Let $a,b$ and $c$ be real numbers such that $| (a-b) (b-c) (c-a) | = 1$. Find the smallest value of the expression $| a | + | b | + | c |$. (K.Satylhanov )

Around the acute-angled triangle $ABC$ ($AC>CB$) a circle is circumscribed, and the point $N$ is midpoint of the arc $ACB$ of this circle. Let the points $A_1$ and $B_1$ be the feet of perpendiculars on the straight line $NC$, drawn from points $A$ and $B$ respectively (segment $NC$ lies inside the segment $A_1B_1$). Altitude $A_1A_2$ of triangle $A_1AC$ and altitude $B_1B_2$ of triangle $B_1BC$ intersect at a point $K$ . Prove that $\angle A_1KN=\angle B_1KM$, where $M$ is midpoint of the segment $A_2B_2$ .

Given natural numbers $a,b$ and function $f: \mathbb{N} \to \mathbb{N} $ such that for any natural number $n, f\left( n+a \right)$ is divided by $f\left( {\left[ {\sqrt n } \right] + b} \right)$. Prove that for any natural $n$ exist $n$ pairwise distinct and pairwise relatively prime natural numbers ${{a}_{1}}$, ${{a}_{2}}$, $\ldots$, ${{a}_{n}}$ such that the number $f\left( {{a}_{i+1}} \right)$ is divided by $f\left( {{a}_{i}} \right)$ for each $i=1,2, \dots ,n-1$ . (Here $[x]$ is the integer part of number $x$, that is, the largest integer not exceeding $x$.)

Let $P(n)$ be the number of ways to split a natural number $n$ to the sum of powers of two, when the order does not matter. For example $P(5) = 4$, as $5=4+1=2+2+1=2+1+1+1=1+1+1+1+1$. Prove that for any natural the identity $P(n) + (-1)^{a_1} P(n-1) + (-1)^{a_2} P(n-2) + \ldots + (-1)^{a_{n-1}} P(1) + (-1)^{a_n} = 0,$ is true, where $a_k$ is the number of units in the binary number record $k$ . source