Discussion on Exam 1

Discussion on Bangladesh National Math Camp
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Masum
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Discussion on Exam 1

Unread post by Masum » Tue Aug 25, 2015 11:07 pm

Ok, the first exam is over. You can discuss on the problems now. Solutions have been posted here. Until then, post your solutions and opinions on the problems or other solutions. And there will be no exam tomorrow. But there will one the next day after that. So be prepared for that.
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Masum
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Re: Discussion on Exam 1

Unread post by Masum » Tue Aug 25, 2015 11:16 pm

Let me give you a hint on how to solve problem 1.2. Though I forgot to mention, but it should be kind of obvious that $a_0>1$.
Hint: If $s(a)$ is the smallest prime divisor of $a$, then $a+s(a)$ is always even.
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rubabredwan
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Re: Discussion on Exam 1

Unread post by rubabredwan » Wed Aug 26, 2015 12:48 am

..........
Last edited by rubabredwan on Wed Aug 26, 2015 5:29 pm, edited 1 time in total.

badass0
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Re: Discussion on Exam 1

Unread post by badass0 » Wed Aug 26, 2015 12:52 am

Can anybody give me hints on how to solve problem 1.4?
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Masum
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Re: Discussion on Exam 1

Unread post by Masum » Wed Aug 26, 2015 1:10 am

rubabredwan wrote:Problem 1.2
$a_{0}$ can't be equal to 1.
let,
$a_{0}=2$ then,
$a_{1} = 4$
$a_{2} = 6/8$
$a_{3} = 8/9/10/12$
$a_{n} = ......$

if $a_{0}=3$ then,
$a_{1} = 6$
$a_{2} = 8/9$
$a_{3} = 10/12$
$a_{n} = ......$

we can see that integers 5, 7, 11 etc. aren't coming in this recurrence. because $a_{n}$ is a multiple of $d(a_{n-1})$ which states that no prime other than $a_{0}$ will come.this recurrence follows a pattern like seive of erastothenes(not fully). if $a_{0}$ is not a prime, then every $a_{n}$ will be composite.

the number of prime numbers are infinite. So there is no such value of $k$.
$a_0$ is a fixed positive integer which is given and greater than $1$. Try it again. And think about the hint too.
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Masum
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Re: Discussion on Exam 1

Unread post by Masum » Wed Aug 26, 2015 1:11 am

badass0 wrote:Can anybody give me hints on how to solve problem 1.4?
$5$ divides $6^n-1$.
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shanto00
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Re: Discussion on Exam 1

Unread post by shanto00 » Wed Aug 26, 2015 5:45 pm

in problem 1.3 we just have to show that they are a primitive Pythagorean triple .

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Re: Discussion on Exam 1

Unread post by Masum » Wed Aug 26, 2015 7:33 pm

No, it is given that they are a Pythagorean Triple. You have to prove $2(b+p)$ is a perfect square.
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shanto00
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Re: Discussion on Exam 1

Unread post by shanto00 » Thu Aug 27, 2015 9:21 pm

Masum wrote:No, it is given that they are a Pythagorean Triple. You have to prove $2(b+p)$ is a perfect square.
i meant to say that if we can prove that they are primitive , then the work is almost done.

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