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1

Part of 2008 Bosnia Herzegovina Team Selection Test

Problems(2)

simple inequality

Source: Bosnian TST 2008, 1.st day

5/17/2008
Prove that in an isosceles triangle △ABC \triangle ABC with AC\equal{}BC\equal{}b following inequality holds b>πr b> \pi r, where r r is inradius.
inequalitiestrigonometrycalculusderivativegeometrygeometry proposed
very simple

Source: Bosnian TST, 2008, 2.nd day

5/18/2008
8 8 students took part in exam that contains 8 8 questions. If it is known that each question was solved by at least 5 5 students, prove that we can always find 2 2 students such that each of questions was solved by at least one of them.
ceiling functionLaTeXcombinatorics proposedcombinatorics