ELMO SL 2018

Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$?

Proposed by Daniel Liu

Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$ Proposed by Carl Schildkraut

Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that $$a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.$$

Proposed by Daniel Liu

Elmo calls a monic polynomial with real coefficients tasty if all of its coefficients are in the range $[-1,1]$ . A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$ .

Proposed by Carl Schildkraut

Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The distance between two cities is the least number of roads on any path between the two cities.)

For which $n$ is it possible for Mark to achieve this?

Proposed by Michael Ren

We say that a positive integer $n$ is $m$ -expressible if it is possible to get $n$ from some $m$ digits and the six operations $+,-,\times,\div$ , exponentiation $^\wedge$ , and concatenation $\oplus$ . For example, $5625$ is $3$ -expressible (in two ways): both $5\oplus (5^\wedge 4)$ and $(7\oplus 5)^\wedge 2$ yield $5625$ .

Does there exist a positive integer $N$ such that all positive integers with $N$ digits are $(N-1)$ -expressible?

Proposed by Krit Boonsiriseth

A windmill is a closed line segment of unit length with a distinguished endpoint, the pivot. Let $S$ be a finite set of $n$ points such that the distance between any two points of $S$ is greater than $c$ . A configuration of $n$ windmills is admissible if no two windmills intersect and each point of $S$ is used exactly once as a pivot.

An admissible configuration of windmills is initially given to Geoff in the plane. In one operation Geoff can rotate any windmill around its pivot, either clockwise or counterclockwise and by any amount, as long as no two windmills intersect during the process. Show that Geoff can reach any other admissible configuration in finitely many operations, where

(i) $c = \sqrt 3$ ,

(ii) $c = \sqrt 2$ .

Proposed by Michael Ren

Let $ABC$ be an acute triangle with orthocenter $H$ , and let $P$ be a point on the nine-point circle of $ABC$ . Lines $BH, CH$ meet the opposite sides $AC, AB$ at $E, F$ , respectively. Suppose that the circumcircles $(EHP), (FHP)$ intersect lines $CH, BH$ a second time at $Q,R$ , respectively. Show that as $P$ varies along the nine-point circle of $ABC$ , the line $QR$ passes through a fixed point.

Proposed by Brandon Wang

Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$ . Let $P$ be the midpoint of $\overline{AH}$ and let $T$ be on line $BC$ with $\angle TAO=90^{\circ}$ . Let $X$ be the foot of the altitude from $O$ onto line $PT$ . Prove that the midpoint of $\overline{PX}$ lies on the nine-point circle* of $\triangle ABC$ .

*The nine-point circle of $\triangle ABC$ is the unique circle passing through the following nine points: the midpoint of the sides, the feet of the altitudes, and the midpoints of $\overline{AH}$ , $\overline{BH}$ , and $\overline{CH}$ .

Proposed by Zack Chroman

Let $A$ be a point in the plane, and $\ell$ a line not passing through $A$ . Evan does not have a straightedge, but instead has a special compass which has the ability to draw a circle through three distinct noncollinear points. (The center of the circle is not marked in this process.) Additionally, Evan can mark the intersections between two objects drawn, and can mark an arbitrary point on a given object or on the plane.

(i) Can Evan construct* the reflection of $A$ over $\ell$ ?

(ii) Can Evan construct the foot of the altitude from $A$ to $\ell$ ?

*To construct a point, Evan must have an algorithm which marks the point in finitely many steps.

Proposed by Zack Chroman

Let $ABCDEF$ be a hexagon inscribed in a circle $\Omega$ such that triangles $ACE$ and $BDF$ have the same orthocenter. Suppose that segments $BD$ and $DF$ intersect $CE$ at $X$ and $Y$ , respectively. Show that there is a point common to $\Omega$ , the circumcircle of $DXY$ , and the line through $A$ perpendicular to $CE$ .

Proposed by Michael Ren and Vincent Huang

Let scalene triangle $ABC$ have altitudes $AD, BE, CF$ and circumcenter $O$ . The circumcircles of $\triangle ABC$ and $\triangle ADO$ meet at $P \ne A$ . The circumcircle of $\triangle ABC$ meets lines $PE$ at $X \ne P$ and $PF$ at $Y \ne P$ . Prove that $XY \parallel BC$ .

Proposed by Daniel Hu

Determine all nonempty finite sets of positive integers $\{a_1, \dots, a_n\}$ such that $a_1 \cdots a_n$ divides $(x + a_1) \cdots (x + a_n)$ for every positive integer $x$ .

Proposed by Ankan Bhattacharya

Call a number $n$ good if it can be expressed as $2^x+y^2$ for where $x$ and $y$ are nonnegative integers. (a) Prove that there exist infinitely many sets of $4$ consecutive good numbers. (b) Find all sets of $5$ consecutive good numbers.

Proposed by Michael Ma

Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$ Proposed by Krit Boonsiriseth

Say a positive integer $n>1$ is $d$ -coverable if for each non-empty subset $S\subseteq \{0, 1, \ldots, n-1\}$ , there exists a polynomial $P$ with integer coefficients and degree at most $d$ such that $S$ is exactly the set of residues modulo $n$ that $P$ attains as it ranges over the integers. For each $n$ , find the smallest $d$ such that $n$ is $d$ -coverable, or prove no such $d$ exists.

Proposed by Carl Schildkraut