Difference between revisions of "2011 IMO Problems/Problem 1"

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Given any set A = {a1, a2, a3, a4} of four distinct positive integers, we denote the sum a1+a2+a3+a4 bysA. Let nA denote the number of pairs (i,j) with 1 i< j 4 for which ai+aj divides sA. Find all sets A of four distinct positive integers which achieve the largest possible value of nA.
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Given any set <math>A = \{a_1, a_2, a_3, a_4\}</math> of four distinct positive integers, we denote the sum <math>a_1+a_2+a_3+a_4</math> by <math>s_A</math>. Let <math>n_A</math> denote the number of pairs <math>(i,j)</math> with <math>1 \leq i < j \leq 4</math> for which <math>a_i+a_j</math> divides <math>s_A</math>. Find all sets <math>A</math> of four distinct positive integers which achieve the largest possible value of <math>n_A</math>.

Revision as of 14:41, 24 July 2011

Given any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers, we denote the sum $a_1+a_2+a_3+a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i,j)$ with $1 \leq i < j \leq 4$ for which $a_i+a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers which achieve the largest possible value of $n_A$.

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