In a lake there are several sorts of fish, in the following distribution: $ 18\%$ catfish, $ 2\%$ sturgeon and $ 80\%$ other. Of a catch of ten fishes, let $ x$ denote the number of the catfish and $ y$ that of the sturgeons. Find the expectation of $ \frac {x}{y \plus{} 1}$

Let $A$ be a set of $65$ integers with pairwise different remainders modulo $2016$. Prove that exists a subset $B=\{a,b,c,d\}$ of set $A$ such that $a+b-c-d$ is divisible with $2016$

Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.

Find a positive integral solution to the equation \[\frac{1+3+5+\dots+(2n-1)}{2+4+6+\dots+2n}=\frac{115}{116}\] $\textbf{(A) }110\qquad\textbf{(B) }115\qquad\textbf{(C) }116\qquad\textbf{(D) }231\qquad\textbf{(E) }\text{The equation has no positive integral solutions.}$

Emerald writes $ 2009^2$ integers in a $ 2009\times 2009$ table, one number in each entry of the table. She sums all the numbers in each row and in each column, obtaining $ 4018$ sums. She notices that all sums are distinct. Is it possible that all such sums are perfect squares?

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 $x, y \in R$.