2018 Ecuador NMO (OMEC)

Day 1

1

Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.

2

During his excursion to the historical park, Pepito set out to collect stones whose weight in kilograms is a power of two. Once the first stone has been collected, Pepito only collects stones strictly heavier than the first. At the end of the excursion, her partner Ana chooses a positive integer $K \ge 2$ and challenges Pepito to divide the stones into $K$ groups of equal weight. a) Can Pepito meet the challenge if all the stones he collected have different weights? b) Can Pepito meet the challenge if some collected stones are allowed to have equal weight?

3

Let $ABCD$ be a convex quadrilateral with $AB\le CD$. Points $E ,F$ are chosen on segment $AB$ and points $G ,H$ are chosen on the segment $CD$, are chosen such that $AE = BF = CG = DH <\frac{AB}{2}$. Let $P, Q$, and $R$ be the midpoints of $EG$, $FH$, and $CD$, respectively. It is known that $PR$ is parallel to $AD$ and $QR$ is parallel to $BC$. a) Show that $ABCD$ is a trapezoid. b) Let $d$ be the difference of the lengths of the parallel sides. Show that $2PQ\le d$.

Day 2

4

Let $k$ be a real number. Show that the polynomial $p (x) = x^3-24x + k$ has at most an integer root.

5

Let $ABC$ be an acute triangle and let $M$, $N$, and $ P$ be on $CB$, $AC$, and $AB$, respectively, such that $AB = AN + PB$, $BC = BP + MC$, $CA = CM + AN$. Let $\ell$ be a line in a different half plane than $C$ with respect to to the line $AB$ such that if $X, Y$ are the projections of $A, B$ on $\ell$ respectively, $AX = AP$ and $BY = BP$. Prove that $NXYM$ is a cyclic quadrilateral.

6

Reduce $$\frac{2}{\sqrt{4-3\sqrt[4]{5} + 2\sqrt[4]{25}-\sqrt[4]{125}}}$$to its lowest form. Then generalize this result and show that it holds for any positive $n$.