Dhaka Higher Secondary 2010/9

Problem for Higher Secondary Group from Divisional Mathematical Olympiad will be solved here.
Forum rules
Please don't post problems (by starting a topic) in the "Higher Secondary: Solved" forum. This forum is only for showcasing the problems for the convenience of the users. You can post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
BdMO
Posts: 134
Joined: Tue Jan 18, 2011 1:31 pm

Dhaka Higher Secondary 2010/9

Unread post by BdMO » Tue Jan 18, 2011 2:06 pm

Out of the digits $1, 2, 3, 4, 5$, three are chosen to form numbers so that their digits are either in increasing or decreasing order. What is the total number of numbers formed? If one number is chosen from each category so that the sum of those two numbers is maximum, what is that sum?

User avatar
leonardo shawon
Posts: 169
Joined: Sat Jan 01, 2011 4:59 pm
Location: Dhaka

Re: Dhaka Higher Secondary 2010/9

Unread post by leonardo shawon » Tue Jan 18, 2011 6:29 pm

decreasing order||| 543 > 432 > 321
increasing order || 123 < 234 < 345

the sum is 888
Ibtehaz Shawon
BRAC University.

long way to go .....

User avatar
Moon
Site Admin
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
Location: Dhaka, Bangladesh
Contact:

Re: Dhaka Higher Secondary 2010/9

Unread post by Moon » Tue Jan 18, 2011 7:22 pm

The second part is correct. :) However, you need to show some logic. Actually we are trying maximize the number. So apparently the largest number $543$ works.

Hint for the first part:
what if we forget that the digits can be increasing or decreasing?
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.

User avatar
leonardo shawon
Posts: 169
Joined: Sat Jan 01, 2011 4:59 pm
Location: Dhaka

Re: Dhaka Higher Secondary 2010/9

Unread post by leonardo shawon » Wed Jan 19, 2011 12:06 am

1,2,3,4,5.. Among this numbers , we can form 60 (5P3) three digits numbers... If we put 5 on left, the rest 2 place can be filled by 4P2 > 12 ways and numbers are 512, 513, 514, 521, 523, 524, 531, 532, 534, 541, 542, 543. And among this 12 numbers, value of 543 is the largest.

Having 1 on left, rest 4 digits can form 12 , 3 digits number. Among them 123 is the lowest.

## what do u think? Did i give some logic?
Ibtehaz Shawon
BRAC University.

long way to go .....

User avatar
Moon
Site Admin
Posts: 751
Joined: Tue Nov 02, 2010 7:52 pm
Location: Dhaka, Bangladesh
Contact:

Re: Dhaka Higher Secondary 2010/9

Unread post by Moon » Tue Feb 01, 2011 11:23 pm

Still no solution for the first part!
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin

Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.

Shifat
Posts: 53
Joined: Sun Jul 31, 2011 12:21 pm
Location: Dhaka, Bangladesh

Re: Dhaka Higher Secondary 2010/9

Unread post by Shifat » Sat Aug 20, 2011 6:20 pm

I think I have a solution, the total number of increasing numbers is and decreasing is 20. Here it is:-
While we are writing them as decreasing numbers, then either the first one should be 5,4,3... Now there is three cases
First Value five:- the second value can be 4,3 or 2 .if the second value is 4 then the third value(3,2,1) can be input in 3P1=3 ways, similarly if the second value is three then the third value can be input in 2P1=2 ways, if the second value is 2 then the third value can be input in only 1 way.. so when the first value is five, the number of decreasing numbers is 3+2+1=6
First value four:- By doing the same we can find the number of decreasing numbers is 1+2=3
first value 3:- we can do the same and the result is 1..
So the total number of decreasing numbers is 10... do the same for the increasing numbers.. the sum is 20. :?:

Ragib Farhat Hasan
Posts: 20
Joined: Sun Mar 30, 2014 10:40 pm

Re: Dhaka Higher Secondary 2010/9

Unread post by Ragib Farhat Hasan » Sat Jan 16, 2016 8:16 pm

Total no. of numbers formed is 12.
The maximum sum is 888.

Post Reply