giáo trình một số điểm cần lưu ý khi dạy học đại số và giải tích 11 nâng cao

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TRIING II
c(} s(}
TUIWffi BS:l; uu#.ilr
@ffih@M
PHAM
oUc
uor
(GV
trudng THCS
NguyOn
Luong
Adng,
Thonh
MiQn, Hoi
Duong)
Trong Sdch gi6o
khoa
llinh hoc
9
t6p
hai
trang
105 c6
bli
to6n sd
9, n6i dung nhu
sau
Sgai
to6n 1.

Cho tam
gidc
ABC vu6ng
d A.
Tr€n canh
AC ldy
m\t didm M vd vd
dtdng
trbn
V,
dudng
kinh
MC. Dudng thdng
BM cdt
V,
tqi D.
Dudng thdng
DA cdt
W,
tqi
S. Chtng
minh
rdng:
a)
ABCD b trt
gidc
ndi tiip.
b)
CA ld tia
phdn

gidc
cil.a
g6c
SCB.
Ldi
gitii.
a) Do tam
gi6c
ABC
vuOng
tai A nOn
6k
=9O
(1)
D nam tr6n dudng
trbn
31
nen
fii
=
90'
(2)
Tt
(1)
vn
(2)
suy
ra tir
gi6c
ABCD nQi tidp

dudng trbn
?"
dudng
kinh BC.
b) Ta c6 frA
=
fiA
GO"
nQi
tidp
cirng ch6n
iE cna
V")
(3)
X6t ba
trudng ho,p:
+)
Trulng hqp S
nam ren cung
nhb frD
G.
1).
X6t dudng trbn
(fr):
frB
=ffi
1"mg
chin
ffi)
(4)

Tt
(3)
vn
(4)
suy ra frA=fr-S .
Hinh I
+)
Trudng hgp D
nh6 fu
$.2).
ndm trOn
cung
X6t
dudng trdn
(v,):
MDA
=
MCS
(ctng
bt vdi
MDS)
(s)
Tt
(3)
va
12
.uv
ra BCA= MCS .
Hinh 2
+)

Trudng
hqp D trtng vdi S. Khi d6 AD
h tidp tuydn cria
dudng trdn
(%).
Ta c6
frA=ffe,=fus.a
Khi
giii
c6u
b) nhidu ban thudng chi x6t mot
trudng
hqp
(tiy
thu6c vio
hinh
v6
ra). Hon
nfia ch6c
it
ban
dat duoc cdc cdu
h6i:
Vi trf
cfra
didm M th€ nio dd didm
D nam trOn cung
nh6 ffi ? Didm
S nim ren cung nnb
frD

t
Didm S trDng vdi
D?
Ogai
1,rl6n 2. Cho tam
gidc
ABC
vudng
tai
A.
Didm
M
di dQng
tr€n canh AC, vd dudng
trdn
ff,
drdng kinh
MC, BM cdt
V,
tqi
D, dudng
thdng
AD cat
fi
tai
S. ){dc
dinh vi
tri cila didm
M
tr€n

canh AC dd:
a) Didm S
ndm ftan cung nhd frD .
b) Didm
D
ndm tr€n
cung nhd ffi
Ldi
gitii.
a)
(h.
3) S thu6c
cung
nh6 frD
yni
vi
chi
khi m
.fr60
.
Do ha.i tam
gi6c
OMS
vd
OSD
ddu cdn tai
O
n6n
suy
ra

OMS
>
OMD
o
OMS
>
AMB
(6)
TOff{
FiA(
'
cfr*di$
si x61
l7-zoo7l
I
D
Hinh
3
Ndu
ggi
giao didm
cia
Vt
v6i
BC
le
/
thi
frie
=

900
din
ddn
tr1
gi6c
ABIM
noi
tidp
duo.c,
nen
frc
=
tfri
(ctng
bir
vot ,qui
)
(7)
Lai
do
ICM
=MCS
(biLi
to6n
1
cau
b)
vi'MC
id
dudng

kinh
cira
dudng
ttdn
Vr
nOn
1
vi S
ddi
xrlng
nhau
qua
MC
c6 ifra
=eES
(8)
Tt
(7)
vn
(8)
d6n
ddn
ffi
=trEa
(e)
Til
(6)
vI
(9)
suy

ra
trEi
rg0'
-m
<> 90"
-
frB
>
90"
-
Tnfu
o
frtw
.frn.
b)
Lap
1u6n
tuong
tu
ta
c6
kdt
qui D
thu6c
cung
nh6
ffi
khi
vd chi
khi

frilrfri.
u
tt
thu6c
canh
AC
n6n
cdn
c6
didu
kiQn
AC
> AB'
A
TiI
bai
bdn2
d6n
ddn
suY
nghi:
Khi
ndo
hai
didm
S
vh D
tring
nhau?
Tt

d6
ta
cd
bii
to6n
sau.
Ogai
to6n
3.
Cho
tam
gidc
ABC
vubng
tai
A,
Didm
M
thuAc
canh
AC,
vd
dtdng
trdn
V1
dudng
kinh
MC,
BM
cdt

Vrtai
D.
){rtc
dinh
vi
tri
cria
didm
M
ffan
canh
AC
dd
AD
ld tidp
ruyAh
cfia
dudng
trdn
ffr.
L6p
luan
tudng
tu
bdi
to6n
2
ta
thdy
AD

la
tidp
tuydn
cfra
dudng
trbn
V,
khi
vd
chi
khi
frfu
=frB
. Dd
cho
M
thuOc
canh
AC
non
cdn
c6 didu
ki6n
AC
> AB.
J
Ddn
day
ta
thdy

d6ng
phii suy-nghr
vd"vi tri
ctra
didm
M.
Do
didlrr.
M
c6
thd
thay
ddi
tron
canh
AC
nOn
nhidu
didm
kh6c
cfing
thay
ddi
theo.
C6c
ban
hdy
quan
s6t
tia AI,

n6
thay
ddi
n6n
giao didm
K
cira
n5
v6i
dudng
trdn Vt
cfing
thay
ddi
theo.
DAn
ta
ddn
bbi
to6n
sau'
T6##t
e+G{
2
Sd
361
(7-2007)
&
cigjige
Ogai

!,rldn
4. Cho
tam
gidc ABC
vuing
tai
A.
Didm
M
thay
ddi
ffan
canh
AC.
Dung
dtdng
trdn
Vt
dtdng
kinh
CM.
Goi
giao
didm
cila
BC
idi
duang
trdn
Wt

ld
I,
giao didm
cfia
AI
vdi
dr.tdng
trbn
Vr
ld K.Tim
tQP
hqP
didm
K'
Ldi
sitii.
(h.4).
Do
hai
tia
AI
vi
AS
ddi
xfng
nhau
qua
duong
kinh
MC

cia
dudng
trdn
V,
vi
A
thu6c
dudng
thing
MC
n€,n
ta
da
dhng
chfng
minh
duoc
K
vi D
ddi xrlng
nhau
qua
AC.
Do
f6d
=
90"
nOn
c5
thd

chi
Hinh
4
ra
duo.
c
quY
tich
didm
D
khi
M
chuydn
ddng
trdn
canh
AC
ld
cung
nh6
tri
ciadudng
trbn
dudng
kinh
BC'
Tt
d6
ta
d1r

do6n
qu! tich
didm
K
khi
M
chuydn
dQng
tr6n
canh
AC
li
hinh
ddi
xung
vdi cung
nh6
iD
qua canh
AC.
Qu!
tfch
niy
li
m6t
cung
trbn
cira
dudng
ttdn V2

ddi
xrlng
vdi dudng
trdn
dudng
kinh
BC
qua canh
AC'
Tt
d6
taxic
dinh
duoc
dudng
lrdn
W,
nhu
sau:
-Ldy
B'
ddi
xrlng
vdi B
qua canh
AC;
-
DUng
dudng
trdn

W,
dudng
kinh
B'C.
Srl
dung
kdt
qui cita
ciic
bli
tdp
tren
ta d6
dlng'ch'rlng
minh
duoc
B',
M,
K
thing
hing
suy
ra
B'
KC
=
MKC
=
90".
V4y

qu! tich
didm
K
khi
M
cbuydn
d6ng
trOn
canhAC
1I
cung
rrho
ID
cira
dudng
trdn
dudng
kinhB'C.
D
Nhu
vay ndl
quan siit
tdt,
nghi0n
crlu
ki
thi
chi
tt
mOt

bii
tap
trong
s6ch
gi6o
khoa,
chring
ta
c6 thd
khai
ih6c
duoc
nhidu
kdt
qui thri
vi'
Chfc
c6c
ban
gat
h6i
duoc
nhi6u
kdt
qui'
rdu
GL\n mrA
flFtrfi
cmot\
Hoc

sNFil
G]roflrdp
s
g(p.766
@hiJllinh
NAM rrQC 2006
-
2007
1Oi
m tfing fiAn THTT
sd 359, thdng 5 ndm 2007)
Cflu
1.
U)
piAu
kiQn x2
-
4
>
o,hay x
>
2hoic x
<
-2.
Fiat
J*
-+
:
y
A

>
0).
Phuong trinh cl5 cho
trd thinh
Ei
-tr
^l+*-v
=8
-(y'+4':t
<> 2rz +y-6=0.
v4
l-r =-
a) Ta c6 A:
16+2^,15
-{(2Vs -3)2
=3.
b)B=
",!t*zJ|+2Ji+z$o
=
J1r
+
Ji
*
Ji),
:
r +
J1*.,6.
c) c
:
[,r$-,r

*
qFr.a
-ry"P](G+r
r)
'
\
6-1
6-22 32
-6
)''-
=
(G-llxG+ll)=-115.
Cffu 2. a)Tathlty
(x+y*r)'.3(x2+y'*r')
o 2x2
+2y2 +222
-Zxy -2y,
-Zzx>
0
e
(x- y)'
+(y
-
r)' +
(z
-
x)2
>
0.
BDT'nAy dring. Ding thric xiy ra khi;r

:
y:
z.
b) Ap dsng k6t
qui
cAu
a)
1.1a,
a1+
Jiy
+t +
^{a,
a1z
<
3(4x+l+4y+1+42
+1)=21.
Suy ra
J4x
+1 +
r!+y
+t
+,{a, 11 <,121 .
Ding thirc x6y ra khi
vi chi
khi
CAu 3.
g)
Ta thdy
x,
y,

z Y,hLc 0. He da
clugc vi6t lai
dudi d4ng
11s
xy12
115
yzlS
1 I 13
zx36
Giei hQ
ndy
tim dugc
;r
=
4,
!
=
6, z
:
9.
PT ndy
vd
d6i
chi5u
v6'i
di0u
kiQn
tr€n ta
3-5
duoc v

=
a.
DAn d6n x:
+:
(thoa
m6n
2 2'
kiQn).
luqn: Phuong trinh tld cho c6 hai nghiQm
55
22
CAu 4. Nhqn xdt: MQt s6 chinh
phuong
16 khi
chia cho
8 sE
c6 15 au la t.
Thflt viy, xdt s5 chinh
phuongld
m2
(m
e Z),
m
lil
s614, dflt m
=
2n
*
1.
Lric nity m2

:
(2n
+
1)2
:
4nz
*
4n
+
I
=
4n(n+ l)
+
1 ohia cho 8 du i.
Trd lai bdi to5n, ta c6 biQt
thric
A
:
b2
-
4ac.
Do
b
16 n6n theo nhfn xdt tr€n b2 chia cho 8
du 1. D[t b2
=8k
+
I
(/r
e Z). L4i vi a,cld

nln ac le, dAt ac
=
2l
-
|
(l
e
Z).
Khi d6 A
= 8k+
I
-
4(21
-
l)
=
8(t- /)
+
5
chia cho
8
du 5.
Tir nhfln xdt tr6n ta th6y A
kh6ng
ph6i
ld
s6
chinh
phuong.
! .

