ELMO SL 2017

Let $0<k<\frac{1}{2}$ be a real number and let $a_0, b_0$ be arbitrary real numbers in $(0,1)$. The sequences $(a_n)_{n\ge 0}$ and $(b_n)_{n\ge 0}$ are then defined recursively by

$$a_{n+1} = \dfrac{a_n+1}{2} \text{ and } b_{n+1} = b_n^k$$

for $n\ge 0$. Prove that $a_n<b_n$ for all sufficiently large $n$.

Proposed by Michael Ma

Let $m$ and $n$ be fixed distinct positive integers. A wren is on an infinite board indexed by $\mathbb Z^2$ , and from a square $(x,y)$ may move to any of the eight squares $(x\pm m, y\pm n)$ or $(x\pm n, y \pm m)$ . For each $\{m,n\}$ , determine the smallest number $k$ of moves required to travel from $(0,0)$ to $(1,0)$ , or prove that no such $k$ exists.

[i]Proposed by Michael Ren

Consider a finite binary string $b$ with at least $2017$ ones. Show that one can insert some plus signs in between pairs of digits such that the resulting sum, when performed in base $2$ , is equal to a power of two.

[i]Proposed by David Stoner

There are $n$ MOPpers $p_1,...,p_n$ designing a carpool system to attend their morning class. Each $p_i$ 's car fits $\chi (p_i)$ people ( $\chi : \{p_1,...,p_n\} \to \{1,2,...,n\}$ ). A $c$ -fair carpool system is an assignment of one or more drivers on each of several days, such that each MOPper drives $c$ times, and all cars are full on each day. (More precisely, it is a sequence of sets $(S_1, ...,S_m)$ such that $|\{k: p_i\in S_k\}|=c$ and $\sum_{x\in S_j} \chi(x) = n$ for all $i,j$ . ) Suppose it turns out that a $2$ -fair carpool system is possible but not a $1$ -fair carpool system. Must $n$ be even?

[i]Proposed by Nathan Ramesh and Palmer Mebane

Let $ABC$ be a scalene triangle with $\angle A = 60^{\circ}$ . Let $E$ and $F$ be the feet of the angle bisectors of $\angle ABC$ and $\angle ACB$ , respectively, and let $I$ be the incenter of $\triangle ABC$ . Let $P,Q$ be distinct points such that $\triangle PEF$ and $\triangle QEF$ are equilateral. If $O$ is the circumcenter of of $\triangle APQ$ , show that $\overline{OI}\perp \overline{BC}$ .

[i]Proposed by Vincent Huang

Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$ . Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs.

[i]Proposed by Daniel Liu

Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\omega$ . Suppose a circle $\omega_B$ is tangent to $BA,BC$ , and internally tangent to $\omega$ at $B_1$ , while a circle $\omega_C$ is tangent to $CA, CB$ , and internally tangent to $\omega$ at $C_1$ . If $B_2, C_2$ are the points opposite to $B,C$ on $\omega$ , respectively, and $X$ denotes the intersection of $B_1C_2, B_2C_1$ , prove that $XA=XI$ .

[i]Proposed by Vincent Huang and Nathan Weckwerth

For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$ , $a_{k+1}^k$ divides $C^ka_1a_2...a_k$ .

[i]Proposed by Daniel Liu