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area of spots inside a circle

Source: Serbia MO 2005 2nd Grade P4

April 11, 2021
circlegeometryarea

Problem Statement

Inside a circle kk of radius RR some round spots are made. The area of each spot is 11. Every radius of circle kk, as well as every circle concentric with kk, meets in no more than one spot. Prove that the total area of all the spots is less than πR+12RR.\pi\sqrt R+\frac12R\sqrt R.
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