Difference between revisions of "1980 USAMO Problems/Problem 1"
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== Solution == | == Solution == | ||
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− | + | The effect of the unequal arms and pans is that if an object of weight <math> x</math> in the left pan balances an object of weight <math>y </math> in the right pan, then <math> x = hy + k</math> for some constants <math>h</math> and <math>k</math>. Thus if the first object has true weight x, then <math> x = hA + k, a = hx + k</math>. | |
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− | + | So <math>a = h^2A + (h+1)k</math>. | |
− | + | Similarly, <math>b = h^2B + (h+1)k</math>. Subtracting gives <math>h^2 = \frac{a-b}{A-B} </math> and so | |
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− | + | <cmath>(h+1)k = a - h^2A = \frac{bA - aB}{A - B}</cmath>. | |
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− | + | The true weight of the third object is thus: | |
− | + | <cmath> | |
+ | hC + k = \\ | ||
+ | \boxed{\sqrt{ \frac{a-b}{A-B}} C + \frac{bA - aB}{(A - B)(\sqrt{ \frac {a-b}{A-B}}+ 1)}} | ||
+ | </cmath>. | ||
− | = | + | More readably: |
− | {{ | + | <cmath> |
+ | \boxed{ h=\sqrt{\frac{a-b}{A-B}} ; | ||
+ | \\ | ||
+ | \text{weight} = hC + \frac{bA - aB}{(A - B)(h + 1)}} | ||
+ | </cmath> | ||
− | + | Credit: John Scholes https://prase.cz/kalva/usa/usoln/usol801.html |
Latest revision as of 13:46, 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
.
The true weight of the third object is thus:
.
More readably:
Credit: John Scholes https://prase.cz/kalva/usa/usoln/usol801.html