There are $n$ knights numbered $1$ to $n$ and a round table with $n$ chairs. The first knight chooses his chair, and from him, the knight number $k+1$ sits $ k$ places to the right of knight number $k$ , for all $1 \le k\le n-1$ (occupied and empty seats are counted). In particular, the second knight sits next to the first. Find all values ​​of $n$ such that the $n$ gentlemen occupy the $n$ chairs following the described procedure.

Each day $289$ students are divided into $17$ groups of $17$. No two students are ever in the same group more than once. What is the largest number of days that this can be done?

The triangle $ABC$ has $\angle BCA=90^{\circ}.$ Bisector of angle $\angle CAB$ intersects the side $BC$ in point $P$ and bisector of angle $\angle ABC$ intersects the side $AC$ in point $Q.$ If $M$ and $N$ are projections of $P$ and $Q$ on side $AB$, find the measure of the angle $\angle MCN.$

Alphonse and Beryl are playing a game. The game starts with two rectangles with integer side lengths. The players alternate turns, with Alphonse going first. On their turn, a player chooses one rectangle, and makes a cut parallel to a side, cutting the rectangle into two pieces, each of which has integer side lengths. The player then discards one of the three rectangles (either the one they did not cut, or one of the two pieces they cut) leaving two rectangles for the other player. A player loses if they cannot cut a rectangle. Determine who wins each of the following games: (a) The starting rectangles are $1 \times 2020$ and $2 \times 4040$. (b) The starting rectangles are $100 \times 100$ and $100 \times 500$.

A rectangular sheet of paper with integer length sides is given. The sheet is marked with unit squares. Arrows are drawn at each lattice point on the sheet in a way that each arrow is parallel to one of its sides, and the arrows at the boundary of the paper do not point outwards. Prove that there exists at least one pair of neighboring lattice points (horizontally, vertically or diagonally) such that the arrows drawn at these points are in opposite directions.

Find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$n!+f(m)!|f(n)!+f(m!)$$ for all $m,n\in\mathbb{N}$ [i]Proposed by Valmir Krasniqi and Dorlir Ahmeti, Albania[/i]

If point $M(x,y)$ lies on the line with equation $y=x+2$ and $1<y<3$, calculate the value of $A=\sqrt{y^2-8x}+\sqrt{y^2+2x+5}$

Let $f:\mathbb{Q}\times\mathbb{Q}\to\mathbb{Q}$ be a function satisfying: [list] [*] For any $x_1,x_2,y_1,y_2 \in \mathbb Q$, $$f\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \leq \frac{f(x_1,y_1)+f(x_2,y_2)}{2}.$$ [*] $f(0,0) \leq 0$. [*] For any $x,y \in \mathbb Q$ satisfying $x^2+y^2>100$, the inequality $f(x,y)>1$ holds.\ Prove that there is some positive rational number $b$ such that for all rationals $x,y$, $$f(x,y) \ge b\sqrt{x^2+y^2} - \frac{1}{b}.$$

Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

Let $n$ be a positive integer. Every square in a $n \times n$-square grid is either white or black. How many such colourings exist, if every $2 \times 2$-square consists of exactly two white and two black squares? The squares in the grid are identified as e.g. in a chessboard, so in general colourings obtained from each other by rotation are different.

In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.

Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.

Find $x$ such that \[\frac{\frac{5}{x-50}+ \frac{7}{x+25}}{\frac{2}{x-50}- \frac{3}{x+25}} = 17.\]

Let $k,n$ be positive integers with $n \geq k \geq 3$. We consider $n+1$ points on the real plane with none three of them on the same line. We colour any segment between the points with one of $k$ possibilities. We say that an angle is a "bicolour angle" iff its vertex is one of the $n+1$ points and the two segments that define it are of different colours. Show that there is always a way to colour the segments that makes more than $n \Big\lfloor{\frac{n}{k}}\Big\rfloor^2 \frac{k(k-1)}{2}$ bicolour angles.

Let $ABCD$ be a rhombus and suppose $E$ and $F$ are the midpoints of $\overline{AD}$ and $\overline{EF}$ are the midpoints of $\overline{AD}$ and $\overline{BC},$ respectively. If $G$ is the intersection of $\overline{AC}$ and $\overline{EF},$ find the ratio of the area of $AEG$ to the area of $AGFB.$

A $n$ by $n$ magic square contains numbers from $1$ to $n^2$ such that the sum of every row and every column is the same. What is this sum?

Find all integers $n \geq 3$ such that the following property holds: if we list the divisors of $n!$ in increasing order as $1 = d_1 < d_2 < \dots < d_k = n!$, then we have \[ d_2 - d_1 \leq d_3 - d_2 \leq \dots \leq d_k - d_{k-1}. \] [i]Proposed by Luke Robitaille.[/i]

Consider five-dimensional Cartesian space $R^5 = \{(x_1, x_2, x_3, x_4, x_5) | x_i \in R\}$, and consider the hyperplanes with the following equations: $\bullet$ $x_i = x_j$ for every $1 \le i < j \le 5$, $\bullet$ $x_1 + x_2 + x_3 + x_4 + x_5 = -1$, $\bullet$ $x_1 + x_2 + x_3 + x_4 + x_5 = 0$, $\bullet$ $x_1 + x_2 + x_3 + x_4 + x_5 = 1$. Into how many regions do these hyperplanes divide $R^5$ ?

Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality \[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4\] holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.

Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.

Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \right)^2 \] can be written as $\tfrac{m}{n} + \sqrt{k}$, where $m$, $n$ and $k$ are positive integers with $\gcd(m,n)=1$, compute $m+10n+100k$. [i]Proposed by Evan Chen[/i]

Assume we have $95$ boxes and $19$ balls distributed in these boxes in an arbitrary manner. We take $6$ new balls at a time and place them in $6$ of the boxes, one ball in each of the six. Can we, by repeating this process a suitable number of times, achieve a situation in which each of the $95$ boxes contains an equal number of balls?

Determine all positive integers $n$ such that $5^n - 1$ can be written as a product of an even number of consecutive integers.

A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatet number of elements that $\mathcal{S}$ can have?

Place eight rooks on a standard $8 \times 8$ chessboard so that no two are in the same row or column. With the standard rules of chess, this means that no two rooks are attacking each other. Now paint $27$ of the remaining squares (not currently occupied by rooks) red. Prove that no matter how the rooks are arranged and which set of $27$ squares are painted, it is always possible to move some or all of the rooks so that: • All the rooks are still on unpainted squares. • The rooks are still not attacking each other (no two are in the same row or same column). • At least one formerly empty square now has a rook on it; that is, the rooks are not on the same $8$ squares as before.