VMO 1978 Problem 2

Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root.

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.