2019 ELMO Problems
Contents
[hide]Day 1
Problem 1
Let be a polynomial with integer coefficients such that , and let be an integer. Define and for all integers . Show that there are infinitely many positive integers such that .
Problem 2
Let be integers. Carl is given marked points in the plane and wishes to mark their centroid.* He has no standard compass or straightedge. Instead, he has a device which, given marked points and , marks the points that divide segment into congruent parts (but does not draw the segment).
For which pairs can Carl necessarily accomplish his task, regardless of which points he is given?
- Here, the centroid of points with coordinates is the point with coordinates .
Problem 3
Let be a fixed integer. A game is played by players sitting in a circle. Initially, each player draws three cards from a shuffled deck of cards numbered . Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.
Let denote the configuration after turns (so is the initial configuration). Show that is eventually periodic with period , and find the smallest integer for which, regardless of the initial configuration, .
Day 2
Problem 4
Carl is given three distinct non-parallel lines and a circle in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line and a point , constructs a new line passing through parallel to . (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle whose sides are parallel to in some order.
Problem 5
Let be a nonempty set of positive integers such that, for any (not necessarily distinct) integers and in , the number is also in . Show that the set of primes that do not divide any element of is finite.
Problem 6
Carl chooses a functional expression* which is a finite nonempty string formed from a set of variables and applications of a function , together with addition, subtraction, multiplication (but not division), and fixed real constants. He then considers the equation , and lets denote the set of functions such that the equation holds for any choices of real numbers . (For example, if Carl chooses the functional equation
then consists of one function, the identity function.
(a) Let denote the set of functions with domain and image exactly . Show that Carl can choose his functional equation such that is nonempty but .
(b) Can Carl choose his functional equation such that and ?
- These can be defined formally in the following way: the set of functional expressions is the minimal one (by inclusion) such that (i) any fixed real constant is a functional expression, (ii) for any positive integer , the variable is a functional expression, and (iii) if and are functional expressions, then so are , , , and .