Vol. 99(7), November 1999
|
|
Vol. 99(7), November 1999 |
|
|
|
|
Solutions should be mailed before January 31, 2000. (Solutions
sent after that date will be considered as deadlines permit.)
4673-S: Student Problem - Only undergraduate
and pre-college students are eligible to submit a solution. Proposed
by Douglas Joy, Guelph, ON, Canada.
Replace the letters in the following cryptarithm by base
10 digits in a one-to-one manner so as to obtain a valid addition operation
in base 10. Also show that your solution is unique, i. e., that your solution
is the only possible solution.
As no solution was submitted, the problem remains open. (Hint: The digit represented by P in PRESTO must result from carrying. Therefore, P represents the digit 1. Next forxus on H and R.)H O C U S
+ P O C U S
P R E S T O
4722: Propsed by L. A. Adelani and J. H. Behe,
St. Louis, MO.
An abundant number is a positive integer for which the
sum of its proper divisors is greater that the number.
a) Show that 1575 is an odd abundant number.
b) If m > 1 is a positive integer
not divisible by 2, 3, 5, or 7, show that 1575m is an odd abundant
number.
4723: Student Problem - Only undergraduate
and pre-college students are eligible to submit a solution.
Proposed by the Editor,
Houston, TX.
In a sport championship two teams, A and B,
play each other until one team wins four games. Assume that team A
wins each game with probability p, 0 < p < 1, and team
B
wins each game with probability 1 - p. Assume each game is independent
of the other games.
a) Find the probability that team A
wins the championship.
b) Find the expected number of games in the
championship series.
4724: OBG: Oldie But Goodie - Proposer Unknown.
A function f is periodic if there exists a non-zero
r such that f(x + r) = f(x) for
all x in the domain. If there is minimum such r, then this minimum
valus is called the period of f.
a) Show that a function cna be periodic but
have no period.
b) If f and g are tow periodic
functions wiht the same domains, is f + g periodic? If not
always, what conditions ensure the sum is periodic?
4725: Proposed by Heinz-Jurgen Seiffert, Berlin,
Germany.
Suppose that a function 
is strictly convex and satifies
Prove the inequality
4726: Proposed by Heinz-Jurgen Seiffert, Berlin,
Germany.
Let be
an interval and suppose that
is
a convex function. Show that if
such
that,
then
where