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SOLVED - $250
The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is equal to $1$.
The answer is yes (in fact with $2$ replaced with any constant $c>1$), proved by Maier and Tenenbaum [MaTe84]. (Tenenbaum has told me that they received \$650 for their solution.)

See also [449].