At what angle do the curves $x^{2}+y^{2}=16$ and $y^{2}=6 x$ intersect? What is the area of the region bounded by the two?
1. The coordinates of the real intersection points of the curves are:

$$
\begin{aligned}
& x_{1}=2 ; \quad y_{1}=2 \sqrt{3} \\
& x_{2}=2 ; \quad y_{2}=-2 \sqrt{3}
\end{aligned}
$$

Let's draw the tangents to both the circle and the parabola at point $P_{1}\left(x_{1}, y_{1}\right)$ (denoted as $t_{1}$ and $t_{2}$), whose equations are:

$$
\begin{gathered}
t_{1} \equiv x x_{1}+y y_{1}=r^{2} \\
t_{2} \equiv y y_{1}=p\left(x+x_{1}\right)
\end{gathered}
$$

Thus,

$$
t_{1} \equiv y=-\frac{1}{\sqrt{3}} x+\frac{8}{\sqrt{3}}
$$

and

$$
t_{2} \equiv y=\frac{\sqrt{3}}{2} x+\sqrt{3}
$$

The tangents of the angles formed by these two tangents are:

$$
\operatorname{tg} \alpha=\frac{5}{3} \sqrt{3}
$$

which means

$$
\alpha=70^{\circ} 53^{\prime} 37^{\prime \prime}
$$

$2^{\circ}$. If the intersection points of the curves are $P_{1}$ and $P_{2}$, and $O$ is the origin of the coordinates, then the common area can be calculated by adding the area of the circular segment bounded by the chord $P_{1} P_{2}$ and the arc $\widehat{P_{1} P_{2}}$ to the area bounded by $P_{1} P_{2}$ and the parabola, i.e., the area bounded by the two:

$$
T=\frac{4}{3} x_{1} y_{1}+\text { area of the circular segment. }
$$

However,

$$
\text { area of the circular segment }=P_{1} O P_{2} \text {, area of the circular sector }-P_{1} O P_{2} \triangle \text {. }
$$

We need to know the angle $P_{1} O P_{2}$. Since

$$
\operatorname{tg} \frac{P_{1} O P_{2}}{2}=\frac{y_{1}}{x_{1}}=\sqrt{3}
$$

it follows that

$$
P_{1} O P_{2}=120^{\circ}
$$

Thus,

$$
\begin{aligned}
\text { area of the circular segment } & =\frac{r^{2} \pi \cdot 120^{\circ}}{360^{\circ}}-2 \cdot \frac{x_{1} y_{1}}{2}= \\
= & \frac{16 \pi}{3}-4 \sqrt{3}
\end{aligned}
$$

and therefore,

$$
\begin{gathered}
T=\frac{16 \sqrt{3}}{3}+\frac{16 \pi}{3}-4 \sqrt{3}=\frac{4}{3}(\sqrt{3}+4 \pi)= \\
19.06 \text { square units. }
\end{gathered}
$$

(Imre Dömény, Budapest.)

The problem was also solved by: Ádámffy E., Bartók I., Braun J., Haar A., Kiss J., Kertész G., Kürti J., Neidenbach E., Pichler S., Rássy P., Rosenberg J., Schöffer J., Szúcs A., Tóth B.