Let $\Delta ABC$ be a triangle with orthocenter $H$ and $\Gamma$ be the circumcircle of $\Delta ABC$ with center $O$. Consider $N$ the center of the circle that passes through the feet of the heights of $\Delta ABC$ and $P$ the intersection of the line $AN$ with the circle $\Gamma$. Suppose that the line $AP$ is perpendicular to the line $OH$. Prove that $P$ belongs to the reflection of the line $OH$ by the line $BC$.

Let $p$ and $q$ be distinct odd primes. Show that $$\bigg\lceil \dfrac{p^q+q^p-pq+1}{pq} \bigg\rceil$$is even.

Let $a_1, a_2, \ldots , a_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = 1$. Prove that $$\dfrac{a_1}{\sqrt{1-a_1}}+\cdots+\dfrac{a_n}{\sqrt{1-a_n}} \geq \dfrac{1}{\sqrt{n-1}}(\sqrt{a_1}+\cdots+\sqrt{a_n}).$$

In the reality show Big Sister Brasil, it is said that there is a treta if two people are friends with each other and enemies with a third one. For audience purposes, the broadcaster wants a lot of tretas. If friendship and enmity are reciprocal relationships, given $n$ people, what is the maximum number of tretas?

Let $\mathbb{Z}_{>0} = \{1, 2, 3, \ldots \}$ be the set of all positive integers. Find all strictly increasing functions $f : \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that $f(f(n)) = 3n$.

Let $A$ be a finite set made up of prime numbers. Determine if there exists an infinite set $B$ that satisfies the following conditions: $(i)$ the prime factors of any element of $B$ are in $A$; $(ii)$ no term of $B$ divides another element of this set.

Let $\Delta ABC$ be a triangle and $L$ be the foot of the bisector of $\angle A$. Let $O_1$ and $O_2$ be the circumcenters of $\triangle ABL$ and $\triangle ACL$ respectively and let $B_1$ and $C_1$ be the projections of $C$ and $B$ through the bisectors of the angles $\angle B$ and $\angle C$ respectively. The incircle of $\Delta ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively and the bisectors of angles $\angle B$ and $\angle C$ meet the perpendicular bisector of $AL$ at points $Q$ and $P$ respectively. Prove that the five lines $PC_0, QB_0, O_1C_1, O_2B_1$ and $BC$ are all concurrent.

A cricket wants to move across a $2n \times 2n$ board that is entirely covered by dominoes, with no overlap. He jumps along the vertical lines of the board, always going from the midpoint of the vertical segment of a $1 \times 1$ square to another midpoint of the vertical segment, according to the rules: $(i)$ When the domino is horizontal, the cricket jumps to the opposite vertical segment (such as from $P_2$ to $P_3$); $(ii)$ When the domino is vertical downwards in relation to its position, the cricket jumps diagonally downwards (such as from $P_1$ to $P_2$); $(iii)$ When the domino is vertically upwards relative to its position, the cricket jumps diagonally upwards (such as from $P_3$ to $P_4$). The image illustrates a possible covering and path on the $4 \times 4$ board. Considering that the starting point is on the first vertical line and the finishing point is on the last vertical line, prove that, regardless of the covering of the board and the height at which the cricket starts its path, the path ends at the same initial height.