MathDB

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Part of 2005 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Numbers of the form n = k + 2[sqrt(k)] + 2 where k in N

Source: XIV Rioplatense Mathematical Olympiad (2005), Level 3

7/26/2011
Find all numbers nn that can be expressed in the form n=k+2⌊k⌋+2n=k+2\lfloor\sqrt{k}\rfloor+2 for some nonnegative integer kk.
floor functionnumber theory unsolvednumber theory
Circumradius ineq PA/(AB*AC) + PB/(BC*BA) + PC/(CA*CB) > 1/R

Source: XIV Rioplatense Mathematical Olympiad (2005), Level 3

7/26/2011
Let PP be a point inside triangle ABCABC and let RR denote the circumradius of triangle ABCABC. Prove that PAAB⋅AC+PBBC⋅BA+PCCA⋅CB≥1R. \frac{PA}{AB\cdot AC}+\frac{PB}{BC\cdot BA}+\frac{PC}{CA\cdot CB}\ge\frac{1}{R}.
geometrycircumcircleinequalities unsolvedinequalities