-brJi
NghiQm PT de cho
(ndu
c6) ld x
:
-
:
4d
oo
JA
ld sti v6 ti, n6n n6u PT cli cho
c6
nghiQm thi c6c nghiQm 6y kh6ng
tfr€ ta
sO
hfru
ti
(ctpcm).
Cflu s.
a)
(h.
1). ra c6 dfrfi
=zIfrH ,
DMH
=2dfrfr
.
Giei
tim
:
dleu

Xet
I
3
cho
rGffi
s+s(
-
cfirdiE6
s6 361
l7-zoo7)
6
Cflu
6.
,DUng
tia
tlep
tuyen
Cx
v6i
dudng
trdn
(O),
tia Cx
nim
tr6n
nria
mAt
phing bd
BC
kh6ng

chria
di6m
A
(h.2).
Khi d6
Hinh
I
Suy
ra dfru
*6ME
:
180o,
d5n
d€n
M, c,
D
thing
hing.
Mat
kh6c
do OM
li dudng
trung
binh
cira
hinh
thang
ABDC
nOn OMllAC,
md

AC I
CD suY
ra
cb
L oM.
NghTa
liL cD
le
ti€p
tuy6n
cira
nira
dudng
trdn
t6m 9.YdY
c6c
<li6m
M, C,
D
cirng
nirn-tr€n
titip tuy6n
ctra
ntra dulng
trdn
(o)tai
M.
b)
.
Ta c6

AC
+
BD:ZOM:2,R
(kh6ng
d6i).
o
Ti
LCMA
<n
L.DBM
(g.g)
=
#=
,rW
Do
d5
AC.BD =
MC.MD
=
'?'
4
c) Tir
OH.OK
:
OIri,
suy
ra OAz
=
OB2 =
OH.OK,

6d=ila
=ffia
=
CxllMN.Do
OC
-]-
Cx
n6n OC
L MN.
o
Tir LCMN
<n
LCBA(g.g)
suy
,^
#=H
U,
vu6ng
tai
N,
c6
ACB=
45o,
(2).
Tri
(1)
vd
(2)
ta thu
dusc

(dpcm).
NGUYEN
rAx
roaN
gP.
Hi chi
Minh)
suu
tAm
vd
gi6i thiQu
NAC
t;
2
Tam
gi6c
^CN
nen
-=
CA
1aN=
4!
JZ
TOAN
HQC
(Ti€p
trang
3o)
T
cHurtll

MnlI
cffirill
TrOn
mOt
bing
c6 25
b6ng
dEn nhu
A hinh
1,
m6i
dEn
c6
miu D6
ho[c
mhu
Xanh.
MQt
bing
didu
khidn
c6
cdc c6ng
t6c
dugc
ghi theo
toa dO
dEn. Khi
dn
c6ng

tic
cfia mdt
dEn nio
ddy
thi
ddn
d6
vdtdt
ch c6c
ddn tidp
xric
v6i dEn
d6 ddu
ddi
miu.
Ching
han
tr6n
hinh
1 ndu
dn
c}rrgtic
D2
thi
c6c ddn
D2,DL,D3,E2,
C2
ddu ddi
miu.
Dinh cho

ban
doc
1) Tr0n
hinh
1 c6
8 dEn xdp
thinh
hinh
vu6ng
miu
D6,
c6c
ddn
cdn lai
miu
Xanh.
Hdy
dat
mdt lOnh
(mQt
d6y
s6p thrl
tu su
dn
c1ngtic
cdc
ddn)
dd
sau khi
thlrc

hiQn
lenh
d6
thi
tdt ch
cic
dBn
ddu
c6 miu
D6.
4
sd36r
(7.2oonrffirHm
2) Trdn
hinh
2
c6 5
ddn xdp
thinh
hinh
chfr
T
(vidt
t6t
cira
To6n Tudi
tr6) miu
D6,
c5c ddn
cdn

lai
mhu
Xanh.
H6y
d{t
mqt
lQnh
dd
sau
khi
thuc
hien
lonh
d5 thi
tdt
ce cdc
ddn
ddu
c6
miu
D6.
Chfi
y:I-fnh
cing
it ldn
dn
cOng
t6c
chng
tdt.

t23
4 5
Hinh
2
PHI PHI
E
D
C
B
A
E
D
C
B
A
K
uffi
m
fifij$
u#p
t,oj
Tr
a:Ulfiffi
pfl
DAI
HOC
C6u
l.
Cho
phuong

trinh
xz
-zxJi
+zJi(Ji
+1)-3
x-l
-0
(1)
a) Tim m dO x:
-1
le
mQt nghiQm cria
phuong
trinh
(1).
b) Tim
rn
d6
phuong
trinh
(l)
vO nghiQm.
CAu
2.
a) Gi6i b6t
phuong
trinh
|
(-x
+ 3)(-r

-r)l-21
,
-
I
l<
x2
-7
.
b) Giei hC
phuong
trinh
IrJ,
+2yJi
=3xJ2x
)
[yJi*
2*Ji
=tyJ41.
CAu
3.
a)
Cho a,
b
ld
hai sti thr,rc
th6a rndn
clleu l(len
a2-3ab+2b2+a-b
=
o'-2ab+b2-5a+7b:0.

Chirng
t6 ring ab
-
12a
+
156
:0.
b) Cho
A
-
d,e
+
q
-zX!'
+J;
+tX.[*, a +2il
*
-2J;
+t
,t.li-A
Hdy tim
tdt ca cac
gi6
tri cta x d6 A
> O.
lllffil.G]
CHI
MINH
inddN
TCIAN

AB
{dd,rafa
aeno
affiffiara
Tod,ra,
Tdra,
f-{, E{6a,
Sdw4a}
Thdi
gi.an ld,m
bd;i: 150
phdt
Cha
4.
Cho tam
gi6c
ABC
nhon
c6 tryc t6m li
H
vit
ilC:
60o. Ggi M, N, P
l6n luqt ld
ch6n
c5c
dudng cao
k0
ti A, B, C cuatam
gi6c

ABC
vi / ld trung cli6m cta BC.
a) Chring minh ring tam
gi6c
1NP d6u.
b) Gqi E vd K lin
luqt
ld trung.di6m cria PB
vd NC. Cht?ng minh rdng c6c di6m I, M, E, K
cr)ng thuQc mdt
dudng trdn.
c)
GiA sri /l ld
ph6n giric
ctta friP. Hay
tinh
s6
do
cta
g6c
BCP.
CAu 5. MQt cdng.ti
may
giao
cho td A may
16800
s6n
phdm,
td B
may 16500 sAn

phAm
vd
bat Oiu thUc hiQn
c6ng viQc cirng mQt lirc. N5u
sau
s6u ngiy, tt5 A dugc h5
trq th6m
10
c6ng
nhdn may
thi ho hoin thdnh
cdng viQc cing
lirc v6i tO
g.
N6u td A duqc h5 trq
th6m
10
c6ng
nhdn may ngay
tir etAu thi hq
s€
hodn
thinh
c6ng.viQc
sdm hon tis B mOt ngiy.H6.y
x5c tlinh
s6 c6ng nhAn
ban tliu cria m5l tO.
Bi5t ring
m5i c6ng nhdn m5i

ngiy may dugc
20
s6n
phAm.
NGUYEN
DUc TAN
(Tp.
Hd chi Minh)
suu tAm vd
gi6i
thiQu
BINH LUAN
(Ti€p
trans
to)
vay
l0
zs=
itr
+x)2ndx-
jf,*
x)2'dx
="r,,'
.r'
.
o
_i
Zn+l
Ta duoc
ili6u cAn chimg

minh.
D
NhQn
xet 1)
C6 thC x6t
P(x)
-(1+x)2"
-(l-a)2"
t
sau ct6 tinh
[fi.x1a"
dt1
suy ra k6t
qui.
0
2) N6u phdi
tinh t6ng
cg, +Lc1, *1"i,
* *rf
,cin
thi
ta x€t P(x)=(l+x)2"
+(l-x)2",
sau d6
I
tinh
Jrlr;or.
0
Bdi
luyQn tSp. Tinh

t6ng
t
=|."r,
**"7, *I"i, *
*-L-c*.
Ipf,f-t
HAC
.,
Gli.rdiF
s6 361
l7.2oa7l
5
ehudn
bi
fhi
vao
Dft:I
nqe
:1
Ntiu
hC
PT
ba
hn x,y,
z kh6ng
thay
ddi
khi
noan
vi

ulng
quurh OOi
,Oi
x,
!,
z
th\
kh6ng
m6t
tinh
t6ng
quit c6
th6
gin thi6t
x
=
max
(x, y,
z).
Nghia
ld x
>
!,
x)
7
(xem
thi
du
3).
ViQc

su
dpng
khio s6t
bi6n
thi6n
cria
HS
dC
eiii
hoic
bie;
ludn
mOt
s5
hC
PT
tgo
n6n
sg
pnong
phf vC
th6
loai
vd
phuong
ph6p
gidi
to6n,
phir hqp
v6i

c5c
ki
thi
tuy6n
sinh
vio
Dai
hqc.
Sau
ddy
ld
mQt
sO
ttri du
minh
hqa.
(1)
(2)
Giei
h0
phucrng
trinh
Orui
dv
1. Gidi
ha
phaong
trinh
(e,
-ey

=X-\
I
t.rri+lo964y3=10
Laigidi.DK
x>0,y>0.
PT
(1)
tlugc
vi6t
lai
dudi
dang
ex_X=ey_J
NGUYEN
ANH
DUNG
(Hd
N6D
MQt
sii
luu
y
chung
1) Phuong
trinhflx)
:
m co
nghiQm
khi
vd

chi
d,i
, tf,rQ"
tflp
gi6
tri
cria
hdm
sti
y:
f(x)
vit
s6
nghiQm
ctra
phuong
trinh
(PT)
li sd
giao
diOm
cria
dO
thi
hdm
sti
(HS)
y
=
flx)

vdi
<ludng
thhngy:
m.
2)
Khi
gap hQ PT
dpng
Ta c6
th6
tim
ldi
gi6i
hu6ng
sau:
Hwdns
1. PT(1)
a
f
(x)-"f(Y)=0
(3)
Tim
c6ch
dua
(3)
vA mOt
PT
tich.
Hthng
2. Xet

HS
y
=
flt).
Ta
thudng
g[P
trudng
h-op
HS
li0n
tr,rc
trong
tfp x6c
dfnh
cfia
n6.
Nr5u
HS
y
=
JU)
don
diQu,
thi
tt
(1),
suy
ra
x

=
y. ftri OO
bii
to6n
dua
v6
gi6i holc
biQn
lufln
PT
(2)
theo
6n
x.
Ntiu
HS
r-
=
lU)
c6
mQt
cgc
tri
tqi
t
=
a thi
n6
thay
cl6i

chi"Au.bi6n
thi6n
mQt
iAn \tri
qva a.
fil-(f
)
suy
ra x =
y
hoflc
x,
y nim
vd
hai
phia
cria
a
(xem
thi
du
2)'
(3)
x6tHS
f
(t)
=
et
-t
,c6f

'(t):
d
-l>
0,
v/>
0-
Do
ct6 Hsflr)
ddng
bii5n
khi
r
>
0.
l fG)= f(v)
Tt(3)suyra
1"' .
"' =x=Y.
-"\_/__J
_
[x>0,y>0
Thay
vdo
(2)
duqc
logr;+
log5
4x3
=10
log2

x-l+2(2+3log2
x)
=10,
haylog2
x
=l
'
HQ
c6 nghiQm
duy
nhAt
@
;
Y)
=
12
;2).
Orni
d+2,
Gidi
hQ
phuong
trinh
Itn(t+x)-tn(l
+Y)=x-Y
0)
{
lzr'
-5ry
+

yz
=g
(2)
Ldi
gidl DK
x
>
-l,y>
-1.
PT(l)
cria
hQ
ttugc
vi6t
tai du6i
d4ng
ln(l+x)-x=ln(1
+Y)-Y
(3)
X6t
HS
f
(t\=ln(l+r)-r,
vdi,
e
(-1;
+o)
c6
l-t
f'(t\=

