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, 2023

Combinatorics Difficulty 8.0 National olympiad, round 2 Prove it Turkey

Any two pupils in a school are either friends or not, the friendship is mutual. For each integer $1 \le l \le 99$ there is a school pupil having exactly $l$ friends in the school. Given that there is no triple of school pupils such that any two of them are friends, find the minimal possible number of pupils in this school.

Solution

6. First $XY \parallel l_0$ implies that $XYBC$ is concyclic. Now let $S$ be the second intersection point of the circle ( $XYZ$ ) with the circumcircle; then the lines $ZS, XY$ and $BC$ are concurrent since they are the pairwise radical axes of the circles ( $XYBC$ ) and ( $XYZS$ ) and the circumcircle (let $P$ denote the point of concurrence). Now the quadruple ( $AP, AZ; AB, AC$ ) is harmonic, hence ( $ZP, ZA; ZB, ZC$ ) is harmonic, thus ( $S, A; B, C$ ) is harmonic, so $S$ is a fixed point.

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