OMCC 2018

There are 2018 cards numbered from 1 to 2018. The numbers of the cards are visible at all times. Tito and Pepe play a game. Starting with Tito, they take turns picking cards until they're finished. Then each player sums the numbers on his cards and whoever has an even sum wins. Determine which player has a winning strategy and describe it.

P.S. Proposed by yours truly :-D

Let $\Delta ABC$ be a triangle inscribed in the circumference $\omega$ of center $O$ . Let $T$ be the symmetric of $C$ respect to $O$ and $T'$ be the reflection of $T$ respect to line $AB$ . Line $BT'$ intersects $\omega$ again at $R$ . The perpendicular to $CT$ through $O$ intersects line $AC$ at $L$ . Let $N$ be the intersection of lines $TR$ and $AC$ . Prove that $\overline{CN}=2\overline{AL}$ .

Let $x, y$ be real numbers such that $x-y, x^2-y^2, x^3-y^3$ are all prime numbers. Prove that $x-y=3$ .

EDIT: Problem submitted by Leonel Castillo, Panama.

Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\frac{r^2-5q^2}{p^2-1}=2$ .

Let $n$ be a positive integer, $1<n<2018$ . For each $i=1, 2, \ldots ,n$ we define the polynomial $S_i(x)=x^2-2018x+l_i$ , where $l_1, l_2, \ldots, l_n$ are distinct positive integers. If the polynomial $S_1(x)+S_2(x)+\cdots+S_n(x)$ has at least an integer root, prove that at least one of the $l_i$ is greater or equal than $2018$ .

A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1,\ldots, 2017$ are marked in a circumference and each couple is placed on a different point. For $i\geq1$ , let $s_i=i\ (\textrm{mod}\ 2018)$ and $r_i=2i\ (\textrm{mod}\ 2018)$ . The dance begins at minute $0$ . On the $i$ -th minute, the couple at point $s_i$ (if there's any) moves to point $r_i$ , the couple on point $r_i$ (if there's any) drops out, and the dance continues with the remaining couples. The dance ends after $2018^2$ minutes. Determine how many couples remain at the end.

Note: If $r_i=s_i$ , the couple on $s_i$ stays there and does not drop out.