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While I was going through past Olympiad math papers, I found this question without any explanation. Here is the question:

Given a function $f(x)=\frac{9^{x}}{9^x+3}$, what is $f(\frac{1}{27})+ f(\frac{2}{27}) + f(\frac{3}{27})+ ...+ f(\frac{26}{27})$?

The answer was 13.

I took a really bad approach and converted $\frac{9^{x}}{9^x+3}$ to $1+\frac{9^{x}}{3}$, which I then noticed was wrong.

I also accidentally multiplied $9^{\frac{1}{27}}$ with $9^{\frac{2}{27}}$, $9^{\frac{2}{27}}$ with $9^{\frac{3}{27}}$, and so on, before realizing that the functions were added and not multiplied.

I suspect that there is something to the power of $\frac{n}{27}$, because 9 is a multiple of 27. However, I am not completely sure.

Is there a law that tells me how I can solve this question? Since this is a Math Olympiad question, there is probably a maximum time limit of five minutes to do this question. This means that I probably won’t have time for tedious mathematical calculations with a calculator and online tools, or something like that. Please give me a quick, fast solution that is probably suitable for an 8th grader, at most a solution at a 10th grader level.

Tyrcnex
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    More similar questions: https://math.stackexchange.com/q/1359094/42969, https://math.stackexchange.com/q/4033568/42969, https://math.stackexchange.com/q/3447113/42969, https://math.stackexchange.com/q/1681980/42969 – all found with Approach0. – Martin R Jul 16 '21 at 11:09
  • @Martin R Ah! Thank you for the links. I now understand the question. But the most fascinating thing is, this was for a 7th grade Math Olympiad competition. Are seventh graders really supposed to be able to understand these magical tricks? From what I’ve seen, they don’t teach that kind of stuff in Math Olympiad practice workbooks. – Tyrcnex Jul 16 '21 at 11:12

1 Answers1

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See I'm a $10^{th} $ grader & this is my APPROACH :

Here we can see , $$f(x)+f(1-x)=1$$

So , I can say , $$\underbrace{\left(f(\frac{1}{27})+f(\frac{26}{27})\right)+.......+\left(f(\frac{13}{27})+f(\frac{14}{27})\right)}_{n=26}$$ OR [ Here $n$ represents number of Terms .] $$\underbrace{1+1+1+.....+1}_{n=13}$$

Hence the answer is $13$ .

Tryhard
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    Note that identical or equivalent questions have been asked and answered before, five possible duplicates targets have already been pointed out. Does this add something new to the existing answers? – Martin R Jul 16 '21 at 11:17
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    Sorry , I didn't knew that . – Tryhard Jul 16 '21 at 11:18
  • Thanks for the reply! I’m sorry that I didn’t catch onto the duplicate targets, I’m pretty sleepy and I have a test tomorrow, so I didn’t have time to type in and check my sources. – Tyrcnex Jul 16 '21 at 11:22