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Difference between revisions of "1980 USAMO Problems/Problem 1"

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<cmath>
 
<cmath>
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hC + k = \\
+
hC + k = \\
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\sqrt{ (a-b)/(A-B)} C
+
\boxed{\sqrt{ (a-b)/(A-B)} C + \frac{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>.
 +
 +
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 13:37, 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 $A$, when placed in the left pan and against a weight $a$, when placed in the right pan. The corresponding weights for the second object are $B$ and $b$. The third object balances against a weight $C$, 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 $x$ in the left pan balances an object of weight $y$ in the right pan, then $x = hy + k$ for some constants $h$ and $k$. Thus if the first object has true weight x, then $x = hA + k, a = hx + k$.

So $a = h^2A + (h+1)k$.

Similarly, $b = h^2B + (h+1)k$. Subtracting gives $h^2 = (a - b)/(A - B)$ and so

\[(h+1)k = a - h^2A = (bA - aB)/(A - B)\].

The true weight of the third object is thus:

\[hC + k = \\ \boxed{\sqrt{ (a-b)/(A-B)} C + \frac{bA - aB}{A - B} \frac{1}{\sqrt{ (a-b)/(A-B) }+ 1}}\].

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

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