- Find the length of the shortest path from the point (3,5) to the point
(8,2) that touches both the
*x*- and*y*-axes.

- A bubble chamber contains three types of subatomic particles: 1998 particles
of type
*X*, 2002 of type*Y*, and 2003 of type*Z*. Whenever an*X*-particle and a*Y*-particle collide, they both become*Z*-particles. Likewise,*Y*- and*Z*-particles collide and become*X*-particles, and*X*- and*Z*-particles become*Y*-particles upon collision. Certainly the total number of particles will never change. Is it possible that they can evolve so that only one type of particle is present?

- Two towns,
*A*and*B*, are connected by a road. At sunrise, Pat begins biking from*A*to*B*along this road, while simultaneously Dana begins biking from*B*to*A*. Each person bikes at a constant speed, and they cross paths at noon. Pat reaches*B*at 5pm while Dana reaches*A*at 11:15pm. When was sunrise?

- A random number generator outputs integers from the set {1, 2, 3, 4, 5, 6, 7, 8},
with each of the eight choices equally likely. If ten such random
integers are created, what is the probability that their product is one
more than a multiple of 8?

- A bug is crawling on the coordinate plane from (7,11) to (-17,-3).
The bug travels at constant speed one unit per second everywhere but quadrant
II (negative
*x*-coordinates and positive*y*-coordinates), where it travels at ½ units per second. What path should the bug take to complete her journey in minimal time?

- Show that the roots of the polynomial equation

*x*^{1998}- 2*x*^{1997}+ 3*x*^{1996}- 4*x*^{1995}+ · · · - 1998*x*+ 1999 = 0

are not all real.

- A
**great circle**is a circle drawn on a sphere that is an "equator," i.e., its center is also the center of the sphere. There are*n*great circles on a sphere, no three of which meet at any point. They divide the sphere into how many regions? - For positive integers
*n*, define*S*to be the minimum value of the sum_{n}

*Sum*_{k=1}^{n}((2*k*- 1)^{2}+*a*_{k}^{2})^{½},

as the*a*_{1},*a*_{2}, . . . ,*a*range through all positive real values such that_{n}*a*_{1}+*a*_{2}+ · · · +*a*= 17. Find_{n}*S*_{1998}.

- Let
*f*:**R ---> R**be continuous, with*f*(*x*)·*f*(*f*(*x*)) = 1 for all*x*in**R**. If*f*(1000) = 999, find*f*(500).

- Prove that
(sin
*x*)/*x*=*Prod*_{n=1}^{infinity}cos(*x*/2).^{n}