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Tags: AMC, USA(J)MO, USAMO, algebra unsolved, algebra

Mrdavid445

26.07.2011 22:17

A two-pan balance is innacurate since its balance arms are of different lengths and its pans are of different weights. Three objects of different weights $A$ , $B$ , and $C$ are each weighed separately. When placed on the left-hand pan, they are balanced by weights $A_1$ , $B_1$ , and $C_1$ , respectively. When $A$ and $B$ are placed on the right-hand pan, they are balanced by $A_2$ and $B_2$ , respectively. Determine the true weight of $C$ in terms of $A_1, B_1, C_1, A_2$ , and $B_2$ .

infiniteturtle

03.12.2013 03:30

Hmm no solution was ever given. Am I right in thinking this is really just basic algebra? Also is this really a combinatorics problem? Moderator says: indeed, not combinatorial at all. Moved.

P_Groudon

15.11.2020 23:45

Let

$L_1$ be the length of the left arm of the balance scale and

$L_2$ be the length of the right arm. Let

$p_1$ be the mass of the pan on the left and

$p_2$ be the mass of the pan on the right. Finally, let

$m_1$ be the combined mass of the weight(s) we put on the left side and

$m_2$ be the combined mass of the weight(s) we put on the right side. To balance the scale, we must have

$(p_1 + m_1) \cdot L_1 = (p_2 + m_2) \cdot L_2$ , since

$p_k + m_k$ represents all the mass on one side of the scale. Distributing and simplifying, we have

$p_1 \cdot L_1 - p_2 \cdot L_2 = L_2 \cdot m_2 - L_1 \cdot m_1$ . The LHS is constant, since we use the same balance every time, so we just let it equal

$k$ . So

$k = L_2 \cdot m_2 - L_1 \cdot m_1$ for any choice of

$(m_1, m_2)$ that balances the scale. For brevity, let

$L_1 = u$ and

$L_2 = v$ . We have

$k = vm_2 - um_1$ . Since

$k, u,$ and

$v$ are constants, we divide through by

$u$ to obtain a new equation with constants

$k'$ and

$v'$ :

$k' = v'm_2 - m_1$ . Now, we plug in

$(m_1, m_2) \in \{(A, A_1), (B, B_1), (C, C_1), (A_2, A), (B_2, B)\}$ and proceed to bash. Should come out to

$C = \frac{BC_1 - AC_1}{B_1 - A_1} + \frac{BA_2 - AB_2}{B - A}$