Team Selection Test for JBMO 2023

Question

33 balls are placed on unit squares of a 10 10 board such that no unit square contains more than one ball. For each empty unit square we calculate the total number of balls located on the same row with this unit square and the total number of balls located on the same column with this unit square and after that the sum of these two numbers we write on this unit square. What is the maximal possible sum of all numbers written on the board?

Accepted Answer

7. The radical axes of the circles (ABC), (BDE), (CDE) must be concurrent at T hence T, D, E are collinear. Moreover, TD TE = TB TK hence the power of T with respect to the circles (ABC), (ADE) are equal and it lies on their radical axis. Since DE and BC are parallel, the radical axis of (ABC), (ADE) is the common tangent at A hence TA is tangent to (ABC).