Best problems from around the world - mathematical olympiads by Cao Minh Quang.

By Cao Minh Quang.

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Find the total area of the green regions divided by the area of the largest circle. 3. S = {1, 2, 3, 5, 8, 13, 21, 34}. Find ∑ max(A) where the sum is taken over all 28 twoelement subsets A of S. 4. Find n such that log10sin x + log10cos x = -1, log10(sin x + cos x) = (log10n - 1)/2. 5. Find the volume of the set of points that are inside or within one unit of a rectangular 3 x 4 x 5 box. 6. Let S be the set of vertices of a unit cube. Find the sum of the areas of all triangles whose vertices are in S.

Find ∠ CMB. 11. The angle x is chosen at random from the interval 0o < x < 90o. Find the probability that there is no triangle with side lengths sin2x, cos2x and sin x cos x. 12. ABCD is a convex quadrilateral with AB = CD = 180, perimeter 640, AD ≠ BC, and ∠ A = ∠ C. Find cos A. 13. Find the number of 1, 2, ... , 2003 which have more 1s than 0s when written in base 2. 14. When written as a decimal, the fraction m/n (with m < n) contains the consecutive digits 2, 5, 1 (in that order). Find the smallest possible n.

ABCDEFGH is a polyhedron. Face ABCD is a square side 12. Face ABFG is a trapezoid with GF parallel to AB and GF = 6, AG = BF = 8. Face CDE is an isosceles triangle with ED = EC = 14. E is a distance 12 from the plane ABCD. The other faces are EFG, ADEG and BCEF. Find EG2. 45 ☺ The best problems from around the world Cao Minh Quang 20th AIME2 2002 1. n is an integer between 100 and 999 inclusive, and so is n' the integer formed by reversing its digits. How many possible values are there for |n-n'|?

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