2009 AMC 10A Problems/Problem 16
Problem
Let , , , and be real numbers with , , and . What is the sum of all possible values of ?
Solution 1
From we get that
Similarly, and .
Substitution gives . This gives . There are possibilities for the value of :
,
,
,
,
,
,
,
Therefore, the only possible values of are , , , and . Their sum is .
Solution 2
If we add the same constant to all of , , , and , we will not change any of the differences. Hence we can assume that .
From we get that , hence .
If we multiply all four numbers by , we will not change any of the differences. (This is due to the fact that we are calculating at the end ~Williamgolly) WLOG we can assume that .
From we get that .
From we get that .
Hence , and the sum of possible values is .
Solution 3
Let Note that we have from which it follows that
Note that must be positive however, the only arrangements of and signs on the RHS which make positive are (There are no cases with or more negative as )
Thus, the answer is
Solution 4
Let .
Then
Thus
Therefore
So and the sum is .
~JH. L
Video Solution
~savannahsolver
See Also
2009 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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