Problem:
The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of
Answer Choices:
A.
B.
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E.
Solution:
First, by adding the numbers on the top row, we know the sum of the rows and columns is . Since the sum of the numbers in the first column is and we know that one of the numbers is , we know the sum of the other two is . This means that the number above is . Since the sum of the numbers in the bottom row is and we know that one of the numbers is , we know the sum of the other two is . This means that the number to the right of is . Since the sum of the numbers in the middle row is and we know that two of them are and , we know the remaining number is . We know that and as is the greatest number. This leads us to know . Since is always true, our only restriction is . This means is our minimum solution.
Thus, the answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions