Exam 2, Online Number Theory Camp, 2015

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rah4927
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Re: Exam 2, Online Number Theory Camp, 2015

Unread post by rah4927 » Fri Aug 28, 2015 4:08 pm

Zawadx wrote:ঘাত - কি Power বুঝাচ্ছে?
Yes.

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Masum
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Re: Exam 2, Online Number Theory Camp, 2015

Unread post by Masum » Fri Aug 28, 2015 4:09 pm

hmm, ghat means power
One one thing is neutral in the universe, that is $0$.

rah4927
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Re: Exam 2, Online Number Theory Camp, 2015

Unread post by rah4927 » Fri Aug 28, 2015 4:44 pm

Should the Vieta Jumping problem be this?

If $a|b^2+1$ and $b|a^2+1$, then show that $a^2+b^2+1=3ab$. Here, $a,b$ are both positive.

I am not making any assumptions about the problem until Masum vai says so, of course, and neither should anyone else. It wouldn't do to spend your energy solving a problem that wasn't on the question paper to begin with.

EDIT: Apparently, Masum vai confused the actual problem with the above one. He has edited the correct version in his first post.
Last edited by rah4927 on Fri Aug 28, 2015 6:40 pm, edited 1 time in total.

rah4927
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Re: Exam 2, Online Number Theory Camp, 2015

Unread post by rah4927 » Fri Aug 28, 2015 6:28 pm

Does the corrected version of the problem use Vieta Jumping as well? It would be bad if somebody is misled to use VRJ when it might use something else.

rah4927
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Re: Exam 2, Online Number Theory Camp, 2015

Unread post by rah4927 » Fri Aug 28, 2015 6:38 pm

Masum wrote:অবশেষে রাত ৯ টা বেজে গেছে। নুরানী পরীক্ষা এখন শুরু করা যায়। সমস্যার সেট নিচে যুক্ত করা হলো। সমাধান জমা দেওয়ার শেষ সময় আজকের রাত ৯ টা। আগের দিন প্রথম ছিল বলে কিছুক্ষণ পরেও নিছি। আজকে থেকে আর নেওয়া হবে না। এখানে মেসেজ ছাড়া অন্য কোন ভাবে সমাধান দিলে এমন ভাবে জমা দিবে যাতে দেখে বুঝা যায় কোনটা কে জমা দিসো। আগের দিনের সমাধান আমি কোনটা কার কিছু বুঝতেছিনা ঠিক মত।

সমস্যা ৪ অন্যটার সাথে মিলে গেসিল। আসলে হবে এটাঃ
প্রমাণ কর যে, $\dfrac{a^2+b^2-1}{ab}$ ভগ্নাংশটি যে কোন ধনাত্মক বদজীন হতে পারে($>1$)।
Do you mean $a,b>1$? Otherwise the problem becomes trivial.

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Masum
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Re: Exam 2, Online Number Theory Camp, 2015

Unread post by Masum » Fri Aug 28, 2015 7:01 pm

rah4927 wrote:Does the corrected version of the problem use Vieta Jumping as well? It would be bad if somebody is misled to use VRJ when it might use something else.
You can use it or not. It can be solved either way. And all integers here are greater than $1$.
One one thing is neutral in the universe, that is $0$.

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Zawadx
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Re: Exam 2, Online Number Theory Camp, 2015

Unread post by Zawadx » Sat Aug 29, 2015 6:07 pm

Couldn't submit answers due to power failure :/ Can anyone give any tips on how to type out solutions faster?

btw, Vieta Jumping can direct you to the solution for number 4 but I didn't have to use it directly in the solution.

rah4927
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Re: Exam 2, Online Number Theory Camp, 2015

Unread post by rah4927 » Sat Aug 29, 2015 6:22 pm

Well, it took me almost forever to type out the solutions. Luckily, I started early. And as to typing out solutions faster, if you are fairly well at formatting using latex, then it shouldn't be too hard. It's just practice.

As to VRJ, yes you are right, it can only motivate you towards the solution. For showing that there are infinite solutions for each $k$, however, you need to to use root jumping.

Epshita32
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Re: Exam 2, Online Number Theory Camp, 2015

Unread post by Epshita32 » Sat Aug 29, 2015 10:42 pm

In problem 4, only possible solution I can find is a= 3 , and b=4 . But then a^2 + b^2 - 1 / ab = 2 . Any hints ? :?

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Masum
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Re: Exam 2, Online Number Theory Camp, 2015

Unread post by Masum » Sat Aug 29, 2015 11:32 pm

আসলে এই সমস্যাটায় যে কোন ধনাত্মক পূর্ণসংখ্যা $k>1$ এর জন্য এমন $a,b>1$ থাকবে যাতে $\dfrac{a^2+b^2-1}{ab}=k$ হয়। এটা প্রমাণ করার চেষ্টা কর। যদি কিছু খুজে না পাও, ভিয়েটা মারার চেষ্টা কর।
হিন্টঃ এমন কিছু বের করার চেষ্টা কর যাতে উপরে নিচে কাটাকাটি যেয়ে শুধু $k$ থাকে। $a|b^2-1$ এটা দেখে কিছু আন্দাজ করা যায় কিনা দেখো।
এখন আমি ঠিক মত অনুবাদ করতে পারতেছি -_-
One one thing is neutral in the universe, that is $0$.

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