CCBC MATH CONTEST

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^thto 215^th 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, Care 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 Dof 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