7. Given a regular prism $K L M N K_{1} L_{1} M_{1} N_{1}$ with base $K L M N$. Planes $\alpha$ and $\beta$ are perpendicular to $L_{1} N$ and pass through vertices $K$ and $N_{1}$, respectively. Let $A$ and $B$ be the points of intersection of planes $\alpha$ and $\beta$ with the diagonal $L_{1} N$, such that $A N < B N$.

a) Find the ratio $L_{1} B: A N$.

b) Suppose additionally that a sphere of radius 2 touches all the lateral faces of the prism, as well as planes $\alpha$ and $\beta$. Find the segment $L_{1} N$ and the volume of the prism $K L M N K_{1} L_{1} M_{1} N_{1}$.
Answer: a) $2: 1$, b) $L_{1} N=2(1+\sqrt{13}), V=32 \sqrt{6+2 \sqrt{13}}$.

Solution. a) By symmetry (with respect to the plane $L N N_{1} L_{1}$), the plane $\alpha$ passes through the point $M$ - and, therefore, through the center $O$ of the face $K L M N$. The segments $L_{1} B$ and $A N$ are projections of the parallel segments $L_{1} N_{1}$ and $N O$ onto the line $L_{1} N$, and $L_{1} N_{1}=2 N O$. Therefore, $L_{1} B: A N=2$.

b) Since the sphere touches all the lateral faces of the prism, its projection onto the base is a circle inscribed in this base. Therefore, $K N=2 r=4$. Moreover, $\alpha$ and $\beta$ are two parallel planes tangent to the sphere, so the distance between them is equal to the diameter of the sphere, which is 4. Since $L_{1} N \perp \alpha$, this distance is the segment $A B$, so $A B=4$.

Let $L_{1} N=d$. Since $N_{1} B$ is the height of the right triangle $L_{1} N_{1} N$, then $L_{1} B \cdot L_{1} N=L_{1} N_{1}^{2}=32$ and, therefore, $L_{1} B=\frac{32}{d}$. Then $A N=\frac{1}{2} L_{1} B=\frac{16}{d}$ and $A B=L_{1} N-L_{1} B-A N=d-\frac{32}{d}-\frac{16}{d}$. We get the equation $4=d-\frac{48}{d}$, from which $d^{2}-4 d-48=0, d=2(1+\sqrt{13})$ (since $d>0$).

Finally, the height of the prism is $h=\sqrt{L_{1} N^{2}-L N^{2}}=\sqrt{4(14+2 \sqrt{13})-32}=2 \sqrt{6+2 \sqrt{13}}$; then the volume of the prism is $V=K L^{2} \cdot h=32 \sqrt{6+2 \sqrt{13}}$.

"Phystech-2016", mathematics, instructions for checking, tickets 9-12
The problem is considered fully solved (and the maximum possible number of points is awarded for it) only if all necessary transformations are provided in the solution text and all logical steps are fully explained, and the answers obtained are reduced to a simplified form.
The presence of a correct answer does not guarantee a positive score for the problem.
1.(6) For each correctly considered case ("base > 1", "base <1") 3 points,
- non-equivalent transformation of the inequality 0 points for all subsequent actions,
- the answer includes a value of $x$ for which the base of the logarithm is 1 ... deduct 1 point,
- the answer includes a value of $x$ for which the base of the logarithm is 0 ... deduct 1 point,
- the answer includes values of $x$ for which the base of the logarithm is negative
no more than 3 points for the problem.
2.(6) It is proved that the minimum of one part of the equation is equal to the maximum of the other ... 1 point,
- only one of the cases of the sign of the expression in parentheses is correctly considered ... 1 point,
3.(6) The substitution of variables (as in the solution) is made ... 1 point,
- a cubic equation is obtained with respect to one of the new variables ... 1 point,
- the cubic equation is solved ... 2 points,
- extraneous solutions are obtained ... deduct 1 point.
4.(8) The ratio of the distances from the centers of the circles to the line $A B$ is found ... 1 point,
- the ratio of the radii of the circles is found ... 4 points.
5.(6) All possible options for the last digit of the number are indicated ... 1 point,
- points are not added for formulating divisibility criteria by 12 (by 75),
- if the answer is obtained by enumeration and is incorrect ... no more than 1 point for the problem,
- the answer is written in the form $9^{k} \cdot 2$ and so on, points are not deducted.
6.(8) The set of points satisfying the first equation is constructed ... 2 points,
From this $^{1}$ :
- for each found value of the parameter ... 3 points,
- 1 extra value of the parameter is obtained ... deduct 1 point,
- 2 extra values of the parameter are obtained ... deduct 3 points.
7.(8) - The ratio of the segments of part a) is found ... 2 points,
- it is indicated that the distance between the planes is equal to the diameter of the sphere ... 1 point,
- the segment of part b) is found ... 3 points,
- the volume of the prism is found ... 2 points.