HMMT-Nov 2022 guts, Problem 7

All positive integers whose binary representations (excluding leading zeroes) have at least as many 1 's as 0 's are put in increasing order. Compute the number of digits in the binary representation of the 200th number.

Solution: Solution: We do a rough estimation. There are 255 positive integers with at most 8 digits and a majority of them, but not more than 200, satisfy the property. Meanwhile, there are 511 positive integers with at most 9 digits, and a majority of them satisfy this property. Thus, the answer must be greater than 8 and at most 9 . $\fbox{9}$.

Answer: 9