MathDB
System with 9 coefficients

Source: Romanian IMO Team Selection Test TST 1996, problem 16

September 27, 2005
functionalgebra proposedalgebrapigeonhole principle

Problem Statement

Let n≥3 n\geq 3 be an integer and let S⊂{1,2,…,n3} \mathcal{S} \subset \{1,2,\ldots, n^3\} be a set with 3n2 3n^2 elements. Prove that there exist nine distinct numbers a1,a2,…,a9∈S a_1,a_2,\ldots,a_9 \in \mathcal{S} such that the following system has a solution in nonzero integers: \begin{eqnarray*} a_1x + a_2y +a_3 z &=& 0 \\ a_4x + a_5 y + a_6 z &=& 0 \\ a_7x + a_8y + a_9z &=& 0. \end{eqnarray*} Marius Cavachi
Back to ProblemsView on AoPS