6. Around an acute isosceles triangle $C L E$ with base $L E$, a circle $\Omega$ is described. Point $N$ is the midpoint of the arc $C E$ not containing point $L$. It is known that the distances from point $N$ to the lines $C E$ and $E L$ are 6 and 9, respectively. Find the radius of the circle $\Omega$ and the area of triangle $C L E$.
Answer: $R=8, S=15 \sqrt{15}$.

Solution. Let $O$ be the center of the circle, $G$ be the intersection point of segments $O N$ and $C E$ (then $G$ is the midpoint of $C E$ and $O G \perp C E$); $C H$ be the height of the triangle ($O \in C H$), $N Q$ be the perpendicular dropped from point $N$ to line $E L$, and $O J$ be the perpendicular dropped from point $O$ to line $N Q$. Let the radius of the circle be $R$, and the angle $E C H$ be $\gamma$.

Then $O G=O C \sin \gamma=R \sin \gamma, G N=O N-O G=R-R \sin \gamma, J N=O N \sin \gamma=R \sin \gamma$. Since triangle $O C E$ is isosceles ($O E=O C=R$), $\angle O E C=\angle O C E=\gamma$, and by the external angle theorem of a triangle, $\angle E O H=2 \gamma$. Therefore, $O H=O E \cos 2 \gamma=R \cos 2 \gamma, J Q=O H=R \cos 2 \gamma$ (since $J Q H O$ is a rectangle). Then $N Q=N J+J Q=R(\sin \gamma+\cos 2 \gamma)=R\left(1+\sin \gamma-2 \sin ^{2} \gamma\right)=R(1-\sin \gamma)(1+2 \sin \gamma)$.

By the problem's condition, $R(1-\sin \gamma)=6, R(1-\sin \gamma)(1+2 \sin \gamma)=9$. Dividing the second equation by the first, we get $1+2 \sin \gamma=\frac{3}{2}, \sin \gamma=\frac{1}{4}$; then $R=\frac{6}{1-\sin \gamma}=8$.

We find the area $S$ of triangle $C L E$:

$S=C H \cdot E H=(C O+O H) \cdot E C=(R+R \cos 2 \gamma) R \sin 2 \gamma=R^{2} 2 \cos ^{2} \gamma \sin 2 \gamma=4 R^{2} \sin \gamma \cos ^{3} \gamma=4 \cdot 64 \cdot \frac{1}{4} \cdot \frac{15 \sqrt{15}}{64}=15 \sqrt{15}$.

"Phystech-2016", mathematics, instructions for checking, tickets 5-8

The problem is considered fully solved (and the maximum possible number of points is awarded) only if all necessary transformations are provided in the solution and all logical steps are fully explained, and the obtained answers are simplified.

The presence of a correct answer does not guarantee a positive score for the problem.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3dd8bb2c54e292d29beg-11.jpg?height=69&width=1882&top_left_y=411&top_left_x=84)

- no selection made ........................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_05_06_d3dd8bb2c54e292d29beg-11.jpg?height=69&width=1884&top_left_y=525&top_left_x=83)

- incorrect trigonometric formulas applied ........................................................ 0 points for all subsequent actions.

3.(6) All possible variants for the last digit of the number are indicated ............................................. 1 point, - for stating the divisibility rules for 12 (for 75) ................................... no additional points, - if the solution by enumeration yields an incorrect answer .................................... no more than 1 point for the problem, - the answer is written in the form $9^{k} \cdot 2$ and so on ................................................................... no points are deducted.

4.(6) A substitution of variables (as in the solution) is made ............................................................................... 1 point,

- a cubic equation in one of the new variables is obtained ............................ 1 point,
- the cubic equation is solved .....................................................................................................
- extraneous solutions are obtained ................................................................................................... deduct 1 point.

5.(8) The set of points satisfying the first equation is constructed .................................... 2 points,

In addition to this:

- 3 points for each value of the parameter found,
- 1 point deducted for each extra value of the parameter obtained,
- 3 points deducted for 2 extra values of the parameter obtained.

6.(7) Only the radius or only the area is found .................................................................................................... 5 points.