MathDB

1

Part of 2007 Korea National Olympiad

Problems(2)

string of length 6 composed of three characters

Source: 2007 Korean MO, 2nd Round, A.M.

8/18/2007
Consider the string of length 6 6 composed of three characters a a, b b, c c. For each string, if two a as are next to each other, or two b bs are next to each other, then replace aa aa by b b, and replace bb bb by a a. Also, if a a and b b are next to each other, or two c cs are next to each other, remove all two of them (i.e. delete ab ab, ba ba, cc cc). Determine the number of strings that can be reduced to c c, the string of length 1, by the reducing processes mentioned above.
geometry3D geometrycomplex numberscombinatorics unsolvedcombinatorics
what is the value of positive constant k?

Source: 2007 Korean MO, 2nd Round, P.M.

8/18/2007
For all positive reals a a, b b, and c c, what is the value of positive constant k k satisfies the following inequality? \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007} .
inequalitiesinequalities unsolved