Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed?

At a maths contest $n$ books are given as prizes to $n$ students (each students gets one book). In how many ways can the organisers give these prizes if they have $n$ copies of one book and $2n+1$ other books each in one copy?

Let $E$ and $F$ be points on side $BC$ of a triangle $\vartriangle ABC$. Points $K$ and $L$ are chosen on segments $AB$ and $AC$, respectively, so that $EK \parallel AC$ and $FL \parallel AB$. The incircles of $\vartriangle BEK$ and $\vartriangle CFL$ touches segments $AB$ and $AC$ at $X$ and $Y$ , respectively. Lines $AC$ and $EX$ intersect at $M$, and lines $AB$ and $FY$ intersect at $N$. Given that $AX = AY$, prove that $MN \parallel BC$.

This problem is generalization of [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=5918]this one[/url]. Suppose $G$ is a graph and $S\subset V(G)$. Suppose we have arbitrarily assign real numbers to each element of $S$. Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.

If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

Let \( m \) and \( n \) be positive integers. Kellem and Carmen play the following game: initially, the number $0$ is on the board. Starting with Kellem and alternating turns, they add powers of \( m \) to the previous number on the board, such that the new value on the board does not exceed \( n \). The player who writes \( n \) wins. Determine, for each pair \( (m, n) \), who has the winning strategy. [b]Note:[/b] A power of \( m \) is a number of the form \( m^k \), where \( k \) is a non-negative integer.

Kelvin the Frog has a pair of standard fair $8$-sided dice (each labelled from $1$ to $8$). Alex the sketchy Kat also has a pair of fair $8$-sided dice, but whose faces are labelled differently (the integers on each Alex's dice need not be distinct). To Alex's dismay, when both Kelvin and Alex roll their dice, the probability that they get any given sum is equal! Suppose that Alex's two dice have $a$ and $b$ total dots on them, respectively. Assuming that $a \neq b$, find all possible values of $\min \{a,b\}$.

A point $M$ is chosen inside the square $ABCD$ in such a way that $\angle MAC = \angle MCD = x$ . Find $\angle ABM$.

There are some people in a meeting; each doesn't know at least 56 others, and for any pair, there exist a third one who knows both of them. Can the number of people be 65?

$n$ symbols line up in a row, numbered as $1,2,...,n$ from left to right. Delete every symbol with squared numbers. Renumber the rest from left to right. Repeat the process until all $n$ symbols are deleted. Let $f(n)$ be the initial number of the last symbol deleted. Find $f(n)$ in terms of $n$ and find $f(2019)$.

In every cell of a board $101 \times 101$ is written a positive integer. For any choice of $101$ cells from different rows and columns, their sum is divisible by $101$. Show that the number of ways to choose a cell from each row of the board, so that the total sum of the numbers in the chosen cells is divisible by $101$, is divisible by $101$.

A natural number $n$ is called "good" if there exists natural numbers $a$ and $b$ such that $a+b=n$ and $ab \mid n^2+n+1$. Show that there are infinitely many "good" numbers

There are $2024$ people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were $0,1, \ldots, 2023$. Every knight answered truthfully, while every liar changed the real answer by exactly $1$. What is the minimal number of liars?

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

Solve the equation: $| z |- 2 = 1 + 2 i$, where $| r |$ is the modulus of a complex number $z$.

Bart, Lisa and Maggie play the following game: Bart colors finitely many points red or blue on a circle such that no four colored points can be chosen on the circle such that their colors are blue-red-blue-red (the four points do not have to be consecutive). Lisa chooses finitely many of the colored points. Now Bart gives the circle (possibly rotated) to Maggie with Lisa's chosen points, however, without their colors. Finally, Maggie colors all the points of the circle to red or blue. Lisa and Maggie wins the game, if Maggie correctly guessed the colors of Bart's points. A strategy of Lisa and Maggie is called a winning strategy, if they can win the game for all possible colorings by Bart. Prove that Lisa and Maggie have a winning strategy, where Lisa chooses at most $c$ points in all possible cases, and find the smallest possible value of $c$. [i]Proposed by Dömötör Pálvölgyi, Budapest[/i]

The only pieces on an $8\times8$ chessboard are three rooks. Each moves along a row or a column without running to or jumping over another rook. The white rook starts at the bottom left corner, the black rook starts in the square directly above the white rook, and the red rook starts in the square directly to the right of the white rook. The white rook is to finish at the top right corner, the black rook in the square directly to the left of the white rook, and the red rook in the square directly below the white rook. At all times, each rook must be either in the same row or the same column as another rook. Is it possible to get the rooks to their destinations?

Let $S$ be the set of all permutations of $\{1, 2, 3, 4, 5\}$. For $s = (a_1, a_2,a_3,a_4,a_5) \in S$, define $\text{nimo}(s)$ to be the sum of all indices $i \in \{1, 2, 3, 4\}$ for which $a_i > a_{i+1}$. For instance, if $s=(2,3,1,5,4)$, then $\text{nimo}(s)=2+4=6$. Compute \[\sum_{s\in S}2^{\text{nimo}(s)}.\] [i]Proposed by Mehtaab Sawhney[/i]

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Let $ f(x) $ be a polynomial, $ n $ - a natural number. Prove that if $ f(x^{n}) $ is divisible by $ x-1 $, then it is also divisible by $ x^{n-1} + x^{n-2} + \ldots + x + $1.

Let $ABC$ be a triangle with incenter $I$. The points $D$ and $E$ lie on the segments $CA$ and $BC$ respectively, such that $CD = CE$. Let $F$ be a point on the segment $CD$. Prove that the quadrilateral $ABEF$ is circumscribable if and only if the quadrilateral $DIEF$ is cyclic. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Prove the following inequality: \[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1} x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in \mathbb{N}.$

Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?

Let $a$, $b$, $c$ be the side lengths of a triangle. Prove that $$ \sqrt[3]{(a^2+bc)(b^2+ca)(c^2+ab)} > \frac{a^2+b^2+c^2}{2}. $$

Each cell of a $1000\times 1000$ table contains $0$ or $1$. Prove that one can either cut out $990$ rows so that at least one $1$ remains in each column, or cut out $990$ columns so that at least one $0$ remains in each row.