2005 Problem of the Week Competition:  Problems  

Final Results!!! 

SJSU Math Dept.

Fall 2005

Problem of the Week Competition

Undergraduate Division 

1st prize   $50   Cuong Dong

2nd prize  $30   Andrew Macheret 

3rd prize   $20   Johan Johansson

Graduate Division 

1st prize   $50   Weidong Shao

2nd prize  $30   Matthew Low 

3rd prize   $20   Helene Payne  

SJMC Division 

1st prize   $50   Powell Akagawa 

2nd prize  $30   Bowei Liu 

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. 

Final Week!!!                                                    11/18/05

Week #13  Mon. Nov. 21 – Fri. Dec. 2 

 Problem 13 (25 points):  The vertices of a 10-dimensional cube are represented by the set of 1024 binary sequences of length 10 (each digit is either 0 or 1).  Let S be any subset of at least 205 binary sequences (vertices).  Show that S will always contain an equilateral triangle.  

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 Fri. Dec. 2 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.  The final results of the POW competition and a correct solution to problem #13 will be posted on Mon. Dec. 5. 

11/2/05

Week #11    Mon. Nov. 7 – Fri. Nov. 11

 Problem #11 (15 points): A sequence (of real numbers) of length  is simply an -tuple . A subsequence of  of length  is an -tuple of the form  for some integers . Such a subsequence is monotone if either  or else .

 (1) Show if  is a positive integer and  is a sequence of length , then  has a monotone subsequence of length .

 (2) Show (by specific example) that if instead  has length , it is possible that the longest monotone subsequence of  has length .

 (3) For all positive integers , let  be the largest integer  such that every sequence of length  has a monotone subsequence of length . Show that  for all , where  denotes the ceiling function.

 10/26/05

Week #10  Wed. Oct. 26 – Fri. Nov. 4 

 Problem 10 (14 points):  Given any 4 real numbers a,b,c,d satisfying a + b + c + d = 0, prove that the sum of their cubes is equal to 0 (a3 + b3 + c3 + d3 = 0) if and only if the sum of their 5th powers is equal to 0 (a5 + b5 + c5 + d5 = 0). 

 Problem Of The Week, 10-24-2005

Problem #9  (13 points)  

 For each number , find the limit; .  

 10/12/05

Week #8  Mon. Oct. 17 – Fri. Oct. 21

 Problem 8  (12 points):  There are 20 chairs around a circular table, numbered from 1 to 20, and 20 people have all been assigned different numbers from 1 to 20. When the people take their seats, they don’t have to sit at the chair with their number, but they must be seated at most one place away from that chair. (For example, person 13 may sit in chair 12 or 13 or 14, and person 20 may sit in chair 19 or 20 or 1). How many seating arrangements are possible?

                                                                  10/5/05

Week #7  Wed. Oct. 5 – Fri. Oct. 14 

 Problem 7 (11 points):  Prove that there exists an integer A such that for every pair of positive integers n,m such that n,m > A, there is a tiling of an n x m rectangle using only 3 x 5 and 4 x 7 rectangles.       

9/28/05

Problem of the Week

Problem #6 (10 points) for the week of October 3, 2005

 A solid, call it S, is the region in three-dimensional space bounded by the planes

              .

(a)    Sketch one of the faces of S carefully using only a compass and a ruler.  I like to see the traces of compass marks.  Take two inches for the unit length.  Indicate the length of each side.  Find the area of the face, and the total surface area of S.

(b)   Find the volume of S.

Week #5  Mon. Sept. 26 – Fri. Sept. 30

Problem 5: (9 points) Let T be any plane triangle.  Construct squares on the edges of T  (external to T ), call these the squares at level 1, and let  denote the sum of their areas. Next form the convex hull of the construction so far by adding 3 new line segments (shown as dotted lines), construct squares on these new segments (external to the construction so far), call these the squares at level 2, and let  denote the sum of their areas. Prove that the ratio  does not depend on T , and determine its value.

 9/14/05

Week #4  Wed. Sept. 14 – Fri. Sept. 25 

Problem 4 (8 points):  Evaluate the limit as n ฎ ฅ  (find the exact value) of the sum

1/(2n+1) + 1/(2n+2) + … + 1/(4n) .

Week #3  Wed. Sept. 7 – Fri. Sept. 16 

Problem 3 (7 points):  Let n be a positive integer, and let p be a prime number.  Prove that  implies .

Week #2  Mon. Sept. 5 – Fri. Sept. 9

Problem 2  (6 points):  There were 5 married couples at a party. One man there asked all 9 of the others there the following question: Before tonight, not counting your spouse, how many of the people here had you met? Surprisingly, he received 9 different answers, no two of the answers he received were the same. What was his wife’s answer?

Week #1  Wed. Aug. 24 – Fri. Sept. 2                  

Problem 1 (5 points):  How many 4-tuples of positive integers {a,b,c,d} are there for which the lowest common multiple of any three integers in the 4-tuple is 3n5m?