1999 Bulgaria National Olympiad

Part of Bulgaria National Olympiad

Subcontests

(3)

2

Bulgaria 1999

The faces of a box with integer edge lengths are painted green. The box is partitioned into unit cubes. Find the dimensions of the box if the number of unit cubes with no green face is one third of the total number of cubes.

Bulgaria 1999 Second Day NT

Find the number of all integers

n

4\le n\le 1023

which contain no three consecutive equal digits in their binary representations.

2

Bulgaria 1999 Geometric NT

The vertices of a triangle have integer coordinates and one of its sides is of length

\sqrt{n}

n

is a square-free natural number. Prove that the ratio of the circumradius and the inradius is an irrational number.

Infinitely many soln.

x^3+y^3+z^3+t^3=1999

has infinitely many soln. over

\mathbb{Z}

2

Bulgaria 1999 Sequence Problem

\{a_n\}

be a sequence of integers satisfying

(n-1)a_{n+1}=(n+1)a_n-2(n-1) \forall n\ge 1

2000|a_{1999}

, find the smallest

n\ge 2

2000|a_n

bulgaria 1999

The vertices A,B,C of an acute-angled triangle ABC lie on the sides B1C1, C1A1, A1B1 respectively of a triangle A1B1C1 similar to the triangle ABC (∠A = ∠A1, etc.). Prove that the orthocenters of triangles ABC and A1B1C1 are equidistant from the circumcenter of △ABC.