HOI
ToAN
HQC
VI~T
NAM
HE
THI
OLYMPIC
ToAN
SINH
VIEN
LAN
THU
XIV
(2011)
Mon:
D.A-I
s6
, { 2 3
n}
"
Call
1.
Chung
minh
rang
h$
eX,
eX
,ex,


Yn+l
=
2x
n
- 2zn
Zn+l
=
Xn
- 2zn
Call
3.
Cho hai rna
tr~n
A
va
B cling
dip
n
va
rna
tr~n
e =
AB
-
BA
giao
hoan
voi ca hai rna
tr~n
A

c§,n
va
du
cua
u
va
v
dEl
da
thuc
sau
cling
co
n nghi$m th\)'c:
P(X)
+ uP'
(x)
+ vpl/(x).
Call
5.
Co hai
b1?on
A
va
B chdi
mQt
tro
chdi
nhu
sau:

cho rna
tr~n
nMn
dUQc
kha
nghich
va
khong
co
diElm
biit
dQng
(tUc
la
t6n
t1?oi
vector v sao cho
Av
= v) . Hoi B
nMn
dinh dung
hay
sai?
VI
sao?
Call
6.
a) TIm di§u ki$n
dEl
M

+
(1
+ b
4
)X4
= 0
(1
+
C)Xl
+
(1
+ C
2
)X2
+
(1
+ C
3
)X3
+
(1
+ C
4
)X4
= 0
(1
+ d)Xl +
(1
+ d
2

THI
OLYMPIC
ToAN
SINH
VIEN
LAN
THU
XIV
(2011)
Mon:
GIAI
TICH
Call
1. Cho
ham
s6
f(x)
=
(x~xl)2
(i)
Chung minh
PT
f(x)
= x
co
nghi$m duy nhiit
tren
[~,
I]
va

vi
x4
+
3x
2
+ 1
-1
Call
3. Cho hai day s6 (x
n
),
(Yn)
tMa
man:
Xn+l
;:;>
xnt
yn
,
Yn+l
;:;>
Vx~ty~,
mQi
s6
tv
nhien
n.
(i)
Chung minh ding cac day
Xn

Jet
-
(3J.
Call
5. Do"n
[m,
n]
la
do"n
t6t
n§u
Ung
voi
a,
b, c
la
cac
s6
thvc
tMa
man
2a +
3b
+ 6c = 0
thl
PT
ax2
+ bx + c = 0
co
nghi$m thuQc

ham
s6
f(x)
kha
vi
trong do"n
[-1,
I]
va:
xf(x)
+
~f(~)
<::
2,
\Ix E
[~,
2].
Chung minh ding:
JE
f(x)
<::
2.1n2.
2
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