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Turkish NMO First Round - 2012 Problem - 34 {Number Theory}

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July 1, 2012

Problem Statement

If 1010 divides the number 1⋅21+2⋅22+3⋅23+⋯+n⋅2n1\cdot2^1+2\cdot2^2+3\cdot2^3+\dots+n\cdot2^n, what is the least integer n≥2012n\geq 2012?
<spanclass=′latex−bold′>(A)</span> 2012<spanclass=′latex−bold′>(B)</span> 2013<spanclass=′latex−bold′>(C)</span> 2014<spanclass=′latex−bold′>(D)</span> 2015<spanclass=′latex−bold′>(E)</span> 2016 <span class='latex-bold'>(A)</span>\ 2012 \qquad <span class='latex-bold'>(B)</span>\ 2013 \qquad <span class='latex-bold'>(C)</span>\ 2014 \qquad <span class='latex-bold'>(D)</span>\ 2015 \qquad <span class='latex-bold'>(E)</span>\ 2016
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