The product of all positive real numbers x x
x 2026 x 20 = 26 x 2 0 x l o g 2 0 2 6 x = 2 6 x
is an integer P P P P
First, we make the substitution x = 2 6 a x = 2 6 a a a x x x x
( 2 6 a ) 2026 x 20 = 2 6 a + 1 26 a ⋅ 2026 x = 2 6 20 a + 20 a ⋅ 2026 x = 20 a + 20 a ⋅ 2026 ( 2 6 a ) = 20 a + 20 a 2 ⋅ 2026 ( 26 ) = 20 a + 20 2 0 ( 2 6 a ) l o g 2 0 2 6 x 2 6 a ⋅ l o g 2 0 2 6 x a ⋅ log 2 0 2 6 x a ⋅ log 2 0 2 6 ( 2 6 a ) a 2 ⋅ log 2 0 2 6 ( 2 6 ) = 2 6 a + 1 = 2 6 2 0 a + 2 0 = 2 0 a + 2 0 = 2 0 a + 2 0 = 2 0 a + 2 0
Note that the solutions to this quadratic are real as the bound 2026 ( 26 ) < 1 log 2 0 2 6 ( 2 6 ) < 1 a 1 a 1 a 2 a 2 P = 2 6 a 1 ⋅ 2 6 a 2 = 2 6 a 1 + a 2 P = 2 6 a 1 ⋅ 2 6 a 2 = 2 6 a 1 + a 2 a 1 + a 2 = 20 2026 26 = 20 26 2026 a 1 + a 2 = log 2 0 2 6 2 6 2 0 = 2 0 log 2 6 2 0 2 6 P = 2 6 20 26 ( 2026 ) = ( 2 6 26 ( 2026 ) ) 20 = 202 6 20 = 2 20 101 3 20 P = 2 6 2 0 l o g 2 6 ( 2 0 2 6 ) = ( 2 6 l o g 2 6 ( 2 0 2 6 ) ) 2 0 = 2 0 2 6 2 0 = 2 2 0 1 0 1 3 2 0 P P 21 ⋅ 21 = 441 2 1 ⋅ 2 1 = 441
The problems and solutions on this page are the property of the MAA's American Mathematics Competitions