High School Challenge




Conversely Speaking


Start with a theorem and then take its converse to make a new hypothesis. Sometimes this converse is true and sometimes it is not. For example: If A is a factor of B, then A^A is a factor of B^B (A and B are whole numbers). If A^A is a factor of B^B, must A be a factor of B?

Either prove that A must be a factor of B, when A^A is a factor of B^B, or give a counter-example.

Professor Tom Butts, University of Texas at Dallas gave this problem at a school awards function and it was submitted by Beth Wilson, Apollo JHS, Richardson, TX.


Correct Solutions:


  1. Xingji Zheng. Abby Senior SS, Abbotsford, BC, Canada
  2. Sorin Ionescu. Ecole Secondaire Dorval, Dorval, Quebec, Canada
  3. Tanvir Prince. Chrishtopher Columbus HS, Bronx NY
  4. Lars Erik Walle. Thor Heyerdahl VGS, Larvik, Norway
  5. Ken Scheiwe. New Trier HS, Wilmette, IL
  6. Yang Liu. Davis HS, Davis, CA
  7. Paul Pollack. Gulf HS, New Port Richey, FL
  8. Ranen Ghosh. Spruce Creek HS, Port Orange, FL
  9. Justin Lam. College Park HS, Pleasant Hill, CA
  10. Jesse Carlin. Salmen HS, Slidell, LA
  11. Randal West. Spruce Creek HS, Port Orange, FL
  12. Ido Yariv. Gan-Nachum School, Rishon LeZion, Israel
  13. Torbjoern Elvsaashagen. Thor Heyerdahl VGS, Larvik, Norway