BdMO 2017 National Round Secondary 7

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Kazi_Zareer
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BdMO 2017 National Round Secondary 7

Unread post by Kazi_Zareer » Fri Feb 10, 2017 9:06 pm

$100$ 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 $k.$
We cannot solve our problems with the same thinking we used when we create them.

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Tasnood
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Re: BdMO 2017 National Round Secondary 7

Unread post by Tasnood » Fri Feb 02, 2018 10:43 pm

Kazi_Zareer wrote:
Fri Feb 10, 2017 9:06 pm
There is a common color in every $20$ pictures.
$Exactly$ $one$ or $at$ $least$ $one$?

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samiul_samin
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Re: BdMO 2017 National Round Secondary 7

Unread post by samiul_samin » Tue Feb 13, 2018 8:32 pm

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?

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Tasnood
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Re: BdMO 2017 National Round Secondary 7

Unread post by Tasnood » Tue Feb 13, 2018 8:33 pm

$K$ may not be $2$

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samiul_samin
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Re: BdMO 2017 National Round Secondary 7

Unread post by samiul_samin » Tue Feb 20, 2018 1:03 am

I am posting my solution.It may be wrong.
Denote
$R$=Red
$B$=Blue
$G$=Green
First Use $B$ & $G$ in $80$ pictures.
Then, Use $G$ & $B$ in $19$ pictures .
At last,Use $R$ & $G$ in $1$ picture.
Now,Pick any $20$ of it,you will get at least one same color and you will not get any particular color in every picture.
$k >1 $ and if $k$ is $3$ then it is easy to color the picture.So,I think the answer is $\fbox 2$

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Tasnood
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Re: BdMO 2017 National Round Secondary 7

Unread post by Tasnood » 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.

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samiul_samin
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Re: BdMO 2017 National Round Secondary 7

Unread post by samiul_samin » Tue Feb 20, 2018 9:00 am

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?

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Tasnood
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Re: BdMO 2017 National Round Secondary 7

Unread post by Tasnood » Tue Feb 20, 2018 9:39 am

If the first case is true, you can't use two colors in $20$ pictures. The answer should be $9$

mac0220
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Re: BdMO 2017 National Round Secondary 7

Unread post by mac0220 » 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.

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samiul_samin
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Re: BdMO 2017 National Round Secondary 7

Unread post by samiul_samin » Wed Mar 07, 2018 10:09 am

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.

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