Olympiad problems

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Found problems: 2

2010 N.N. Mihăileanu Individual, 1

Tags: calculus , real analysis , inequalities , lipschitz , integral inequalities , definite integral , integration

Let $ m:[0,1]\longrightarrow\mathbb{R} $ be a metric map. [b]a)[/b] Prove that $ -\text{identity} +m $ is continuous and nonincreasing. [b]b)[/b] Show that $ \int_0^1\int_0^x (-t+m(t))dtdx=\int_0^1 (x-1)(x-m(x))dx. $ [b]c)[/b] Demonstrate that $ \int_0^1\int_0^x m(t)dtdx -\frac{1}{2}\int_0^1 m(x)dx\ge -\frac{1}{12} . $ [i]Gabriela Constantinescu[/i] and [i]Nelu Chichirim[/i]

2006 Victor Vâlcovici, 1

Tags: real analysis , lipschitz , fixed point , short map , nonexpansive map , composition , function

Let be a nondegenerate and closed interval $ I $ of real numbers, a short map $ m:I\longrightarrow I, $ and a sequence of functions $ \left( x_n \right)_{n\ge 1} :I\longrightarrow\mathbb{R} $ such that $ x_1 $ is the identity map and $$ 2x_{n+1}=x_n+m\circ x_n , $$ for any natural numbers $ n. $ Prove that: [b]a)[/b] there exists a nondegenerate interval having the property that any point of it is a fixed point for $ m. $ [b]b)[/b] $ \left( x_n \right)_{n\ge 1} $ is pointwise convergent and that its limit function is a short map.