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BdMO Online Forum • View topic - Dhaka Regional '16 P8

Dhaka Regional '16 P8

For students of class 9-10 (age 14-16)
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Dhaka Regional '16 P8

Post Number:#1  Unread postby Kazi_Zareer » Thu Jan 19, 2017 2:55 am

How many eight digit number can be formed by using the digits $1, 2, 3, 4, 5, 6, 7, 8$ so that each number has $6$ digits in such place where that digit is less than the next digit?
Example: In number $2314$; $2, 1$ are two digits such that each of them is less than
the next digit.

Translated into Bengali:
$1, 2, 3, 4, 5, 6, 7, 8$ অংকগুলো ব্যবহার করে $8$ অংকের এমন কতগুলো সংখ্যা বানানো যায় যাতে করে প্রতিটি সংখ্যায় $6$ এমন টি অংক থাকবে যারা প্রত্যেকে তাদের পরের অংকটি থেকে ছোট?

যেমনঃ $2314$ সংখ্যাটিতে $2, 1$ এমন এমন দুটি অংক যারা প্রত্যেকে তাদের পরের অংক থেকে ছোট।
We cannot solve our problems with the same thinking we used when we create them.
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Re: Dhaka Regional '16 P8

Post Number:#2  Unread postby Thanic Nur Samin » Thu Jan 19, 2017 1:15 pm

$1 2 3 4 5 6 8 7$
$1 2 3 4 5 7 6 8$
$1 2 3 4 5 7 8 6$
$1 2 3 4 5 8 6 7$
$1 2 3 4 6 5 7 8$
$1 2 3 4 6 7 5 8$
$1 2 3 4 6 7 8 5$
$1 2 3 4 6 8 5 7$
$1 2 3 4 7 5 6 8$
$1 2 3 4 7 8 5 6$
$1 2 3 4 8 5 6 7$
$1 2 3 5 4 6 7 8$
$1 2 3 5 6 4 7 8$
$1 2 3 5 6 7 4 8$
$1 2 3 5 6 7 8 4$
$1 2 3 5 6 8 4 7$
$1 2 3 5 7 4 6 8$
$1 2 3 5 7 8 4 6$
$1 2 3 5 8 4 6 7$
$1 2 3 6 4 5 7 8$
$1 2 3 6 7 4 5 8$
$1 2 3 6 7 8 4 5$
$1 2 3 6 8 4 5 7$
$1 2 3 7 4 5 6 8$
$1 2 3 7 8 4 5 6$
$1 2 3 8 4 5 6 7$
$1 2 4 3 5 6 7 8$
$1 2 4 5 3 6 7 8$
$1 2 4 5 6 3 7 8$
$1 2 4 5 6 7 3 8$
$1 2 4 5 6 7 8 3$
$1 2 4 5 6 8 3 7$
$1 2 4 5 7 3 6 8$
$1 2 4 5 7 8 3 6$
$1 2 4 5 8 3 6 7$
$1 2 4 6 3 5 7 8$
$1 2 4 6 7 3 5 8$
$1 2 4 6 7 8 3 5$
$1 2 4 6 8 3 5 7$
$1 2 4 7 3 5 6 8$
$1 2 4 7 8 3 5 6$
$1 2 4 8 3 5 6 7$
$1 2 5 3 4 6 7 8$
$1 2 5 6 3 4 7 8$
$1 2 5 6 7 3 4 8$
$1 2 5 6 7 8 3 4$
$1 2 5 6 8 3 4 7$
$1 2 5 7 3 4 6 8$
$1 2 5 7 8 3 4 6$
$1 2 5 8 3 4 6 7$
$1 2 6 3 4 5 7 8$
$1 2 6 7 3 4 5 8$
$1 2 6 7 8 3 4 5$
$1 2 6 8 3 4 5 7$
$1 2 7 3 4 5 6 8$
$1 2 7 8 3 4 5 6$
$1 2 8 3 4 5 6 7$
$1 3 2 4 5 6 7 8$
$1 3 4 2 5 6 7 8$
$1 3 4 5 2 6 7 8$
