SJSU Math Problem of the Week Competition

Final Problem! 

Week #13  Mon. Nov. 17 - Fri. Nov. 21

Problem 13 (25 points) 

Let n1 < n2 < n3 < ... be an infinite sequence of positive integers.  Prove there exists either an infinite subsequence in which no integer divides another, or an infinite subsequence where each integer is a multiple of the preceding one.  

 

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 #13 should be submitted to the Math office by Mon. Dec. 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 #13 and the final standings of the Problem of the Week competition will be posted on Thurs. Dec. 4.  

 

Week #12  Mon. Nov. 10 - Fri. Nov. 14

Problem 12 (20 points) 

Suppose the points in the xy-plane are partitioned into three subsets.  Given any

d > 0, show that there exist two points which belong to the same partition subset and which are distance d apart. 

 

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 #12 should be submitted to the Math office by Wed. Nov. 19  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 #12 and the current standings of the Problem of the Week competition will be posted on Wed. Nov. 26.  

 

Week #11  Mon. Nov. 3 - Fri. Nov. 7

Problem 11 (15 points) 

A quadrilateral ABCD with diagonals AC and BD meeting at point E in the interior is shown below.  Assume that angle BDA is 80 degrees, angle CDB is 20 degrees, angle DCA is 50 degrees and angle BCA is also 50 degrees.  What is the measure of angle CAB. Hint: It's not just angle-chasing. 

  

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 #11 should be submitted to the Math office by Wed. Nov. 12  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 #11 and the current standings of the Problem of the Week competition will be posted on Wed. Nov. 19.  

 

 

SJSU Math Problem of the Week Competition

Week #10  Mon. Oct. 27 - Fri. Oct. 31

Problem 10 (14 points) 

 

2008problem10.pdf

 

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 #10 should be submitted to the Math office by Wed. Nov. 5  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 #10 and the current standings of the Problem of the Week competition will be posted on Wed. Nov. 5.  

 

Week #9  Mon. Oct. 20 - Fri. Oct. 24

 

Problem 9 (13 points)  How many of the positive factors of the number 729,000,000 are not a perfect square and not a perfect cube? 

  

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 #9 should be submitted to the Math office by Wed. Oct. 29.  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 #9 and the current standings of the Problem of the Week competition will be posted on Wed. Nov. 5.  

 

Week #8  Mon. Oct. 13 - Fri. Oct. 17

 

Problem 8 (12 points)  Find an expression for the continued radical

C = (m + (m + (m + ... )^.5)^.5)^.5 in terms of m that does not involve a continued radical.  Then determine all positive integers m so that C is a positive integer.  Here n^.5 denotes the square root of n.

  

Week #7  Mon. Oct. 6 - Fri. Oct. 10  

 

Problem 7  (11 points)  In triangle ABC, the length of side CB is a, the length of side CA is b, and the measure of the angle ACB between these sides is 120 degrees.  The bisector of angle ACB is drawn and its intersection point with the opposite side AB is labeled D.  Express the length of CD in terms of a and b. 

 

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).