I
-

l+r
l+t
I
ft*t=
ftt)
(l)
|s(x,y)=0
(2)
theo
mQt
trong
hai
6
_
T@J HA(
sd 361
(z-2002)
&
quditl6
Tath6y
f'(t)=0e/=0.
HS
l(0
d6ng bii5n trong
(-1
;
0) vd nghich

bi€n
trong
(0
;
+oo).
Ta c6
(3)
e
/(.r)
=
f
(y).
Lric d6 x
:
y
hoic
xy
<
O
(n6u
x,
y
thuQc cing mQt khoing
clon
diQu thi x
:
!,
trong trudng hqp ngugc lai thi
xy
<

0).
Nt5u xy
<
0 thi v6
trAi crta
Q)
lu6n
duong. PT
kh6ng th6a mdn.
N6u x
:
y,
thay vdo PT
(2),
ta dugc nghiQm
ctra hQ ld x
:
y:
0.
Ofni
d,V3.
Gidi hQ
phaong
trinh
l;c-3x2+5x+1=4y
I
lyt-3y'+5y+l=42
I
lz3
-322

+52+l
=4x
Ldi
gidi.
X6t
HS
f(t)
=
t3
-3t2
+5/+1, c5
.f
'(t)
=
3t2
-6t
+5>0,Vr.
Do
d6
HSI(/)
lu6n tl6ng
bii5n.
I
f(r)
=
+v
I
He PT
c6 d4ng
1 f{y)

=
q,
-t-
lf
(z)
=
+*
Vi
hQ
kh6ng thay
d6i khi ho6n
vi vdng
quanh
!4.
tl6i v6i x,y,z
ndnc6th6gi6thi0t
"r)y,x)2.
N6u x
>
y
thi
/(x)
>
fu)
-
y
>
z
=
fu)

>
flz)
) z
>.x.
MAu thuAn.
Tuong
tg n6u x
)
z
ta
cfing
di
di5n m0u thuin,
suy ra x:
y:
z.
Tri
mQt PT trong
hQ, ta c6
x3
-3xz
+.r + I
=
0
<+(x-l)(xz
-2x-l)=0
Ta duoc
nghiQm cta
hQ: x-
y

-
z
-1;
x=!=z=l+Ji.
NhQn xit. Xdt
he PT c6
dang
N6u c6c nS
flf),
g(r) ctrng ddng bitin
(ho[c
cirng nghich biOn) thi li lufln
nhu
tr€n.
ta suy
ra x:
y
:
z.
BiQn lu$n hQ
phucrng
trinh
fifni dq 4. Tim m dA ne
phaong
trinh sau
co ngnr?m
f#.t
+^li-r=*
1-
[Jy*t+J3-x=m

Ldi
gidi.
DK
-1
<
x,y
<3
.
Tr* theo v6
cria
(1)
cho
(2)
vd chuy6n v6,
a
dugc:
J;+1
-$ =JF-J3-r,
nC trr6y HS
/(r)=Jr*t-J:-raOng
tr6n
(-l
;
3)
n6n
tt
(3)
suy
ra
x

=
y
.
Khi d6 tar
(1)
c6
Jx
+t +.{3-}
=
* .
Xdt HS
gQ)
=".6.
t *
JZ
-
",
ta c6
g(x)
tuc
tr6n
[-t
;:]
va
8'(x)
=-+ +,
g'(.r)
=
0 <> x
-

l.
2'lx +1 2tl3
-
x
Ta c6
s(-1)=2,
S(l)=2Ji, BQ)=2.
Tir
d6 2<
s(x)<zJ,
Vdy he
c6 nghiQm khi 2 < m <2J2 .
Otni
dq 5. Chilmg minh ring
vdi moi m
)
0,
hQ
phuong
trinh sau cd nghiQm
duy nhiit
fs*'y-2y2
-m=o
trrr,
-2x2
-m=o
(l)
(2)
Ldi
gidi.

Ni5u
y -<
0 thi v6 trSi cria
(l)
6m,
PT kh6ng
thoA
mdn,
suy ra
y
>
0. Tuong tg c6
x>0.
Trir theo
v6
cria
(1)
cho
(2),
ta dugc
3x2y
-3y2x
+2x2
-2y2
=0
<+
(x
-
y)(3xy
+2x +2y)

=
0
(l)
(2)
(3)
bii5n
li6n
lfk)=so)
]rrrl
=
se)
[/(z)
=
s(x).
TCffi\8 HOC
"
cfuonfe
sd 361
l7-2oo7l
7
Y\ x,
y
>
0 n6n 3ry+2x+2y>
0. Ta dugc
x-Y:0haYx:Y.
Khi d6,
tt PT(1) c6 3x3
-2x2
=

m .
Xdt
HS
f
(x)=3x3
-2x2
,
c6
f
'(x)=9x2
-4x.
Tath6y
f'(x)=Oe,r=0
hoic *=1.
,9
HS ddng bi6n
trong
/
nghich
bi6n trong
|
0;
\
;
o)
vd
(;,-,*),
.fco: fl})
:
0,

(-oo
4)
s)'
(n\
fc,:
f
[,.J
'o
VSy vdi
mqim
>
0,
PTI(x):0
c6 mOt
nghiQm
x>0.
Do d5 v6im> 0,
hQ c6 nghiOm
duy nhAtx
:
y>
0
(bpn
doc
tu vE d6 thi
ho4c iQp bnng bi6n
thiOn
cira
HS AC Uam
tra k6t

quA tr6n).
Bii luyQn t$p
1. Ciei
cric hQ
phuong trinh
(t
l.or*
=l-L
[x=siny
l.ory
=tr-L
[z
=ri*
\4
Itnx-lny=1-,
c)
{
x+r
lz,*t.3i
=36
2. Tim m d€ cilc
hC PT sau c6 nghiQm
(
ut lr-)=cosx-cosy
'
l2sinx-3cos.v=rt
[.os * * coS'r.
=
-l
b){

'
[cos
3x
+
cos 3y
=
*.
3.
Gie
st x,
y
ld nghiQm cria hQ
phuong
trinh
I *+r=*
I
lx2
+Y2
=Yn.
Hdy tim
girl
tri
l6n nh6t vi nh6 nhAt
cfia
3-1
x
-rv
I
,rru,
,r'rooaw},l#ffi

X
hbi ?
Y,7, trf,
lhii.
%ii
- dflity',
* *'.*'*'*s,*
*.n'.*.
l.(7.07).
Trong
mot.ldi
giii
bhi to6n
c6
cdc
bidn ddi
J;.1*{6-r
IIIII

:-+0
2X

(J-*t-r
r+V7-r)
-
ltlllt
T-
|
x+o\
2x

2x
)
Jr*r-r
l+t[J
:
lrm-
r ll[1
r+0
2X
,->0
2X
C6
f
kidn cho
rlng lim
nhu trOn
li sai
vi
Jr*r-r.t+Vx-t,,
.,
o
2_
nu
2*
luc
oo co oang

Xin h6i
ldi
giii

bidn ddi nhu
tren
dring hay
sai?
(N.N.,
I lAt, THPT
Lang Giong,
Bdc Giong).
Mong
nhAn
duoc
!
kidn
tri ldi
cia cdc ban
gitip
2 ban N.N.
vd,ln
-
dtatfoq*
1.
(3.07).
Y t<ign
3 li chua
chinh x6c.
Ta ncn
v€
tru6c bang
brit chi rdi
sau t6

lai bang
brit muc.
Ndu kh6ng
lfc ddu
vE bang
n6t drlt nhfrng
ch6
dudng khudt. Lim
ddn cdu
sau m6i
xem x6t
vE
dudng n6t lidn
cho chinh x6c.
(NhO
Vdn Vlnh,
t tAt
,
\HPT
TAn YOn tt, Bdc
Giang)
DAT
MUA
tAP
cHiroAN
HQc
vA
rud
rnl
oAt

u4trt TAt
cAc
co
sd Buu
DIF:N
rRorvc
cA aa6c
\
fiinhlufrn
ui
dTfu
WMm
pi
thi TSEH m6n Todn tnAt a ndm nay sdt
vdi
chaong
trinh
phd
th6ng. Ti lQ
gifra
cdc
cdu dd,
trung
binh,
kh6 t.d hqp li. H7c sinh khd,
gi6i
vdi kidn thil:c
chdc chdn mdi cd th€ dqt
daqc didm
cao.

Vi(c
giai
day di vd chinh
xdc
cdc
cdu 11.2, 1V.2 kh6ng
phdi
don
gidn.
Sau ddy ld
ldi
gitii
md
sii cau
trong de thi vd m\t sij nhqn xil
vA
cdc ldi
giai
d6.
*ceu
L2. Cho hdm s6
NGQC THAO
(Hd
NoD
Ldi
gidi.
DK x
>
1
.

Phuong trinh trdn tuong
duong vli mJ x
+ I
=
-}J-v -
1 + z1l7' t .
Chia ci
hai
v6 cta
PT cho
Jx
+ I
>
0
,
ta dugc
t; _,
^=-tJ*-!-
oz
*l!
(r)
Vx+l Vx+l
Ddt t=
Vix>l,n6n0</<1.
Tt
(l),
ta dugc m
=
-3t2
-r2t

=
E(t)
.
I
Tac6g'{t)=-6t+2;
g'(t):0khi
t=1
.
Lip bang bi€n thi€n
cta him
g(r)
v6i 0
<
t
<l ta
,11
thdy
-l<g(r)<;,
suy ra
-l<m
(:.O
33
NhQn xdt. l) Khi dAt di6u
kiQn
cho
r,
dE
mac
sai lAm h chi th6y t
>

0
md kh6ng th6y
0
<
r < 1 Tir d6 d6n tdi k5t
qun
sailitm=1frl
J
2) Khi
gap
PT dpng auz
+buv+cv2
=0,
c6ch
giii
chung
li :
N6u v;r 0, chia ci
hai vi5 cria PT cho
,',
sau
d6 d6t 7
=!
,
tudugc mQt
PT
bdc
hai dtii v6i r
v
(trudng

hgp v
:
0 dugc x6t ri6ng).
3)
N6u PT c6
d4ng
m
:
/(t),
thi PT
ndy c6
nghiQm khi rn thuQc tip
gi6
tri cria HSI(0.
Bdi luyQn
fip.
Bien ludn
sii
nghiQm cita
PT
Jr{
+ +**,[7-z*
* z +
(m
+lJx
-z
=
O .
v'
x2 +2(m + 1)x + m2 + 4m

x+2
(l)
m ld tham
si5.
Ti* m dii hdm
sa
(t)
c6
cvc dqi
cqrc tidu, d67tg thdi cdc diiim cqc tri cila d6 thi"
cirng
vdi g6c
tog d0 O tqo thdnh mAt tum
gidc
vu6ng tai
O.
x2
+4x+4-m2
LOt
gtot.
la co
y'3

(x
+ 2)z
Hdm sd c6 cgc tri khi m
*0
.
Gi6 su A vit B li c5c di6m cgc trf cira him sti
(1).

