International Mathematics Competition
for University Students
2019

IMC2019: Day 1, Problem 2

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: If the solution of the system is not unique then \(\displaystyle \det\begin{pmatrix} Y & E & A & R \\ R & Y & E & A \\ A & R & Y & E \\ E & A & R & Y \end{pmatrix}=0\).

  Solution