A finite set $M$ of real numbers is called special if $M$ has at least two elements and the following condition is true: If $a$ and $b$ are distinct elements of $M$ then $5\sqrt{|a|}-\frac{2b}{3}$ is also a element of $M$. a) Determine if there is a special set with (exactly) two elements. b) Determine if there is a special set with three (or more) elements such that all elements are positive.

Find all positive integers $b$ for which there exists a positive integer $a$ with the following properties: - $a$ is not a divisor of $b$. - $a^a$ is a divisor of $b^b$

The tangent lines to the circumcircle of triangle ABC passing through vertices $B$ and $C$ intersect at point $F$. Points $M$, $L$ and $N$ are the feet of the perpendiculars from vertex $A$ to the lines $FB$, $FC$ and $BC$ respectively. Show that $AM+AL \geq 2AN$

There are $300$ apples in a table and the heaviest apple is not heavier than three times the weight of the lightest apple. Prove that the apples can be splitted in sets of $4$ elements such that no set is heavier than $\frac{3}{2}$ times the weight of any other set.

Determine all integers $k$ such that the equation: $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{k}{xyz}$$has an infinite number of integer solutions $(x,y,z)$ with gcd$(k,xyz)=1$.

Find all functions $f : R \to R$ such that $$f(x + y) \ge xf(x) + yf(y)$$, for all $x, y \in R$ .

Let $x_0,x_1,\dots, x_{n-1}$ be real numbers such that $0<|x_0|<|x_1|<\dots<|x_{n-1}|$. We will write the sum of the elements of each one of the $2^n$ subsets of $\{x_0,x_1,\dots,x_{n-1}\}$ in a paper. Prove that the $2^n$ written numbers are consecutive elements of a arithmetic progression if and only if the ratios $$|\frac{x_i}{x_j}|, 0\leq j<i\leq n-1$$are equal(s) to the ratio(s) obtained with the numbers $2^0,2^1,\dots,2^{n-1}$. Note: The sum of the elements of the empty set is $0$.