Olympiad problems

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2006 MOP Homework, 4

Assume that $f : [0,1)\to R$ is a function such that $f(x)-x^3$ and $f(x)-3x$ are both increasing functions. Determine if $f(x)-x^2-x$ is also an increasing function.

2011 District Olympiad, 1

Tags: Increasing function , algebra , equations , easy , district olympiad

Let $ a,b,c $ be three positive numbers. Show that the equation $$ a^x+b^x=c^x $$ has, at most, one real solution.

1999 Poland - Second Round, 1

Tags: function , Increasing function , algebra

Let $f : (0,1) \to R$ be a function such that $f(1/n) = (-1)^n$ for all n ∈ N. Prove that there are no increasing functions $g,h : (0,1) \to R$ such that $f = g - h$.

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