OMCC 2013

Juan writes the list of pairs $(n, 3^n)$ , with $n=1, 2, 3,...$ on a chalkboard. As he writes the list, he underlines the pairs $(n, 3^n)$ when $n$ and $3^n$ have the same units digit. What is the $2013^{th}$ underlined pair?

Around a round table the people $P_1, P_2,..., P_{2013}$ are seated in a clockwise order. Each person starts with a certain amount of coins (possibly none); there are a total of $10000$ coins. Starting with $P_1$ and proceeding in clockwise order, each person does the following on their turn:

• If they have an even number of coins, they give all of their coins to their neighbor to the left.
• If they have an odd number of coins, they give their neighbor to the left an odd number of coins (at least $1$ and at most all of their coins) and keep the rest.

Prove that, repeating this procedure, there will necessarily be a point where one person has all of the coins.

Let $ABCD$ be a convex quadrilateral and let $M$ be the midpoint of side $AB$ . The circle passing through $D$ and tangent to $AB$ at $A$ intersects the segment $DM$ at $E$ . The circle passing through $C$ and tangent to $AB$ at $B$ intersects the segment $CM$ at $F$ . Suppose that the lines $AF$ and $BE$ intersect at a point which belongs to the perpendicular bisector of side $AB$ . Prove that $A$ , $E$ , and $C$ are collinear if and only if $B$ , $F$ , and $D$ are collinear.

Ana and Beatriz take turns in a game that starts with a square of side $1$ drawn on an infinite grid. Each turn consists of drawing a square that does not overlap with the rectangle already drawn, in such a way that one of its sides is a (complete) side of the figure already drawn. A player wins if she completes a rectangle whose area is a multiple of $5$ . If Ana goes first, does either player have a winning strategy?

Let $ABC$ be an acute triangle and let $\Gamma$ be its circumcircle. The bisector of $\angle{A}$ intersects $BC$ at $D$ , $\Gamma$ at $K$ (different from $A$ ), and the line through $B$ tangent to $\Gamma$ at $X$ . Show that $K$ is the midpoint of $AX$ if and only if $\frac{AD}{DC}=\sqrt{2}$ .

Determine all pairs of non-constant polynomials $p(x)$ and $q(x)$ , each with leading coefficient $1$ , degree $n$ , and $n$ roots which are non-negative integers, that satisfy $p(x)-q(x)=1$ .