## BdMO 2017 National Round Secondary 7

Kazi_Zareer
Posts: 86
Joined: Thu Aug 20, 2015 7:11 pm
Location: Malibagh,Dhaka-1217

### BdMO 2017 National Round Secondary 7

\$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.

Tasnood
Posts: 73
Joined: Tue Jan 06, 2015 1:46 pm

### Re: BdMO 2017 National Round Secondary 7

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\$?

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO 2017 National Round Secondary 7

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?

Tasnood
Posts: 73
Joined: Tue Jan 06, 2015 1:46 pm

### Re: BdMO 2017 National Round Secondary 7

\$K\$ may not be \$2\$

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO 2017 National Round Secondary 7

I am posting my solution.It may be wrong.

Tasnood
Posts: 73
Joined: Tue Jan 06, 2015 1:46 pm

### Re: BdMO 2017 National Round Secondary 7

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.

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO 2017 National Round Secondary 7

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?

Tasnood
Posts: 73
Joined: Tue Jan 06, 2015 1:46 pm

### Re: BdMO 2017 National Round Secondary 7

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

mac0220
Posts: 4
Joined: Mon Feb 05, 2018 3:45 am

### Re: BdMO 2017 National Round Secondary 7

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.

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO 2017 National Round Secondary 7

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.