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

By Cao Minh Quang.

Similar mathematics books

Einstein Manifolds: reprint of the 1987 edition, with 22 figures

Einstein's equations stem from normal Relativity. within the context of Riemannian manifolds, an self sustaining mathematical idea has built round them. lately, it has produced numerous remarkable effects, that have been of significant curiosity additionally to physicists. This Ergebnisse quantity is the 1st booklet which provides an updated assessment of the state-of-the-art during this box.

Monopoles and Three-Manifolds

Originating with Andreas Floer within the Eighties, Floer homology presents an invariant of three-d manifolds and 4-dimensional cobordisms among them. It has proved to be a good device in tackling many vital difficulties in 3- and 4-dimensional geometry and topology. This booklet offers a accomplished remedy of Floer homology, in accordance with the Seiberg-Witten equations.

Extra info for Best problems from around the world - mathematical olympiads

Example text

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'|?