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BdMO 2017 National Round Secondary 7
Posted: Fri Feb 10, 2017 9:06 pm
by Kazi_Zareer
$100$ pictures of BdMO math campers were painted by Urmi. Exactly $k$ colors were used in each picture. There is a common color in every $20$ pictures. But, there is no common color in all $100$ pictures. Find the smallest possible value of $k.$
Re: BdMO 2017 National Round Secondary 7
Posted: Fri Feb 02, 2018 10:43 pm
by Tasnood
Kazi_Zareer wrote: ↑Fri Feb 10, 2017 9:06 pm
There is a common color in every $20$ pictures.
$Exactly$ $one$ or $at$ $least$ $one$?
Re: BdMO 2017 National Round Secondary 7
Posted: Tue Feb 13, 2018 8:32 pm
by samiul_samin
If I can show an example by setting $k=2$then the lowest value of $k$ is 2.But,is it enogh to show just one example to get the full solution?
Re: BdMO 2017 National Round Secondary 7
Posted: Tue Feb 13, 2018 8:33 pm
by Tasnood
$K$ may not be $2$
Re: BdMO 2017 National Round Secondary 7
Posted: Tue Feb 20, 2018 1:03 am
by samiul_samin
I am posting my solution.It may be wrong.
Re: BdMO 2017 National Round Secondary 7
Posted: Tue Feb 20, 2018 8:08 am
by Tasnood
Kazi_Zareer wrote: ↑Fri Feb 10, 2017 9:06 pm
There is a common color in every $20$ pictures.
My confusion is here. Does it mean
just one common color in every 20 pictures or
at least one among 20?
If the first case is true, your solution is wrong. Because in $80$ pictures, you will get $2$
(B,G) common color. The answer should be $9$
If the second case is true, this solution is just like mine!
[As a problem $7 of secondary level, it might not be very easy.
Re: BdMO 2017 National Round Secondary 7
Posted: Tue Feb 20, 2018 9:00 am
by samiul_samin
Tasnood wrote: ↑Tue Feb 20, 2018 8:08 am
Kazi_Zareer wrote: ↑Fri Feb 10, 2017 9:06 pm
There is a common color in every $20$ pictures.
My confusion is here. Does it mean
just one common color in every 20 pictures or
at least one among 20?
If the first case is true, your solution is wrong. Because in $80$ pictures, you will get $2$
(B,G) common color. The answer should be $9$
If the second case is true, this solution is just like mine!
[As a problem $7 of secondary level, it might not be very easy.
What is the solution if the first case is true?
Re: BdMO 2017 National Round Secondary 7
Posted: Tue Feb 20, 2018 9:39 am
by Tasnood
If the first case is true, you can't use two colors in $20$ pictures. The answer should be $9$
Re: BdMO 2017 National Round Secondary 7
Posted: Wed Feb 28, 2018 3:34 am
by mac0220
I think the condition is for ,any 20 pics share a common colour, but not all of them share the same colour.
If that so ,then the answer is 3.
Re: BdMO 2017 National Round Secondary 7
Posted: Wed Mar 07, 2018 10:09 am
by samiul_samin
mac0220 wrote: ↑Wed Feb 28, 2018 3:34 am
I think the condition is for ,any 20 pics share a common colour, but not all of them share the same colour.
If that so ,then the answer is 3.
But,I have proved $k=2$ for this case.