Olympiad problems

2018 Centroamerican and Caribbean Math Olympiad, 1

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

2012 CentroAmerican, 1

Tags: sum of digits , number theory , Centroamerican , digit sum

Find all positive integers that are equal to $700$ times the sum of its digits.

2018 Centroamerican and Caribbean Math Olympiad, 4

Tags: number theory , Centroamerican

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

2018 Centroamerican and Caribbean Math Olympiad, 3

Tags: number theory , Centroamerican , prime numbers

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.

2018 Centroamerican and Caribbean Math Olympiad, 5

Tags: algebra , Polynomials , Centroamerican , polynomial

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 $2018$.

2018 Centroamerican and Caribbean Math Olympiad, 2

Tags: geometry , Centroamerican , geometric transformation , reflection

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 $\overline{CN}=2\overline{AL}$.

2018 Centroamerican and Caribbean Math Olympiad, 6

Tags: combinatorics , Centroamerican

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.