1980 USAMO Problems/Problem 1

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

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A balance scale will balance when the torques exerted on both sides cancel out. On each of the two sides, the total torque will be [some constant amount] (due to the weight, and distribution of the weight, of the arm itself) plus [the length of the arm] times [the weight of what is sitting in the pan]. Thus, the information we have tells us that, for some constants x, y, z, u:

 x + yA = z + ua
 x + yB = z + ub
 x + yC = z + uc

In fact, we don't exactly care what x,y,z,u are. By subtracting x from all equations and dividing by y, we get:

 A = (z-x)/y + (u/y)a
 B = (z-x)/y + (u/y)b
 C = (z-x)/y + (u/y)c

We can just give the names X and Y to the quantities (z-x)/y and (u/y).

 A = X + Ya
 B = X + Yb
 C = X + Yc

Our task is to compute c in terms of A, a, B, b, and C. This can be done by solving for X and Y in terms of A,a,B,b and eliminating them from the implicit expression for c in the last equation. Perhaps there is a shortcut, but this will work:

 A = X + Ya
 => X = A - Ya
 B = X + Yb
 => B = A - Ya + Yb
 => Y(b-a) = B-A
 => Y = (B-A)/(b-a)
 => X = A - (B-A)/(b-a) * a
 
 C = X + Yc
 => Yc = C - X
 => c = (C-X)/Y
 => c = (C - [A - (B-A)/(b-a) * a]) / [(B-A)/(b-a)]
 => [simplify numerator]
 c = (C - A + a(B-A)/(b-a)) / [(B-A)/(b-a)]
 => [multiply numerator and denominator by (b-a)]
 c = (C(b-a) - A(b-a) + a(B-A)) / (B-A)
 => [distribute numerator]
 c = (Cb - Ca - Ab + Aa + Ba - Aa) / (B-A)
 => [cancel Aa's]
 c = (Cb - Ca - Ab + Ba) / (B-A)

So the answer is: (Cb - Ca - Ab + Ba) / (B-A).

[Someone else feel free to clean up the formatting here.]

See Also

1980 USAMO (Problems • Resources)
Preceded by First Question	Followed by Problem 2
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions