Let $n\geq 1$ be a positive integer. We say that an integer $k$ is a [i]fan [/i]of $n$ if $0\leq k\leq n-1$ and there exist integers $x,y,z\in\mathbb{Z}$ such that \begin{align*} x^2+y^2+z^2 &\equiv 0 \pmod n;\\ xyz &\equiv k \pmod n. \end{align*} Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.

i) Prove that for every sequence $(a_{n})_{n\in \mathbb{N}}$, such that $a_{n}>0$ for all $n \in \mathbb{N}$ and $\sum_{n=1}^{\infty}a_{n}<\infty$, we have $$\sum_{n=1}^{\infty}(a_{1}a_{2} \cdots a_{n})^{\frac{1}{n}}< e\sum_{n=1}^{\infty}a_{n}.$$ ii) Prove that for every $\epsilon>0$ there exists a sequence $(b_{n})_{n\in \mathbb{N}}$ such that $b_{n}>0$ for all $n \in \mathbb{N}$ and $\sum_{n=1}^{\infty}b_{n}<\infty$ and $$\sum_{n=1}^{\infty}(b_{1}b_{2} \cdots b_{n})^{\frac{1}{n}}> (e-\epsilon)\sum_{n=1}^{\infty}b_{n}.$$

$p(x)$ is a polynomial of degree $n$ with leading coefficient $c$, and $q(x)$ is a polynomial of degree $m$ with leading coefficient $c$, such that \[ p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1 \] Show that $p'(x) = nq(x)$.

Let $a,b$ and $c$ be real numbers such that $|a+b|+|b+c|+|c+a|=8.$ Find the maximum and minimum value of the expression $P=a^2+b^2+c^2.$

A group $ G $ of order at least $ 4 $ has the property that there exists a natural number $ n\not\in\{ 1,|G| \} $ such that $ G $ admits exactly $ \binom{|G|-1}{n-1} $ subgroups of order $ n. $ Show that $ G $ is commutative. [i]Marius Tărnăuceanu[/i]

A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\sqrt7$. Notes: ''Using a ruler construct points $\ldots$'' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler.

A line, which passes through the incentre $I$ of the triangle $ABC$, meets its sides $AB$ and $BC$ at the points $M$ and $N$ respectively. The triangle $BMN$ is acute. The points $K,L$ are chosen on the side $AC$ such that $\angle ILA=\angle IMB$ and $\angle KC=\angle INB$. Prove that $AM+KL+CN=AC$. [i]S. Berlov[/i]

Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$

Graph $G$ has $n\geq 2$ vertices. Find the largest $m$ such that one of the following is true for always: 1. There exists a cycle with $k\geq m$ vertices. 2. There exists an independent set with $m$ vertices.

Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius $72$ and rolls along the surface toward the smaller tube, which has radius $24$. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance $x$ from where it starts. The distance $x$ can be expressed in the form $a\pi+b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c.$

a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges? b) 2 white and 3 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 11, to another black cat is 7 and to the third black cat is 9. The sum of distances from the black cats to one white cat is 12, to the other white cat is 15. What cats are sitting on the edges? [size=85](Kyiv mathematical festival 2014)[/size]

Let $O$ be the circumcenter of an acute $\vartriangle ABC$ which has altitude $AD$. Let $AO$ intersect the circumcircle of $\vartriangle BOC$ again at $X$. If $E$ and $F$ are points on lines $AB$ and $AC$ such that $\angle XEA = \angle XFA = 90^o$ , then prove that the line $DX$ bisects the segment $EF$.

At the Rice Mathematics Tournament, 80% of contestants wear blue jeans, 70% wear tennis shoes, and 80% of those who wear blue jeans also wear tennis shoes. What fraction of people wearing tennis shoes are wearing blue jeans?

Let $P_1P_2\dotsb P_{100}$ be a cyclic $100$-gon and let $P_i = P_{i+100}$ for all $i$. Define $Q_i$ as the intersection of diagonals $\overline{P_{i-2}P_{i+1}}$ and $\overline{P_{i-1}P_{i+2}}$ for all integers $i$. Suppose there exists a point $P$ satisfying $\overline{PP_i}\perp\overline{P_{i-1}P_{i+1}}$ for all integers $i$. Prove that the points $Q_1,Q_2,\dots, Q_{100}$ are concyclic. [i]Michael Ren[/i]

Let $ABC$ be a triangle such that $AB=2$, $CA=3$, and $BC=4$. A semicircle with its diameter on $BC$ is tangent to $AB$ and $AC$. Compute the area of the semicircle.

Let $n$ be a positive integer, where $n \geq 3$ and let $a_1, a_2, ..., a_n$ be the lengths of sides of some $n$-gon. Prove that $$a_1 + a_2 + ... + a_n \geq \sqrt{2 \cdot (a_1^2 + a_2^2 + ... + a_n^2)} $$