Difference between revisions of "1980 USAMO Problems/Problem 1"
Line 36: | Line 36: | ||
== See Also == | == See Also == | ||
{{USAMO box|year=1980|before=First Question|num-a=2}} | {{USAMO box|year=1980|before=First Question|num-a=2}} | ||
+ | {{MAA Notice}} | ||
[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] |
Revision as of 19:09, 3 July 2013
Problem
A balance has unequal arms and pans of unequal weight. It is used to weigh three objects. The first object balances against a weight , when placed in the left pan and against a weight
, when placed in the right pan. The corresponding weights for the second object are
and
. The third object balances against a weight
, when placed in the left pan. What is its true weight?
Solution
A balance scale will balance when the torques exerted on both sides cancel out. On each of the two sides, the total torque will be . Thus, the information we have tells us that, for some constants
:
In fact, we don't exactly care what are. By subtracting
from all equations and dividing by
, we get:
We can just give the names and
to the quantities
and
.
Our task is to compute in terms of
,
,
,
, and
. This can be done by solving for
and
in terms of
,
,
,
and eliminating them from the implicit expression for
in the last equation. Perhaps there is a shortcut, but this will work:
So the answer is: .
See Also
1980 USAMO (Problems • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.