MathDB

1

Part of 2006 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Express n as the sum of k distinct divisors of n

Source: XV Rioplatense Mathematical Olympiad (2006), Level 3

8/9/2011
(a) For each integer k≥3k\ge 3, find a positive integer nn that can be represented as the sum of exactly kk mutually distinct positive divisors of nn. (b) Suppose that nn can be expressed as the sum of exactly kk mutually distinct positive divisors of nn for some k≥3k\ge 3. Let pp be the smallest prime divisor of nn. Show that 1p+1p+1+⋯+1p+k−1≥1.\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.
factorialinductioninequalitiesnumber theory unsolvednumber theory
Concurrent: line thru orthocenter & circumcenter, BC & other

Source: XV Rioplatense Mathematical Olympiad (2006), Level 3

8/9/2011
The acute triangle ABCABC with AB≠ACAB\neq AC has circumcircle Γ\Gamma, circumcenter OO, and orthocenter HH. The midpoint of BCBC is MM, and the extension of the median AMAM intersects Γ\Gamma at NN. The circle of diameter AMAM intersects Γ\Gamma again at AA and PP. Show that the lines APAP, BCBC, and OHOH are concurrent if and only if AH=HNAH = HN.
geometrycircumcircleparallelogramgeometric transformationreflectionperpendicular bisectorBritishMathematicalOlympiad