**Problems**

1. If

*ABC*is an equilateral triangle with each sidelength 10 and*P*is a point inside the triangle what is the average value of the sum of the distances of*P*from the three sides?
A) 5Ö2 B) 5Ö3 C)
10 D) 12

2. A sphere of radius 1 is packed in a
cube with each side of length 2. Eight more smaller spheres are packed tightly
at the eight corners of the cube. The radius of each smaller sphere is

A) 1/4 B) 2 - Ö3 C) 1 - 1/Ö3 D)1/Ö3

3. The sum of the digits from the 200

^{th}to 215^{th}places (inclusive) in the decimal expansion of 1/17 is
A) 45 B)
48 C) 72 D)85

4. If

*n*_{1}= the sum of all divisors of 17424 and*n*_{2}= the sum of squares of all the divisors of 17424 what is*n*_{2}/*n*_{1}?
A) 2665
B) 8547 C)
14400 D) 17424

5. If

*r*is the remainder when 2006^{100}is divided by 153 then the sum of the digits of*r*is
A) 1
B) 8 C) 10 D) 17

6.

*A*is a fixed point on a plane and*B*,*C*are two points at a constant positive distance from each other moving on the plane in such a manner that the perimeter of the triangle*ABC*remains constant. If additionally the line segment*BC*is always parallel to a fixed line*l*on the plane, the curve traced out by the midpoint*D*of the line segment*BC*is
A) a line
segment B) a triangle
C) a circle D) an
ellipse

8. Young Andy claims he can run at
least twice as fast as his grandfather George. To prove his claim he challenges
George to a race across a field and back. They both start at one end except
Andy starts when George is already halfway across the field but he catches up
with his grandfather at a point 50 ft from the other end. Andy continues to the
end and turns back and meets George only 20 ft from the end. He continues
running to the starting point and starts running back again to the other end
while George reaches the other end and starts his return trip. Assuming they
were running at constant speeds, how far from the starting point do they meet
for the third time?

A) 10ft B)
70ft C) 150ft D) 180ft

9. 10. 2006 can be written as 2

*a*^{2}+*b*^{2}where*a*and*b*are positive integers in several ways. What is the sum of*a*+*b*from all possible ways?
A) 100 B)
120 C) 145 D) 180

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