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## IMC2019: Problems on Day 1
\(\displaystyle \prod_{n=3}^{\infty}\frac{(n^3+3n)^2}{n^6-64} .\) Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan
\(\displaystyle \begin{aligned} Yx+Ey+Az+Rw &= Y \\ Rx + Yy + Ez + Aw &= E \\ Ax + Ry + Yz + Ew &= A \\ Ex + Ay + Rz + Yw &= R \end{aligned} \) of linear equations in the variables \(\displaystyle x,y,z\) and \(\displaystyle w\) has at least two solutions. Find all very good YEARs in the 21st century. (The 21st century starts in 2001 and ends in 2100.) Proposed by Tomáš Bárta, Charles University, Prague
\(\displaystyle {2f'(x)+xf''(x)\geq1} \quad\text{for \(\displaystyle x\in(-1,1)\)}. \) Prove that \(\displaystyle \int_{-1}^1xf(x)\,\mathrm{d}x\geq\frac{1}{3}. \) Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karim Rakhimov, Scuola Normale Superiore and National University of Uzbekistan
\(\displaystyle a_0=1, \quad a_1=2, \quad (n+3)a_{n+2}=(6n+9)a_{n+1}- na_n \quad \text{for \(\displaystyle n\ge 0\).} \) Prove that all terms of this sequence are integers. Proposed by Khakimboy Egamberganov, ICTP, Italy
(1) \(\displaystyle \det(B)=1\); (2) \(\displaystyle AB=BA\); (3) \(\displaystyle A^4+4A^2B^2+16B^4=2019I\). (Here \(\displaystyle I\) denotes the \(\displaystyle n\times n\) identity matrix.) Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan | |||||||||||||||||||

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