Mayhem Problems: M363-M368

396
MATHEMATICAL MAYHEM
Mathematial Mayhem began in 1988 as a Mathematial Journal for and by
High Shool and University Students. It ontinues, with the same emphasis,
as an integral part of Crux Mathematiorum with Mathematial Mayhem.
The Mayhem Editor is Ian VanderBurgh (University of Waterloo). The
other sta members are Monika Khbeis (Asension of Our Lord Seondary
Shool, Mississauga), Eri Robert (Leo Hayes High Shool, Frederiton), Larry
Rie (University of Waterloo), and Ron Lanaster (University of Toronto).
Mayhem Problems
Please send your solutions to the problems in this edition by 15 January 2009.
Solutions reeived after this date will only be onsidered if there is time before
publiation of the solutions.
Eah problem is given in English and Frenh, the oÆial languages of Canada.
In issues 1, 3, 5, and 7, English will preede Frenh, and in issues 2, 4, 6, and 8,
Frenh will preede English.
The editor thanks Jean-Mar Terrier of the University of Montreal for translations of the problems.
M363. Proposed by Brue Shawyer, Memorial University of Newfoundland, St. John's, NL.
Suppose that A is a six-digit positive integer and B is the positive integer formed by writing the digits of A in reverse order. Prove that A − B
is a multiple of 9.
M364.
Proposed by the Mayhem Sta.
A semi-irle of radius 2 is drawn with diameter AB . The square P QRS
is drawn with P and Q on the semi-irle and R and S on AB . Is the area of
the square less than or greater than one-half of the area of the semi-irle?
M365.
Proposed by Alexander Gurevih, student, University of Waterloo,
Waterloo, ON.
Let D be the family of lines of the form y = nx + n2 , with n ≥ 2
a positive integer. Let H be the family of lines of the form y = m, where
m ≥ 2 is a positive integer. Prove that a line from H has a prime y -interept
if and only if this line does not interset any line from D at a point with an
x-oordinate that is a non-negative integer.
397
M366.
Proposed by John Grant MLoughlin, University of New Brunswik,
Frederiton, NB.
The roots of the equation x3 + bx2 + cx + d = 0 are p, q, and r. Find
a quadrati equation with roots (p2 + q2 + r2 ) and (p + q + r).
M367.
Proposed by George Tsapakidis, Agrinio, Greee.
For the positive real numbers a, b, and
have a +
√ c we √
√b + c
Determine the maximum possible value of a bc + b ac + c ab.
= 6.
M368.
Proposed by J. Walter Lynh, Athens, GA, USA.
An innite series of positive rational numbers a1 + a2 + a3 + · · · is
the fastest onverging innite series with a sum of 1, a1 = 12 , and eah ai
having numerator 1. (By \fastest onverging", we mean that eah term an
is suessively hosen to make the sum a1 + a2 + · · · + an as lose to 1 as
possible.) Determine a5 and desribe a reursive proedure for nding an .
.................................................................
M363. Propose par Brue Shawyer, Universite Memorial de Terre-Neuve,
St. John's, NL.
Soit A un entier positif de six hires et B l'entier positif forme des
hires de A erits
dans l'ordre inverse. Montrer que A − B est un multiple
de 9.
M364.
Propose par l'Equipe
de Mayhem.
Dans un demi-erle de rayon 2 et de diametre
AB , on dessine un arre
P QRS ave P et Q sur le demi-erle, et R et S sur AB . L'aire du arre estelle plus petite ou plus grande que la moitie de elle du demi-erle ?
M365.
Propose par Alexander Gurevih, etudiant,
Universite de Waterloo, Waterloo, ON.
Soit D la famille des droites de la forme y = nx + n2 , ave n ≥ 2,
un entier positif. Soit H la famille des droites de la forme y = m, ave
m ≥ 2, un entier positif. Montrer que l'ordonnee
du point d'intersetion
d'une droite de H ave l'axe des y est un nombre premier si et seulement si
ette droite ne oupe auune droite de D en un point d'absisse entiere
non
negative.
M366.
Propose par John Grant MLoughlin, Universite du NouveauBrunswik, Frederiton, NB.
Soit p, q et r les raines de l'equation
x3 + bx2 + cx + d = 0. Trouver
une equation
quadratique dont les raines sont (p2 + q2 + r2 ) et (p + q + r).
398
M367.
Propose par George Tsapakidis, Agrinio, Gree.
Soit a, b et c trois nombres reels
positifs
que a +√b +
√ tels √
Determiner
la valeur maximale possible de a bc + b ac + c ab.
c = 6.
M368.
Propose par J. Walter Lynh, Athens, GA, E-U.
Une serie
innie de nombres rationnels positifs a1 + a2 + a3 + · · ·
est la serie
la plus rapidement onvergente de somme 1, ave a1 = 12 et
haque ai de numerateur
1. (Par \la plus rapidement onvergente", on entend que haque terme an est tour a tour hoisi de maniere
a e que la somme
a1 + a2 + · · · + an soit aussi prohe de 1 que possible.) Determiner
a5 et
derire
une proedure
reursive
pour trouver an .
Mayhem Solutions
The Editor would like to thank Emily Saltstone, Gravenhurst High Shool,
Gravenhurst, ON, for her help in preparing this month's solutions.
M325. Proposed by Brue Shawyer, Memorial University of Newfoundland, St. John's, NL.
Let a, b, and c be non-zero digits. A student takes the fration ab
,
ca
where ab and ca represent the two-digit integers 10a + b and 10c + a, and
applies a (false) anellation law, anelling the a from the numerator with
the a from the denominator. For example, if a = 6, b = 5, and c = 2, the
student would obtain 65/26 = 5/2 (by `anelling' the 6s!).
Determine all triples (a, b, c) for whih this student atually obtains the
orret answer.
Solution by Samuel Gomez
Moreno, Universidad de Jaen,
Jaen,
Spain, modied by the editor.
If
10a + b
b
= , then c(10a + b) = b(10c + a) or 10c(a − b) = b(a − c)
10c + a
c
so 2 · 5 · c(a − b) = b(a − c). Therefore, 5 must be a divisor of the right hand
side.
Sine 5 is a prime number, then 5 must be a divisor of either b or a − c.
Sine a, b, and c are integers suh that 1 ≤ a, b, c ≤ 9, then either a − c = 0
(whih would imply that a − b = 0), b = 5, or a − c = 5.
Case I: We have a − c = 0 and a − b = 0.
Here a = b = c, and we get nine triples (a, a, a), where a ∈ {1, 2, . . . , 9}.