Ta c6
O)
:
(2-m
;
-2),
OB
=
(_2*m;4m_2).
LOAB
vu6ng
tai
O
khi vd chi khi
OL.OE
=0 ,
hay
m2
+82-8=0
e m
=
-4ilJ6(th6a
mdn m * 0). A
Nhfin x/r. N6u str dpng dinh
li Pythagore
cho
tam
gi6c
vu6ng OAB th\ lli
gi6i

sE dii, tinh
toSn
phirc
t4p
hon.
Bdi
tuyQn
tfrp. Chto hdm si5
x2 +(2m-3)x+3m+5
v-
x-1
vd didm PQ;
-l).
Tim m ae ltam s6 c6 cqrc
.:
-:
,
:
dqt cuc tteu, dong tnot cac dtem c$c tr! cua
dO thi cilng vdi diAm P
@o
thdnh mQt tam
gidc
nhgn
tqi dinh P.
*Cau ll.2 Tim m dii
phuong
trinh sau cd
nghiQm thqrc
3*r{ +*J**t=21f

*z
-1
TO#H
HO(
-
6ludiua
s6 361
17-2007)
g
*Ceu
lY.l.
Tinh
di€n
ttch
cila
hinh
phdng
gidi hqn boi
cdc
dadng
y=(e+1)x,y=Q+d)x.
l n
Ldi
giai.
PT
(e
+ l)x
=
(1
+ e')x*

L;
=;
I
Ta
c6 s
=
jlte
+ 1)x
-
(1+
e')xl
d;c
Tinh dugc
Iz
=1,
suy
ra
S
=:
-
1
(dvdt).
tr
"2
NhQnxdr.
Ni5u
HS
dudi
dAu
tich

ph6n
c6
dang
p(x).e*
,
trong
d6
p(x)
ln
mQt
da
thf'c,
thi ta
giAi
bdi
to6n
b6ng
phuo'ng
ph6p
tich
ph6n tirng
'
l"
=
P(x)
phAn
voi
ph6p d[t
{
t

!
[dv=e'dx.
Bdi
tuyQn
tfip.
T{nh di€n
tich
cua
hinh
phdng
gioi
hqn bni
cdc
dudng
y
=
-:=
;
1t
=
0;x
=
0; x
=
1.
g'l3x+l
''
*Cau
LY.2.
Cho

x,
y,
z ld cdc
sii
thttc
duong
thay
doi
vd
thoa
mdn xyz =
l. Tim
gid
tri
nhd
nhat cua bidu
th*c
,_
x2(y+z)
-
y2(z+x)
*
1-
t
^
t t,+Zx.Jx
yly+zzlz
z!z
Ldi
gi,fii. Ap,dpng.BDT

Cauchy
cho
hai
s6
duong
tr6n
m6i
tu
s6, vd
tir xyz
:
1,
ta dugc
^
r
".v^lv
zzJi
p>
!"1'
:-
1'"'
: (l).
y,ty +2z"lz
z'lz
+Zx.,lx
x^,lx
+2Y"tY
Ddt
a=*Ji,
b=yJ-!,

"=rJ,
th\a,b,c>0;
abc =
l.
BET
(1)
trd
thdnh
2a 2b
2c
P>

+
.r
(2)
b+2c
c+2a
a+2b
Ap
dqng
BET
Bunyakovski
cho
hai b6
s6
(J;@
. k);
$@
+ 2,);
G@

r2D)
t" )
l_ I
\
a+zu
)'
0
l1
=
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-
[xe.
.dx
=
It
-
Iz
.
00
I
Tathhy
11=
[ex.er=;.
0
(u=x
fdu=dx
Edt I
+{
'

[d,
=
e*dx
[v:
e'.
tadugc
(a+b+c)z
(
a
b
c
)
<3(ab+bc+cdl:-:-+ :
+
^
l(3)
-
-\
'
\b
+2c
c
+2a
a
+2b
)
Lai
c6 3(ab
+ bc
+ ca)

<
(a
+ b
+ c)2,
ndn tir
(3)
suv
,u
o
*
b.
','-
J-
21
(4)
\-
'
'J
h+?t
c+2a
a+2b
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vd
(4),
ta
c6
P>2.Tir
d6
min
P

:
2
khix:y:z:LA
NhPn
xdt.
C6i
kh6
cira
bdi
toSn
o
ch6
kh6o
sil
dpng
BDT
Bunyakovski
dO
chirng
minh
BDT
(4).
Bdi
tuyQn
tQp. Cho
a, b,
I
vd
m, n
ld cdc

si5
thryc
duong.
Chang
minh
ring
abc-3
mb+nc
mc+na
ma+nb
m+n
*cau
Ya.2. Ch*ng
minh
riing
)rrr,.Ic'r,
*+crr,
*
n*r;r-'
=+i
.
Ldi
gidi.
Ta
ggi
bi6u
thric
trong
vi5 trSi
ld S.

2n
Xdt da
thr?c
P(x)
=(I+
x)2'
=lc\,xk
.
k=0
Luu
yi ting
tt6u k
ln s6 chin
thi
10
!c5,*ua*
-
0
Khi
k le
thi
1
lc5,*oa*
-
0
Tir d6
l0
[c5,rod,
=o
.

t
0-1
)c\,xt'ax
=;
,c5,.
,
A
TI
-t
Jr1,;4,
-
J
r1,;a*
0-l
(Xem
tidp
tuang
5)
z2(x+
y)
aTiy$

2(
!c\
+
L
ct
+
1cl-
+

*
l.;r-,
)
\2
4
'"
6
'"
2n )
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10
Tffiffie,B
*{Gs{
Sd
361
(7-2007)
&
e6ffiffiR
PHUONG
PHAP
Trudc hdt xin nhac
lai kh6i ni€m GTLN,
GTNN
cria him s6.
Cho him sdfl.r)
x6c dinlitren midn D
(DcR).
Gii srl M, m ldn
lugt li GTLN,
GTNN cira

hdm/(x) trOn
midn D.
BAI TOAhI
DINH TINH
ffiWWMMffi
TRAN TUYET THANH
(GV
Hgc vi6n Phdng kh6ng
-
Kh6ng
qudn)
.
^,
\ lf7l>m,YxeD
m=mrn / l-rl<><
-,.o't*t
-
l:roeDsaochof
(x6)=m.
Sau dAy ld m6t sd bdi to6n minh hoa.
finai
todn
t. cho hdm sd
f
(*)
=l+ax3
+ 2bx2
+
(t
-

so) *
-
ul
trong d6
a, b
ld cdc
sd
thuc tily
i,.
Gqi
T
,
=,:tTrl
f
(x).
Chrtng minh rdng M
-+
Ldi
gitii.
Trudc hdt ta c6 cdck€t
qui
sau:
max
{a,
O\
>f,@ *
O)
(1)
min
{a,

pI
.;@ *
f)
l"l+lfl>a
- B
D.
f,ua
-frr,uQc
do4n
[-1
;
1]
,
ndn
22
ta c6
(
t,
M
=ntzx
f
(
x\ol/tx)
< M'
Yx e
D
-,Ji'
r"t
-
L=ro

eDsao cho
f
(xs)=Al
(2)
(3)
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-,(+
1.6
ul
I
z
'zl'
*,-r(i)=
=
l*(-f)'.
*(-+)' ,-,,,[-f)-,1
rcftl FIOC
.
crudiue
s6 361
17-20071
11
2rl
+o-zaf-ul
frnei to6n
3.
Gid
sft
M
ld

gid
tri
lon
nhdt
cua
lbl
sao
cho
4bx3
+(a-lU)
x
31,
vdi
moi
gid tri
crta
x.[-t;\)
vd
vdi moi
sd
thuc
a'
Ch{rns
minh
rdng
M
<1.
Ldi
gidi.
Dat/("r)

=
4bx3
*("-3b)x.
Tn
gii
thidt
ta c6
Jf
{tl
=4b+a-3b=a+b<t
f
f
(-t)
=
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-(a
-3b)
=
-a
-
b
al
Suyra
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toin
tudng
tu
ta

c6
Ir(r)
=!*(,-3b).
|
-o
-b<l
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z'
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2
lr(-r)
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-(a-3b\.1
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z)
z
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2
TiI
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vh
(5),
ta

nhdn
duoc
{t=o*u:.'
^
=-3
<3b
<l=lal<r.
l-z<-o
+2b<2
suy
ra
.ux
lal
<
1.
vay M
< l.
a
Dd
cirng
cd
phuong
ph6p,
mdi
c6c
ban
glbi cdc
bdi
to6n
sau:

onai
h6n
4' cho
hdm
sd
f
(x)=ox2
tbx*c
'
trong
dd
a,
b,
c ld
cdc
tham
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thod
mdn
la+b+c<1
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I
c<l
t
GQi
M=
nm><,f(r)
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minh
rdng

tW
t2
'
8
Onai to6n
5. cho
hdm
sd
f
(*)
=cos2x
+ acos
(x
+ a)
,
trong
d6
a,
a
ld
cdc
tham
sd
thuc
tuy
y'
Gid
stl
M
=max

f(x);
m=min
f(x).
Chftng
minh
rdng
M2
+mz
>2.
l-6
-sl
I
z'21'
rrd6suyra
M>max{r[f)
,[
f)]
=maxllf.ql,
l 6
bll
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2ll
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ket
qui
(3))
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z)
[
2
2))
Y?tv M=€.
o
2
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m,6n2.
cho
hdm
sd
f
(')
=
2x2
-7
+ ax
-
zoo1 ll
4'
'
trong
d6
a ld

tham
sd
thuc
tilY
Y.
Goi
M
=,Ii3,1,1 f
(*);
r=,S-il,l
t(-)
lu>t
.t
Chftns
minh
rdng
t'
2007
l*= E
Ldi
gitii.Ta c6
[*r-t(l)=t*o
l*.t(-l)=t-o.
*
r:(/trl
*
/(-1))
=
I
(dpcm)

tu
ta c5
(
t ) o
2oO7
=rlo)=T E
(
l)
a
2oo7
,rl-fi)=-o E
;[r(#).r[
#)]
('cheo
kd'l
qui';z)
Suy
ra
Tuong
l*
l
I
l*
)m4
Ydy
m<
-
.D
2007
'12

tr{ }ff}c
cryl'ldiw
12
sd
361
(7-2002)
&
PhdG GrfiAm Gh@mn
tI
THUYET
SO
NOUYEN TO
6t
trong nhfrng
ngudi duoc
giii
thu&ng
Fields
2006,
Terence Tao, di ddy viec
nghiOn cts ciLc
sd nguyOn
td di xa
hon, bang
c6ch
dua
ra mot dinh
li mdi.
Nhi
to6n hoc

trd Terence
Tao, &
trudng
D+i hoc
Los
Angeles,
ngudi
nam ngo6i
duoc
nhdn Huy
chuong
Field vdi
c6c
kdt
qui
nghi0n crlu
vd cilc
s6
nguyOn td,
dd
lim cho li thuydt
c6c
sd nguy0n
td
phdt
tridn
hon
nfia. Cing vdi
ban ddng
su Tamar

Ziegler
b
trudng
Dai hoc
Michigan,
Terence
Tao dd chirng
minh
rang ddy sd
nguy6n
td chrla
nhfrng
"cdp
s6 da thric
dhi
tilY
1i"
rr).
56' nguydn
t6'ld
s6'tu
nhi€n l6n
hon
I chi chia
h€i
cho
I vd
chinh
s6' d6. Ydi
dinh nghia

cuc
ki
don
giin
d6,
ngudi
ta dd
xAy dung
nhfing vdn
dd to6n
hoc
rdt
phrlc
tap.
Tir
thdi
vdn
minh
cd dai,
ngudi
ta dd,
bi6t c6c
sd
nguyOn td
tao
thlnh
m6t d6y v6
han.
Nhftng
sd nguy6n

td
ddu ti6n
li
2, 3, 5,7,
ll,
L3,
17,
19,23
Khi c6c
sd
cing
l6n thi vi6c
x6c dinh
c6c
sd nguyOn
t6
cdng
kh6
khan
hon.
Dudng
nhu
khi ci1c
sd cing
l6n thi
sd nguy0n
td
clng
hidm. Hi0n
nay

cdc
nhi
to6n
hoc chua
thd
m6
tA 16 rlng
c6ch
thri'c
md c6c
sd
nguyOn
td duoc
phdn
bd trong
ddy
c6c
sd
rU nhi0n.
Nam
2004,
cing vdi
Ben
Green,
6 trudng
Dai
hoc
Cambridge,
Terence
Tao dd chrlng

minh
mOt trong nhtng
kdt
qui
dd
mang lai
cho 6ng
Huy chuong
Fields: Tdp hqp cdc
sd
nguy€n td
chtta nhfrng
"cdp
s6'cing c6 dO ddi tiy
y"
ttc
ld sd sd hang l6n tny
y.
Vi du, d6y 5,
Il, 17,
23 lh mdt cdp sd c6 4 sd nguy6n td hon k6m
nhau6(5+6= 11, 11+6=
17,17
+6=23).
Day
ln m6t cdp sd c6ng c6 cOng sai lh 5.
Phii chang tdn tai
nhfrng d6y sd khOng
phii
ln

