AT-OEMO-FED1 2021

Let $a,b,c\geq 0$ and $a+b+c=1.$ Prove that $$\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.$$ (Marian Dinca)

Let $ABC$ denote a triangle. The point $X$ lies on the extension of $AC$ beyond $A$ , such that $AX = AB$ . Similarly, the point $Y$ lies on the extension of $BC$ beyond $B$ such that $BY = AB$ . Prove that the circumcircles of $ACY$ and $BCX$ intersect a second time in a point different from $C$ that lies on the bisector of the angle $\angle BCA$ .

(Theresia Eisenkölbl)

Let $n \ge 3$ be an integer. On a circle, there are $n$ points. Each of them is labelled with a real number at most $1$ such that each number is the absolute value of the difference of the two numbers immediately preceding it in clockwise order. Determine the maximal possible value of the sum of all numbers as a function of $n$ .

(Walther Janous)

On a blackboard, there are $17$ integers not divisible by $17$ . Alice and Bob play a game. Alice starts and they alternately play the following moves: $\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$ $\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$ . Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps. Prove that Alice has a winning strategy.

(Daniel Holmes)