AoPS Academy
Leningrad 1965-92, 98 selected problems (besides Leningrad MO) Moscow Mathematical Circles - 99 Selected Problems (MMCircles) All Soviet Union Math Competitions 1961 - 1987 EN translated by S/W engineer Vladimir Pertsel (link) All Soviet Union Mathematical Olympiad 1961-1992 EN with solutions, by John Scholes (Kalva) (link) Всероссийских математических олимпиад школьников 1993–2006 (pdf) 60-odd YEARS of MOSCOW MATHEMATICAL OLYMPIADS, Edited by D. Leites (pdf) Московские математические олимпиады. 1935 - 1957 г. (pdf here) Московские математические олимпиады. 1958 - 1967 г. г (pdf here) Московские математические олимпиады. 1981 - 1992 г. (pdf here) International Mathematical Tournamenent of Towns Book 1, 1980-1984 (AMT) International Mathematical Tournamenent of Towns Book 2, 1984-1989 (AMT) International Mathematical Tournamenent of Towns Book 3, 1989-1993 (AMT) International Mathematical Tournamenent of Towns Book 4, 1993-1997 (AMT) International Mathematical Tournamenent of Towns Book 5, 1997-2002 (AMT) Кишиневские математические олимпиады Ю.М.Рябухин 1983 Санкт-Петербургские математические олимпиады by Фомин Д. В. (1961-93)
, 
such that 
and 
are the roots of the second trinomial, 
and 
are the roots of the first.
and 
not exceeding 
, prove that
such that 
, there is such natural 
, which 
rubles. Petya has 
rubles. Does Petya have enough money for ice cream?
are distinct reals. The number 
is written in position 
of a 
array. The product of the numbers in each column is . Show that the product of the numbers in each row is 
.
are distinct reals. The number is written in position of a 
array. The product of the numbers in each column is the same. Show that the product of the numbers in each row is also the same.
, 
and , 
. It is known that the square trinomials 
and 
have a common root (not necessarily integer). Prove that 
is divisible by 
.
is equal to 
. Prove that
integers 
such that 
(
) and whose sum is even. Find whether it is possible to divide them into two groups so that the sum of numbers in one group is equal to the sum of numbers in the other group.
of nonzero real numbers, such that for any natural number a polynomial of degree no less than with coefficients in , all of whose roots are real and belong to ?
defined on the segment [![$-1,1]$](http://latex.artofproblemsolving.com/9/8/8/988b44fd7f73ec681f289049ed9a181549f3590a.png)
(including the ends) such that 
for all x? (A function is called piecewise linear if its graph is the union of a finite set of points and intervals; it may be discontinuous).
be the lengths of the sides of a triangle and 
, the opposite angles. Prove that 
