VWO 2007

The numbers $1,2, \ldots$ are placed in a triangle as following:

$$\begin{matrix} 1 & & & 2 & 3 & & 4 & 5 & 6 & 7 & 8 & 9 & 10 \ldots \end{matrix}$$

What is the sum of the numbers on the $n$ -th row?

Given is a half circle with midpoint $O$ and diameter $AB$ . Let $Z$ be a random point inside the half circle, and let $X$ be the intersection of $OZ$ and the half circle, and $Y$ the intersection of $AZ$ and the half circle.

If $P$ is the intersection of $BY$ with the tangent line in $X$ to the half circle, show that $PZ \perp BX$ .

Let $ABCD$ be a square with side $10$ . Let $M$ and $N$ be the midpoints of $[AB]$ and $[BC]$ respectively. Three circles are drawn: one with midpoint $D$ and radius $|AD|$ , one with midpoint $M$ and radius $|AM|$ , and one with midpoint $N$ and radius $|BN|$ . The three circles intersect in the points $R, S$ and $T$ inside the square. Determine the area of $\triangle RST$ .

If $f,g: \mathbb{R} \to \mathbb{R}$ are functions that satisfy $f(x+g(y)) = 2x+y$ $\forall x,y \in \mathbb{R}$ , then determine $g(x+f(y))$ .