Let $a$, $b$, $c$ be integers with $a^3 + b^3 + c^3$ divisible by $18$. Prove that $abc$ is divisible by $6$. (Karl Czakler)

Let $x$, $y$ be positive real numbers with $xy = 4$. Prove that $$\frac{1}{x+3} + \frac{1}{y+3} \le \frac{2}{5}$$ For which $x$ and $y$ does equality hold? (Walther Janous)

Anton chooses as starting number an integer $n \ge 0$ which is not a square. Berta adds to this number its successor $n+1$. If this sum is a perfect square, she has won. Otherwise, Anton adds to this sum, the subsequent number $n+2$. If this sum is a perfect square, he has won. Otherwise, it is again Berta's turn and she adds the subsequent number $n+3$, and so on. Prove that there are infinitely many starting numbers, leading to Anton's win. (Richard Henner)

Let $k_1$ and $k_2$ be internally tangent circles with common point $X$. Let $P$ be a point lying neither on one of the two circles nor on the line through the two centers. Let $N_1$ be the point on $k_1$ closest to $P$ and $F_1$ the point on $k_1$ that is farthest from $P$. Analogously, let $N_2$ be the point on $k_2$ closest to $P$ and $F_2$ the point on $k_2$ that is farthest from $P$. Prove that $\angle N_1 X N_2 = \angle F_1 X F_2$. (Robert Geretschläger)