stock@uwvax.ARPA (daniel L. stock) (09/25/83)
Here are three puzzles for net.math fans. Except as noted, these
were made up by me within the last ten years or so (references to
older sources are welcome). PLEASE DO NOT ANSWER THESE ON THE NET;
send answers to me, and I will post the best answers that I get in
a month or so. I will also list all solvers at that time.
People who answer these on the net before I post the answers will
be raised to the 1/pi'th power and taken in the limit to zero 8-) !
1. a. (Appears in a book of Soviet math olympiad problems, whose
title I forget) X and Y are positive, rational, unequal, and
satisfy
X**Y = Y**X ("**" is the exponentiation operator).
Characterize all such pairs (X,Y).
b. Find the largest integer I and the smallest integer J such
that if X and Y satisfy the conditions of part (a), then
I < X**Y <= J .
2. Three rhombi (A rhombus is a parallelogram with all sides having
equal length) are inscribed in a scalene triangle (i.e., a
triangle having no two sides with a common length) in such a
way that each rhombus has one vertex that coincides with one
vertex of the triangle, and the other three vertices of each rhombus
lie on the three sides of the triangle.
A-----J---F---G------B (In the figure, the triangle is ABC;
| / /| | / the rhombi are AGDH, BIEJ, and CKFL).
| / / | | /
| / / | | / Show that the smallest of the three
| / / | | / rhombi is the one which contains the
|/ / | | / vertex of the triangle with the angle
E---+-----+---+I of intermediate size (i.e, neither the
H--+------+---D biggest nor the smallest of the three
| / | / angles).
|/ | /
K |/
| L
| /
| /
| /
| /
| /
| /
| /
| /
|/
C
3. Characterize those positive integers x such that the number represented
by a sequence of x ones considered as a unary number divides the number
represented by a sequence of x ones considered as a decimal number.
[Less romantically, find those positive integers x such that x divides
(10**x - 1) / 9 .
We shall say x is UDD (unary-decimal divisible) if this holds
(Please, no UDDerly ridiculous puns 8-) ).
I have not yet solved this problem totally, so partial results
are welcome. The results I have so far include:
a) If x is UDD and is greater than one, then 3 divides x.
b) All (non-negative integer) powers of three are UDD.
c) If x is UDD, then (10**x - 1) / 9 is UDD.
Any partial solutions to this problem should be at least
as strong as (a), (b), or (c)., or should shed light where
(a), (b), and (c) do not.]
Have fun with these!
-- daniel stock
stock@uwisc
...!seismo!uwvax!stock