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,
,
eX
dQc
I~p
tuyen
tlnh
trong
khong
gian cac
ham
s6
dUdng.
Call
2. Cho day
s6
(xn), (Yn), (zn) thOa man:
Xo
=
Yo
=
Zo
va:
Tfnh
X2011.
Xn+l
= 4xn -
Yn
-
5z
n
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
va
B.
Chung
minh
ding
t6n
t1?oi
s6
nguyen dUdng m
sao cho
em
=
o.
Call
4. Cho
da
thuc
P(x)
co
b~c
n
va
co
n nghi$m th\)'c (co
thEl
phan
bi$t
hoi;Lc
bQi).
TIm di§u ki$n
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:
Tren
mQt
bang
0 vuong n x
n,
A di§n
VaG
0 d vi
trf
(i,
j)
mQt
s6
nguyen dUdng
nao do.
B1?on
B
co
thEl
giu nguyen
s6
do
hoi;Lc
tang, giam
s6
do 1 ddn vi. B khl1ng
dinh ding
co
thEl
lam
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$n
dEl
M
sau
co
nghi$m duy nhiit:
(1
+
a)xl
+
(1
+ a
2
)x2 +
(1
+ a
3
)x3 +
(1
+ a
4
)x4 = 0
(1
+ b)Xl +
(1
+ b
2
)X2
+
(1
+ b
3
)X3
+
(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
)X2
+
(1
+ d
3
)X3
+
(1
+ d
4
)X4
= 0
[
11
-11].
b) Cho rna
tr~n
A =
Tfnh
A
2012
* * *
WWW.VNMATH.COM***
1
HOI
ToAN
HQC
VI~T
NAM
HE
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$m duy nhiit
tren
[~,
I]
va
ham
f'(x)
d6ng
bi§n.
(ii) Chung minh day (un) voi
Ul
= I,U
n
+l
=
f(u
n
)
co
cac
pMn
tu
d§u thuQc
do"n
[~,
I].
Call
2. Tfnh tfch phan:
1
J-
J
dx
1 + x + x
2
+
vi
x4
+
3x
2
+ 1
-1
Call
3. Cho hai day s6 (x
n
),
(Yn)
tMa
man:
Xn+l
;:;>
xnt
yn
,
Yn+l
;:;>
Vx~ty~,
mQi
s6
tv
nhien
n.
(i)
Chung minh ding cac day
Xn
+
Yn,
Xn-Yn
tang.
(ii) N§u cho truoc hai day (xn),
(Yn)
bi cMn. Chung minh hai day nay cling
hQi
t1J
v§
mQt
di§m.
Call
4. Cho
et,
(3
tMa
man
biit
d11ng
thuc:
(1
+ t)n+a < e <
(1
+ t)n+
i3
,
mQi
n
nguyen dUdng. TIm min cua
Jet
-
(3J.
Call
5. Do"n
[m,
n]
la
do"n
t6t
n§u
Ung
voi
a,
b, c
la
cac
s6
thvc
tMa
man
2a +
3b
+ 6c = 0
thl
PT
ax2
+ bx + c = 0
co
nghi$m thuQc
[m,
n].
TIm
do"n
t6t
co
dQ
dai
nM
nhiit.
Call
6.
(i)
TIm tiit ca cac
ham
s6
f(x)
tMa
man: (x -
y)f(x
+ y) - (x +
y)f(x
- y) =
4xy(x
2
- y2),
mQi
x,
y.
(ii) Cho
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
* * * WWW.VNMATH.COM * * *
2
. 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. tr~n A = Tfnh A 2012 * * * WWW.VNMATH.COM*** 1 HOI ToAN HQC VI~T NAM HE THI OLYMPIC ToAN SINH VIEN LAN THU XIV (2011) Mon: GIAI TICH Call 1. Cho ham s6 f(x) = (x~xl)2