MACAZINE
MATHEMATICS
376
perpendicularsfrom 1 to lines M B andMC, respectively.Provethat the value of
IE+IF
AM
is independentof the positionof M.
1810. Proposedby Greg Oman,OtterbeinCollege,Westerville,OH'
Let R be a ring. For elements.r, y € R we say x divides y on the right if and only
if thereis a z e R with xz : y.(We denotethis by xl'y.) An elementp € R is a
pl,x!,theneither pl,x ot pl,y. Provethatifevery
rightprimeif andonlyif whenever
elementof R is right prime, then R is a division ring, that is, the nonzeroelementsof
R form a group undermultiplication.(Note: R is not assumedto be commutativenor
is it assumedthat R hasa multiplicativeidentity.)
.
(.urcKres
t-
Answersto the Quickies are on page 381.
Q985. Proposedby Ovidiu Furdui, Campia-Turzii,Cluj, Romania.
LeI x be a real number.Evaluatethe sum
x2
x
S,/"
- ,t)
"'\
1 ,-\n -r- l!- 2t-
Q9S6. Proposed by Peter Ross,Santa Clara University, Santa Clara, CA.
Prove that in a given ellipse, there exist infinitely many inscribed triangles of maximal
area.
Solutions
December 2007
Growth of 1, -'l
!781. Proposedby Paul Bracken,Universityof Texas,Edinburg,TX.
Let y be Euler's constantand for positiveintegern define
y":
*I_ I
qn:2n(y,-y).
and
Li-rotn
,. monotonicallyincreasingandboundedabove.In addiProvethat ttr" ."qu".r"Jl,,
tion, determinelim,,-- 0n.
Solutionby AngetPlaza, Universityof Las Palmasde Gran Canaria,Las PalmasG.C.,
Spain.
1
Forn > 1, definethesequence
{f"}by Fn: Yn- y - }.fnen,2nBn - cv,,- 1. For
n > l, we have
|
fl u r - f,:;j 1
n
r\
/
1
tos( t *;) *
/
'\
l\
^,.,
1
n)
2n(n* 1)