2001 AMC 10 Problems/Problem 10
Contents
[hide]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.
~idk12345678
See Also
2001 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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All AMC 10 Problems and Solutions |
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