### International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

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### Problems on Day 1

July 23, 2018

Problem 1. Let $\displaystyle (a_n)_{n=1}^{\infty}$ and $\displaystyle (b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent:

(1) There is a sequence $\displaystyle (c_n)_{n=1}^{\infty}$ of positive numbers such that $\displaystyle \displaystyle\sum\limits_{n=1}^{\infty} \dfrac{a_n}{c_n}$ and $\displaystyle \displaystyle\sum\limits_{n=1}^{\infty} \dfrac{c_n}{b_n}$ both converge;

(2) $\displaystyle \displaystyle\sum\limits_{n=1}^{\infty} \sqrt{\dfrac{a_n}{b_n}}$ converges.

(Proposed by Tomáš Bárta, Charles University, Prague)

Problem 2. Does there exist a field such that its multiplicative group is isomorphic to its additive group?

(Proposed by Alexandre Chapovalov, New York University, Abu Dhabi)

Problem 3. Determine all rational numbers $\displaystyle a$ for which the matrix

$\displaystyle \begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{pmatrix}$

is the square of a matrix with all rational entries.

(Proposed by Daniël Kroes, University of California, San Diego)

Problem 4. Find all differentiable functions $\displaystyle f:(0,\infty)\to\RR$ such that

$\displaystyle f(b)-f(a)=(b-a)f'\left(\sqrt{ab}\right) \quad \text{for all} \quad a,b>0. \tag2$

(Proposed by Orif Ibrogimov, National University of Uzbekistan)

Problem 5. Let $\displaystyle p$ and $\displaystyle q$ be prime numbers with $\displaystyle p<q$. Suppose that in a convex polygon $\displaystyle P_1P_2\dots P_{pq}$ all angles are equal and the side lengths are distinct positive integers. Prove that

$\displaystyle P_1P_2+P_2P_3+\dots+P_kP_{k+1}\geq \dfrac{k^3+k}2$

holds for every integer $\displaystyle k$ with $\displaystyle 1\le k\le p$.

(Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Berlin)