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2010 USAMO #5:Prove that if $frac{1}{p}-2S_q = frac{m}{n}$ for integers $m$ and $n$, then $m - n$ is divisible by $p$.

Mathematics Asked by Shubhangi on September 7, 2020

Let $q = frac{3p-5}{2}$ where $p$ is an odd prime, and let $S_q = frac{1}{2cdot 3 cdot 4} + frac{1}{5cdot 6 cdot 7} + cdots + frac{1}{q(q+1)(q+2)}
$

Prove that if $frac{1}{p}-2S_q = frac{m}{n}$ for coprime integers $m$ and $n$, then $m – n$ is divisible by $p$.

My Progress till now: $$2S_q = 2sum_{x=1}^{frac{q+1}{3}} frac{1}{(3x-1)(3x)(3x+1)} = sum_{x=1}^{frac{p-1}{2}} left[frac{1}{3x(3x-1)}-frac{1}{3x(3x+1)}right]\
=sum_{x=1}^{frac{p-1}{2}} left[ frac{1}{3x-1} – frac{2}{3x} +frac{1}{3x+1}right]\
=sum_{x=1}^{frac{p-1}{2}}left[ frac{1}{3x-1} + frac{1}{3x} +frac{1}{3x+1}right] – sum_{x=1}^{frac{p-1}{2}} frac{1}{x} $$

With the help of @user10354138 , I have got $frac{1}{p} – 2S_q = frac{1}{p} + frac{1}{1} – sum_{k=frac{p+1}{2}}^{frac{3p-1}{2}}frac{1}{k} = frac{m}{n}$

But then I am stuck.

Please give me some hints rather than a solution.
Thanks in advance.

PS: I didn’t post it in AOPS, because there we don’t get any guidance.

2 Answers

(Original) Hint: You are almost there with the simplification. Note that you are summing over $frac1n$ from $n=2$ to $frac{3p-1}2$ in the first. So $$ 2S_q+1=sum_{n=(p+1)/2}^{(3p-1)/2}frac1n $$ If you tweak the RHS slightly, you would be summing over $frac1n$ as $n$ runs through representative of each of the nonzero residue classes mod $p$. So ...


Addendum (2020-07-29): As discussed in the comments, begin{align*} frac1p-2S_q-1&=-left(sum_{n=(p+1)/2}^{p-1}frac1n+sum_{n=p+1}^{p+(p-1)/2}frac1nright)\ &=-sum_{i=1}^{(p-1)/2}left(frac1{p-i}+frac1{p+i}right) end{align*} and now $$ frac1{p-i}+frac1{p+i}=frac{p}{(p-i)(p+i)} $$ so the numerators are divisible by $p$ and the denominators are not. So putting everything over a common denominator, we see $$ frac{m-n}{n}=-sum_{i=1}^{(p-1)/2}frac{p}{(p-i)(p+i)}=frac{ptimes text{some integer}}{text{some integer not divisible by }p}. $$ That is, every representation of $frac{m-n}{n}$ must have more factors of $p$ in the numerator than in the denominator, hence $m-n$ is divisible by $p$.

Correct answer by user10354138 on September 7, 2020

With the help of @user10354138 's hints, I think I got the solution.I will be grateful if someone proof reads it.

Note that $$2S_q = 2sum_{x=1}^{frac{q+1}{3}} frac{1}{(3x-1)(3x)(3x+1)} = sum_{x=1}^{frac{p-1}{2}} left[frac{1}{3x(3x-1)}-frac{1}{3x(3x+1)}right]\ =sum_{x=1}^{frac{p-1}{2}} left[ frac{1}{3x-1} - frac{2}{3x} +frac{1}{3x+1}right]\ =sum_{x=1}^{frac{p-1}{2}}left[ frac{1}{3x-1} + frac{1}{3x} +frac{1}{3x+1}right] - sum_{x=1}^{frac{p-1}{2}} frac{1}{x}$$ .

Proceeding further we get that,$$frac{1}{p} - 2S_q = frac{1}{p} + frac{1}{1} - sum_{k=frac{p+1}{2}}^{frac{3p-1}{2}}frac{1}{k} = frac{m}{n}$$

or we get that $$- 2S_q = frac{1}{1} - sum_{k=frac{p+1}{2}}^{frac{3p-1}{2}}frac{1}{k} = frac{m}{n}-frac{1}{p}$$

Now, note that $$sum_{k=frac{p+1}{2}}^{frac{3p-1}{2}}frac{1}{k}equiv sum_{k=1}^{p-1}frac1k equiv sum_{k=1}^{p-1}k equiv 0$$ mod $p$

So we get that $$frac{m}{n}equiv 1$$ mod $p$ .

Hence we have $$1-frac{m}{n}equiv 0$$ mod p $$implies m-n equiv 0 $$ mod $p$.

And we are done!

Answered by Shubhangi on September 7, 2020

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