Putnam 52/B3 solution

Problem B3

Let A(x) = be the matrix

 0    a-x   b-x

-a-x   0    c-x

-b-x  -c-x   0

For which (a, b, c) does det A(x) = 0 have a repeated root in x?

Solution

Multiplying out, we find that det A(x) = -2x³ + 2(bc - ac + ab)x. This has roots 0 and ±√(ab + bc - ac). This has a repeated root iff ac = ab + bc.