cdp sd c6ng nhung c6 thd m6 ti
quy
luAt
vd
khoing cdcb
grtra
hai sd nguyOn td li0n tidp
thu6c ddy?
Dinh
li mdi m6 rOng kdt
qui
tru6c cho nhfrng
cdp
sd
"da
thfc", trlc ld nhtng
cdp sd c6
khoing c6ch
gifra
ci{c sd hang 1i0n tidp kh6ng
nhdt thidt li mdt
hdng
s6, nhu trong trudng
ho.
p
cdc cdp sd c6ng.
Vi
du, d6y
3, 5, 13,
31 le

m6t cdp so c6 bon
sd
nguyOn t6, c6 khoing
c6ch
gifia
sd hang
thrl
n
vh
sd hang
thrl
n
+
I
bang 2n2. Dinh li
Terence Tao vh Tamar
Ziegler chi
16 rdng
c6
thd dat
duoc nhfing cdp
sd
c5
nhidu sd hang
hon
nfia
(kh6ng
h?n
chO), c6
khoAng

c6ch
c[ra
m6i
sd
hang
v6i
s6
hang tidp
sau
duoc
x6c
dinh
bang mQt c6ng
thrlc
cing loai, trong d6
chi
dDng ddn
hing r
vh
c6c
liy thira
cfia n6. Su
chri'ng
minh
dinh
li d6 dua tr6n
cirng nhfrng
f
tu&ng
dd cho

ph6p
Ben
Green vd
Terence Tao
giAi
quy6t
trudng
hgp cr{c cAp
s6 cdng.
TAp hqp
nhfrng sd
nguy€n td
duo. c nhAn dinh
nhu ld m6t
Qp
hcr,p
con cira tap
ho,p
nhfrng sd
nguy6n
goi
lh
"gdn
nhu
nguy0n
td".
Tr0n tap hap d6,
ngudi
ta
xdy

dqrng
mOr ham dic
bi6t cho
ph6p
lim
ndi rd
m6t sd tinh
chdt d[c
bi6t cia
cdc sd nguyOn
td.
Nhu vdy
c6c sd
nguy6n
td dd
16 ra th€m chrit
it
nhfrng
didu
bi mAt cria
chring.
NGUYEN
VAN
THIEM
(theo
"La
Recherche" thdng
I-2007)
tf l
fJ|^"

",
f .Ziegler.
i
DAI
e5
nGt
2
-"*** *' "
.".
. .
r
IORN
HOC
*
gltroiUe
sd
361
lz-zoo7\
t5
nAm
gilng day
thi
didm
tai
gdn 50
trudng
C6c kidn
thfc
v<i td
hop

vd xdc
sudt
dang
THPT
vi
tidp
thu
c6c
f
kidn
t0m
huydt
cira
rdt
nhidu
c6n b6,
gi6o vi0n
trong
cA nu6c.
Ir)I-Ii?:hJ
mE-&
6ch
916o
khoa
(SGK)
Dai
sd
vd Gidi
ttch
11

ndng
cao
dugc
biOn
soan
theo
chuong
trinh
mOn
Toiin
Trung
hgc
Phd
th6ng
ban
hdnh
theo
Quydt
dinh
sd
L6120061QD-BGDDT
ngiy
05-5-2006
cira
86
tru6ng
B0
Gi6o
duc
vd

Dio
tao,
tron
co
s&
tdng
kdt
cdc
uu
vh nhuoc
didm
cira
cudn
SGK
thi didm
Dai
sd
vh Giii
tich
11
ban
Khoa
hoc
Tu
nhi6n
(bO
1)
sau
hai
3ffiffiru

*{W{
Sd
361
i7-2007i
&
€fr.{trUe
Trong
khu6n
khd
cira
bii
b6o
ndy,
v6i
ciic
quan didm
nOu
tr0n,
chring
t6i
trinh
bdy
nhfrng
,en
ad
trong
ydu
nhdt
md
m6i

gi6o vien
vh hoc
sinh
cdn
luu
f
khi
sil
dung
cudn
SGK
niy
trong
nam
hoc
sap
tdi,
d5c
bi0t
la
nhfrng
didm
mdi
vd kh6c
so
v6i
chuong
trinh
vd SGK
ldp

11
chinh
li
ho.
p
nhdt
nam
2000
(SGK2000).
r.
vB
NOr
DLrNG
cHUoNG
rRiNH
So
v6i SGK2000,
SGK
Dai
sd
vd Gidi
tich
11
ndng
cao
c6
nhfing
didm
kh6c
bi6t

sau
ddy:
-
Y6,
luong
gidc,hoc
sinh
d6
lim
quen
vdi
c6c
kidn
thrlc
m&
ddu
vd 1uo.
ng
giSc b ldp
10'
Do
d6
phdn luo.ng
giSclop
11
chi
cbn
vdn
dd
cdc

hinn
id
tuong
gidc vit
phuong
trinh
luong
gidc'
Chuongirinh
kh6ng
tli
cQp
cdc
vdn
dd'
vd'
bdt
phuong
trinh
vd h€
phaong
trinh
lrtong
gidc'
iiJl
f#i;1:,
A
@
@
@ N

W
[[a[[g
qAO
ddi
sdng,
thuc
ti6n
khoa
hoc.
Truc
quan, tfc ld
coi
truc
quan ld
phuong
phrip chir
dao
trong
vi0c
tidp
cdn
c6c
kh6i
nigr,
to6n hoc
;
dAn
d6t
hoc
sinh

nhdn
thrlc
tir
truc
quan sinh
dOng
ddn
tu duy
trtu
tugng.
Nhe
nhhng,
tri'c tri
x6c
dinh
nhtng
yOu
cdu
vta
phii AOi
vOi hoc
sinh
;
trdnh
him
ldm
;
cd
g6n[ trinh
biy

van dd ng6n
gon, sfc
tich,
khdng
g?ty
cdng
thing
cho
ngudi
hoc.
Odi
mOi,
trlc li
c6ch
tan
cilch
trinh
bly,
ndng
cao
tinh
su
pham
;
g6p
phdn ddi
m6i
phuong
ph6p
day

hoc
vi
phuong
ph6p
dSnh
gi6.
ngiy
chng
tri
n0n
quan trong
ddi
vdi
m6i
con
ngu-Oi
trorg
xd
h6i
hiOn
dai.
Vi.
vAY,
A
nhidu
nudc,
nam
1995).
-
Cdc

vAn
dd
va hdm
sd
mfi
vd
hiim
sd
lAgarit
duoc
chuydn
sang
l6p
12,
dlLnh
thdi
luong
cho
chuong
"Dao
hdm"
(gdm
kh6i
niOm
dao
hhm
vd
c6c
c6ng
thfc

tinh
dao
hlm,
chi
trir
c6ng
thrlc
dao
him
cira
him
s6
mfi
vi him
sd
l6garit).
Chuong
trinh
quy dinh
khdng
tlua
vio
kh6i
nidm
dao
hdm
m6t
b€n.
Vi0c
dua

dao
hdm
cirng
vdi
c6c
n6i
dung
-td
hqp
vd
xdc
sudt
vdo l6p
11
lI
m6t.d4c
didm
.&i,
ndi
bdt
cira
chuong
trinh.
Didu
d6
cbn
nham
cung
cdp
nhfing

cOng
cu
to6n
hoc
cdn
348)
th
:
nhung
v6i
mrlc
do
rdt
khdc
nhau.
S6t
thdC,
trlC
NGUYEN
HUY
DOAN
6 NU6C
td,
th
gdn vdi thuc
(ChI
bi6n
SGK
Dgisd
vd

Giditich
l l
Ndng
coo)
day ln
ldn
ilai
a+v hqc
&
ddu
tion
xric
;;e
;dod'nhrm
ndng
cao
rinh
khi
thi
cira
sudt
dttoc
dua
vho
chuong
trinh
phd th6ng
"t
oo.rg
trinh

vi SGK
;6i
;
tidp
ctn
thuc
ti6n
(khong
kd
ddn
chuong
trinh
phAn
ban
thi
didm
14
thidt chudn
bi cho
viOc hoc
t6t mot sd m6n
hoc
kh6c
(nhu
Vat li, Sinh hoc,
H6a hoc,
Dia Ii, ),
thd hien
tinh lion mdn
trong

toin bO Chuong
trinh
gido
duc
phd
thdng.
tt.
vi PHUoNG
PHAP
rdp cAN
vA MOc
o0 vBu
cAu
Chuong
|
(Hdm
sd
luong
gidc
vd
phuong
trinh
luong
gidc)
ld
phdn
ndi tidp
cira
phdn
luong

gi6c
l6p
10. Khi khdo
sdt su bidn
thi€n
ctn cdc
hdm sd luong
gi6c,
SGK
Dar sd
vd
Gidi
tich
11 ndng cao
srl dung
phuong
ph6p
truc
quan,
trlc li
khio s6t su
chuydn
d6ng c[ra c6c
didm dd suy
ra su tang-giim
cfia
c6c him sd
luong
gi6c.
Phuong

ph6p
tinh tidn
dd thi
(dA
hoc & l6p 10)
cflng duoc
nh6c lai
vi sir dung
dd
vE dd thi ctra c6c hdm
sd luo. ng
gi6c.
Kh6i
ni€m hdm sd
tudn hodn
khOng coi
li nOi
dung trong
ydu.
Do d6 SGK
gi6i
thiOu
dinh
ngfr,a
hdm sd tudn hodn
rdi dua ra mOt
vii vi
du nhu
dd
khdi

qudt
cdc
tinh chdt dac trung
vd
tinh tuen
hoin ctra cdc hdm
sd luong
gi6c
di
hoc
tru6c d6.
Ya
phuong
trinh luong
gidc,
SGK d6 c6
g6ng
giim
nhe
c6c
y€u
cdu vi
ki ning nhu: kh6ng
xdt cdc
phuong
trinh luong
gi6c
chita tham sd,
loai
b6 ciic dang

phuong trinh c5 bi6n ddi
hoac
phAi
x6t didu kiOn
phrlc
tap, khdng
gidi
'r)
thiOu
phuong ph6p
dat dn
phu
r
=
tan
I
'
\2)
Vi0c dua
cr{c
ki hi6u arcsin,
arccos,
arctan
v}r
arccot nham
girip
gi6o viOn vi
hoc sinh
trinh
bdy

ldi
grii
cdc
phuong
trinh luo. ng
gi6c
duo.
c
gon,
khOng nham
gidi
thiOu c6c him sd luong
gi6c
nguoc.
Muc
dich
cira
chuong
Il
(Td
hop vd )fic sudt)
li dd hoc sinh lim
quen v6i
nhtng
vdn dd don
glin
c6
n6i
dung td hgp thudng
gap

trong ddi
sdng vi khoa hoc. Do d6 hdu
hdt c6c vi du
trong SGK ddu duoc ldy tt thuc td cuQc sdng.
Hoc
sinh cdn hidu
vd ph6n
biOt
duoc c6c kh6i
ni6m, nh6
vi vdn dung duoc c6c
quy
tltc, cdc
c6ng
thrlc vio
nhfrng bhi
to6n don
giin,
khOng
dbi
h6i suy
luAn
qua
nhiCu
budc trung
gian.
Dac biOt,
cdn
g6n v6i
nhflng

bii to6n thuc td
nhu
vdn dd chon c6n bO l6p,
vin dd didm s6,
vdn dd an tohn
giao
thdng,
van dd din sd,
NOi dung
chuong
lll
(Ddy
sd,
cdp
sd c1ng
vd
cdp
sd nhdn)
vd
chuong IY
(Gi6i
han)
vd,
co
bin
cfrng
gidng
nhu
tru6c
ddy. Tuy

nhiOn
c6
mdt tliiu
kh6c biOt
rdt
quan
trong
h
kh6i
nram
gidi
han
vd cltc
dttoc
hidu
theo nghia
gidi
han
hoac
bang
+co,
hodc
-oo,
kh6ng
ch0p
nhxn
gi6i
han
bang
co

chung
chung nhu
trudc
day. Ching
han,
ding
thrlc lim(-l)nn
=
a
nay
kh6ng
duoc
chdp nhAn,
mic
dil n6
dfng
theo
dinh nghia
cira
SGK2000.
Sr,r
thay
ddi d5
c6
nhidu
uu didm
vd
phi
ho.
p

