Herzog Competition - 2023

The traditional Herzog competition of year 2023 took place on November 4. Twenty MSU undergraduate students had three hours to solve six problems. The winners are

First prize:	Erik Brodsky
Second prize:	Minh Do Quang, Zhan (Jenny) Zhan
Third Prize:	Aamierul Harith Helmi, Hoang Phuc Nguen
Honorable:	William Carvalho, Andrew Harms, Thinh Quang Ly

Here is the list of problems:

Question 1

Find

\sum_{n = 1}^{\infty} \frac{1}{(n + 2) \sqrt{n} + n \sqrt{n + 2}}

Question 2

Given that $\sum \frac{1}{a_{n}} = \infty$ and $\sum \frac{1}{b_{n}} = \infty$ , where $(a_{n})$ and $(b_{n})$ are nondecreasing sequences of positive real numbers, is it true that $\sum \frac{1}{a_{n} + b_{n}} = \infty$ ?

Question 3

Consider the power series expansion (defined for sufficiently small $| x |$ )

\frac{1}{1 - 2 x - x^{2}} = \sum_{n = 0}^{\infty} a_{n} x^{n}

Prove that for any integer $n \geq 0$ there exists an integer $m$ such that ${a_{n}}^{2} + {a_{n + 1}}^{2} = a_{m}$ .

Question 4

Define the function $C (n)$ on the set of natural numbers recurrently:

$C (1) = 1$ ; $C (2 n) = {(- 1)}^{n} C (n)$ , $C (2 n + 1) = C (n)$ . Find

\sum_{n = 1}^{2023} C (n) C (n + 2)

Question 5

Let us have four circles $C_{i}$ , each of which is tangent to the (outer) unit circle centered at the point $O$ and to its two neighbor circles as shown in the figure. Let $A_{i}$ , $i = 1, 2, 3, 4$ be the points of intersection of the circles $C_{i}$ with the unit circle. Denote by $φ_{i, j}$ angles $A_{i} O A_{j}$ . Prove that

\sin (\frac{φ_{1, 2}}{2}) \sin (\frac{φ_{3, 4}}{2}) = \sin (\frac{φ_{2, 3}}{2}) \sin (\frac{φ_{4, 1}}{2})

A large outer circle centered at O contains four mutually neighboring smaller circles. Four blue radii extend from O to points A1 at the bottom, A2 at the upper right, A3 at the upper left, and A4 at the left edge of the outer circle. — Circle configuration for Question 5, as printed in the 2023 examination.

Question 6

Let $a$ , $b$ , and $c$ be integer numbers. Prove that there exists a natural number $n$ such that $\sqrt{n^{3} + a n^{2} + b n + c}$ is not an integer.