Školjka

%V0 $(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy $x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$ Prove that: $(a)$ None of $x_1, x_2, x_3$ equals $1$. $(b)$ Exactly one of these numbers is less than $1.$

%V0 $(USS 1)$ Prove that for a natural number $n > 2, (n!)! > n[(n - 1)!]^{n!}.$

%V0 $(SWE 4)$ Let $a_0, a_1, a_2, \cdots$ be determined with $a_0 = 0, a_{n+1} = 2a_n + 2^n$. Prove that if $n$ is power of $2$, then so is $a_n$

%V0 Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$

%V0 $(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$

%V0 $(NET 6)$ A curve determined by $y =\sqrt{x^2 - 10x+ 52}, 0\le x \le 100,$ is constructed in a rectangular grid. Determine the number of squares cut by the curve.