International Mathematics Competition
for University Students
2019

IMC2019: Problems on Day 1

Problem 1. Evaluate the product

\(\displaystyle \prod_{n=3}^{\infty}\frac{(n^3+3n)^2}{n^6-64} .\)

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and

Karen Keryan, Yerevan State University and American University of Armenia, Yerevan

  Hint    Solution  

Problem 2. A four-digit number \(\displaystyle YEAR\) is called very good if the system

\(\displaystyle \begin{aligned} Yx+Ey+Az+Rw &= Y \\ Rx + Yy + Ez + Aw &= E \\ Ax + Ry + Yz + Ew &= A \\ Ex + Ay + Rz + Yw &= R \end{aligned} \)

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

  Hint    Solution  

Problem 3. Let \(\displaystyle f:(-1,1)\to\RR\) be a twice differentiable function such that

\(\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

  Hint    Solution  

Problem 4. Define the sequence \(\displaystyle a_0,a_1,\ldots\) of numbers by the following recurrence:

\(\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

  Hint    Solution  

Problem 5. Determine whether there exist an odd positive integer \(\displaystyle n\) and \(\displaystyle n\times n\) matrices \(\displaystyle A\) and \(\displaystyle B\) with integer entries, that satisfy the following conditions:

    (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

  Hint    Solution