Team Selection Test for the Selection Team of 55th IMO
Lincoln, Nebraska
Day I 1:00 PM - 5:30 PM
June 21, 2013
1. Let ABC be a triangle and D, E, F be the midpoints of arcs BC, CA, AB on the circumcircle.
Line `a passes through the feet of the perpendiculars from A to DB and DC. Line ma passes
through the feet of the perpendiculars from D to AB and AC. Let A1 denote the intersection
of lines `a and ma . Define points B1 and C1 similarly. Prove that triangles DEF and A1 B1 C1
are similar to each other.
2. A finite sequence of integers a1 , a2 , . . . , an is called regular if there exists a real number x
satisfying
bkxc = ak for 1 ≤ k ≤ n.
Given a regular sequence a1 , a2 , . . . , an , for 1 ≤ k ≤ n we say that the term ak is forced if the
following condition is satisfied: the sequence
a1 , a2 , . . . , ak−1 , b
is regular if and only if b = ak . Find the maximum possible number of forced terms in a regular
sequence with 1000 terms.
3. Divide the plane into an infinite square grid by drawing all the lines x = m and y = n for
m, n ∈ Z. Next, if a square’s upper-right corner has both coordinates even, color it black;
otherwise, color it white (in this way, exactly 1/4 of the squares are black and no two black
squares are adjacent). Let r and s be odd integers, and let (x, y) be a point in the interior
of any white square such that rx − sy is irrational. Shoot a laser out of this point with slope
r/s; lasers pass through white squares and reflect off black squares. Prove that the path of
this laser will from a closed loop.
Committee on the American Mathematics Competitions,
Mathematical Association of America
1
Team Selection Test for the Selection Team of 55th IMO
Lincoln, Nebraska
Day II 1:00 PM - 5:30 PM
June 23, 2013
4. Circle ω, centered at X, is internally tangent to circle Ω, centered at Y , at T . Let P and S be
variable points on Ω and ω, respectively, such that line P S is tangent to ω (at S). Determine
the locus of O – the circumcenter of triangle P ST .
5. Let p be a prime. Prove that any complete graph with 1000p vertices, whose edges are labelled
with integers, has a cycle whose sum of labels is divisible by p.
6. Let N be the set of positive integers. Find all functions f : N → N that satisfy the equation
f abc−a (abc) + f abc−b (abc) + f abc−c (abc) = a + b + c
for all a, b, c ≥ 2.
(Here f 1 (n) = f (n) and f k (n) = f (f k−1 (n)) for every integer k greater than 1.)
Committee on the American Mathematics Competitions,
Mathematical Association of America
2
Team Selection Test for the Selection Team of 55th IMO
Lincoln, Nebraska
Day III 1:00 PM - 5:30 PM
June 25, 2013
7. A country has n cities, labelled 1, 2, 3, . . . , n. It wants to build exactly n − 1 roads between
certain pairs of cities so that every city is reachable from every other city via some sequence
of roads. However, it is not permitted to put roads between pairs of cities that have labels
differing by exactly 1, and it is also not permitted to put a road between cities 1 and n. Let
Tn be the total number of possible ways to build these roads.
(a) For all odd n, prove that Tn is divisible by n.
(b) For all even n, prove that Tn is divisible by n/2.
8. Define a function f : N → N by f (1) = 1, f (n + 1) = f (n) + 2f (n) for every positive integer n.
Prove that f (1), f (2), . . . , f (32013 ) leave distinct remainders when divided by 32013 .
9. Let r be a rational number in the interval [−1, 1] and let θ = cos−1 r. Call a subset S of the
plane good if S is unchanged upon rotation by θ around any point of S (in both clockwise and
counterclockwise directions). Determine all values of r satisfying the following property: The
midpoint of any two points in a good set also lies in the set.
Committee on the American Mathematics Competitions,
Mathematical Association of America
