10.5

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

x^2+3y, y^2+3x perfect squares (II Soros Olympiad 1995-96 R1 10.5)

Source:

Find all pairs of natural numbers

x

y

x^2+3y

y^2+3x

are simultaneously squares of natural numbers.

number theoryPerfect SquaresPerfect Square

broken line via vertices of cube (II Soros Olympiad 1995-96 R2 10.5)

Source:

Is there a six-link broken line in space that passes through all the vertices of a given cube?

geometry3D geometrycombinatoricscombinatorial geometry

equilaterals on laterals of trapezoid (II Soros Olympiad 1997-98 R3 10.5)

Source:

Each of the lateral sides of the trapezoid, whose bases are equal to

a

b

, serves as a side of a regular triangle. One triangle is located entirely outside the trapezoid, and the other has common points with it. Find the distance between the centers of these triangles.

geometrytrapezoid