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 200th
to 215th places (inclusive) in the decimal expansion of 1/17 is
A) 45 B)
48 C) 72 D)85
4. If n1 = the sum of
all divisors of 17424 and n2 = the sum of squares of all the
divisors of 17424 what is n2/n1 ?
A) 2665
B) 8547 C)
14400 D) 17424
5. If r is the remainder when
2006100 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 2a2
+ b2 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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