MathDB

2

Part of 2010 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Minimum and maximum value of \frac{a}{b}+\frac{c}{d}

Source: XIX Olimpíada Matemática Rioplatense (2010)

7/22/2011
Find the minimum and maximum values of S=ab+cd S=\frac{a}{b}+\frac{c}{d} where aa, bb, cc, dd are positive integers satisfying a+c=20202a + c = 20202 and b+d=20200b + d = 20200.
inequalitiesnumber theory unsolvednumber theory
Ratio in acute triangle with lines through feet of altitudes

Source: XIX Olimpíada Matemática Rioplatense (2010)

7/22/2011
Acute triangle ABPABP, where AB>BPAB > BP, has altitudes BHBH, PQPQ, and ASAS. Let CC denote the intersection of lines QSQS and APAP, and let LL denote the intersection of lines HSHS and BCBC. If HS=SLHS = SL and HLHL is perpendicular to BCBC, find the value of SLSC\frac{SL}{SC}.
ratiogeometry unsolvedgeometry