Don't make mistakes! #3
There are three positive integers a , b and c that satisfy:
- a b c + a b + b c + c a = 1 1 6 ,
- a + b + c = 1 8 .
Find the value of a 2 + b 2 + c 2 .
This problem is a part of <Don't make mistakes!> series .
The answer is 204.
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1 solution
The condition a = b = c is not really necessary to solve the problem. In fact, I have a feeling it might confuse some people because they might misinterpret it as all the numbers being distinct. What do you think?
(Great problem and solution, btw!)
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That was the very essence of this problem! It is to make people realize that a = b = c does not mean a = c . That is why I put the condition! :D
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But my issue is that people might misinterpret the condition and believe these is no solution when in fact there is one. Plus, there is only one possible solution to the problem anyway (excluding permutations), so it's not needed to solve the problem.
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@Steven Yuan – THat's a good point. We can indeed remove that condition. We have ( a + 1 ) ( b + 1 ) ( c + 1 ) = 1 1 6 and a + b + c = 1 8 , which forces { a , b , c } = { 2 , 2 , 1 4 } .
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@Calvin Lin – Hmm how about making the problem be
"Find the value of 3 a 2 + b 2 − c 2 "
Then the condition a = b = c is needed.
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@Boi (보이) – What we're trying to do is simplify the question, so that it is easier to focus in on what makes the problem interesting, instead of having to take care of the nitty gritty details.
Your suggestion would complicate the question further, which makes it slightly distracting.
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@Calvin Lin – Ah okay that makes sense. I deleted a = b = c !
According to the problem,
a b c + a b + b c + c a + a + b + c + 1 = 1 1 6 + 1 8 + 1 = 1 3 5 .
( a + 1 ) ( b + 1 ) ( c + 1 ) = 1 3 5 .
Since a , b and c are all positive integers, we see that neither a + 1 , b + 1 or c + 1 is equal to 1 .
There are two ways of splitting 1 3 5 into 3 parts. 3 × 5 × 9 and 3 × 3 × 1 5 .
Case 1.
If we're splitting 1 3 5 into 3 × 5 × 9 , a + b + c = 2 + 4 + 8 = 1 4 , which does not satisfy the condition a + b + c = 1 8 .
Case 2.
If we're splitting 1 3 5 into 3 × 3 × 1 5 , we can see that a + b + c = 1 8 . This satisfies the conditions.
From Case 1 and Case 2 , we see that a 2 + b 2 + c 2 = 2 2 + 1 4 2 + 2 2 = 2 0 4 .