Olympiad problems

2012 Argentina Cono Sur TST, 1

Sofía colours $46$ cells of a $9 \times 9$ board red. If Pedro can find a $2 \times 2$ square from the board that has $3$ or more red cells, he wins; otherwise, Sofía wins. Determine the player with the winning strategy.

2021 Argentina National Olympiad, 3

Tags: geometry , Angle Chasing , Argentina

Let $ABCD$ be a quadrilateral inscribed in a circle such that $\angle ABC=60^{\circ}.$ a) Prove that if $BC=CD$ then $AB= CD+DA.$ b) Is it true that if $AB= CD+DA$ then $BC=CD$?

2021 Argentina National Olympiad, 1

Tags: Sequence , combinatorics , contests , Argentina

An infinite sequence of digits $1$ and $2$ is determined by the following two properties: i) The sequence is built by writing, in some order, blocks $12$ and blocks $112.$ ii) If each block $12$ is replaced by $1$ and each block $112$ by $2$, the same sequence is again obtained. In which position is the hundredth digit $1$? What is the thousandth digit of the sequence?

2021 Argentina National Olympiad, 5

Tags: number theory , prime numbers , contests , Argentina

The sequence $a_n (n\geq 1)$ of natural numbers is defined as $a_{n+1}=a_n+b_n,$ where $b_n$ is the number that has the same digits as $a_n$ but in the opposite order ($b_n$ can start with $0$). For example, if $a_1=180,$ then $a_2=261, a_3=423.$ a) Decide if $a_1$ can be chosen so that $a_7$ is prime. b) Decide if $a_1$ can be chosen so that $a_5$ is prime.

2011 Argentina Team Selection Test, 3

Tags: geometry , trapezoid , circumcircle , Argentina , TST

Let $ABCD$ be a trapezoid with bases $BC \parallel AD$, where $AD > BC$, and non-parallel legs $AB$ and $CD$. Let $M$ be the intersection of $AC$ and $BD$. Let $\Gamma_1$ be a circumference that passes through $M$ and is tangent to $AD$ at point $A$; let $\Gamma_2$ be a circumference that passes through $M$ and is tangent to $AD$ at point $D$. Let $S$ be the intersection of the lines $AB$ and $CD$, $X$ the intersection of $\Gamma_1$ with the line $AS$, $Y$ the intesection of $\Gamma_2$ with the line $DS$, and $O$ the circumcenter of triangle $ASD$. Show that $SO \perp XY$.

2021 Argentina National Olympiad, 4

Tags: algebra , Argentina , contests

Find the real numbers $x, y, z$ such that, $$\frac{1}{x}+\frac{1}{y+z}=\frac{1}{2}, \frac{1}{y}+\frac{1}{z+x}=\frac{1}{3}, \frac{1}{z}+\frac{1}{x+y}=\frac{1}{4}.$$

2021 Argentina National Olympiad, 2

Tags: number theory , prime numbers , Perfect Squares , positive integers , contests , Argentina

Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$