Math GRE - #9

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

1. Some sector contains at least 4 of the points.
2. Some sector contains at most 3 of the points.
3. Some pair of adjacent sectors contain a total of at least 9 of the points.

• 1 only.
• 3 only.
• 1 and 2.
• 1 and 3.
• 1, 2, and 3.

Solution :

Statement 1 is true. If all sectors contained less than 5 points (i.e. max 4 points in each sector), there would only be a maximum of 5*4 = 20 points in the circle.

Statement 2 is false. {4, 4, 4, 4, 5} is a counter example.

Statement 3 is true. We can label the number of points in our sectors as {a, b, c, d, e}. Suppose no pair of adjacent sectors contain more than 9 points. This means that:

a + b <= 8

b + c <= 8

c + d <= 8

d + e <= 8

e + a <= 8

Adding all of the above inequalities together, we obtain:

2(a + b + c + d + e) <= 40

a + b + c + d + e <= 20

which is a contradiction as a + b + c + d +e = 21.