2020 CIME I Problems/Problem 12

Problem 12

Define a sequence $a_0, a_1, a_2, ...$ by \[a_i=\underbrace{1\ldots1}_{2^{i}\text{ 1's}}\underbrace{0\ldots0}_{(2^i-1)\text{ 0's}}1_2,\] where $a_i$ is expressed in binary. Let $S$ be the sum of the digits when $a_0 a_1 a_2 \cdots a_{10}$ is expressed in binary. Find the remainder when $S$ is divided by $1000$.


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See also

2020 CIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All CIME Problems and Solutions

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