[Solutions] International Mathematics Competition for University Students 2018

1. Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent
a) There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\displaystyle\sum_{n=1}^{\infty}{\frac{a_n}{c_n}}$ and $\displaystyle\sum_{n=1}^{\infty}{\frac{c_n}{b_n}}$ both converge;
b) $\displaystyle\sum_{n=1}^{\infty}{\sqrt{\frac{a_n}{b_n}}}$ converges.
2. Does there exist a field such that its multiplicative group is isomorphism to its additive group?
3. Determine all rational numbers $a$ for which the matrix $$\begin{pmatrix}a & -a & -1 & 0 \\a & -a & 0 & -1 \\1 & 0 & a & -a\\0 & 1 & a & -a\end{pmatrix}$$ is the square of a matrix with all rational entries.
4. Find all differentiable functions $f:(0,\infty) \to \mathbb{R}$ such that $$f(b)-f(a)=(b-a)f’(\sqrt{ab}),\,\forall a,b>0.$$
5. Let $p$ and $q$ be prime numbers with $p<q$. Suppose that in a convex polygon $P_1,P_2,…,P_{pq}$ all angles are equal and the side lengths are distinct positive integers. Prove that $$P_1P_2+P_2P_3+\cdots +P_kP_{k+1}\geqslant \frac{k^3+k}{2}$$holds for every integer $k$ with $1\leqslant k\leqslant p$.
6. Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exists $k$ nonzero vectors $v_1,v_2,…,v_k$ in $\mathbb{R}^n$ such that for every pair $i,j$ of indices with $|i-j|>1$ the vectors $v_i$ and $v_j$ are orthogonal.
7. Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that $a_0=0$ and $$a_{n+1}^3=a_n^2-8,\, n=0,1,2,…$$ Prove that the following series is convergent $$\sum_{n=0}^{\infty}{|a_{n+1}-a_n|}.$$
8. Let $\Omega =\{ (x,y,z)\in \mathbb{Z}^3:y+1\geqslant x\geqslant y\geqslant z\geqslant 0\}$. A frog moves along the points of $\Omega$ by jumps of length $1$. For every positive integer $n$, determine the number of paths the frog can take to reach $(n,n,n)$ starting from $(0,0,0)$ in exactly $3n$ jumps.
9. Determine all pairs $P(x),Q(x)$ of complex polynomials with leading coefficient $1$ such that $P(x)$ divides $Q(x)^2+1$ and $Q(x)$ divides $P(x)^2+1$.
10. For $R>1$ let $\mathcal{D}_R =\{ (a,b)\in \mathbb{Z}^2: 0<a^2+b^2<R\}$. Compute $$\lim_{R\rightarrow \infty}{\sum_{(a,b)\in \mathcal{D}_R}{\frac{(-1)^{a+b}}{a^2+b^2}}}.$$
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