A balance problem

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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?

Answered question
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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 313947d12720286d9aab4cd9dd82693fa9e8d77f. Thus, the information we have tells us that, for some constants eafe5cdf19385919172858416a355b85f1c194b7:

1af6a9374021ffb043d18b0abb8d771b5d1311e415340a88bc18303142f0b1334c83e8f26bd71435648687d88e8a986a35afc243b31e23b3c9930ef7

In fact, we don’t exactly care what f32956a5cd4b4bd1ec7a414017b7643d172acd02 are. By subtracting 26eeb5258ca5099acf8fe96b2a1049c48c89a5e6 from all equations and dividing by 092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c, we get:

e58c096b55eb98091bfb59be454222144a07f502ea55121cf120f2d49450369f185cabe63e2dc3e6a0f8407f526aa4fdf9444931b8af5b0acc19ec13

We can just give the names 6a47ca0fe7cb276abc022af6ac88ddae1a9d6894 and ce58e4af225c93d08606c26554caaa5ae32edeba to the quantities 793a70489767a2d287609d864de62215c67cb591 and db1d7f53807976de7112a10f599c533e2a29632e.

e730933b73293d555310bf98793c8a3ee4e989422ce988e326ebcaae2e621d37689a905f54c83aa25019b1fed38e6dc38657a32152a78020f43fcf9f

Our task is to compute 3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc in terms of 019e9892786e493964e145e7c5cf7b700314e53bc7d457e388298246adb06c587bccd419ea67f7e8ff5fb3d775862e2123b007eb4373ff6cc1a34d4e8136a7ef6a03334a7246df9097e5bcc31ba33fd2, and c3355896da590fc491a10150a50416687626d7cc. This can be done by solving for 6a47ca0fe7cb276abc022af6ac88ddae1a9d6894 and ce58e4af225c93d08606c26554caaa5ae32edeba in terms of 019e9892786e493964e145e7c5cf7b700314e53b,c7d457e388298246adb06c587bccd419ea67f7e8,ff5fb3d775862e2123b007eb4373ff6cc1a34d4e,8136a7ef6a03334a7246df9097e5bcc31ba33fd2 and eliminating them from the implicit expression for 3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc in the last equation. Perhaps there is a shortcut, but this will work:

f5f37f1ea2a7ae57b451147c500de3061b9492a57b5171a4c417e1bb724287cf38900ba8de51384a6d5861126c4f6d9bb04991a9914c8005c68a814e

So the answer is:8084ecfa518d7b7a3a3abcab389b205fc2fe3ba5

Answered question