## Clarification on 0568 #45

Forum for the GRE subject test in mathematics.
newb
Posts: 6
Joined: Wed Oct 29, 2008 4:02 am

### Clarification on 0568 #45

Here is 45 on GRE 0568:

A circular region is divided by 5 radii into sectors as shown above. 21 points are chosen in the circular region, none of which is on any of the 5 radii. Which of the following statements must be true?

1. Some sector contains at least 5 of the points.
2. Some sector contains at most 3 of the points.
3. Some pair of adjacent sectors contains a total of at least 9 of the points.

A. 1. only
B. 3. only
C 1. and 2. only
D. 1. and 3. only
E. 1., 2., 3. only

Can't all 21 points be found in one sector? Nothing in the question sounds like it's preventing this from happening... So 2. could be true, no?
What am I missing o_O ?

a huge thank you with a ! in advance

Nameless
Posts: 128
Joined: Sun Aug 31, 2008 4:42 pm
Use the pigeonhole principle, since we have 21 points, and 5 regions so at least one region must contain 5 points. There is at least one of the other four regions must contain 4 points. So there at least two adjacent regions must contains 9 point. Therefore the answer is D

ngoc
Posts: 6
Joined: Mon Nov 03, 2008 5:54 am
So if 21 points is in 1 sector (call that sector A), then the number points in sector A and its neighbor would be 21 > 9, so (3) is true.

Nameless
Posts: 128
Joined: Sun Aug 31, 2008 4:42 pm
So if 21 points is in 1 sector

there is no "IF" here. what happens if "YOUR IF" is not true?

ngoc
Posts: 6
Joined: Mon Nov 03, 2008 5:54 am
I'm just pointing to the newb guy that his counter example doesn't work. (Hence it's an IF, because it considers a particular case). Of course the logic argument relies on Pigeon hole.

newb
Posts: 6
Joined: Wed Oct 29, 2008 4:02 am
Thanks very much for the help
I totally understand it now.
(Feeling really stupid at the moment.)