On the parabola $y = x^2$ lie three different points $P, Q$ and $R$. Their projections $P', Q'$ and $R'$ on the $x$-axis are equidistant and equal to $s$ , i.e. $| P'Q'| = | Q'R'| = s$. Determine the area of $\vartriangle PQR$ in terms of $s$

In triangle $\vartriangle ABC$, $\angle A = 50^o, \angle B = 60^o$ and $\angle C = 70^o$. The point $P$ is on the side $[AB]$ (with $P \ne A$ and $P \ne B$). The inscribed circle of $\vartriangle ABC$ intersects the inscribed circle of $\vartriangle ACP$ at points $U$ and $V$ and intersects the inscribed circle of $\vartriangle BCP$ at points $X$ and $Y$. The rights $UV$ and $XY$ intersect in $K$. Calculate the $\angle UKX$.

In a closed rectangular neighborhood there are: $S$ streets (these are straight roads of maximum length), $V$ four-arm intersections ( ), $H$ city blocks (these are rectangular areas bounded by four streets, which are no be intersected by another street) and $T$ represents the number of $T$-intersections ( ). For example, in the neighborhood below, there are $15$ streets, $8$ four-arm intersections, $20$ city blocks and $22$ $T$-intersections. Prove that in each district $S + V = H + 3$.

For every natural number $n$ we define the derived number $n'$ as follows: $\bullet$ $0' = 1' = 0$ $\bullet$ if $n$ is prime, then $n' = 1$ $\bullet$ if $n = a \cdot b$, then $n' = a' b + a b'$ . For example: $15' = 3' 5 + 3 5' = 1\cdot 5 + 3\cdot 1 = 8$. Determine all natural numbers $n$ for which $n = n'$.