18.314  Fall 2007

Problem Set 1.  Due Thursday, September 20.


1.  We are given 17 points inside a square with a side
length 1.  Prove that there are two points among them 
with the distance between them less than 0.36.


2.  15 squirrels gathered 100 nuts. Prove that some pair of 
squirrels gathered the same number of nuts.


3.  Prove that the sequence 7002, 70027002, 700270027002, ...
has a element divisible by 2007.


4.   We select n+1 different integers from the set {1,2,...,2n}.
Prove that there will always be two among the selected integers
whose largest common divisor is 1.


5.  A spider cannot tolerate another spider at distance <= 1 inch.
What is the maximal number of spiders that can piecefully coexist
on this web?

  |    |    |
--|----|----|--
  |    |    |
--|----|----|--  
  |    |    | 
--|----|----|--
  |    |    |

Here each small square has side length 1 inch and each of
the 12 edges on the boundary of the web has length 1/2 inch.

(Clarifications:  We assume that each spider is a point on the web. 
The "distance" refers to the distance along the web.)


6.  The sequence A(n) is given by A(1)=1 and 
A(n+1) = A(1) + A(2) + ... + A(n), for n >= 1. 
Find a closed expression for A(n) and prove it.


7.  Prove that for a positive integer n, we have

   1^3 + 2^3 + ... + n^3 = (1 + 2 + ... + n)^2


8.  Find a closed expression for 

  1 C(n,1) + 2 C(n,2) + 3 C(n,3) + ... + n C(n,n)

and prove it.  Here C(n,k) is the binomial coefficient 
"n choose k".


9.  Find the number of ways to place 8 nonattacking rooks
on this rhombic chessboard?

        []
      [][][]
    [][][][][]
  [][][][][][][]
[][][][][][][][][]
  [][][][][][][]
    [][][][][]
      [][][]
        []


(Here [] represents a box of the board.)


10.  Prove that the number 11111...111 (with 2187 ones) is 
divisible by 2187.