Difference between revisions of "2001 AMC 10 Problems/Problem 10"
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Revision as of 11:10, 4 July 2013
Contents
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
.
See Also
2001 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.