SILK-ROAD 2015

Prove that there is no positive real numbers $a,b,c,d$ such that both of the following equations hold. $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6 , \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}=32$$ .

Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$ , at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right)$ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$ , for any natural $n$ .

Let $B_n$ be the set of all sequences of length $n$ , consisting of zeros and ones. For every two sequences $a,b \in B_n$ (not necessarily different) we define strings $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$\varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} (0 \leq i \leq n-1).$$ . Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)}$ . .

Let O be a circumcenter of an acute-angled triangle ABC. Consider two circles ω and Ω inscribed in the angle BAC in such way that ω is tangent from the outside to the arc BOC of a circle circumscribed about the triangle BOC; and the circle Ω is tangent internally to a circumcircle of triangle ABC. Prove that the radius of Ω is twice the radius ω.