Several positive integers $a_0 , a_1 , a_2 , ... , a_n$ are written on a board. On a second board, we write the amount $b_0$ of numbers written on the first board, the amount $b_1$ of numbers on the first board exceeding $1$, the amount $b_2$ of numbers greater than $2$, and so on as long as the $b$s are still positive. Then we stop, so that we do not write any zeros. On a third board we write the numbers $c_0 , c_1 , c_2 , ...$. using the same rules as before, but applied to the numbers $b_0 , b_1 , b_2 , ...$ of the second board. Prove that the same numbers are written on the first and the third boards. (H. Lebesgue - A Kanel)

Let $a_n$ denote an angle from the interval for each $\left( 0, \frac{\pi}{2}\right)$ , the tangent of which is equal to $n$ . Prove that $$\sqrt{1+1^2} \sin(a_1-a_{1000}) + \sqrt{1+2^2} \sin(a_2-a_{1000})+...+\sqrt{1+2000^2} \sin(a_{2000}-a_{1000}) = \sin a_{1000} $$

$\bullet$ Determine a natural $n$ such that the constant sum $S$ of a magic square of $ n \times n$ (that is, the sum of its elements in any column, or the diagonal) differs as little as possible from $1992$. $\bullet$ Construct or describe the construction of this magic square.

Prove that $\sum_{n=1}^{2022^{2022}} \frac{1}{\sqrt{n^3+2n^2+n}}<\frac{19}{10}$.

If $a_1 \le a_2 \le .. \le a_n$ are natural numbers ($n \ge 2$), show that the inequality $$\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0$$ holds for all $n$-tuples $(x_1,...,x_n) \ne (0,..., 0)$ of real numbers if and only if $a_2 \ge 2$.

Let a positive integer $n$ be given. Determine, in terms of $n$, the least positive integer $k$ such that among any $k$ positive integers, it is always possible to select a positive even number of them having sum divisible by $n$.

Let $m, n$ be positive integers. Consider a sequence of positive integers $a_1, a_2, ... , a_n$ that satisfies $m = a_1 \ge a_2\ge ... \ge a_n \ge 1$. Then define, for $1\le  i\le  m$, $b_i =$ # $\{ j \in \{1, 2, ... , n\}: a_j \ge i\}$, so $b_i$ is the number of terms $a_j $ of the given sequence for which $a_j  \ge i$. Similarly, we define, for $1\le   j \le  n$, $c_j=$ # $\{ i \in \{1, 2, ... , m\}: b_i \ge j\}$ , thus $c_j$ is the number of terms bi in the given sequence for which $b_i \ge j$. E.g.: If $a$ is the sequence $5, 3, 3, 2, 1, 1$ then $b$ is the sequence $6, 4, 3, 1, 1$. (a) Prove that $a_j = c_j $ for $1  \le j  \le n$. (b) Prove that for $1\le  k \le m$: $\sum_{i=1}^{k} b_i = k \cdot b_k + \sum_{j=b_{k+1}}^{n} a_j$.

Can You fill the squares of the infinite cross-lined paper with integers so, that the sum of the numbers in every $4\times 6$ fields rectangle would be a) $10$? b) $1$?

Given a positive integer $n$, show that the set $\{1,2,...,n\}$ can be partitioned into $m$ sets, each with the same sum, if and only if m is a divisor of $\frac{n(n + 1)}{2}$ which does not exceed $\frac{n + 1}{2}$.

Each positive integer should be colored red or green in such a way that the following two conditions are met: - Let $n$ be any red number. The sum of any $n$ (not necessarily different) red numbers is red. - Let $m$ be any green number. The sum of any $m$ (not necessarily different) green numbers is green. Determine all such colorings.

Dene the set $F$ as the following: $F = \{(a_1,a_2,... , a_{2006}) : \forall i = 1, 2,..., 2006, a_i \in \{-1,1\}\}$ Prove that there exists a subset of $F$, called $S$ which satises the following: $|S| = 2006$ and for all $(a_1,a_2,... , a_{2006})\in F$ there exists $(b_1,b_2,... , b_{2006}) \in S$, such that $\Sigma_{i=1} ^{2006}a_ib_i = 0$.

A tourist office was investigating the numbers of sunny and rainy days in a year in each of six regions. The results are partly shown in the following table: Region , sunny or rainy , unclassified $A \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 336 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,29$ $B \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 321 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,44$ $C \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 335 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,30$ $D \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 343 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,22$ $E \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 329 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,36$ $F \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 330 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,35$ Looking at the detailed data, an officer observed that if one region is excluded, then the total number of rainy days in the other regions equals one third of the total number of sunny days in these regions. Determine which region is excluded.

Positive numbers $a_1,a_2,...,a_n, b_1, b_2,...,b_n$ satisfy the condition $a_1+a_2+...+a_n=b_1+ b_2+...+b_n=1$. Find the smallest possible value of the sum $$\frac{a_1^2}{a_1+b_1}+\frac{a_2^2}{a_2+b_2}+...+\frac{a_n^2}{a_n+b_n}$$ (V.Kolbun)

Write $102$ as the sum of the largest number of distinct primes.

In each box of a $ 1 \times 2009$ grid, we place either a $ 0$ or a $ 1$, such that the sum of any $ 90$ consecutive boxes is $ 65$. Determine all possible values of the sum of the $ 2009$ boxes in the grid.

Let $a, b, c$ be positive real numbers with $abc = 1$. Find all possible values ​​of the expression $$\frac{1 + a}{1 + a + ab}+\frac{1 + b}{1 + b + bc}+\frac{1 + c}{1 + c + ca}$$ can take.

A $x>2$ real number is given. Bob has got $1997$ labels and writes one of the numbers $"x^0, x^1, x^2 ,\dotsm x^{1995}, x^{1996}"$ each labels such that all labels has distinct numbers. Bob puts some labels to right pocket, some labels to left pocket. Prove that sum of numbers of the right pocket never equal to sum of numbers of the left pocket.

Prove that some (or one) of any $100$ integers can always be chosen so that the sum of the chosen integers is divisible by $100$.

Consider a sequence of positive integers with total sum $2019$ such that no number and no sum of a set of consecutive num bers is equal to $40$. What is the greatest possible length of such a sequence? (Alexandr Shapovalov)

Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

Determine the natural number $a =\frac{p+q}{r}+\frac{q+r}{p}+\frac{r+p}{q}$ where $p, q$ and $r$ are prime positive numbers.

Let $S$ be the sum of the greatest odd divisors of the natural numbers $1$ through $2^n$. Prove that $3S = 4^n + 2$.

Find all natural numbers that have exactly six divisors whose sum is $3500$.

Consider a triple $(a, b, c)$ of pairwise distinct positive integers satisfying $a + b + c = 2013$. A step consists of replacing the triple $(x, y, z)$ by the triple $(y + z - x,z + x - y,x + y - z)$. Prove that, starting from the given triple $(a, b,c)$, after $10$ steps we obtain a triple containing at least one negative number.