ELMO 2016

Cookie Monster says a positive integer $n$ is $crunchy$ if there exist $2n$ real numbers $x_1,x_2,\ldots,x_{2n}$ , not all equal, such that the sum of any $n$ of the $x_i$ 's is equal to the product of the other $n$ of the $x_i$ 's. Help Cookie Monster determine all crunchy integers.

Yannick Yao

Oscar is drawing diagrams with trash can lids and sticks. He draws a triangle $ABC$ and a point $D$ such that $DB$ and $DC$ are tangent to the circumcircle of $ABC$ . Let $B'$ be the reflection of $B$ over $AC$ and $C'$ be the reflection of $C$ over $AB$ . If $O$ is the circumcenter of $DB'C'$ , help Oscar prove that $AO$ is perpendicular to $BC$ .

James Lin

In a Cartesian coordinate plane, call a rectangle $standard$ if all of its sides are parallel to the $x$ - and $y$ - axes, and call a set of points $nice$ if no two of them have the same $x$ - or $y$ - coordinate. First, Bert chooses a nice set $B$ of $2016$ points in the coordinate plane. To mess with Bert, Ernie then chooses a set $E$ of $n$ points in the coordinate plane such that $B\cup E$ is a nice set with $2016+n$ points. Bert returns and then miraculously notices that there does not exist a standard rectangle that contains at least two points in $B$ and no points in $E$ in its interior. For a given nice set $B$ that Bert chooses, define $f(B)$ as the smallest positive integer $n$ such that Ernie can find a nice set $E$ of size $n$ with the aforementioned properties. Help Bert determine the minimum and maximum possible values of $f(B)$ .

Yannick Yao

Big Bird has a polynomial $P$ with integer coefficients such that $n$ divides $P(2^n)$ for every positive integer $n$ . Prove that Big Bird's polynomial must be the zero polynomial.

Ashwin Sah

Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set $S$ of $n>1$ collinear points. Then, for every unordered pair of points $\{X,Y\}$ in $S$ , Elmo draws the circle with diameter $XY$ so that each pair of circles which intersect at two distinct points are drawn in different colors. Count von Count then wishes to count the number of colors Elmo used. In terms of $n$ , what is the minimum number of colors Elmo could have used?

Michael Ren

Elmo is now learning olympiad geometry. In triangle $ABC$ with $AB\neq AC$ , let its incircle be tangent to sides $BC$ , $CA$ , and $AB$ at $D$ , $E$ , and $F$ , respectively. The internal angle bisector of $\angle BAC$ intersects lines $DE$ and $DF$ at $X$ and $Y$ , respectively. Let $S$ and $T$ be distinct points on side $BC$ such that $\angle XSY=\angle XTY=90^\circ$ . Finally, let $\gamma$ be the circumcircle of $\triangle AST$ .

(a) Help Elmo show that $\gamma$ is tangent to the circumcircle of $\triangle ABC$ .

(b) Help Elmo show that $\gamma$ is tangent to the incircle of $\triangle ABC$ .

James Lin