Exceedingly Luck-based Math Olympiad
Day 1
1. Determine all (not necessarily finite) sets S of points in the plane such that given any
four distinct points in S, there is a circle passing through all four or a line passing
through some three.
2. Let r and s be positive integers. Define a0 = 0, a1 = 1, and an = ran−1 + san−2 for
fn
n ≥ 2. Let fn = a1 a2 · · · an . Prove that is an integer for all integers n and k
fk fn−k
such that 0 < k < n.
3. Let n > 1 be a positive integer. A 2-dimensional grid, infinite in all directions, is given.
Each 1 by 1 square in a given n by n square has a counter on it. A move consists
of taking n adjacent counters in a row or column and sliding them each by one space
along that row or column. A returning sequence is a finite sequence of moves such that
all counters again fill the original n by n square at the end of the sequence.
(a) Assume that all counters are distinguishable except two, which are indistinguish-
able from each other. Prove that any distinguishable arrangement of counters in
the n by n square can be reached by a returning sequence.
(b) Assume all counters are distinguishable. Prove that there is no returning sequence
that switches two counters and returns the rest to their original positions.
Exceedingly Luck-based Math Olympiad
Day 2
4. Determine all strictly increasing functions f : N → N satisfying nf (f (n)) = f (n)2 for
all positive integers n.
5. 2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and
a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010
are chosen. On the red slip of paper, each MOPper writes the remainder when the
product of A and his or her number is divided by 2011. On the blue slip of paper, he
or she writes the remainder when the product of B and his or her number is divided
by 2011. The MOPpers may then perform either of the following two operations:
• Each MOPper gives his or her red slip to the MOPper whose number is written
on his or her blue slip.
• Each MOPper gives his or her blue slip to the MOPper whose number is written
on his or her red slip.
Show that it is always possible to perform some number of these operations such that
each MOPper is holding a red slip with his or her number written on it.
6. Let ABC be a triangle with circumcircle ω, incenter I, and A-excenter IA . Let the
incircle and the A-excircle hit BC at D and E, respectively, and let M be the midpoint
of arc BC without A. Consider the circle tangent to BC at D and arc BAC at T . If
T I intersects ω again at S, prove that SIA and M E meet on ω.
