2005 Indonesia MO Problems/Problem 6
Find all triples of integers which satisfy
By using the Distributive Property on all three equations, we get Add all three equations, rearrange, and factor to get Notice that since all the variables are integers, the squares must also be integers. The only combination of three integer squares that sum to six is .
Therefore, , , and , so , , and . By rearrangement, and .
Substitute and solve to get Therefore, one solution is , and plugging that value back in satisfies the system. Since any permutation is also a solution, the solutions are , and plugging all the values back into the original system satisfies the system.
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