Proving a certain determinant $\left|\det A\right|$ is complete square
It was a problem posed at an annual contest for undergraduates called IMC (International Mathematics Contest). The official site of the site http://www.imc-math.org . It was the fifth problem of the second day (year 2008), which means it is pretty hard to solve. Here is a reference of the official site http://www.imc-math.org.uk/imc2008/day2_solutions.pdf . Also there is a discussion here at artofproblemsolving forum http://www.artofproblemsolving.com/Forum/viewtopic.php?f=79&t=217339 .