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BdMO Online Forum • View topic - BdMO 2017 junior/4

BdMO 2017 junior/4

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BdMO 2017 junior/4

Post Number:#1  Unread postby Thamim Zahin » Sun Feb 12, 2017 11:43 pm

(a) Can two consecutive numbers $n$ and $n-1$ both be divisible by $3$?
(b) Determine the smallest integer $n > 1$ such that $n^2(n-1)$ is divisible by $1971$. Note:
$1971 = 3^3 \times 73$.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.
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Re: BdMO 2017 junior/4

Post Number:#2  Unread postby Thamim Zahin » Sun Feb 12, 2017 11:46 pm

And I don't know how i missed the part (b) in the actual exam. And i not posting this solution. Left for the others.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.
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Re: BdMO 2017 junior/4

Post Number:#3  Unread postby dshasan » Mon Feb 13, 2017 12:12 am

I also messed up part(b) during actual exam. However, I was able to find an answer using brute force then.
I got $n = 730$.

Am I correct?..
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Re: BdMO 2017 junior/4

Post Number:#4  Unread postby ahmedittihad » Mon Feb 13, 2017 11:46 pm

Apply crt to get $n=585$.
Frankly, my dear, I don't give a damn.
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Re: BdMO 2017 junior/4

Post Number:#5  Unread postby Thamim Zahin » Fri Feb 17, 2017 1:39 am

By judging the 6,7,8 ..... This one is kinda hard.... Rather than 8,9.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.
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Re: BdMO 2017 junior/4

Post Number:#6  Unread postby jayon_2 » Wed Apr 12, 2017 2:23 pm

ahmedittihad wrote:Apply crt to get $n=585$.

what is crt?
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Re: BdMO 2017 junior/4

Post Number:#7  Unread postby Thamim Zahin » Wed Apr 12, 2017 2:31 pm

jayon_2 wrote:
ahmedittihad wrote:Apply crt to get $n=585$.

what is crt?

Chinese remainder theorem :mrgreen:
http://s3.amazonaws.com/aops-cdn.artofp ... theory.pdf

Take a look at Page no.40 for further information.
Last edited by Thamim Zahin on Wed Apr 12, 2017 9:11 pm, edited 2 times in total.
I think we judge talent wrong. What do we see as talent? I think I have made the same mistake myself. We judge talent by the trophies on their showcases, the flamboyance the supremacy. We don't see things like determination, courage, discipline, temperament.
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Re: BdMO 2017 junior/4

Post Number:#8  Unread postby ahmedittihad » Wed Apr 12, 2017 7:14 pm

So basically you do different cases on the primes dividing $n^2$ or $n-1$. And these cases yield different divisibility conditions. And CRT is the method to find the smallest number that suppression every congruence in a system. Look it up. The notation might be scary but it's quite natural and intuitive.
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