2001 AMC 10 Problems/Problem 10
Problem
If , , and are positive with , , and , then is
Solution 1
The first two equations in the problem are and . Since , we have . We can substitute into the third equation to obtain and . We replace into the first equation to obtain .
Since we know every variable's value, we can substitute them in to find .
Solution 2
These equations are symmetric, and furthermore, they use multiplication. This makes us think to multiply them all. This gives . We divide by each of the given equations, which yields , , and . The desired sum is , so the answer is .
Solution 3(strategic guess and check)
Seeing the equations, we notice that they are all multiples of 12. Trying in factors of 12, we find that , , and work.
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See Also
2001 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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