MAΘ Problem Set 2
January 12, 2004
The Mississippi School for Mathematics and Science
1. To the nearest percent, what percent of the volume of a cone is below the
point halfway between the tip and the base?
2. For how many positive integers n < 100 is it true that 14n + 1 is divisible
by 13?
3. The sum of any three terms of an arithmetic sequence is divisible by 3.
For what other numbers n is it true that the sum of any n terms in an
arithmetic sequence is divisible by n?
4. Someone is trying to place 143 cubes each with edge length 3 inches inside
800 × 800 × 800 boxes. How many such boxes will he need to contain all the
cubes?
5. An angle bisector of a certain triangle bisects the triangle into two regions
of equal area. Prove that the triangle must have at least two congruent
angles.
6. The notation bxc means the greatest integer that is less than or equal to
x. Find bln(e2003 + 9e2004 )c. (No calculator solutions).