Number theory · Diophantine equations · Symmetric equations · Natural numbers
a) Suppose the ordered quadruple \((x, y, z, w)\) is a solution of the equation
\[ x^2 + y^2 + z^2 + w^2 = xyzw . \]
Prove that \((yzw - x,\, y,\, z,\, w)\) is a solution of the same equation.
b) Prove that the equation
\[ x^2 + y^2 + z^2 + w^2 = xyzw \]
has infinitely many solutions in natural numbers.
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Serbian Municipal Competition 2015, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source