Australian Olympiad Problem 5

Algebra Level 4

There are two positive integers a a and b b such that:

1 a + 1 b = 1 20 \frac{1}{a} + \frac{1}{b} = \frac{1}{20}

Now, what is the highest value of a + b a + b ?


The answer is 441.

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Shrenik Jobanputra by Shrenik Jobanputra , 22, Australia

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2 solutions

4 Helpful 0 Interesting 0 Brilliant 0 Confused

@Shrenik Jobanputra Your solution doesn't explain why it's the highest value...

Satvik Golechha - 6 years, 9 months ago

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The thing is that I just did the competition just 2 days ago. I know the answer but I still don't know the procedure of working it out. If possible @Satvik Golechha , can U PLEASE also write a solution.

Shrenik Jobanputra - 6 years, 9 months ago

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@Shrenik Jobanputra my solution was listing out all the possible pairs and crossing out. Not bash-free though.

Satvik Golechha - 6 years, 9 months ago

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@Satvik Golechha – That is actually what I would also do.

Shrenik Jobanputra - 6 years, 9 months ago

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@Shrenik Jobanputra – Well I failed that pretty badly. Tried to find denominators from 20 to 40 that worked out. I found as many as I could and picked the pair with the highest sum. Unfortunately that didn't really work. Looking back on it, I think I over complicated it a bit, although I'm not sure what a simpler way would be.

Ryan Schrier - 6 years, 3 months ago
Rohit Sachdeva
Apr 18, 2015

Lets start with 20x2/1=40

1/40 + 1/40 = 1/20

Now 20x3/2=30

1/30 + 1/60 = 1/20

20x4/3 is not an integer

20x5/4=25

1/25 + 1/100 = 1/20

& so on till 20x21/20=21

So 1/21 + 1/420 = 1/20

The trick here is that closer we are to 1/20, smaller the value(& hence greater the denominator) we will have to add to reach 1/20

21 is the closest to 20 & hence the result.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

There is an easier way to reach that final stage. Use the identity: 1/x = 1/(x+1) + 1/(x)(x+1). This can be proven by either using the Principle of Mathematical Induction or expanding the RHS. This means that 1/20 = 1/21 + 1/(20)(21), i.e. 1/20 = 1/21 + 1/420. Like you said in your solution, the closer you are to 20, the smaller the value and hence the greater the denominator which we will need to add to obtain 1/20/ So a+b = 21+ 420 = 441

A Former Brilliant Member - 6 years, 1 month ago

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