v6i SGK
cira
nhidu
nu6c
trOn thd
gi6i.
Chac
ch6n
n6 sE
gdy
ra
m6t
so kh6
khan
cho
gi6o vi6n b&i
n6 dbi
h6i
gi6o
vi6n
phii thay ddi
trong mdt
sd
c6ch trinh
bdy
ldi
gi6i.
Ching han:
-
Khdng

vidt
lim/(r)
md
ph6i vidt
1im
/(,r)
hoic
,\T-/tr);
-
Kh6ns viet fiml=
oo
rnh nhAi
viei [m
1=
+*
'
i-o
a
'
x-o'
;g
vilim
l=-oo.
x +0-
A
Ngoii nhflng
dinh li
v6
gi6i
han md

c6c SGK
trudc diy
thudng dd
cdp, SGK
cbn cung
cdp
nhfrng
quy
tac
chir
ydu
nham
gifp hoc sinh
bidt xdc dinh
xem m6t
gidi
han
vd cuc khi
nio
thi
bdng
+"o,
khi nio
thi bang
o
.
Do
c6 thay ddi nhu
thd,
gi6o

vi6n cdn nghiOn
criu ki SGK
dd tr6nh c6c
so suat
do th6i
quen
cfi dd lai.
Dao hdm li n6i
dung kh6
quen thudc trong
chuong trinh to6n
phd
thOng.
Didu d6ng
luu
f
Id su tldi
mdi c6ch
tidp cAn mdt
sd
vdn dr)
nhu:
vi du m& ddu
ddn ddn khdi
ni€m
dao
hdm,
y
nghTa
hinh hoc

cia dao hhm,
dao him
cira hdm
sd hop,
vi
phdn.
Ching
han,
vd
f
nghia hinh hoc
ctra dao hhm,
SGK
c5 trinh bhy
16
cdch x6c
dinh ddu li
"vi
tri
gi6i
han
cia cdt
ttrydn Mslvl
khi M
di chuydn
doc theo dd
thi
ddn ddn Ms"
md trong ciic
s6ch trudc

dd kh6ng
dd cflp.
Cdc tdc
gi|l6t
mong
nhAn
duoc c6c
f
kidn xay
dung cria
c5c thdy
gii{o,
cO
gi6o,
cdc nhi
khoa
hoc
vd
ciacdc
em hoc
sinh dd SGK
ngiy
cdng
t6t hon.
T#IN HOC
-
GfudiUe
s6 361
l7-20o7) 15
nt

Ffr
,tl
//t(
BAi
T5i361.
Giei
he
Phuong
trinh
CAC
LOP
THCS
Bni
T1/361.(Lfp
6).
56
1h;svOi
x
e
{0,
1,
2, ,9),
vi6t
trong
h0
thflp
phAn
c6
bao
nhi6u

chir
sii?
VU HOU
CIiiN
(GV
THCS
Hing
Bitng,
Hdi
Phdng)
tsir-
T2l36L
(Lfp 7).
Gqi
.I/
ld
giao
diOm
ba
dulng
cao
ctia
mQt
tam
gi6c
nhqn
ABC
vd M
lir
trung

di6m
canh
BC'
Dudng
thing
vg6ng
g6c v6i
MH
b
H cbt
hai
duhng
thdng
AB
vit
AC
tuong
ring
o
P
vd
Q.
Chring
minh
ring
HP =
IIQ.
NGUYEN
MINH
HA

(GV DHSP
Hd
N1i)
Bni
T3/361.
Chring
minh
ring
n6u
a,
b,
c,
d
ld
c5c
sl5
nguy6n
duong
ddi
m6t
khSc
nhau
sao
cho
bi6u
thric
-+.+*J-*
d
a+b
b+c

c+d
d+a
li
mQt
sO
nguy6n
thi
tich
abcd
litmQt
s5
chinh
phuong.
CU
HUY
TOAN
(Khoa
C6ng
nlh€
Vqt
li€u
02,
kh6a
50'
DHBK
Hd
NAi)
Bni
T4l351.
So

sSnh
gi6 tri
crta
cbc
s6
I
vd
B sau:
I
=
min
max(xz
+
yx)
lyl<l
txl<l
B
=
max
min(.x2
+
yx).
lxl<
I
tYl<
I
rE
ruemr
HAt
(SV

Khai
thdc,
K44,
DH
Md
- Dia
chiit'
Cdm
Phd,
Qudng
Ninh)
t-ll18
./x+./y
+'!z- -
r-
r T=;
r/x
tlY
lz
5
1
I
I
118
x+y+z+-+-!-=-
'xyz9
rr-1111728
xvx+yiY+ziz
-
T=;'

xVx
Y\l
Y
zxlz
Lt
DANG
QUdC
CHA}I
(GV THCS
Nam
Son,
Nam
Trqc,
Nam
Dlnh)
Bni
T61361.
Cho
hinh
binh
hdnh
ABCD'
Di€'m
M
nim
ctng
m[t
Phing
vli
ABCD

sao
cho
frD)
=
fril
. Chirng
minh
ring
hai
tam
giSc
MAB
vit
MCD
c6
cirng
trYc
t6m'
NGUYEN
DANG
KHOI
(GV
THCS
Bach
LiAu,
YAn
Thdnh,
NghQ
An)
Be[

T71361
.
Gqi
AD,
BE,
CF
theo
thir
t'u
ld
c6c
duhng
ph6n
gi6c
trong
tam
gi6c ABC
(D
e
BC'
E
e CA,
F
e AB);
O
vi
R
theo
thrl
tU

le
tam
vd
b6n
kinh
dudng
trdn
ngopi
ti6p
tam
gi6c
d6.
Gqi
Ot,
Oz
vd
O:
theo
thf
tu
li
tim
dudng
trdn
ngopi
ti6p
c5c
tam
giSc ABD,
BCE

vit
ACF.
Ch'bng
minh
r6ng
?
1n<oQ+ooz+oo3<2R'
2
LA
VAN
THINH
(SV
K4g,
AIT,
DHKHTN,
DHQG
Hd
NOi)
CAC IdP
THPT
Bni
T8/361
.
Cho
a,
b,
c
ld
c6c
s5

thgc
duong
1 I
1
.,.: t-2
'
th6a
man
l+i+:<l.
Tim
gi6 tri
nh6
nhdt
abc
cta
bi6u
thrlc
P =
fa
*
bl
+
lb
+
cl
+
lc
+
al,
trong

d6
ki
hiQu
[x]
ln
phAn
nguy6n
cira
x'
vO IrdNC
PHONG
(GV
THPT
TiAn
Du
I
'
TiAn
Du'
Bdc
Ninh)
Bni
T91361.
Tim
tAt
ch
cdc
him
s5
I

R
-+ R
thoa
mdn
di6u
kiQn
lU'
-
y)
+
2yQf2@)
+
Y')
:
11Y
+
flx))
vdi
mgix,ye
IR.
DUONG
CHAU
DINH
(GV THPT
chuvAn
LA
Qu!
D6n'
Qudng
Tri)

I
I
t:
rctr{
Fpc
16
sd
36]
(7-2007)
&
qirdi6
Bni T10/361. Chring minh
ring v6i mgi tam
gi6c
ABC
ta tlOu c6
-l
<
6cosl
*
3cos,B
+
ZcosC
<
7.
NGUYEN
VAN MAU, NGUYEN THU
H,i.NG
(DHKHTN,
DHSG Hd N1i)

Bni T11/361.
Cho tam
gi6c
ABC nQi ti6p
<luong
trdn
(O
;
R). Gqi
E li trung cli6m ctia
AB. TrAn cqnh
AChy diOm F sao cho
+
=:
AC3
Dr,mg
hinh binh hdnhl EMF. Chtmg
minh
ring
MA+ MB+ MC
<fiG'4M\
Gi6 sri
(O
,'
R) li cludng
trdn ci5 <linh.
H6y
dqng
tam
giilc

ABC
nQi titip
(O
;
R) sao cho
14tr
+
MB
+
MC
=
,1lt1n, -OU1
.
NGUYEN
VAN DUC
(GV
khoa Todn, DH
Vinh, NghQ An)
Bni
T121361. Cho tit diQn
ABCD c6 DA, DB,
DC vudng
g6c
tring
tl6i mQt. Ki hiQu x,
y,
z
theo thri tg
ld
g6c

nhi diQn canh
AB, BC,
CA
cria tf diQn
ABCD. Chring
minh ring
(2
+
tan2x)(Z
+
tanzy)(Z
+
tan2 z)
>
64.
HOANG
THANH
VAN
(GV
THPT
Dinh Hda, Thdi NguyAn)
CAC DE VAT LI
Bni LU361.
Trong
m4ch dao <lQng
ZC
tr6n hinh v€,
khi
kh6a
K ngit, diQn

tich
tren tp tht
K
nh6t c6 <tiQn
dung
C1 bing
4e,
cdn tp
thir
hai c6 diQn
dung C2
khdng tich
di6n. H6i bao l6u
sau
khi kh6a
K d6ng eliQn tich
tr6n tv Cz
dat
gi6
tri cuc
dai? 86
qua
diQn
tr& thuin
cria
mach.
NGUYfiN
XUAN
QUANG
(GV

THPT chuyAn
Hd NQi
-
Amsterdam)
BitiL2t36L
MQt
qu6
cAu el6ng
ch6t b6n
kinh R
dugc th6 vdo
trong mQt ch6t l6ng.
Khi
qui cAu
& vi tri cdn
bing
thi tam
quA
ciu cSch
mdt
thoSng cria chSt
long
mQt
doqn H.
Chung t6
ring
qui
cAu
sC dao dQng
di6u hda

niiu n6 lQch
kh6i vi tri
cdn bing mQt do4n
nho theo
phucrng
thlng dirng.
Tim chu
ki
dao
tlQng cria
qu6
cAu.
NGUYEN
DUC HIPP
(GV
THPT chuyAn
Trdn Eqi
Nghia,
TP.
HCl4)
PROBTEMS I]I
TIIIS ISSUE
FOR
LOWER
SECONDARY
SCHOOLS
Tll36t.
(For
6tr
grade).

The number
(9r)t,
where x e
{0,
1,2, ,9}
contains how
many
digits
when written
in deoimal
form?
T2136l.
(For
7tr
grade).
Let Ilbe
the orthocenter
of
an
acute triangle
ABC
and denote
by M the
midpoint
of
BC.
The
perpendicular
lineto
MH

through
11
meets AB
and. AC at
P and
Q
respectively.
Prove
that HP
=
HQ.
T3/361.
Prove thatif
a, b, c and
d are distinct
positive
integers
such that
the sum
abcd
=*
+ *-;-
lS
also an rnteger,
a+b
b+c
c+d d+a
then the product
abcd is a
perfect

square.
T4136l.
Compare the values of the following
two numbers
I
:
min max(x2 +
yx)
lyj<1 lxl<1
.B: max
min(x2 +
yx).
lxl<1
lyl<1
T5/361.
Solve
the
system of equations
-rr-ll18
r/x+a7Y+lz-
=-
'
{*
'!Y
J;
3
1 I I 118
x+y+z+-+-+
xyz9
*Ji+yr[y *rJ,

-
1
r
xlx
11728
+
=-
yJy
z^l z
27
(Xem
tidp trang
3l)
rQtu
rfr?c
.
qudi$A
sd 361
(7-zoo7l
17
*nai
ry357.
cho
hai binh,
mdi binh
cd
dung
tich
1 lit.
Binh

thrir
nhtit
dryng
ddy
nudc
va iinh
thilr
hai
kh6ng
dqrng
gi
ci.
Ban
ddu,
I
ngrdi
ta
rdt
;
luOng
nwdc
trong
binh
thu
nhdt
sang
bin'h
thdr
hai,
ti€p

d6
lcti
rat
I
3
luqng
nadc
trong
binh
thie
hai sang
binh
tha
nhtit, sau
d6
lqi r6t
!
hrqng
naoc
c6
trong
4
binh
thrt
nhdt
sang
binh
thw
hai
vd

qud trinh
.lll
)ot
nhu
thd
thi sau
tdn
rdt
tha 2007
c6
bao
nhiAu
nadc
trong
mdt binh
?
Ldi
gidi. Ta
c6
si5
nu6c
trong
m6i binh
nhu sau:
Ldn 1:Binh
thrir
nh6t
c6
I
(D,

2"
DU
do6n:
Lugng
nudc
trong
binh
thri
nhAt
sau
m5i
tAn
r6t
thri
le le
!
(0,
2"
That vay,
gi6
sri
sau
l6n
r6t thr?
2n
-
|
(n
e N,
n

>
l),
trong
mdi binh
c6
I
tO.
Th6
thi
sau
2"
lAn
r6t
thir
2n,
trong
binh
thri
nhAt
c6
1
I
1
n+\
+
_.
_
=
___
(/).

