47-th International Mathematical Olympiad
Ljubljana, Slovenia, July 6–18, 2006
First Day – July 12
1. Let
ABC be a triangle with incenter I. A point P in the interior of thetriangle satisfies
\
PBA + \PCA = \PBC + \PCB.Show that
AP AI, and that equality holds if and only if P = I.(South Korea)
2. Let
P be a regular 2006-gon. A diagonal of P is called good if its endpointsdivide the boundary of
P into two parts, each composed of an odd numberof sides of
P. The sides of P are also called good.Suppose
P has been dissected into triangles by 2003 diagonals, no twoof which have a common point in the interior of
P. Find the maximumnumber of isosceles triangles having two good sides that could appear in
such a configuration.
(Serbia)3. Determine the least real number
M such that the inequalityab
(a2 ? b2) + bc(b2 ? c2) + ca(c2 ? a2) M(a2 + b2 + c2)2holds for all real numbers
a, b and c. (Ireland)Second Day – July 13
4. Determine all pairs (
x, y) of integers such that1 + 2
x + 22x+1 = y2.(United States of America)
5. Let
P(x) be a polynomial of degree n > 1 with integer coefficientsand let
k be a positive integer. Consider the polynomial Q(x) =P
(P(. . . P(P(x)) . . . )), where P occurs k times. Prove that there are atmost
n integers t such that Q(t) = t. (Romania)6. Assign to each side
b of a convex polygon P the maximum area of a trianglethat has
b as a side and is contained in P. Show that the sum of the areasassigned to the sides of
P is at least twice the area of P. (Serbia)1
The IMO Compendium Group,
D. Djuki´c, V. Jankovi´c, I. Mati´c, N. Petrovi´c
www.imo.org.yu
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