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Number of solutions divisible by 10

Source: AIMO 5/2, German TST 2010

July 16, 2011
algebra unsolvedalgebra

Problem Statement

We are given m,n∈Z+.m,n \in \mathbb{Z}^+. Show the number of solution 4−4-tuples (a,b,c,d)(a,b,c,d) of the system
\begin{align*} ab + bc + cd - (ca + ad + db) &= m\\ 2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) &= n \end{align*}
is divisible by 10.
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