This is an initially mind-boggling problem from the 1995 American Invitational Mathematics Exam (AIME).

“Find the last three digits of the product of the positive roots of

”

See Log Lunacy

This is an initially mind-boggling problem from the 1995 American Invitational Mathematics Exam (AIME).

“Find the last three digits of the product of the positive roots of

”

See Log Lunacy

This is a fun problem from the 1989 American Invitational Mathematics Exam (AIME).

“Two skaters, Allie and Billie, are at points *A* and *B*, respectively, on a flat, frozen lake. The distance between *A* and *B* is 100 meters. Allie leaves *A* and skates at a speed of 8 meters per second on a straight line that makes a 60° angle with *AB*. At the same time Allie leaves *A*, Billie leaves *B* at a speed of 7 meters per second and follows the straight path that produces the earliest possible meeting of the two skaters, given their speeds. How many meters does Allie skate before meeting Billie?”

See the Skating Rendezvous Problem

This is a slightly challenging problem from the 1993 American Invitational Mathematics Exam (AIME).

“Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Find *t*, the amount of time in seconds, before Jenny and Kenny can see each other again.”

See the Parallel Stroll Problem

This is a challenging problem from the 1986 American Invitational Mathematics Exam (AIME).

“Let triangle *ABC* be a right triangle in the *xy*-plane with a right angle at *C*. Given that the length of the hypotenuse *AB* is 60, and that the medians through *A* and *B* lie along the lines *y = x* + 3 and *y* = 2*x* + 4 respectively, find the area of triangle *ABC*.”

I have included a sketch to indicate that the sides of the right triangle are not parallel to the Cartesian coordinate axes.

The AIME (American Invitational Mathematics Examination) is an intermediate examination between the American Mathematics Competitions AMC 10 or AMC 12 and the USAMO (United States of America Mathematical Olympiad). All students who took the AMC 12 (high school 12^{th} grade) and achieved a score of 100 or more out of a possible 150 or were in the top 5% are invited to take the AIME. All students who took the AMC 10 (high school 10^{th} grade and below) and had a score of 120 or more out of a possible 150, or were in the top 2.5% also qualify for the AIME.

See the Challenging Triangle Problem.

Here is another engaging problem from Presh Talwalkar.

“___________**Triangle Area 1984 AIME**

Point P is in the interior of triangle ABC, and the lines through P are parallel to the sides of ABC. The three triangles shown in the diagram have areas of 4, 9, and 49. What is the area of triangle ABC?”

See the Pinwheel Area Problem