$1 3 4 5 6 2 7 8$
$1 3 4 5 6 7 2 8$
$1 3 4 5 6 7 8 2$
$1 3 4 5 6 8 2 7$
$1 3 4 5 7 2 6 8$
$1 3 4 5 7 8 2 6$
$1 3 4 5 8 2 6 7$
$1 3 4 6 2 5 7 8$
$1 3 4 6 7 2 5 8$
$1 3 4 6 7 8 2 5$
$1 3 4 6 8 2 5 7$
$1 3 4 7 2 5 6 8$
$1 3 4 7 8 2 5 6$
$1 3 4 8 2 5 6 7$
$1 3 5 2 4 6 7 8$
$1 3 5 6 2 4 7 8$
$1 3 5 6 7 2 4 8$
$1 3 5 6 7 8 2 4$
$1 3 5 6 8 2 4 7$
$1 3 5 7 2 4 6 8$
$1 3 5 7 8 2 4 6$
$1 3 5 8 2 4 6 7$
$1 3 6 2 4 5 7 8$
$1 3 6 7 2 4 5 8$
$1 3 6 7 8 2 4 5$
$1 3 6 8 2 4 5 7$
$1 3 7 2 4 5 6 8$
$1 3 7 8 2 4 5 6$
$1 3 8 2 4 5 6 7$
$1 4 2 3 5 6 7 8$
$1 4 5 2 3 6 7 8$
$1 4 5 6 2 3 7 8$
$1 4 5 6 7 2 3 8$
$1 4 5 6 7 8 2 3$
$1 4 5 6 8 2 3 7$
$1 4 5 7 2 3 6 8$
$1 4 5 7 8 2 3 6$
$1 4 5 8 2 3 6 7$
$1 4 6 2 3 5 7 8$
$1 4 6 7 2 3 5 8$
$1 4 6 7 8 2 3 5$
$1 4 6 8 2 3 5 7$
$1 4 7 2 3 5 6 8$
$1 4 7 8 2 3 5 6$
$1 4 8 2 3 5 6 7$
$1 5 2 3 4 6 7 8$
$1 5 6 2 3 4 7 8$
$1 5 6 7 2 3 4 8$
$1 5 6 7 8 2 3 4$
$1 5 6 8 2 3 4 7$
$1 5 7 2 3 4 6 8$
$1 5 7 8 2 3 4 6$
$1 5 8 2 3 4 6 7$
$1 6 2 3 4 5 7 8$
$1 6 7 2 3 4 5 8$
$1 6 7 8 2 3 4 5$
$1 6 8 2 3 4 5 7$
$1 7 2 3 4 5 6 8$
$1 7 8 2 3 4 5 6$
$1 8 2 3 4 5 6 7$
$2 1 3 4 5 6 7 8$
$2 3 1 4 5 6 7 8$
$2 3 4 1 5 6 7 8$
$2 3 4 5 1 6 7 8$
$2 3 4 5 6 1 7 8$
$2 3 4 5 6 7 1 8$
$2 3 4 5 6 7 8 1$
$2 3 4 5 6 8 1 7$
$2 3 4 5 7 1 6 8$
$2 3 4 5 7 8 1 6$
$2 3 4 5 8 1 6 7$
$2 3 4 6 1 5 7 8$
$2 3 4 6 7 1 5 8$
$2 3 4 6 7 8 1 5$
$2 3 4 6 8 1 5 7$
$2 3 4 7 1 5 6 8$
$2 3 4 7 8 1 5 6$
$2 3 4 8 1 5 6 7$
$2 3 5 1 4 6 7 8$
$2 3 5 6 1 4 7 8$
$2 3 5 6 7 1 4 8$
$2 3 5 6 7 8 1 4$
$2 3 5 6 8 1 4 7$
$2 3 5 7 1 4 6 8$
$2 3 5 7 8 1 4 6$
$2 3 5 8 1 4 6 7$
$2 3 6 1 4 5 7 8$
$2 3 6 7 1 4 5 8$
$2 3 6 7 8 1 4 5$
$2 3 6 8 1 4 5 7$
$2 3 7 1 4 5 6 8$
$2 3 7 8 1 4 5 6$
$2 3 8 1 4 5 6 7$
$2 4 1 3 5 6 7 8$
$2 4 5 1 3 6 7 8$
$2 4 5 6 1 3 7 8$
$2 4 5 6 7 1 3 8$
$2 4 5 6 7 8 1 3$
$2 4 5 6 8 1 3 7$
$2 4 5 7 1 3 6 8$
$2 4 5 7 8 1 3 6$
$2 4 5 8 1 3 6 7$
$2 4 6 1 3 5 7 8$
$2 4 6 7 1 3 5 8$
$2 4 6 7 8 1 3 5$
$2 4 6 8 1 3 5 7$
$2 4 7 1 3 5 6 8$
$2 4 7 8 1 3 5 6$
$2 4 8 1 3 5 6 7$
$2 5 1 3 4 6 7 8$
$2 5 6 1 3 4 7 8$
$2 5 6 7 1 3 4 8$
$2 5 6 7 8 1 3 4$
$2 5 6 8 1 3 4 7$
$2 5 7 1 3 4 6 8$
$2 5 7 8 1 3 4 6$
$2 5 8 1 3 4 6 7$
$2 6 1 3 4 5 7 8$
$2 6 7 1 3 4 5 8$
$2 6 7 8 1 3 4 5$
$2 6 8 1 3 4 5 7$
$2 7 1 3 4 5 6 8$
