1980 USAMO Problems/Problem 1
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?
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: .
|1980 USAMO (Problems • Resources)|
|1 • 2 • 3 • 4 • 5|
|All USAMO Problems and Solutions|