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P02 [Number Theory] - Turkish NMO 1st Round - 2005

Source:

October 26, 2013
modular arithmetic

Problem Statement

Let a1,a2,…,ana_1, a_2, \dots, a_n be positive integers such that none of them is a multiple of 55. What is the largest integer n<2005n<2005, such that a14+a24+⋯+an4a_1^4 + a_2^4 + \cdots + a_n^4 is divisible by 55?
<spanclass=′latex−bold′>(A)</span> 2000<spanclass=′latex−bold′>(B)</span> 2001<spanclass=′latex−bold′>(C)</span> 2002<spanclass=′latex−bold′>(D)</span> 2003<spanclass=′latex−bold′>(E)</span> 2004 <span class='latex-bold'>(A)</span>\ 2000 \qquad<span class='latex-bold'>(B)</span>\ 2001 \qquad<span class='latex-bold'>(C)</span>\ 2002 \qquad<span class='latex-bold'>(D)</span>\ 2003 \qquad<span class='latex-bold'>(E)</span>\ 2004
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