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## IMC2018: Problems on Day 1
(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)
(Proposed by Alexandre Chapovalov, New York University, Abu Dhabi)
\(\displaystyle \begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{pmatrix} \) is the square of a matrix with all rational entries. (Proposed by Daniël Kroes, University of California, San Diego)
\(\displaystyle f(b)-f(a)=(b-a)f'\left(\sqrt{ab}\right) \quad \text{for all} \quad a,b>0. \tag2 \) (Proposed by Orif Ibrogimov, National University of Uzbekistan)
\(\displaystyle P_1P_2+P_2P_3+\dots+P_kP_{k+1}\geq \dfrac{k^3+k}2\) holds for every integer \(\displaystyle k\) with \(\displaystyle 1\le k\le p\). (Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Berlin) | |||||||||||

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