Dillingerová)
Hint. How ma... | 2 | 97 | 1 |
math | 9. If the internal angles $A, B, C$ of $\triangle A B C$ satisfy $\sin A=\cos B=\tan C$, find the value of $\cos ^{3} A+\cos ^{2} A-\cos A$. | \frac{1}{2} | 53 | 7 |
math | Example 1. For what integer values of $x$ is $f(x)=x^{2}+5 x-20$ a perfect square? | x=3,4,7,24,-8,-9,-12,-29 | 32 | 20 |
math | Example 4 Real numbers $x_{1}, x_{2}, \cdots, x_{2} 0001$ satisfy
$$
\begin{array}{c}
\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001, \\
\text { let } y_{k}=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+x_{k}\right), k=1,2, \cdots,
\end{array}
$$
2001 . Find the maximum possible value of $\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\r... | 2000 | 174 | 4 |
math | (from The Good Soldier Svejk) Senior military doctor Bautze exposed $abccc$ malingerers among $aabbb$ draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not taken by a stroke at the very moment when the doc... | 10999 | 132 | 5 |
math | 2. $y=\arcsin ^{2} x-2 \arcsin x-2, y_{\max }=$ $\qquad$ | \frac{\pi^{2}}{4}+\pi-2 | 33 | 14 |
math | 5. Over three days, juice was sold in the cafeteria. On the first day, 1 large and 4 medium cans were sold, on the second day - 2 large and 6 liter cans, and on the third day - 1 large, 3 medium, and 3 liter cans. How many liters of juice were sold over 3 days, if the same amount of juice was sold each day? | 54 | 86 | 2 |
math | What is the unit digit of the number $3^{1}+3^{2}+3^{3}+3^{4}+\cdots 3^{2013}$?
Remember, for example, that $3^{2}=3 \times 3=9$. And $3^{3}=3 \times 3 \times 3=27$. Also remember that the unit digit of a number is the last one on the right. For example, the unit digit of 376564534539 is 9. | 3 | 119 | 1 |
math | 1. How many 4-digit numbers exist where the digit in the thousands place is greater than the digit in the hundreds place? | 4500 | 26 | 4 |
math | 10. Let $x+y+z=1, p=2 x^{2}+3 y^{2}+z^{2}$, find $x$ when $p$ takes its minimum value. | \frac{3}{11} | 43 | 8 |
math | Allowing $x$ to be a real number, what is the largest value that can be obtained by the function $25\sin(4x)-60\cos(4x)?$ | 65 | 40 | 2 |
math | The product of several distinct positive integers is divisible by ${2006}^{2}$. Determine the minimum value the sum of such numbers can take. | 228 | 32 | 3 |
math | 10. (This question is worth 20 points) Given that $f(x)$ is an odd function on $\mathbf{R}$, $f(1)=1$, and for any $x<0$, $f\left(\frac{x}{x-1}\right)=x f(x)$. Find the value of $f(1) f\left(\frac{1}{100}\right)+f\left(\frac{1}{2}\right) f\left(\frac{1}{99}\right)+f\left(\frac{1}{3}\right) f\left(\frac{1}{98}\right)+\c... | \frac{2^{98}}{99!} | 168 | 13 |
math | Let $a$ be a real number. Find all real-valued functions $f$ such that
$$\int f(x)^{a} dx=\left( \int f(x) dx \right)^{a}$$
when constants of integration are suitably chosen. | f(x) = k Ce^{kx} | 57 | 11 |
math | 14.5.19 ** The numbers 1 and 2 are written on the blackboard. Now, new numbers are written on the blackboard according to the following rule: if the numbers $a$ and $b$ are already on the blackboard, then the number $ab + a + b$ can be written. Can the following numbers be obtained by such a rule:
(1) 13121, (2) 12131? | 13121 | 100 | 5 |
math | 3. For every 7 boys in a school, there are 8 girls, and for every 9 boys, there is one teacher. In this school, there are a total of 675 students. How many teachers are there in this school? | 35 | 53 | 2 |
math | ## Task 35/78
In the USSR, it is customary to check in urban public transport, after tearing off the ticket control section, whether one has a "lucky ticket." This is the case when the sum of the first three digits equals the sum of the last three digits of the six-digit control number.
What is the probability that a... | 0.001296 | 85 | 8 |
math | Problem 3. Borche's mother is three times older than Borche, and his father is four years older than Borche's mother. How old is each of them if together they are 88 years old? | Borche:12,Mother:36,Father:40 | 45 | 16 |
math | Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
(y+1) f(x)+f(x f(y)+f(x+y))=y
$$
for all $x, y \in \mathbb{R}$. | f(x)=-x | 62 | 5 |
math | 15. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{n}+S_{n}=n$, the sequence $\left\{b_{n}\right\}$ satisfies $b_{1}=a_{1}$, and $b_{n}=a_{n}-$ $a_{n-1}(n \geqslant 2)$.
