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

From the Preface:

This is the 1st entire compilation of the issues from Moscow Mathematical Olympiads with

solutions of ALL difficulties. it's according to prior 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 options. The publication [GT] strives to assemble formulations of all (cf. historic feedback) problems

of Olympiads 1–49 and ideas or tricks to so much of them.

For whom is that this booklet? The luck of its Russian counterpart [Le], [GT] with their a million copies

sold aren't decieve us: a great deal of the luck 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 be extra limited.
However, we handle it to ALL English-reading lecturers of arithmetic who may recommend the e-book to their
students and libraries: we gave comprehensible recommendations to ALL difficulties.

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**Extra info for 60 Odd Years of Moscow Mathematical Olympiads**

**Example text**

2. Two triangular pyramids have common base. One pyramid contains the other. Can the sum of the lengths of the edges of the inner pyramid be longer than that of the outer one? 3. Arrange 81 weights of 12 , 22 , . . , 812 (all in grams) into three piles of equal mass. 4. Solve the equation √ x+3−4 x−1+ √ x + 8 − 6 x − 1 = 1. 5. We are given n circles O1 , O2 , . . , On , passing through one point O. Let A1 , . . , On with O1 , respectively. , until we get a point Bn on On . We draw the line segment through Bn and An to the second intersection with O1 at Bn+1 .

Find the sum of these vectors. 3. 3. 4. Find all solutions of the system consisting of 3 equations: 1 1 1 +y 1− x 1 − 2n 2n + 1 + z 1 − 2n + 2 = 0 for n = 1, 2, 3. OLYMPIAD 17 (1954) 45 Figure 20. (Probl. 5. 1. 4. Prove that if x40 + a1 x30 + a2 x20 + a3 x0 + a4 = 0 and 4x30 + 3a1 x20 + 2a2 x0 + a3 = 0, then . x4 + a1 x3 + a2 x2 + a3 x + a4 .. (x − x0 )2 . 2. Delete 100 digits from the number 1234567891011 . . 9899100 so that the remaining number were as big as possible. 3. Given 100 numbers a1 , .

5. Two legs of an angle α on a plane are mirrors. Prove that after several reflections in the ◦ mirrors any ray leaves in the direction opposite the one from which it came if and only if α = 90 n for an integer n. Find the number of reflections. 1. 35 Find all positive rational solutions of the equation xy = y x (x = y). 2*. What is the radius of the largest possible circle inscribed into a cube with side a? 3. How many different integer solutions to the inequality |x| + |y| < 100 are there? 4.

### 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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