AT-OEMO-BEG 2014

Determine all solutions of the diophantine equation $a^2 = b \cdot (b + 7)$ in integers $a\ge 0$ and $b \ge 0$ .

(W. Janous, Innsbruck)

All empty white triangles in figure are to be filled with integers such that for each gray triangle the three numbers in the white neighboring triangles sum to a multiple of $5$ . The lower left and the lower right white triangle are already filled with the numbers $12$ and $3$ , respectively. Find all integers that can occur in the uppermost white triangle.

(G. Woeginger, Eindhoven, The Netherlands)

Let $a, b, c$ and $d$ be real numbers with $a < b < c < d$ . Sort the numbers $x = a \cdot b + c \cdot d, y = b \cdot c + a \cdot d$ and $z = c \cdot a + b \cdot d$ in ascending\order and prove the correctness of your result.

(R. Henner, Vienna)

Consider a triangle $ABC$ . The midpoints of the sides $BC, CA$ , and $AB$ are denoted by $D, E$ , and $F$ , respectively. Assume that the median $AD$ is perpendicular to the median $BE$ and that their lengths are given by $AD = 18$ and $BE = 13.5$ . Compute the length of the third median $CF$ .

(K. Czakler, Vienna)