Conference rusure::math

780.0. "Recent USENET problems" by CLT::GILBERT (Builder) Fri Oct 30 1987 10:25

Several good problems have been appearing on USENET lately.  Have at 'em!
Half are from Chris Long of Rutgers University, who is "Seeking people to
exchange problems/puzzles by e-mail, esp. on the order of difficulty of
the Putnam exam", and half are from Ambati Jayakrishna of John Hopkins.
I've collected and renumbered a few of their posted problems.


Path: decwrl!labrea!rutgers!topaz.rutgers.edu!clong
Subject: Re: Easy Problems to Do Over Lunch
Organization: Rutgers Univ., New Brunswick, N.J.
Xref: decwrl sci.math:2267 rec.puzzles:618
 
(1)	Show that ( 2^(2^m) + 1, 2^(2^n) + 1) = 1 if m<>n.

(2)	Find all solutions to:

		a^2 + b^2 + c^2 = d^2,

	where a,b,and d are prime.

(3)	Let R = M2(Z/(p)) , i.e. the set of 2x2 matrices with entries in
	Z mod p, p prime.  Show that if a member A of R is invertible,
	then A^q = 1 if q = (p^2-1)(p^2-p).  Show also that A^(q+2) =
	A^2 for all A's in R.

	This can be found on page 105 in Jacobson's book, and a similar
	problem can be found somewhere in Herstein.

(3)	Prove than n does not divide 2^n-1, unless n=1.  Generalize.

(4)	Calculate the Stuyvesant Town laundry function:
 
	One load of wash or wash or dry costs $1.  Each dryer can handle
	the equivalent of one and a half loads of wash.  How much does
	n loads of wash cost to wash and dry?
 
	This must be of the form W(n) = ....  No multi-line definitions allowed.

Chris Long
Rutgers University


Path: decwrl!decvax!ucbvax!ucbcad!ames!hao!oddjob!mimsy!aplcen!jhunix!ins_aajk
Subject: A few problems
Organization: Johns Hopkins
 
 
(5)	Find all 2-dimensional closed convex sets such that the locus of 
	midpoints of all chords of a fixed length l is a circle.
 
	             n
	            ---       k
(6)	Let S    =  >     a  i    ,  where n,k are positive integers,
	     n,k    ---    i                                         
	            i=1

	and a  = (+ or -) 1.
	     i          
 
	What is the smallest non-negative value of S    ?
	                                            n,k
 
	Can the a  be chosen such that S    remains finite as n,k --> infinity?
	         i                      n,k                     
 
 
(7)	What is the probability p  that the plane triangle formed by 3 points, 
	                         n
 
	chosen at random, independently, and uniformly with respect to surface 
 
	                          n  --   n+1
	content, on an n-sphere, S  (    R     , is acute?  BTW, P  = 0.25.
	                             --                           1
	                             --
 
 
(8)	Find all non-trivial solutions to the simultaneous set of equations 
	over the complex number field:
 
		x  + x  + x   =  a x  x  x
		 1    2    3        1  2  3
	 
		  2      2      2         2   2   2
		x   +  x   +  x    =  b x   x   x
		 1      2      3         1   2   3
	 
		  3      3      3         3   3   3
		x   +  x   +  x    =  c x   x   x     .
		 1      2      3         1   2   3
 
 
Ambati Jayakrishna
Dept. of Electrical and Computer Engineering &
Dept. of Mathematical Sciences
The Johns Hopkins University

    Does anybody else think the wash-dry problem is too easy, or am I
    missing something?
    
    I thought the solution given on the Usenet to the first problem,
    showing that gcd(2^(2^m)+1,2^(2^n)+1) = 1 if m <> n, was too long, so
    here is my version: 
    
    Both 2^(2^m) and 2^(2^n) must be congruent to -1 modulo some supposed
    common factor larger than 1, k, but one of them is some power of the
    square of the other, so it is also congruent to (-1^2)^p = 1, and -1
    and 1 are congruent only if k is two, which is obviously not the case. 
    
    
    				-- edp 

    Yes, the wash-dry problem does sound too easy.
    
    Isn't W(n) = 2, 4, 5, 7, 9, 10, 12, ...
    
    for     n  = 1, 2, 3, 4, 5, 6, 7, ... ?
    
    If that's the case, floor() does the trick.
    
    John