[Suggested by W. Martin] A bag contains 101 pieces of candy. Lisa and Eric play a game, starting with Lisa. In rotation (first Lisa, then Eric) they take from 1 to 10 pieces of candy from the bag. When the bag becomes empty, they each count the number of pieces they have. If these numbers are relatively prime (that is, have no common divisors other than 1), then Lisa is the winner. Otherwise, Eric is the winner. Who should win the game and what must the winning strategy be? [Suggested by W. Martin] A lucky year is one in which at least one date, when written in the form month/day/year, has the following property: The product of the month times the day is equal to the last two digits of the year. For example, 2001 was a lucky year because 1/1/2001 gives 1 x 1 = 01. How many lucky years will there be during the century that began on 1/1/2000? [Suggested by J. Calvo] Given a triangle ABC, let A', B', and C' be the midpoints of sides BC,AC, and AB, respectively. Then the line segments AA', BB', and CC' meet at a single point; they divide ABC into 6 smaller triangles. Note that, in most cases, no two of these triangles are congruent. Nevertheless, prove that all six triangles have equal area. [Suggested by I. Kornfeld] N is a positive integer. The decimal representation of N consists only of 0's and 1's. Moreover, the digit 1 appears exactly 3 times. Prove that N cannot be a perfect square. Problems will be due on November 15, 2004. Check www.ndsu.edu/math/talent for solutions and more problems! Send your answers, along with this form to: Talent Search, Dept. of Mathematics, NDSU, Fargo, ND 58105. Your Name: __________________________________     Set: 1 Your Address: __________________________________ 1. _____ __________________________________ 2. _____ __________________________________ 3. _____ Your High School: __________________________________ 4. _____ Grade: _____ Total: _____