Problem 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 .
C M
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I P A B
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Let Ω be the circumcircle of triangle ABC . It is a well-known fact that the centre of ω is the midpoint M of the arc BC of Ω. This is also the point where the angle bisector AI intersects Ω. From triangle AP M we have AP + P M ≥ AM = AI + IM = AI + P M. Therefore AP ≥ AI . Equality holds if and only if P lies on the line segment AI , which occurs if and only if P = I .
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Problem 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. Problem 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 .
Probleem 3. Bepaal die kleinste re¨ ele getal M waarvoor d ab(a2 − b2 ) + bc(b2 − c2 ) + ca(c2 − a2 ) vir alle re¨ ele getalle a, b en c geld.
M(
Contents
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47th International Mathematical Olympiad Slovenia 2006
12 Julie 2006
Problems with Solutions
Probleem 1. Laat I die middelpunt van die ingeskrewe sir ’n punt binne die driehoek sodat ˆ + P CA ˆ = P BC ˆ + P CB. ˆ P BA Bewys dat: • AP AI ;
Problems Solutions Problem Problem Problem Problem Problem Problem 5 7 7 7 8 10 10 11
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Problems
Problem 1. Let ABC be a triangle with incentre I . A point P in the interior of the triangle satisfies ∠P BA + ∠P CA = ∠P BC + ∠P CB. Show that AP ≥ AI , and that equality holds if and only if P = I . Problem 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. Problem 3. Determine the least real number M such that the inequality ab(a2 − b2 ) + bc(b2 − c2 ) + ca(c2 − a2 ) ≤ M a2 + b2 + c2 holds for all real numbers a, b and c. Problem 4. Determine all pairs (x, y ) of integers such that 1 + 2x + 22x+1 = y 2 .
• gelykheid geld as en slegs as P = I .
Probleem 2. Gegee ’n re¨ elmatige 2006-hoek P . ’n Diagona as sy eindpunte die rand van P in twee dele verdeel wat elk P bestaan. Die sye van P word ook goed genoem. Nou word P opgedeel in driehoeke deur 2003 diagonale, wa skaplike punt binne P het nie. Vind die grootste aantal gelyk goeie sye wat op hierdie wyse kan ontstaan.
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Solutions
Problem 1.
Let ABC be a triangle with incentre I . A point P in the interior of the triangle satisfies ∠P BA + ∠P CA = ∠P BC + ∠P CB. Show that AP ≥ AI , and that equality holds if and only if P = I . Solution. Let ∠A = α, ∠B = β , ∠C = γ . Since ∠P BA + ∠P CA + ∠P BC + ∠P CB = β + γ , the condition from the problem statement is equivalent to ∠P BC + ∠P CB = (β + γ )/2, i. e. ∠BP C = 90◦ + α/2. On the other hand ∠BIC = 180◦ − (β + γ )/2 = 90◦ + α/2. Hence ∠BP C = ∠BIC , and since P and I are on the same side of BC , the points B , C , I and P are concyclic. In other words, P lies on the circumcircle ω of triangle BCI .
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8 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. Solution 1. Call an isosceles triangle good if it has two odd sides. Suppose we are given a dissection as in the problem statement. A triangle in the dissection which is good and isosceles will be called iso-good for brevity. Lemma. Let AB be one of dissecting diagonals and let L be the shorter part of the boundary of the 2006-gon with endpoints A, B . Suppose that L consists of n segments. Then the number of iso-good triangles with vertices on L does not exceed n/2.