2018 TSTST Problems
Contents
[hide]Day 1
Problem 1
As usual, let denote the set of single-variable polynomials in with integer coefficients. Find all functions such that for any polynomials , , and if then divides .
Problem 2
In the nation of Onewaynia, certain pairs of cities are connected by one-way roads. Every road connects exactly two cities (roads are allowed to cross each other, e.g., via bridges), and each pair of cities has at most one road between them. Moreover, every city has exactly two roads leaving it and exactly two roads entering it.
We wish to close half the roads of Onewaynia in such a way that every city has exactly one road leaving it and exactly one road entering it. Show that the number of ways to do so is a power of greater than (i.e.\ of the form for some integer ). Solution
Problem 3
Let be an acute triangle with incenter , circumcenter , and circumcircle . Let be the midpoint of . Ray meets at . Denote by and the circumcircles of and , respectively. Line meets at and , while line meets at and . Assume that lies inside and .
Consider the tangents to at and and the tangents to at and . Given that , prove that these four lines are concurrent on . Solution
Day 2
Problem 4
For an integer , denote by the set of integers for which the polynomial has an integer root.
Let denote the set of integers for which contains two consecutive integers. Show that is infinite but
Prove that there are infinitely many positive integers such that contains three consecutive integers.
Problem 5
Let be an acute triangle with circumcircle , and let be the foot of the altitude from to . Let and be the points on with and . The tangent to at intersects lines and at and respectively; the tangent to at intersects lines and at and respectively. Show that the circumcircles of and are congruent, and the line through their centers is parallel to the tangent to at .
Problem 6
Let , and for every positive integer defineDetermine which have the following property: if we color any elements of red, then at least half of the -tuples in have an even number of coordinates with red elements. Solution
Day 3
Problem 7
Let be a positive integer. A frog starts on the number line at . Suppose it makes a finite sequence of hops, subject to two conditions:
The frog visits only points in , each at most once.
The length of each hop is in . (The hops may be either direction, left or right.)
Let be the sum of the (positive) lengths of all hops in the sequence. What is the maximum possible value of ?
Problem 8
For which positive integers do there exist infinitely many positive integers such that divides ? Solution
Problem 9
Show that there is an absolute constant with the following property: whenever is a polygon with area in the plane, one can translate it by a distance of in some direction to obtain a polygon , for which the intersection of the interiors of and has total area at most . Solution