Week #1  Mon. Aug. 25 – Wed. Sept. 3 

Problem 1 (5 points):  Find the exact sum of the following infinite series, 1³/1! + 2³/2! + 3³/3! + … n³/n! + … .

Week #3  Mon. Sept. 8 – Wed. Sept. 17 

Problem 3 (7 points):  a) Do there exist 14 consecutive integers each of which is divisible by one or more of the primes p = 2,3,5,7,11?  

b) Do there exist 21 consecutive integers each of which is divisible by one or more of the primes p = 2,3,5,7,11,13?

Posted Mon. Sept. 22

Fall 2003 SJSU Problem of the Week Competition

Week #5  Mon. Sept. 22 – Wed. Oct. 1 

Problem 5 (9 points):  Prove that the roots of the polynomial p(x) = x⁵ + ax⁴ +bx³ +cx² +dx + e = 0 cannot all be real if 2a² < 5b. 

Posted Mon. Oct. 6

Fall 2003 SJSU Problem of the Week Competition

Week #7  Mon. Oct. 6 – Wed. Oct. 15 

Problem 7 (11 points):  In an election to recall the dogcatcher Davis Gray there are 1002 yes votes and 1001 no votes.  The votes are counted one at a time in a random order and a subtotal is computed after each vote is tallied.  What is the probability that the yes votes always outnumber the no votes in each subtotal?

Posted Mon. Oct. 20

Fall 2003 SJSU Problem of the Week Competition

Week #9  Mon. Oct. 20 – Wed. Oct. 29 

Problem 9 (13 points):  For each positive decimal integer with n² digits (no leading zeros) we take the determinant of the matrix obtained by writing the digits in order across the rows.  For example, we associate the det  é 4 3 ù = -2, to the

                                 ë 2 1 û

integer 4321.  Define f(n) to be the sum of all the determinants associated with the n²-digit integers.  Which is larger f(9) or f(10)? 

Posted Mon. Nov. 17

Fall 2003 SJSU Problem of the Week Competition

Week #11  Mon. Oct. 20 – Wed. Oct. 29 

Problem 11 (15 points):  Consider cards 1,2, … ,n in a pile.  When the top card is m, we reverse the order of the first m cards.  The process stops only when card 1 is at the top of the pile.  Prove that the process always stops after a finite number of steps, regardless of the initial order of the cards.

Posted Mon. Nov.  17  Final Problem!!!

Fall 2003 SJSU Problem of the Week Competition

Week #13  Mon. Nov. 17 – Mon. Dec. 1 

Problem 13 (25 points):  Let S be any set of 2003 distinct positive integers, none of which have a prime divisor greater than 23.  Prove or disprove: 1) There exist 2 distinct integers in S whose product is the square of an integer. 

2) There exist 4 distinct integers in S whose product is the fourth power of an integer. 

3) There exist 5 distinct integers in S whose product is the fifth power of an integer.