Download e-book for iPad: 60 Odd Years of Moscow Mathematical Olympiads by D. Leites (ed.), G. Galperin, A. Tolpygo, P. Grozman, A.

By D. Leites (ed.), G. Galperin, A. Tolpygo, P. Grozman, A. Shapovalov, V. Prasolov, A. Fomenko

From the Preface:

This is the 1st whole compilation of the issues from Moscow Mathematical Olympiads with
solutions of ALL difficulties. it's in keeping with past Russian choices: [SCY], [Le] and [GT]. The first
two of those books comprise chosen difficulties of Olympiads 1–15 and 1–27, respectively, with painstakingly
elaborated recommendations. The ebook [GT] strives to assemble formulations of all (cf. ancient feedback) problems
of Olympiads 1–49 and options or tricks to so much of them.

For whom is that this publication? The luck of its Russian counterpart [Le], [GT] with their a million copies
sold aren't decieve us: a great deal of the good fortune is because of the truth that the costs of books, especially
text-books, have been increadibly low (< 0.005 of the bottom salary.) Our viewers will likely be extra limited. However, we handle it to ALL English-reading academics of arithmetic who may perhaps recommend the publication to their students and libraries: we gave comprehensible strategies to ALL difficulties.

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Sample text

3. Consider 13 weights of integer mass (in grams). It is known that any 6 of them may be placed onto two pans of a balance achieving equilibrium. Prove that all the weights are of equal mass. 4. The midpoints of alternative sides of a hexagon are connected by line segments. Prove that the intersection points of the medians of the two triangles obtained coincide. 36 MOSCOW MATHEMATICAL OLYMPIADS 1 – 59 Figure 10. (Probl. 2) Figure 11. (Probl. 5. Prove that some (or one) of any 100 integers can always be chosen so that the sum of the chosen integers is divisible by 100.

What is the minimal possible number of sides of the polygons? 3. The side AD of a parallelogram ABCD is divided into n equal segments. The nearest to A division point P is connected with B. Prove that line BP intersects the diagonal AC at point Q such that AQ = nAC + 1 ; see Fig. 5. 4. Segments connect vertices A, B, C of ABC with respective points A1 , B1 , C1 on the opposite sides of the triangle. Prove that the midpoints of segments AA1 , BB1 , CC1 do not belong to one straight line. 1. Solve in integers the equation xy + 3x − 5y = −3.

A  99 − 3a100 + 2a1 ≥ 0,   a100 − 3a1 + 2a2 ≥ 0, prove that the numbers are equal. 4. Consider ABC and a point S inside it. Let A1 , B1 , C1 be the intersection points of AS, BS, CS with BC, AC, AB, respectively. Prove that at least in one of the resulting quadrilaterals AB1 SC1 , C1 SA1 B, A1 SB1 C both angles at either C1 and B1 , or C1 and A1 , or A1 and B1 are not acute. 5. Do there exist points A, B, C, D in space, such that AB = CD = 8, AC = BD = 10, and AD = BC = 13? 1. Find all real solutions of the equation x2 + 2x · sin(xy) + 1 = 0.

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60 Odd Years of Moscow Mathematical Olympiads by D. Leites (ed.), G. Galperin, A. Tolpygo, P. Grozman, A. Shapovalov, V. Prasolov, A. Fomenko

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