2
2n+1
2
Zn+l
"
Sau
lAn
r6t
thri
Zn*l,trong
binh
thri
nh6t
c6
n+l
I
n+l
2"+l
-
2n+2
2".I
n+l(-
I
\
t.
2n+l\
2n+2)
2"
Di6u
dg

doSn
trOn
dd clugc
chring
minh.
Vfly
sau
lin
r6t
thft
2OO7
sl5
nu6c
trong
m5i
1
binh
ld
:
lit.
D
2
(Nhao
x6t.
l)
HAu
h6t c6c
ban
tt€u
tir

mOt
s5
l6n
r6t
eAu,
rrit
ra nhf,n
x6t t6ng
qu6t,. r6i
tt
d6
kdt lufln
mir
kh6ng
qua
bu6c
chring
minh
t6ng
qu6t. Lam
-nhu.
YAJ
la
kh6n!
ch4t
chE.
K6t
quA
du
do6n

trong
mQt
s6 bdi
to6n
c6 th€
kh6ng
dring
!
2)
CScban
sau
d6y
c6
ldi
gi6i
t6t:
Hn NQi:
Nguydn
Quiic
Hodn,
6416,
THCS
Gi6ng
v6,
Q:Ba
5inh,
Ed
Tr*dng
Son,
6Gl,

THCS
Marie
Cuiiel
Vint
?hfrct
Nguydn
Thi
Thanh
Thdo,
6A1,
TH
Trung
Vuong,
M€
Linh;
Hh
Nam:
Nguy€n
Hoa
Qulnh,
O-Al,
fg6S
Nguy6n
Khuy6n,
Blnh
Lqc;
Nam
51nn:
ra Thi
Phuong

Thanh,7A4,
THCS
Trdn
Drng
Ninh,
TP Nam
Dinh;
Hii
Phdng:
Hodng
Th!
QuyAy,
6D,
THCS
Hod
Blnh,
Vinh.Bdol
Hi
Tinh:
Nguy€n
Thi
Kim Oanh,
6C,
THCS
Bdc
H6ng,
TX HOnq
Ljn^h;
Bgc
Li6u:

Trdn
Quang
Minh,
6ll,
THCS
Trdn
Huj,nh,
TX
Bac
Li€u.
NGUYEN
ANH
DONG
*gei
'121357.
Cho
tam
gidc ABC
vu6ng
tsi
A.
Goi
I ld
giao didm
cdc
dudrlS
phdn
gidc
trong;
E, F

ldn
laqt
ld hinh
chi€u
cila
A
tr\n
BI,
CE. Chtlng
minh
riing
2EFz
=
AI2.
binh
thf
hai c6
Ldn
2:
Binh thri
nh6t c6
1
:
(/).
z
11
-+
ZJ
,)
binh thri hai

c6
I
-:
-
^
J
1
2
I
J
2
=T,,,
(D.
3
I
2
z_l
34
=2,
{')'
(D
Ldn i: Binh
thrt
nhAt
c6
t
binh thri
hai c6
I
-

-
=
2
Ni5u tinh
tii5p,
ta
thAy
luqng
nu6c trong
binh
thf
nh6t sau
lAn
r6t
thf 5,7,
d6u
ld
I
Ur'
LN
girti
18
saso,
trzooaw
Trong
chc tam
gi6c
vu6ng IAE, IAF
ta c6
I

iii * triE
=eo"
I
i,qE *iE)
+ iZE
=eo"
lsl
l^ l^
IIAF
+
AIF
=90"
IIAF
+
ICA+ IAC
=90o
(^
I
^
I
uz +:CBA+ 45o
=9oo
la
.<]
L
-l^
r
^
lnq,r+-BCA+45o=9oo
lz

l^
=
IAE + IAF +
-
(CBA+
BCA)=90'
2'
=iIr
+45o=9oo
=ilF
=45o
(1)
Gqi O li trung
eli6rn cria AI. vl
ifi)=iftr=gg"
(
oE
=o.q
n6n ta c6
{
loF
=oA.
Do
cl6 c6c tam
gi6c
OAE, OAF cdntai
O.
vav Edr
=fdr
*fri

=
zdZd+2il)
=
ZilF=90'
(theo
1)
Suy
ra
tam
giSc
OEF cdn tai O. Tt d6
EF
= OE
+
OP:ZO?(vtOE=
OF= OD
='
rt
oiot:!.tn
=
zEF
:
AP .d
22
(xn4o
x6t:
l) Tdt ce
clc b4n ctdu nhSn th6y
ttA bli
kh6ng chuAn:

CE cdn
phdi
duqc s*a thdnh
CL
D5i
vdi hgc sinh
ldp 7, ddy ld
bdi to6n kh6.
2) Xin
n6u t€n c6c
ban c6 ldi
giAi
dtng:
Vinh Phfc: LA
Vdn Til,
'l{t,
THCS
YOn L4c, YCn
Lac. Thdi Blnhz
Nguydn Vdn Pha,10A3,
THPT Bic
Ki6n Xuong.
NghQ An Phan Phdt
Nguy€n, 7A,
THCS
Ly Nhlt
Quang,
D6 Luong; TP HO
Chi
Minh: Br)i

Trdn Long,9A5,
THCS Chu VIn An,
Qr.
Phf YGn:
Hu)nh Xhanh Vdn,8E,
THCS TrAn
Qu6c
Todn, TP. Tuy Hda.
NGUYEN MINH HA
*nei
T31357,
Tim tiit
cd cdc
tQp hop hfru
hgn A
(l
c
N-
)
sao cho
tin tqi
tQp hqp hfiu
hgn B
(.8
c
N-
)
thoa mdn
Ac. B
vd tiing

cdc
si|. cila tQp hqp
B bdng tiing
binh
phactng
cdc sd cua
tQp
hW
A
Ldi gidi.
G1i m ld s6 lcyn nh6t trong
A.
Ta
xet
ba trudng hop:
l)
m
=
l. Khi d6 A:
{l}.
Ta
chgn B
=
{l}
y6u
cAu
bii
to6n
dugc th6a m6n.
2) m

=
2. Trong trudng hqp ndy ta c6 ho{c
A:
{2}hoicA: {1;2}.
o
X6t trudng hqp A
:
{2\.Khi
d6
|
x2
=
4
.
xeA
Nhu
thr5
n6u t6n
t4i tfp .B th6a
mdn c6c diAu
ki6n
cta bii to6n thi
Z
*
=
4
-
2:2.
Tt d6
xeB\A

B
\ A
=
{2},
mdu
thu5n,
vi 2e A.
o
Xdt trulng t'tqp A
=
{1;
21. Tuong tg
nhu
tr€n ta cfing chring minh <lugc ring kh6ng t6n
tai tfp B th6a mdn
c6c di6u
kiQn
cria bdi to6n,
3)m>2.
X6t s5
y=
Z@,
-x).
xeA
Ta c6
y
:
Z@2
-x)
>

m'
-
m
:
m(m- 1) >
xeA
2m>
m.
Nhuth6
y
e
A.X6t tfp hqp
B
:
A v
{t}.
Tac6
Ir=fIr)
+y=Zx2,do
d6tfpB
xeB
\re,4 J
xeA
th6a mdn
y6u
cAu bdi to6n.
Vfly t6t
cit citc tap hqp A th6a m6n cl6 bei lA
A
=

{l }
hoflc A ld t$p ht?u han
tiry
f
md
phAn
tu lon nh6t
ctia n6 l6n hon 2. D
(Nn4"
x6t: 1) Bdi
to6n tuong d6i hay vd kh6
vA
mlt
lQp
lu{n logic. MQt
s6 ban
xdt di
ba tru}ng
hqp nhung
trong kdt
luf,n
lo4i bo truong hqp I
=
{
I
}.
2) C6c b4n c6 ldi
gidi
ttring ld:
NghQ An: Vfi

Dinh Long,8A, THCS Li Nhflt
Quang,
D6
Luong; Phf YGn: Phgm
Quang
Thinh, SH,THCS
Hing Vuong, TP.
Tuy Hda.
NGUYEN
XUAN BINH
*nai
T41357.
Chirng minh
riing v6i
mdi sii
tu nhiAn
n
(n>
2\
ta
cd
rgHfri
Hffi
.
qr.di$
Sd 361
l7-2oo7l
19
r.ffi*
I

<n+-
2
2) C6c b4n
sau ttAy
c6 ldi
giAi
tOt
:
Bic
Ninh:
Nguydn
Hiru
Tru&ng,
8A,
THCS
Y€n
Phong,
Nguydn
Vdn Pho,9A.l,
THCS
Thi trdn
Chd,
Y€n Phong;
Vinh
Phric:
NguyAn
Thdnh
T[n,
9A,
THCS

Lfp
Thach;
Nam D!nh:
Trdn Vit
Trung,
9A9,
THCS
Phing
Chi
Ki6n;
Hi Nam;
Trdn Trung
KiAn,
98,
THCS
trAn
Phri,
Trdn
Thi
Thu!' Hoan,
8D,
THCS
TrAn
Qu6c
Toirn,
Phri
Ly;
Thanh
H.ohz
LA

Hing
Qudn,88,
THCS
Nguy6n
Chich,
E6ng
Son,
Hodng Kiln
An.9E,
THCS
Bic Sqn,
Sdm
Son;
NghQ
An:
Trdn
vi,h Thdnh,
9A2
THCS
Trd
Ldn,
Con
Cu6ng,
Tdng
Vdn Binh,
98,
TH_CS
Iti
ryhQt.Quang,
D6 Luong,

HA Hfru
Qudn,-
9C,
THCS
Hd
Xudn
Huong,
Qulnh
Luu;
TP.
H6 Chi
Minh:
Bili
Trdn
Long,
9A5,
THCS
Chu
Vdn An;
Cfln
Tho:
LA
Dqi
Thdnh,
9A4,
THCS
Trd
N6c.
PHA,M
THI

BACH
NGOC
*nei T51357.
Cho
hinh
vu6ng
ABCD
n6i
tidp duong
trdn
(O).
TrAn cung
nhd
Fi
liiy
diilm M
Uiit *i
(M
khdc
B, C)
. Cdc
drdng
thiing
CM vd
DB cdt
nhau
tqi di6m
E;
cdc
dadng

thdng
DM
vd AB cdt
nhau
tsi diAm
F.
Chfing
minh
rdng hai tam
gidc ABE
vd DOF
c6 di€n
t{ch bdng
nhau.
trong
d6 kt hi€u
nt.
=
1.2.3 n.
Ldi
gidi.
*Tru6c
htit,
nh6n
x6t
ring vdi x
>
0
vd ke N-
(k

>2)
lu6n
c6
(l+x)ft
>l+lu
(1)
Th4t
vf,y,
voi k:2
ta c6
(1+x)2
=l+2x+
x2
>l+2x.
Vflv
(1)
dirng
khi
k
=
2.
Gi6
su
(1)
dring
v1i k
(k>2)
ttrc
lit
(1+x)fr

>l+lu.
Khi d6
(1
+ x)k*l
=
(1
+
x)ft
(1
+ I)
>
(1
+ h)(1 + x)
:
1 +(k+
l)x+ kx2
>7+
(k
+ l)x.
VflV
(1)
dring v6i
k
+i.
Theo
nguy6n
li
quy
npp thi
(1)

dring
v6i mgi sti
tg nhi6n
k
(k>-2).
Tir
(1)
suy
ra
*ll.
t*
<
1+ x
.
V6ix=
k
tu"6
(fr+1)!
k
< l+-
(k+1)!
r ,
nuvplt*'.k '.t*l-
1
Q)
'!
(fr+1)t
k!
(k+1)!
+

