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2009 Bosnia Herzegovina Team Selection Test

Part of Bosnia Herzegovina Team Selection Test

Subcontests

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Bosnia 2009 Problem 3

a1,a2,…,a100a_{1},a_{2},\dots,a_{100} are real numbers such that:a1≥a2≥⋯≥a100≥0 a_{1}\geq a_{2}\geq\dots\geq a_{100}\geq0 a12+a22≥100 a_{1}^{2}+a_{2}^{2}\geq100 a32+a42+⋯+a1002≥100 a_{3}^{2}+a_{4}^{2}+\dots+a_{100}^{2}\geq100 What is the minimum value of sum a1+a2+⋯+a100.a_{1}+a_{2}+\dots+a_{100}.

Bosnia 2009 Problem 6

Let nn be a positive integer and xx positive real number such that none of numbers x,2x,…,nxx,2x,\dots,nx and none of 1x,2x,…,⌊nx⌋x\frac{1}{x},\frac{2}{x},\dots,\frac{\left\lfloor nx\right\rfloor }{x} is an integer. Prove that ⌊x⌋+⌊2x⌋+⋯+⌊nx⌋+⌊1x⌋+⌊2x⌋+⋯+⌊⌊nx⌋x⌋=n⌊nx⌋ \left\lfloor x\right\rfloor +\left\lfloor 2x\right\rfloor +\dots+\left\lfloor nx\right\rfloor +\left\lfloor \frac{1}{x}\right\rfloor +\left\lfloor \frac{2}{x}\right\rfloor +\dots+\left\lfloor \frac{\left\lfloor nx\right\rfloor }{x}\right\rfloor =n\left\lfloor nx\right\rfloor
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