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)
, 
an integer, that lie within the polygon is equal to the sum of the lengths of the segments of lines of the form 
, 
an integer, that lie within the polygon.
, there is a function 
, different from a constant, such that
intersects a circle at exactly two points 
and 
. If their tangents at coincide, must their tangents at also coincide?
is extended beyond to the point 
such that 
. The bisectors of the exterior angles at vertices and 
intersect at the point 
. Prove that quadrilateral 
is cyclic. (4)
ascertain whether among its first 
terms there is a number that ends with four zeros.
straight lines in general position are drawn on the plane. Prove that among the parts into which they divide the plane, there are at least 
triangles
and 
such that 
pupils is organised into 
clubs , so that each club contains 
pupils and no two clubs have the same composition. Prove that there are two clubs which have exactly one common member.
of its volume is below the water level, while more than half of its surface area is above the water level?
red points divide a blue circle into arcs such that their lengths are all positive integers from 
to in an arbitrary order. Prove that there exist two perpendicular chords with red endpoints.
be a polynomial of odd degree. Prove that the equation 
has at least as many different real roots as the equation 
as 
and 
as 
, we conclude from Intermediate Value Theorem that is surjective.
be roots of 
. From surjectivity, there exists distinct 
such that 
. But then, 
, so any is a root of 
.
