1993 UNCO Math Contest II Problems/Problem 3
A student thinks of four numbers. She adds them in pairs to get the six sums What are the four numbers? There are two different solutions.
Let the numbers be , , , and . Writing in terms of these, we have:
Adding these up, we have , so . Adding the first two equations, we get . Subtracting this, we obtain . Adding to the third equation, we have , so . Substituting, we have . , and as one of our solutions.
- second solution
Let the numbers be , , , and . We have the following 6 equations from the problem statement:
Adding all of these up: gives : call this equation A
Adding first two equations: : call this equation B
Subtracting equation B from A gives us : call this equation C
Adding C to the 4th equation gives
Substituting w in the original equations gives: , ,
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