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Turkey NMO 2007 1st Round - P34 (Number Theory)

Source:

October 5, 2012
modular arithmeticquadratics

Problem Statement

For how many primes pp less than 1515, there exists integer triples (m,n,k)(m,n,k) such that m+n+k≡0(modp)mn+mk+nk≡1(modp)mnk≡2(modp). \begin{array}{rcl} m+n+k &\equiv& 0 \pmod p \\ mn+mk+nk &\equiv& 1 \pmod p \\ mnk &\equiv& 2 \pmod p. \end{array}
<spanclass=′latex−bold′>(A)</span> 2<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 4<spanclass=′latex−bold′>(D)</span> 5<spanclass=′latex−bold′>(E)</span> 6 <span class='latex-bold'>(A)</span>\ 2 \qquad<span class='latex-bold'>(B)</span>\ 3 \qquad<span class='latex-bold'>(C)</span>\ 4 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ 6
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