The product of all positive real numbers x satisfying the equation
20xlog2026x=26x
is an integer P. Find the number of positive integer divisors of P.
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Note that the solutions to this quadratic are real as the bound log2026(26)<1 reveals that the discriminant is positive. If the two solutions to the quadratic are a1 and a2, then, P=26a1⋅26a2=26a1+a2. From Vieta's, we see that a1+a2=log20262620=20log262026. Thus P=2620log26(2026)=(26log26(2026))20=202620=220101320. Then, P has 21⋅21=441 divisors.