ÈÊÇ Ä ÅË

325
PROBLEMS
Solutions to problems in this issue should arrive no later than 1 Marh 2010.
An asterisk (⋆) after a number indiates that a problem was proposed without a
solution.
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. In the solutions' setion, the problem will be stated in
the language of the primary featured solution.
Problems 3451, 3452, 3453, and 3454 below are dediated by the proposers to
the memory of Jim Totten.
The editor thanks Jean-Mar Terrier of the University of Montreal for translations of the problems.
3451.
Proposed by Jose
Luis Daz-Barrero, Universitat Politenia
de
.
Let (X, < ·, · >) be a real or omplex inner produt spae and let x, y ,
and z be nonzero vetors in X . Prove that
Catalunya, Barelona, Spain
X ||x|| 1/2
X < z, x > < x, y > 1/2
≤
|< y, z >| .
||x||
||y|| ||z||
yli
yli
3452.
Proposed by D.J. Smeenk, Zaltbommel, the Netherlands
.
Prove the following and generalize these results.
(a) tan2 36◦ + tan2 72◦ = 10,
(b) tan4 36◦ + tan4 72◦ = 90,
() tan6 36◦ + tan6 72◦ = 850,
(d) tan8 36◦ + tan8 72◦ = 8050.
3453.
.
Proposed by Sott Brown, Auburn University, Montgomery, AL,
USA
Triangle ABC has side lengths a = BC , b = AC , c = AB ; and
altitudes ha , hb , hc from the verties A, B , C , respetively. Prove that

8
X
yli

h2a (hb + hc ) + 16ha hb hc


X
√
√
≤ 3 3
a2 (b + c) + 6 3abc .
yli
326
3454.
.
Proposed by Rihard Hoshino, Government of Canada, Ottawa,
ON
Let a, b, c, and d be positive real numbers suh that a + b + c + d = 1.
Prove that
b3
c3
d3
1
a3
+
+
+
≥
.
b+c
c+d
d+a
a+b
8
3455.
.
Find the minimum value of x2 + y 2 + z 2 over all triples (x, y, z) of real
numbers suh that
Proposed by Mihel Bataille, Rouen, Frane
13x2 + 40y 2 + 45z 2 − 36xy + 12yz + 24xz ≥ 2009
and haraterize all the triples at whih the minimum is attained.
3456.
Proposed by Mihel Bataille, Rouen, Frane
3457.
Proposed by Mihel Bataille, Rouen, Frane
.
Given a triangle ABC with irumirle Γ, let irle Γ′ entred on the
line BC interset Γ at D and D′ . Denote by Q and Q′ the projetions of D
and D′ on the line AB , and by R and R′ their projetions on AC ; assume
that none of these projetions oinide with a vertex of the triangle.
Show that if Γ′ is orthogonal to Γ, then BQ′ = CR′ . Does the onverse
BQ
CR
hold?
.
Let An =
⌊n/2⌋
P
j=0
(−3)j
aj ,
2j + 1
where aj =
n
P
k=2j
k k+1
and n is a posi2j
2k
tive integer. Prove that A1 + A2 + · · · + An ≥ n with equality for innitely
many n.
3458.
Proposed by Brue Shawyer, Memorial University of Newfound-
.
Determine the entral angle of a setor, suh that the square drawn with
one vertex on eah radius of the setor and two verties on the irumferene,
has area equal to the square of the radius of the setor.
land, St. John's, NL
3459.
.
Let a, b, c and p, q , r be positive real numbers. Prove that if q 2 ≤ pr
2
and r ≤ pq , then
Proposed by Zafar Ahmed, BARC, Mumbai, India
a
b
c
3
+
+
≤
.
pa + qb + rc
pb + qc + ra
pc + qa + rb
p+q+r
When does equality hold?
327
3460.
Proposed by Tran Quang Hung, student, Hanoi National Univer-
.
The triangle ABC has irumentre O, orthoentre H , and irumradius R. Prove that
sity, Vietnam
3461.
3R − 2 OH ≤ HA + HB + HC ≤ 3R + OH
Proposed by Tran Quang Hung, student, Hanoi National Univer-
.
Let I be the inentre of triangle ABC and let A′ , B ′ , and C ′ be the
intersetions of the rays AI , BI , and CI with the respetive sides of the
triangle. Prove that
sity, Vietnam
IA + IB + IC ≥ 2(IA′ + IB ′ + IC ′ ) .
3462.
.
Let x, y , and z be positive real numbers suh that
Proposed by Sotiris Louridas, Aegaleo, Greee
Prove that
x3 + z 3 − y 3
y 3 + x3 − z 3
x3 + y 3 + z 3 + 3xyz
Y
yli
≤ 3
z 3 + y 3 − x3
x3 + y 3 − z 3 + xyz
> 0.
Y q
3
4
x4 (x2 + yz) .
yli
.................................................................
3451.
Propose
par Jose
Luis Daz-Barrero, Universite
Polytehnique de
.
Soit (X, < ·, · >) un espae vetoriel reel
ou omplexe muni d'un produit salaire et soit x, y et z des veteurs non nuls de X . Montrer que
Catalogne, Barelone, Espagne
X < z, x > < x, y > 1/2
≤
||x||
ylique
3452.
(a)
(b)
()
(d)
X ylique
||x||
||y|| ||z||
1/2
|< y, z >| .
.
Montrer les egalit
es
suivantes et gen
eraliser
es resultats.
2
2
◦
◦
tan 36 + tan 72 = 10,
tan4 36◦ + tan4 72◦ = 90,
tan6 36◦ + tan6 72◦ = 850,
tan8 36◦ + tan8 72◦ = 8050.
Propose
par D.J. Smeenk, Zaltbommel, Pays-Bas
328
3453.
Propose
par Sott Brown, Universite
Auburn, Montgomery, AL,
.
USA
Soit a = BC , b = AC , c = AB les longueurs des ot
^ es
du triangle
ABC , de hauteurs respetives ha , hb , hc issues des sommets A, B , C . Mon-
trer que

