Indra has a bag for bringing flowers for her grandmother. The first day she brings $n$ flowers. From the second day Indra tries to bring three times plus one with respect to the number of flowers of the previous day. However, if this number is greater or equal to $40$, Indra substracts multiples of $40$ until the remainder is less than this number, since her bag cannot containt so many flowers. For which value of $n$ Indra will bring $30$ flowers the day $2016$?

Let $A$ be an infinite set of real numbers containing at least one irrational number. Prove that for every natural number $n > 1$ there exists a subset $S$ of $A$ with n elements such that the sum of the elements of $S$ is an irrational number.

A circle $\omega$ with center $O$ and radius $r$ is constructed. A point $P$ is chosen on the circumference $\omega$ and a point A is taken inside it, such that is outside the line that passes through $P$ and $O$. Point $B$ is constructed, the reflection of $A$ wrt $O$. and $P'$ is another point on the circumference such that the chord $PP'$ is perpendicular to $PA$. Let $Q$ be the point on the line $PP'$ that minimizes the sum of distances from $A$ to $Q$ and from $Q$ to $B$. Show that the value of the sum of the lengths $AQ+QB$ does not depend on the choice of points $P$ or $A$

Points $A, B, C$ and $D$ lie on a circle, in that order clockwise, such that there is a point $E$ on segment $CD$ with the property that $AD = DE$ and $BC = EC$. Prove that the intersection point of the bisectors of the angles $\angle DAB$ and $\angle ABC$ is on the line $CD$.

Determine all real solutions of the following system of equations: $$x+y^2=y^3$$$$y+x^2=x^3$$

The vertices of a regular polygon with $2016$ sides are colored gold or silver. Prove that there are at least $512$ different isosceles triangles whose vertices have the same color.