$2 7 8 1 3 4 5 6$
$2 8 1 3 4 5 6 7$
$3 1 2 4 5 6 7 8$
$3 4 1 2 5 6 7 8$
$3 4 5 1 2 6 7 8$
$3 4 5 6 1 2 7 8$
$3 4 5 6 7 1 2 8$
$3 4 5 6 7 8 1 2$
$3 4 5 6 8 1 2 7$
$3 4 5 7 1 2 6 8$
$3 4 5 7 8 1 2 6$
$3 4 5 8 1 2 6 7$
$3 4 6 1 2 5 7 8$
$3 4 6 7 1 2 5 8$
$3 4 6 7 8 1 2 5$
$3 4 6 8 1 2 5 7$
$3 4 7 1 2 5 6 8$
$3 4 7 8 1 2 5 6$
$3 4 8 1 2 5 6 7$
$3 5 1 2 4 6 7 8$
$3 5 6 1 2 4 7 8$
$3 5 6 7 1 2 4 8$
$3 5 6 7 8 1 2 4$
$3 5 6 8 1 2 4 7$
$3 5 7 1 2 4 6 8$
$3 5 7 8 1 2 4 6$
$3 5 8 1 2 4 6 7$
$3 6 1 2 4 5 7 8$
$3 6 7 1 2 4 5 8$
$3 6 7 8 1 2 4 5$
$3 6 8 1 2 4 5 7$
$3 7 1 2 4 5 6 8$
$3 7 8 1 2 4 5 6$
$3 8 1 2 4 5 6 7$
$4 1 2 3 5 6 7 8$
$4 5 1 2 3 6 7 8$
$4 5 6 1 2 3 7 8$
$4 5 6 7 1 2 3 8$
$4 5 6 7 8 1 2 3$
$4 5 6 8 1 2 3 7$
$4 5 7 1 2 3 6 8$
$4 5 7 8 1 2 3 6$
$4 5 8 1 2 3 6 7$
$4 6 1 2 3 5 7 8$
$4 6 7 1 2 3 5 8$
$4 6 7 8 1 2 3 5$
$4 6 8 1 2 3 5 7$
$4 7 1 2 3 5 6 8$
$4 7 8 1 2 3 5 6$
$4 8 1 2 3 5 6 7$
$5 1 2 3 4 6 7 8$
$5 6 1 2 3 4 7 8$
$5 6 7 1 2 3 4 8$
$5 6 7 8 1 2 3 4$
$5 6 8 1 2 3 4 7$
$5 7 1 2 3 4 6 8$
$5 7 8 1 2 3 4 6$
$5 8 1 2 3 4 6 7$
$6 1 2 3 4 5 7 8$
$6 7 1 2 3 4 5 8$
$6 7 8 1 2 3 4 5$
$6 8 1 2 3 4 5 7$
$7 1 2 3 4 5 6 8$
$7 8 1 2 3 4 5 6$
$8 1 2 3 4 5 6 7$

Number of numbers is $247$.
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.
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Thanic Nur Samin
 
Posts: 176
Joined: Sun Dec 01, 2013 11:02 am

Re: Dhaka Regional '16 P8

Post Number:#3  Unread postby Kazi_Zareer » Thu Jan 19, 2017 6:54 pm

I have to solve it like this!? :(
We cannot solve our problems with the same thinking we used when we create them.
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Kazi_Zareer
 
Posts: 86
Joined: Thu Aug 20, 2015 7:11 pm
Location: Malibagh,Dhaka-1217

Re: Dhaka Regional '16 P8

Post Number:#4  Unread postby Thanic Nur Samin » Thu Jan 19, 2017 8:00 pm

No. Experimenting with smaller values yield $2^n-n-1$ as a conjecture and it can be finished off with a recursion argument.
Hammer with tact.

Because destroying everything mindlessly isn't cool enough.
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Thanic Nur Samin
 
Posts: 176
Joined: Sun Dec 01, 2013 11:02 am


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