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58th Ukrainian National Mathematical Olympiad

Ukraine geometry

Problem

Given is circle with center in point and diameter . is a square, is the second point of intersection of and circle , is the middle of the segment . Find the value of the angle .
Solution
Since is a diameter, then , and as the middle segment. Therefore, and quadrilateral is inscribed. Hence, .
Final answer
45°

Techniques

Cyclic quadrilateralsAngle chasing