Romanian Olympiad 2004 IMO Team Selection Tests 1st Test - April ...

Romanian Olympiad 2004
IMO Team Selection Tests
2nd Test - May 1, 2004
1. A disk is partitioned in 2n equal sectors. Half of the sectors are colored in blue,
and the other half in red. We number the red sectors with numbers from 1 to n
in counter-clockwise direction, and then we number the blue sectors with numbers
from 1 to n in clockwise direction. Prove that one can find a half-disk which contains
sectors numbers with all the numbers from 1 to n.
2. Let a, b be two positive integers, such that ab ≠ 1. Find all the integer values that
f(a, b) can take, where
f(a, b) = a2 + ab + b 2
.
ab − 1
3. Let a, b, c be 3 integers, b odd, and define the sequence {x n } n≥0 by x 0 = 4, x 1 = 0,
x 2 = 2c, x 3 = 3b and for all positive integers n we have
x n+3 = ax n−1 + bx n + cx n+1 .
Prove that for all positive integers m, and for all primes p the number x p m is divisible
by p.
Călin Popescu
4. Let Γ be a circle, and let ABCD be a square lying inside the circle Γ. Let C a be a
circle tangent interiorly to Γ, and also tangent to the sides AB and AD of the square,
and also lying inside the opposite angle of ∠BAD. Let A ′ be the tangency point of the
two circles. Define similarly the circles C b , C c , C d and the points B ′ , C ′ , D ′ respectively.
Prove that the lines AA ′ , BB ′ , CC ′ and DD ′ are concurrent.
Work time: 4 hours.
TeX (c) 2004 Valentin Vornicu - MathLinks.ro