OliverA

[1]

We will count the number of it $\< 2^{11}=2048$ (Error compiling LaTeX. Unknown error_msg) instead of $2003$ (In other words, the length of the base-2 representation is at most $11$ . If there are even digits, $2n$ , then the leftmost digit is $1$ , the rest, $2n-1$ , has odd number of digits. In order for the base-2 representation to have more $1$ 's, we will need more $1$ in the remaining $2n-1$ than $0$ 's. Using symmetry, this is equal to $\frac{2^9+2^7+..+2^1}{2}$ Using similar argument where there are odd amount of digits. The remaining even amount of digit must contains the number of $1$ 's at least as the number of $0$ 's. So it's equal to $\frac{\binom{10}{5}+2^{10}+\binom{8}{4}+2^8+\binom{6}{3}+2^6+...+\binom{0}{0}+2^0}{2}$ Summing both cases, we have $\frac{2^0+2^1+..+2^{10}+\binom{0}{0}+..+\binom{10}{5}}{2} = 1199$ . There are $44$ numbers between $2004$ and $2047$ inclusive that satisfy it. So the answer is $1199-44=1\boxed{155}$

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OliverA