Week #6  Mon. Sept. 29 - Fri. Oct. 3  

 

Problem 6  (10 points)  Let H_n = 1/1 + 1/2 + 1/3 + ... + 1/n be the nth Harmonic number.  For which x > 1 does the sum 1/(x^{H_1}) + 1/(x^{H_2}) +... + 1/(x^{H_n}) converge as n goes to infinity (where x^y represents the yth power of x).   

 

Copies of this problem (and answers to the old problems) are available from the shelves in front of the Math office, MH 308, or online at http://www.math.sjsu.edu/~jackson.  Solutions to problem #6 should be submitted to the Math office by Wed. Oct. 8.  Remember that a correct solution should also contain an explanation of the answer.  Make sure that your name, student ID number, and the division you are participating in (undergraduate or graduate or SJMC) are printed legibly at the top of the page.  A correct solution to problem #6 and the current standings of the Problem of the Week competition will be posted on Tues. Oct. 14.  

 

 

Week #5  Mon. Sept. 22 - Fri. Sept. 26  

 

Problem 5  (9 points)  Let a,b, and c be real numbers satisfying a^2 + b^2 + c^2 = 1.  Prove the inequalities -1/2 <= ab + bc + ca <= 1.  (where x^2 represents the square of x and <= represents less than or equal to). 

 

Copies of this problem (and answers to the old problems) are available from the shelves in front of the Math office, MH 308, or online at http://www.math.sjsu.edu/~jackson.  Solutions to problem #5 should be submitted to the Math office by Wed. Oct. 1.  Remember that a correct solution should also contain an explanation of the answer.  Make sure that your name, student ID number, and the division you are participating in (undergraduate or graduate or SJMC) are printed legibly at the top of the page.  A correct solution to problem #5 and the current standing of the Problem of the Week competition will be posted on Tues. Oct. 7.  

 

 

Week #4  Mon. Sept. 15 - Fri. Sept. 19  

 

Problem 4  (8 points)  Two players play a game on a 5 x 5 checkerboard.  In turn, each player places a 1 x 2 domino so that it exactly covers two uncovered squares of the checkerboard.  The last player able to place a domino wins the game.  Determine (with justification) the length of the shortest possible game. 

 

Week #3  Tues. Sept. 8 - Fri. Sept. 12  

 

Problem 3  (7 points)  Given any set S of ten distinct positive integers less than 100, prove that there are two disjoint subsets X,Y of S such that the sum of the elements of X is equal to the sum of the elements of Y.   

 

Week #2  Tues. Sept. 2 - Fri. Sept. 5  

 

Problem 2 (6 points) Find the limit as n goes to infinity of

 

{[(1 + 1/n)^n]/e}^n (where x^n represents x to the nth power).