Difference between revisions of "1980 USAMO Problems/Problem 1"
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+ (bA - aB)/(A - B) \frac{1}{\sqrt{ (a-b)/(A-B) }+ 1} | + (bA - aB)/(A - B) \frac{1}{\sqrt{ (a-b)/(A-B) }+ 1} | ||
</cmath>. | </cmath>. | ||
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| + | Credit: John Scholes https://prase.cz/kalva/usa/usoln/usol801.html | ||
== See Also == | == See Also == | ||
Revision as of 13:27, 26 March 2023
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
The effect of the unequal arms and pans is that if an object of weight 
in the left pan balances an object of weight 
in the right pan, then 
for some constants 
and 
. Thus if the first object has true weight x, then 
.
So 
.
Similarly, 
. Subtracting gives 
and so
![\[(h+1)k = a - h^2A = (bA - aB)/(A - B)\]](http://latex.artofproblemsolving.com/d/c/a/dca5326b3c6427574d818e6971d38e2b2f18ada0.png)
.
The true weight of the third object is thus:
![\[hC + k = \\ \sqrt{ (a-b)/(A-B)} C + (bA - aB)/(A - B) \frac{1}{\sqrt{ (a-b)/(A-B) }+ 1}\]](http://latex.artofproblemsolving.com/1/d/9/1d93bd3e9650fe00ffaa6dba8c10c7993b9016ec.png)
.
Credit: John Scholes https://prase.cz/kalva/usa/usoln/usol801.html
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. 
