2

Part of 2016 Middle European Mathematical Olympiad

Problems(2)

numbers on a board, non-degenerate triangle

Source: MEMO 2016 I2

n \ge 3

positive integers written on a board. A move consists of choosing three numbers

a, b, c

written from the board such that there exists a non-degenerate non-equilateral triangle with sides

a, b, c

and replacing those numbers with

a + b - c, b + c - a

c + a - b

.
Prove that a sequence of moves cannot be infinite.

combinatoricstriangle inequality

functional equation on reals with y-x

Source: MEMO 2016 T2

\mathbb{R}

denote the set of the reals. Find all

f : \mathbb{R} \to \mathbb{R}

f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2)

x, y

algebrafunctional equationalgebra proposed