Problems for 2016

1.Let a , b , c be the lenghts of the edges of a right parallelepiped and d be the the lenght og its diagonal .Prove that $(ab)^2+(bc)^2+(ca)^2 \geq abcd\sqrt3$ .
2.Prove that for 1$\leq x_k\leq 2$ , k = 1………….n the inequality
$\displaystyle{(\sum_{k=1}^{n}}x_k)$$\displaystyle{(\sum_{k=1}^{n}}\frac{1}{x_k})^2$$\leq n^3.$
3.Prove that for every integer n $\geq$ 2 , the number $2^{2^n-2}+1$ is not a prime .
4.Show that there exist infinetely many $a$ such that for any $n$ the number $n^4 + a$ is not an prime .
5.Prove that for any integer n $\geq$ 1 , the number $n^12 + 64 $ can be written as the product of four distinct integers .
6.$Hermit$$Identity$ : Prove that the function $f:\mathbb{R}$$\rightarrow \mathbb{N}$ ,

where f(x) = $\lfloor x \rfloor +$$\lfloor{x+\frac{1}{n}}\rfloor+$………+$\lfloor{x+\frac{n-1}{n}}\rfloor$$\lfloor{nx}\rfloor$.
One can easily check that f(x) = 0 for x$\in$ $\big{[0,\frac{1}{n})}$ .
prove that $f$ is a periodic function with period $\frac{1}{n}$

7.If a function $f:\mathbb{R}\rightarrow\mathbb{R} $ is not an one -to -one function and there exist a function $g:\mathbb{R}\times\mathbb{R} $ such that $f(x+y) = g(f(x) , y) $ for all x,y $\in$ $\mathbb{R}$.Show that f is a periodic function .
8.Let $f:\mathbb{R}\rightarrow\mathbb{R}$ is a $periodic$ function such that the set {$f(n)|n\in \mathbb{N}$} has infinetely many elements .Prove that the period off $f$ is irrational .


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