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5

Part of 2003 Irish Math Olympiad

Problems(2)

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12/14/2005
show that thee is no function f definedonthe positive real numbes such that : f(y)>(yāˆ’x)f(x)2f(y) > (y-x)f(x)^2
functioninequalitiesalgebra unsolvedalgebra
Combinatorics (from Irish MO)

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12/17/2015
(a) In how many ways can 10031003 distinct integers be chosen from the set {1,2,...,2003}\{1, 2, ... , 2003\} so that no two of the chosen integers di ffer by 10?10? (b) Show that there are (3(5151)+7(1700))1017(3(5151) + 7(1700)) 101^7 ways to choose 10021002 distinct integers from the set {1,2,...,2003}\{1, 2, ... , 2003\} so that no two of the chosen integers diff er by 10.10.
combinatoricscombinatorics unsolved