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Found problems: 2

2005 Gheorghe Vranceanu, 3
Tags: function , calculus , integration , leibniz-newton , FTC , real analysis , periodic function
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that: $$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$

2006 Cezar Ivănescu, 3
Tags: function , real analysis , FTC , leibniz-newton
[b]a)[/b] Let $ h:\mathbb{R}\longrightarrow\mathbb{R} $ he a function that admits a primitive $ H $ such that the function $ h/H $ is constant. Prove that there is a real number $ \gamma $ such that $ h(x)=\gamma\cdot\exp \left( x\cdot\frac{h}{H} (x) \right) , $ for any real number $ x. $ [b]b)[/b] Find the functions $ f,g:\mathbb{R}\longrightarrow\mathbb{R} $ that admit the primitives $ F,G, $ respectively, that satisfy $ f=\frac{G+g}{2},g=\frac{F+f}{2} $ and $ f(0)=g(0)=0. $