Difference between revisions of "1980 USAMO Problems/Problem 1"
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| + | \boxed{ h=\sqrt{\frac{a-b}{A-B}} ; | ||
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| + | \text{weight} = hC + \frac{bA - aB}{(A - B)(h + 1)}} | ||
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Credit: John Scholes https://prase.cz/kalva/usa/usoln/usol801.html | Credit: John Scholes https://prase.cz/kalva/usa/usoln/usol801.html | ||
Latest revision as of 14:46, 26 March 2023
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
![\[(h+1)k = a - h^2A = \frac{bA - aB}{A - B}\]](http://latex.artofproblemsolving.com/c/9/6/c96aa637cac47ed01a61df6e6bf95dff89639774.png)
.
The true weight of the third object is thus:
![\[hC + k = \\ \boxed{\sqrt{ \frac{a-b}{A-B}} C + \frac{bA - aB}{(A - B)(\sqrt{ \frac {a-b}{A-B}}+ 1)}}\]](http://latex.artofproblemsolving.com/b/6/4/b64bcc1c61b364ac6519ee29e2a1d71a9eb998e9.png)
.
More readably:
![\[\boxed{ h=\sqrt{\frac{a-b}{A-B}} ; \\ \text{weight} = hC + \frac{bA - aB}{(A - B)(h + 1)}}\]](http://latex.artofproblemsolving.com/2/c/5/2c5c482240148b75788baf2e38f541cd87597dd6.png)
Credit: John Scholes https://prase.cz/kalva/usa/usoln/usol801.html
