Difference between revisions of "OliverA"

Revision as of 11:17, 26 June 2020

Testing:

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

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Difference between revisions of "OliverA"

Revision as of 11:17, 26 June 2020