IMO 1989 LL COL9

Let m be a positive integer and define f(m) to be the number

imolonglistmathematicsolympiad

IMO 1989 LL COL9

Origin: COL

Problem

Let m be a positive integer and define f(m) to be the number of factors of 2 in m! (that is, the greatest positive integer k such that 2k | m!). Prove that there are infinitely many positive integers m such that m −f(m) = 1989.