(1) Find the general term formula for $\left\{a_{n}\right\}$;
(2) Find the general term formula for $\left\{b_{n}\right\}$;
(3) Find the sum ... | 1-(\frac{1}{2})^{n} | 158 | 12 |
math | 4. Find all positive integers $n$, such that
$$
\frac{n^{3 n-2}-3 n+1}{3 n-2} \in \mathbf{Z} \text {. }
$$ | 1 | 46 | 1 |
math | Let $d_1, d_2, \ldots , d_{k}$ be the distinct positive integer divisors of $6^8$. Find the number of ordered pairs $(i, j)$ such that $d_i - d_j$ is divisible by $11$. | 665 | 59 | 3 |
math | 11. Let $a$ and $b$ be integers for which $\frac{a}{2}+\frac{b}{1009}=\frac{1}{2018}$. Find the smallest possible value of $|a b|$. | 504 | 55 | 3 |
math | 10. Problem: Let $S$ be the locus of all points $(x, y)$ in the first quadrant such that $\frac{x}{t}+\frac{y}{1-t}=1$ for some $t$ with $0<t<1$. Find the area of $S$. | \frac{1}{6} | 61 | 7 |
math | Each of the positive integers 2013 and 3210 has the following three properties:
(i) it is an integer between 1000 and 10000 ,
(ii) its four digits are consecutive integers, and
(iii) it is divisible by 3 .
In total, how many positive integers have these three properties? | 66 | 75 | 2 |
math | Let $I_{m}=\textstyle\int_{0}^{2 \pi} \cos (x) \cos (2 x) \cdots \cos (m x) d x .$ For which integers $m, 1 \leq m \leq 10$ is $I_{m} \neq 0 ?$ | m = 3, 4, 7, 8 | 74 | 13 |
math | Example 1 In an $8 \times 8$ chessboard, how many ways are there to select 56 squares such that: all the black squares are selected, and each row and each column has exactly seven squares selected? ? ${ }^{[1]}$
(2014, Irish Mathematical Olympiad) | 576 | 67 | 3 |
math | 3. Solve the inequality $5^{\frac{x+5}{x+4}} \cdot 3^{x+4} \geq 75$. | x\in(-4;-3]\cup[\log_{3}5-4;+\infty) | 34 | 22 |
math | 5. If a number can be expressed as the sum of the digits of some multiple of 91, then this number is called a "harmonious number". Then, among $1,2, \cdots, 2008$, the number of harmonious numbers is $\qquad$ | 2007 | 63 | 4 |
math | In a certain competition, each player plays exactly one game with every other player. The winner of each game gets 1 point, the loser gets 0 points. If it's a tie, each gets $\frac{1}{2}$ point. After the competition, it is found that each player's score is exactly half from the games played against the ten lowest-scor... | 25 | 106 | 2 |
math | 1. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}=2 a_{n}-1(n=1,2, \cdots)$, and the sequence $\left\{b_{n}\right\}$ satisfies $b_{1}=3$, $b_{k+1}=a_{k}+b_{k}(k=1,2, \cdots)$. Find the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$. | 2^{n}+2n-1 | 122 | 9 |
math | 6. On six oaks, there were 129 birds. At one moment, 6 flew away from the first, 11 from the second, 8 from the third, 10 from the fourth, 7 from the fifth, and 9 from the sixth oak. Then, an equal number of birds remained on each oak. How many birds were there on each oak at the beginning? | 19,24,21,23,20,22 | 85 | 17 |
math | ## Task B-2.1.
Calculate $\sqrt{1-A^{-1}}$ if $A=\frac{\sqrt{2022}+1}{\sqrt{2022}-1}-\frac{2}{\sqrt{2022}+1}$. | \frac{2}{45} | 61 | 8 |
math | There is exactly one isosceles triangle that has a side of length 10 and a side of length 22 . What is the perimeter of this triangle? | 54 | 35 | 2 |
math | For real numbers $a$ and $b$, we define $a \nabla b=a b-b a^{2}$. For example, $5 \nabla 4=5(4)-4\left(5^{2}\right)=-80$. Determine the sum of the values of $x$ for which $(2 \nabla x)-8=x \nabla 6$. | \frac{4}{3} | 86 | 7 |
math | A spider web has the form of a grid of $100 \times 100$ nodes (in other words, it is a grid of 99×99 cells). In one of its corners sits a spider, and in some 100 nodes, flies are stuck to the web. In one move, the spider can move to any adjacent node. Can the spider guarantee to eat all the flies, spending no more than... | 1980 | 111 | 4 |
math | 8. In $\triangle A B C$, it is known that the three angles $A, B, C$ form an arithmetic sequence. Suppose the sides opposite to these angles are $a, b, c$, respectively, and $c-a$ equals the height $h$ from vertex $A$ to side $AC$, then $\sin \frac{C-A}{2}=$ $\qquad$ . | \frac{1}{2} | 83 | 7 |
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