8
X

ylique
3454.
.
h2a (hb + hc ) + 16ha hb hc


X
√
√
≤ 3 3
a2 (b + c) + 6 3abc .
ylique
Propose
par Rihard Hoshino, Gouvernement du Canada, Ottawa,
ON
Soit a, b, c et d des nombres reels
non negatifs
ave a + b + c + d = 1.
Montrer que
a3
b+c
3455.
+
b3
c+d
+
c3
d+a
+
d3
a+b
Propose
by Mihel Bataille, Rouen, Frane
≥
1
8
.
.
Trouver la valeur minimale de x2 +y 2 +z 2 pour tous les triplets (x, y, z)
de nombres reels
tels que
13x2 + 40y 2 + 45z 2 − 36xy + 12yz + 24xz ≥ 2009
et arateriser
tous les triplets pour lesquels le minimum est atteint.
3456.
Propose
par Mihel Bataille, Rouen, Frane
.
Etant
donne un triangle ABC et Γ son erle ironsrit, soit Γ′ un
erle entre sur la droite BC et oupant Γ en D et D′ . Designons
par Q et Q′
′
′
les projetions de D et D sur la droite AB , et soit R et R leurs projetions
sur AC ; supposons qu'auune de es projetions ne oinide ave un sommet
du triangle.
Montrer que si Γ′ est orthogonal a Γ, alors BQ′ = CR′ . La reiproque
BQ
CR
est-elle valide ?
3457.
.
Propose
par Mihel Bataille, Rouen, Frane
⌊n/2⌋
P
n
P
k k+1
et n est un entier
2j
2k
j=0
k=2j
positif. Montrer que A1 + A2 + · · · + An ≥ n ou l'on a egalit
e pour une
innite de n.
Soit An =
(−3)j
aj ,
2j + 1
ou aj =
329
3458.
Propose
by Brue Shawyer, Universite
Memorial de Terre-Neuve,
.
Determiner
l'angle au entre d'un seteur tel que le arre,
onstruit ave
un sommet sur haque rayon du seteur et deux sommets sur la ironferene,
possede
une aire egale
au arre du rayon du seteur.
St. John's, NL
3459.
.
Soit a, b, c et p, q , r des nombres reels
positifs. Montrer que si q 2 ≤ pr
et r2 ≤ pq , alors
Propose
par Zafar Ahmed, BARC, Mumbai, Inde
a
pa + qb + rc
+
b
+
pb + qc + ra
c
pc + qa + rb
≤
3
p+q+r
.
Quand y a-t-il egalit
e ?
3460.
Propose
par Tran Quang Hung, etudiant,
Universite
Nationale de
.
Soit O le entre du erle ironsrit du triangle ABC , de rayon R , et
H son orthoentre. Montrer que
Hanoi, Vietnam
3R − 2 OH ≤ HA + HB + HC ≤ 3R + OH
3461.
Proposed by Tran Quang Hung, etudiant,
Universite
Nationale de
.
Soit I le entre du erle ironsrit du triangle ABC et soit A′ , B ′
et C ′ les intersetions des rayons AI , BI et CI ave les ot
^ es
respetifs du
triangle. Montrer que
Hanoi, Vietnam
IA + IB + IC ≥ 2(IA′ + IB ′ + IC ′ ) .
3462.
.
Soit x, y et z trois nombres reels
positifs tels que
Propose
par Sotiris Louridas, Aegaleo, Gree
x3 + z 3 − y 3
Montrer que
y 3 + x3 − z 3
x3 + y 3 + z 3 + 3xyz
Y
ylique
≤ 3
z 3 + y 3 − x3
> 0.
x3 + y 3 − z 3 + xyz
Y q
3
4
x4 (x2 + yz) .
ylique