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45th Mongolian Mathematical Olympiad

Mongolia geometry

Problem

circles with radius were fit into a square without any overlaps. Suppose that maximum value of is . Find the minimum value of such that .

(proposed by Davii-Od)
Solution
Answer . Observe that thus position (big circle) better than this position (small circle). Hence by easy calculation about circles and empty areas of square, we get satisfying above condition .
Final answer
29

Techniques

Optimization in geometryColoring schemes, extremal arguments