Difference between revisions of "1980 USAMO Problems/Problem 1"
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− | + | hC + k = \\ | |
− | + | \boxed{\sqrt{ (a-b)/(A-B)} C + \frac{bA - aB}{A - B} \frac{1}{\sqrt{ (a-b)/(A-B) }+ 1}} | |
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</cmath>. | </cmath>. | ||
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+ | More readably: <math>\boxed{ h=\sqrt{\frac{a-b}{b-a}} ; \text{weight} = hC + \frac{bA - aB}{A - B} \frac{1}{h + 1}</math> | ||
Credit: John Scholes https://prase.cz/kalva/usa/usoln/usol801.html | Credit: John Scholes https://prase.cz/kalva/usa/usoln/usol801.html |
Revision as of 14:37, 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
.
The true weight of the third object is thus:
.
More readably: $\boxed{ h=\sqrt{\frac{a-b}{b-a}} ; \text{weight} = hC + \frac{bA - aB}{A - B} \frac{1}{h + 1}$ (Error compiling LaTeX. Unknown error_msg)
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.