[Announcement for admission to the boarding school at Leningrad State University]

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1. Prove that there is an infinite set of natural numbers $a$ with the following properties: the number $z = n^4 + a$ is not prime or for which natural number $n$ (GDR, 5 points).

2. Let $a_1, a_2, \dots, a_n$ be real constants, $x$ be a real variable and
   $$f(x) = \cos(a_1+x) + \frac{\cos(a_2+x)}{2} + \frac{\cos(a_3+x)}{2^2} + \dots + \frac{\cos(a_n+x)}{2^{n-1}}.$$
   Prove that from $f(x_1) = f(x_2) = 0$ it follows that $x_1 - x_2 = m$, where $m$ is an integer (Veneria, 7 points).

3. For each value $k = 1, 2, 3, 4, 5$, find the necessary and sufficient conditions that must be satisfied by the number $a > 0$ in order for there to exist a terraer whose $k$ edges have length $a$, and the remaining $6 - k$ edges have length 1 (Polysha, 7 points).

On the second day the following tasks were proposed:

4. The semicircle $\gamma$ is built on the diameter $AB$. The point $C$ lies on $\gamma$ and distinguishes $A$ and $B$. We denote the orthogonal projection of $C$ onto $AB$ by $D$. Consider three circles $\gamma_1, \gamma_2, \gamma_3$ having $AB$ as a common tangent: of them $\gamma_1$ is inscribed in the triangle $ABC$, $\gamma_2$ and $\gamma_3$ both tangent to the segment $CD$ and $\gamma$. Prove that $\gamma_1, \gamma_2, \gamma_3$ have a second common tangent (Holland, 6 points).

5. There are $n > 4$ points in the data plane, and no three lie on the same straight line. Show that it is possible to find at least $C_n^2 - 3$ convex quadrilaterals with vertices at four of the given points (Mongolia, 7 points).

6. Prove that if $x_1 > 0, x_2 > 0$ and $x_1y_1 - z_1^2 > 0, x_2y_2 - z_2^2 > 0$, then
   $$\frac{8}{(x_1+x_2)(y_1+y_2)-(z_1-z_2)^2} \leq \frac{1}{x_1y_1 - z_1^2} + \frac{1}{x_2y_2 - z_2^2}.$$
   Establish necessary and sufficient conditions under which equality occurs in this inequality (CCCP, 8 points).
   N. S. Petrakov

The boarding school at Leningrad State University accepts those who have successfully completed the intermediate and eighth grade of a general education school and who have demonstrated ability to study natural subjects, living in the Arkhangelsk, Volgodsk, Kaliningrad, Kirov, Leningrad, Murmansk, Novgorod, Pskov regions, Karelian and Komi Autonomous Soviet Socialist Republics, Latvian, Lithuanian and Estonian SSRs.

To enter the eighth grade of a boarding school, you must take a competitive exam in mathematics (written) and pass a test with the admissions committee (with a physicist, mathematician or biologist).

Students who live in rural areas, working-class villages, and small towns in the North-West are admitted to the eighth grade.

Admission to ninth grade will include a written exam in mathematics and an oral exam in physics. Admission to the physico-mathematical specialization is based on mathematics, and to the chemical-biological specialization - either chemistry or biology.

Entrance exams are held during the spring holidays simultaneously with the regional (national) round of the Physics, Mathematics, Chemistry and Biology Olympiad, as well as in June in the regional and national centers among candidates selected by public education organizations.

Desire to participate in competitive examinations of subjects for public education authorities (OBLONO or the Ministry of Education of the Republic):

A recommendation from the teaching council to your school.

Characteristics (approved by the pedagogical council).

Statement from the parent or their surrogate.

Medical certificate (form No. 26).

The issue of admission to a boarding school is decided by the admissions committee in Leningrad. The decision of the selection committee is communicated by July 10.

Our address: Leningrad, P-228, Savushkina Street, 61. Specialized boarding school at Leningrad University.