IMO Shortlist 2006

[...]... Further rows will periodically repeat this pattern and there will be no row of zeros 21 C2 A diagonal of a regular 2006- gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides Sides are also regarded as odd diagonals Suppose the 2006- gon has been dissected into triangles by 2003 nonintersecting diagonals Find the maximum possible number of isosceles... are no more than 2006/ 2 iso-odd triangles in all, unless XY Z is one of them But in that case XZ and Y Z are odd diagonals and the corresponding inequalities are strict This shows that also in this case the total number of iso-odd triangles in the dissection, including XY Z, is not greater than 1003 This bound can be achieved For this to happen, it just suffices to select a vertex of the 2006- gon and draw... selected one Since 2006 is even, the line closes This already gives us the required 1003 iso-odd triangles Then we can complete the triangulation in an arbitrary fashion 22 Solution 2 Let the terms odd triangle and iso-odd triangle have the same meaning as in the first solution Let ABC be an iso-odd triangle, with AB and BC odd sides This means that there are an odd number of sides of the 2006- gon between... problem statement A triangle in the dissection which is odd and isosceles will be called iso-odd 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-odd triangles with vertices on L does not exceed n/2 Proof This is obvious for n = 2 Take n with 2 < n ≤ 1003 and .