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Read e-book online The IMO compendium a collection of problems suggested for PDF

By Dušan Djukić, Vladimir Janković, Ivan Matić, Nikola Petrović

ISBN-10: 0387242996

ISBN-13: 9780387242996

The foreign Mathematical Olympiad (IMO) has inside its virtually 50-year-old background turn into the most well-liked and prestigious festival for high-school scholars drawn to arithmetic. merely six scholars from each one partaking nation are given the distinction of partaking during this pageant each year. The IMO represents not just an excellent chance to take on attention-grabbing and tough arithmetic difficulties, it additionally bargains a fashion for top institution scholars to degree up with scholars from the remainder of the world.

The IMO has sparked off a burst of creativity between lovers in developing new and fascinating arithmetic difficulties. In an exceptionally stiff pageant, basically six difficulties are selected every year to seem at the IMO. the whole variety of difficulties proposed for the IMOs as much as this element is spectacular and, as a complete, this number of difficulties represents a worthy source for all highschool scholars getting ready for the IMO.

Until now it's been nearly very unlikely to acquire an entire choice of the issues proposed on the IMO in booklet shape. "The IMO Compendium" is the results of a 12 months lengthy collaboration among 4 former IMO contributors from Yugoslavia, now Serbia and Montenegro, to rescue those difficulties from previous and scattered manuscripts, and convey the last word resource of IMO perform difficulties. This publication makes an attempt to assemble all of the difficulties and recommendations showing at the IMO, in addition to the so-called "short-lists", a complete of 864 difficulties. additionally, the booklet includes 1036 difficulties from a number of "long-lists" through the years, for a grand overall of 1900 problems.

In brief, "The IMO Compendium" is the last word selection of not easy high-school-level arithmetic difficulties. it is going to be a useful source, not just for high-school scholars getting ready for arithmetic competitions, yet for a person who loves and appreciates math.

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2. (HUN) Find all real numbers x for which √ √ 1 3−x− x+1> . 2 3. (CZS) A cube ABCDA B C D is given. The point X is moving at a constant speed along the square ABCD in the direction from A to B. The point Y is moving with the same constant speed along the square BCC B in the direction from B to C . Initially, X and Y start out from A and B respectively. Find the locus of all the midpoints of XY . Second Day 4. (ROM) Solve the equation cos2 x + cos2 2x + cos2 3x = 1 . 5. (BUL) On the circle k three points A, B, and C are given.

53. (USS, 1966) Prove that in every convex hexagon of area S one can draw a diagonal that cuts off a triangle of area not exceeding 61 S. 54. (USS, 1966) Find the last two digits of a sum of eighth powers of 100 consecutive integers. 55. (USS, 1966) Given the vertex A and the centroid M of a triangle ABC, find the locus of vertices B such that all the angles of the triangle lie in the interval [40◦ , 70◦ ]. 8 IMO 1966 41 56. (USS, 1966) Let ABCD be a tetrahedron such that AB ⊥ CD, AC ⊥ BD, and AD ⊥ BC.

BUL) Find digits x, y, z such that the equality √ xx · · · x − yy · · · y = zz · · · z 2n n n holds for at least two values of n ∈ N, and in that case find all n for which this equality is true. 13. (YUG) Let a1 , a2 , . . , an be positive real numbers. Prove the inequality ⎞2 ⎛ n 2 i

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