## A problem seminar |

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algebraics angle approach average best possible binary category argument change variables choose coins concurrent lines contains contradiction convex convex cone convex set course decreasing definite integral derivative determine equal estimate exactly expected number extreme point fact failure probability function form functional equation given gives Hence induction inequality interval lattice points leads least length limit line segments log log look lower bound mean value theorem monic polynomial monotonicity multiples Namely non-negative Note number of lattice observe obtain odd number perfect square polynomial positive integers positive numbers prime factors prove rational real number real zeros recurrence formula recurrence relation recurrent sequence result Rolle's theorem rows Second Solution shows side simply solved Stirling's formula success probability Suppose Taylor series thereby tion toss total number triples unit square upper bound vertices wlog write