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 the

triangle 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 endpoints

divide the boundary of P into two parts, each composed of an odd number

of sides of P. The sides of P are also called good.

Suppose P has been dissected into triangles by 2003 diagonals, no two

of which have a common point in the interior of P. Find the maximum

number 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 inequality

ab(a2 ? b2) + bc(b2 ? c2) + ca(c2 ? a2)

 M(a2 + b2 + c2)2

holds for all real numbers a, b and c. (Ireland)

Second Day – July 13

4. Determine all pairs (x, y) of integers such that

1 + 2x + 22x+1 = y2.

(United States of America)

5. Let P(x) be a polynomial of degree n > 1 with integer coefficients

and 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 at

most 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 triangle

that has b as a side and is contained in P. Show that the sum of the areas

assigned 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

المپیاد ریاضی2006
المپیلد ریاضی 2006
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