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International Mathematics Olympiad Problems - 2006 - Olympiad Preparation for Class 10

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12 July 2006
Problem 1. Let ABC be a triangle with incentre I. A point P in the interior of the
triangle satis?es
6 PBA+
6 PCA =
6 PBC +
6 PCB.
Show that AP =AI, and that equality holds if and only if P =I.
Problem 2. LetP bearegular2006-gon. AdiagonalofP iscalled good ifitsendpoints
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 con?guration.
Problem 3. Determine the least real number M such that the inequality


ab(a
2
-b
2
)+bc(b
2
-c
2
)+ca(c
2
-a
2
)


=M(a
2
+b
2
+c
2
)
2
holds for all real numbers a, b and c.
Page 2


12 July 2006
Problem 1. Let ABC be a triangle with incentre I. A point P in the interior of the
triangle satis?es
6 PBA+
6 PCA =
6 PBC +
6 PCB.
Show that AP =AI, and that equality holds if and only if P =I.
Problem 2. LetP bearegular2006-gon. AdiagonalofP iscalled good ifitsendpoints
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 con?guration.
Problem 3. Determine the least real number M such that the inequality


ab(a
2
-b
2
)+bc(b
2
-c
2
)+ca(c
2
-a
2
)


=M(a
2
+b
2
+c
2
)
2
holds for all real numbers a, b and c.
13 July 2006
Problem 4. Determine all pairs (x,y) of integers such that
1+2
x
+2
2x+1
= y
2
.
Problem 5. Let P(x) be a polynomial of degree n > 1 with integer coe?cients 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.
Problem6. AssigntoeachsidebofaconvexpolygonP themaximumareaofatriangle
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.
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11 videos|36 docs|201 tests
11 videos|36 docs|201 tests
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