12/11/07 

Final Results!!! 

Fall 2007 SJSU Math Dept.

Problem of the Week Competition 

Undergraduate Division 

1st prize   $100   Siddhartha Kanungo 

($50 extra for a perfect score)

2nd prize  $30   Cuong Dong  

3rd prize   $20   Paul Craciunoiu 

 

Graduate Division 

1st prize   $50   Michael Pejic

2nd prize  $30   Ryan Flarity 

3rd prize   $20   Tomoki Tsuchida  

 

SJMC Division 

1st prize   $50   Sahana Vasudevan 

 

Congratulations to Siddhartha Kanungo who got the first ever perfect score in the Problem of the Week competition. Stop by the math office to make sure that we have your correct student ID number.  Your checks can be picked up after Feb. 1. 

 

11/13/07 

Final Problem!  

 

Week #13  Mon. Nov. 26 - Fri. Nov. 30  

Problem 13 (25 points)  

Starting with a set S in [0,1], and iteratively applying (in any order) the operations of taking the complement or the closure, is it possible to obtain infinitely many subsets of [0,1]?  If possible, give a set S which generates infinitely many subsets.  Otherwise, give a set S which generates the largest possible (finite) number of subsets.  

 

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 Wed. Dec. 5 at noon.  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 results of the Problem of the Week competition will be posted on Mon. Dec. 10.  

 

11/13/07 

Week #12  Mon. Nov. 12 - Fri. Nov. 16  

Problem 12 (20 points)  

An n x n matrix has 0's on the main diagonal and +1 or -1 in the remaining positions.  

a) Show that if n is even, then M is nonsingular.  

b) Does a) still hold if n is odd?  

 

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. 21 at noon.  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 will be posted on Mon. Nov. 26.  

 

11/06/07 

Week #11  Mon. Nov. 5 - Fri. Nov. 9  

 

Problem 11 (15 points) 

Show that there exist 2007 consecutive positive integers, each of which is divisible by some number of the form a2007 (the 2007th power of a), where a is some positive integer greater than 1.  

 

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. 14 at noon.  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 will be posted on Mon. Nov. 19.  

 

10/30/07 

Week #10  Mon. Oct. 29 - Fri. Nov. 2  

 

Problem 10 (14 points) 

A regular n-gon, with vertices v1,v2, ... , vn, is inscribed in a circle of radius 1.  Vertex v1 of the n-gon is then joined by a line segment to each of the other n-1 vertices.  Show that the product of the lengths of these line segments |v1 v2| |v1 v3| ... |v1 vn| is equal to n.  

 

10/25/07 

Week #9  Mon. Oct. 22 - Fri. Oct. 26  

 

Problem 9 (13 points) 

Suppose that 2007 points are chosen at random in a square that is 20 inches x 20 inches.  Show that it is always possible to choose three of the points, which form a triangle (possibly degenerate) whose perimeter is less than 3.  

 

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. 31 at noon.  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 will be posted on Mon. Nov. 5.  

 

10/17/07 

Week #8  Mon. Oct. 15 - Fri. Oct. 20  

Problem 8 (12 points) 

Let S be any set of 7 different integers chosen at random.  Show that S always has a subset of 4 integers whose sum is divisible by 4.  

 

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 #8 should be submitted to the Math office by Wed. Oct. 24 at noon.  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 #8 will be posted on Mon. Oct. 29.  

 

10/10/07 

Week #7  Mon. Oct. 8 - Fri. Oct. 13  

 

Problem 7 (11 points) 

For positive integers n and k such that 1 < k < n, let S = {1,2,...,n}.  Consider all of the k-subsets of S.  Each of these subsets has a largest element.  Let M(n,k) denote the arithmetic mean of these largest numbers.  Prove that M(n,k) = k(n+1)/(k+1).  

 

10/01/07 

Week #6  Mon. Oct. 1 - Fri. Oct. 5  

 

Problem 6 (10 points) 

Determine the smallest positive integer of the form  33a - 7b, where a and b are positive integers. 

 

Corrected Version 

9/28/07  

Week #5  Mon. Sept. 24 - Fri. Sept. 28  

Problem 5 (9 points) 

Let f(n) be a function defined for all positive integers n which takes on only nonnegative integer values.  In addition, suppose that f(m + n) - f(m) - f(n) = 0 or 1 for all positive integers m,n.  If f(2) = 0, f(3) > 0 and f(9999) = 3333, determine f(2007).   

 

9/18/07  

Week #4  Mon. Sept. 17 - Fri. Sept. 21  

Problem 4 (8 points) 

Two players play a game by taking turns rolling a fair n-sided die (each number 1,2, ... ,n is equally likely to occur).  On each roll a player who fails to roll a number higher than every previous number that has appeared, loses the game.  What is the probability that the second player wins the game?   

 

9/11/07  

Week #3  Mon. Sept. 10 - Fri. Sept. 14  

Problem 3 (7 points) 

Let f(x) = exp(x2).  Find an open interval I and a non-zero function g(x) on I such that (fg)' = f'g' on I, or prove that they do not exist.  

 

9/04/07 

Week #2  Mon. Sept. 3 - Fri. Sept. 7  

Problem 2 (6 points) 

A spherical ball is placed on the interior of the surface (paraboloid) defined by the equation z = x2 + y2.  The ball is then rolled toward the vertex of the paraboloid (0,0,0).  What is the largest diameter of a ball that will be able to reach the vertex without getting stuck? 

 

08/13/07  

 

Week #1  Mon. Aug. 27 - Fri. Aug. 31  

Problem 1 (5 points) Let d(0), d(1), d(2), ... be a sequence of decimal integers defined as follows: d(0) = 0; d(1) = 1; and for n > 1, d(n+2) is obtained by writing the digits of d(n+1) immediately followed by those of d(n).  Thus d(2) = 10, d(3) = 101, d(4) = 10110, and so on.  For which values of n is d(n) divisible by 11.