AT-OEMO-REG 2015

Determine all triples $(a,b,c)$ of positive integers satisfying the conditions $$\gcd(a,20) = b$$ $$\gcd(b,15) = c$$ $$\gcd(a,c) = 5$$ (Richard Henner)

Let $x$ , $y$ , and $z$ be positive real numbers with $x+y+z = 3$ . Prove that at least one of the three numbers $$x(x+y-z)$$ $$y(y+z-x)$$ $$z(z+x-y)$$ is less or equal $1$ .

(Karl Czakler)

Let $n \ge 3$ be a fixed integer. The numbers $1,2,3, \cdots , n$ are written on a board. In every move one chooses two numbers and replaces them by their arithmetic mean. This is done until only a single number remains on the board.

Determine the least integer that can be reached at the end by an appropriate sequence of moves.

(Theresia Eisenkölbl)

Let $ABC$ be an isosceles triangle with $AC = BC$ and $\angle ACB < 60^\circ$ . We denote the incenter and circumcenter by $I$ and $O$ , respectively. The circumcircle of triangle $BIO$ intersects the leg $BC$ also at point $D \ne B$ .

(a) Prove that the lines $AC$ and $DI$ are parallel. (b) Prove that the lines $OD$ and $IB$ are mutually perpendicular.

(Walther Janous)