MathDB

1

Part of 2018 Caucasus Mathematical Olympiad

Problems(2)

Easy algebra

Source: III Caucasus Mathematical Olympiad

3/17/2018
Let aa, bb, cc be real numbers, not all of them are equal. Prove that a+b+c=0a+b+c=0 if and only if a2+ab+b2=b2+bc+c2=c2+ca+a2a^2+ab+b^2=b^2+bc+c^2=c^2+ca+a^2.
algebra
Easy problem

Source: III Caucasus Mathematical Olympiad

3/17/2018
A tetrahedron is given. Determine whether it is possible to put some 10 consecutive positive integers at 4 vertices and at 6 midpoints of the edges so that the number at the midpoint of each edge is equal to the arithmetic mean of two numbers at the endpoints of this edge.
combinatoricsParity