Ggi P
ld
v6
tr6i cria
BDT cAn
chring minh
vd
vfln
dpng BDT
(2)
voi k:2,3,
,
n,
cho
n-\
si5
hpng cta
P, ta c6
p
.( t*l_l) +( r
*l_l)
* *
\'
zt
3t)
\
3!
4t)
*(r*1
-

1
):r*1-
1
.n*!.
I
r!
(n+r)t)
2
(n+r)t
2
(xn6o
x6t. l)
D€ ddn di5n BDT
kll+
'-
<l+
'-
tJ''tr+l)!
-''(t+l)!'
nhidu ban
tl6 srl dung BDT Cauchy
cho
fr s5 khOng
6m
gdm
(&-l)
s6
li I vd t so
ta 1*
-!-

tliAu ndy
(/r
+
l)!
vugt
qu6
chuong
trinh cria
c6p THCS,
ho{c sr} dqng
BDT
Bernoulli
md kh6ng chimg minh.
Ldi
gi,rti.
o
Cdch
l. Ta c6
FED=dfu=
45o.
Suy
ra
idF=fu=
4so
n6n trl
gi6c
MBEF
nOi ti6p
duqc.
Tir d6

ffiF =
tggo-
fu:90o.
Nhu vfly
tam
giSc
BEF
vudng
c6n t4i
E. Ta
c6
rl
BE: EF n6n
lx
OAxBE=
-xODxEF
.
22
V{y hai tam
gi6c
ABE vit
DOF
c6
diQn tich
bing
nhau.
o
Cdch
2. Ta c6
sdffi

=
l(raD
-"difu)
2
sdBM
=
45o
2
^
=
45"
-rdfu
Tuong
t.u sdBFM
=
z
20
T0ft-l Fffi
s6 361
17-2007l,
a
qfu,SiES
!*,qBxEH
=!*Ji*oo*
22
=
!*ODx
EF . A
2
NCn ffi

=6Ffu
hay tri
gi6c
BMFE nQi
ti6p
dusc. Tir
d6 6iF
=l8oo
-dMF
=90o.
Mat kh6c
aEF=IED=45'
n6n tam giSc
BEF
vu6ng
c6n t4i E.
Ke EH L BF.
Ta
c6
(J;
-r)k
=
A- BJ;
=J,42
-JBr"
=JV
-
Trong
trudng hqp
ndy, khing

dinh
o
dAu
bdi
dugc
chring minh,
n6u
ta chgn ar
=
12
e
N*.
o
Trudnq
hW 2. fr le.
Khi d6
A2
-
B2n
=
-(n -
l)ft, suy ra
A2
=
En-(n-1/
.
vdv
(Ji-l)o
=
-0-Ji)t

=
al-n
-,.q
:
JEtrn-JA,
=,f-gzn
-Jaz"-<"ly
Trong trudng
hgp niy, khing
dinh d dAu
bdi
<lugc chr?ng minh,
n6u chgn
a1,= ,B2n e
N*.
T6m lqi; Tri
hai trudng
hgp tr6n
ta thdy s5
d;
-l)e
lu6n dugc
bi6u diSn
du6i dpng
J* -{;-C:*
,vbi
a1,e
N D
(Nn6,
x6t.

l) B4n Nguydn
Et)rc
C6ng,10Al, THPT
DO
Luong I, NghQ
An
dd chi ra bdi T61357
ld trudng
hqp.ri€ng
cria bii
to6n 5, trang
3l trong
cu6n s6ch
Tuy?n chgn
theo chuyAn
d€ Todn h7c vd Tu6i
trd,
Quy€n
2 v6i
nQi
dung nhu sau:
Cho a"k ld
cdc s6 nguyAn
dtrong Ch*ng.ninh
riing
vdi
m6i s6
try nhiAn n
bdt
ki

ludn t6n tai
sd tu nhiAn m
sao cho
(.{r.
k
-
JA'
='[*
-
7
-
1;
.
Cho
a = l, k
=
n
-
l, n
:
k, m
+
li
:
oo, ta c6 bdi
T6/3s7.
2)
Cdcb4n
sau c6 ldi
gi6i

t6t:
Hn
NQi: Nguydn
Thi Thanh
Hoa, l1A1
To6n,
DHKHTN,
DHQG
Hd
N0i; Vinh Phric:
Phan
Q.uiic
Khdnh,gc,
THiS
Tam
Duong, Tam
Duong,
N{"ydn
Huy
Hodng, Mqc
Thd Trudn[,l0Al,
THpi
c[uy6n
Vinh Phric,
D6 Thi
Tuy€t, 10A6
chuy6n
Vdn, THPT
chuyCn
Vinh

Phric; IIii
Phdng:
Trdn Hodng_Bd,
ll
Tojin,
THPT
NK TrAn
Phri; Bf,c
Ninh:
Nguydn Minh
Hdng,
llA0, THPT
Y6n
Phong I,
Ydn Phong;
Hda
Blnhz
Vil
ViQt Dilng,
12 Todn,
THPT chuy€n
Hoing
Vdn
Thg, TP.
Hda Binh;
Nghp An:,
Thdn Trong
Dilrc,
Vii
Thanh Thily,10A1,

THPT
chuy6n Phan
EQi
Ch6u,
Nguy€n
Hodng
Hdi, llAl,
THPT
LC
Vi6t
ThuAt,
TP. Vinh;
Hi Tinh:
Nguydn
Thi
Cdm Hiing, 1l
To6n,
THPT
chuy€n
Hd finh,
Vd Td
Son,
l2Al,
THPT Trdn
Phri, Dric
Thg; Di
Ning: Nguydn
Nhr
D*c Trung,llAl,_THPT
chuy6n

Ld
Quf
D6n; Binh
Dinh:
Trdn
Qu6c
ThuQn,
l0T, THPT
chuydn L€
Quf
D6n,
Quy
Nhon; Bi Ria
-
Yf,ng
Tdut, Duong Vdn
An, llAl,
THPT
Chdu Thinh,
TX. Be Pria,
Dinh
Ngpc Thdi,
ll To6n,
THPT chuydn Ld
Quj
E6n, TP.
HO
QUANG
VINH
EF

l2
(Nr,6o
x6t.
1) Nhi€u
b4n
gidi
bdy ndy
r6t
dei. C6
nhtng
b4n
phdi
ding dtin him
s6
luqng
gi6c.
2) CLcban c6 ldi
gidi
t6t:
Phri Thg: Tq
Diic Thdnh,9A3,
Nguydn
Ngoc Trung,
9Al, THCS
L6m Thao;
Vinh Phric:
Mqc Thi
Thu
HuQ,

7A, THCS
Ddng
Qur5,
LAp Thach,
Nguydn
Thdnh
TIn,
9A, THCS Ldp
Thach;
Hhi Dvtngt
Vfi
Th.anh
Til, 10AJ,
THPT
Ke
SAt, Binh
Giang;
Btrc Ninh:
Nguy€n
Vdn Pho,9Al,
THCS
TTr.
Chd,
Yen Phong;
H) NQi:
Phqm
Duy Long,9H,
THCS
Le
Quf

.E6n,
Cdu GiAy;
Nam
Dinh:
Nguydn Thanh
Quy1t,
9A2,
THCS Trdn
Edng
Ninh, TP.
Nam Dinh,
Trdn Vil
Trung,
9A9; THCS Phirng
Chi Ki6n; Thanh
H6at, Ddrn
TriQu Dqt,9A,
THCS
Hgp
Thdnh, Tri6u
Son; Ngh$
Anz
Vd Vdn
C6ng,98,
THCS
L), Nhat
Quang,
D6
Luong;
Quing

Ngfii:.
Trinh
Nggc Thuy€n,
9D, THCS
Hulnh Thric
Kh6ng,
Nghia
Hdnh; Phrri
YAn:
Phan Long
Tri
Y€n,
8H, THCS Htrng
Vuong,
Tuy Hda.
VO
KIM THUY
*nai
T61357.
Ch*ng minh
ring v6i
mdi
cfip
sii nguyan
duong n vd
k,
t6
(J;
-t)o
auo"

biau didn
dwdi
dqng
Jn
-
c7, eN*
.
Ldi
gidi.
Ttr
c6ng
thric khai
triiSn
nhi th*c
Newton,
ta
c6
Q-Ji)P
=
A- B.Ji
Q+
'1;)k
=
A+ BJi,v'i
A,.B e
N
Ti d6
A2
-
Bzn:

(l
-
Jb,o.(t+Ji)o
:
(rr*
Ji>.g*^l
)o:
?r)k.(n-l)ft.
C6 hai trudng
hop
cAn x6t,
tt6 li:
.
Trudng
h.op 1.
k chin. L:6c
nityA2-Pn
=
(n
-l)k,
suy ra
B2n
-
A2
-
(n-
l)t
. Do d6
-
(n

-11t'*
ap-(n-l)k
T.Of;]{
lfi
.'
GIUdiW
s6
?4_v_-200?)
2l
*Bii
T71357.
Cho
rang
nli
JVJ
tam
gidc ABC.
Chang
minh
2"or4.or4.or
9
222
> 5(l)
Dting
tht:rc
xdy
ra
khi
ndo 7
Ldi

gi,rti. Ki hiQu
"f(48,C)
=
316+
2sinlsinBsinC
-tocos{cor{"or$
.
222
A6t
Aing
thric
(BDT)
(1)
tuong
duong
vdi
BDT
J(A,
B, C)
>
0
(2)
(4)
=
3.6+2sin2
A*B
,inc-locos2
nlB
"orC
2

'-
4
2
=
3.8+2cos
rc
.2"*c
*r'
-r(*"'!u*tl*r9
2' 2
2
\
2
)
2
:
tJi
+Jt-*,
.(4x(t-xz)-5(x+1))
=
3Ji-g@)
(s)
trong
d6
g(x)=J-xz
(4x3 +x+5) .
Lai chri
g
.,lt-*.g'(x)
=

(l-x2)(l2xz
+l)-x(4x3
+x+5)
:
-16xa
+10x2
-5x+l
=
110x2
-5x)+(1-(2;r)a)
:
(2x
-1)(5
x
-l-Zx
-
4x2
-8r3
)
=
(2x-1)(-x-(Zx-l)z
-8x3)
>
0
v6imoi 0.r.!.
,2
Nhu
viy
hdm s6
g(x) v6i

0
<
x
<
i
dOng
bl6n
thuc
su.
Boi v6y
v6i
mgi
0
<
x
<
:
,ta
c6
Do
<16 tir
(3)
suy
ra
f(An,ort(#,#,r)
Ding
thric
xity
ra
khi

vd chi
khi
A =
B
Ti6p theo
ta
s€ chimg
*irrh
f(+,#
Edt x=sin9.Vi
0
<C<
I
=0<x
23
. .(
.q+a
A+B
^)
Tac6 ll cl
"\
2
2
)
g(x)<'(;)=[,8)
6=311
Do d6
theo
(5)
thi

t(ry,#,c):3$-g(x)
>o
f6t
hqp
v6i
(a)
a c6
flA,
B, C)
>-
0, tirc
ld
BDT
(1)
ditng.
A.B,C
+ Ssrn-srn-sln-
')))
,cl>0.
I
2
Do
vai trd
cria
A,
B, C
ld nhu
nhau
trong
BDT

(1),
n6n kh6ng
m5t tinh
t6ng
quSt
ta
c6 thiS
gi6
tYrietC<!.
a
J
ra c6
f(A,
B,c)
- f(
+!,4*{,.'l
\z
2
)
(
^,4+B\. ^
I
zsn
,lsin
B
-
2sin2
'-
-
lsin

C
r2)
-
s( 2.o.4.o,
!-2ror"l+a).or9
-\ z z
4
)
2
:
(cos(l
-
B)
-cos(l
+ B)
+cos(l
+ B)
-l)
x
CC
x
lsln-cos-
-
22
(
,q-a
A+B
A+B
.)
c

-
5l cos-*CoS-
-cos l
lCoS-
-\ z z
2
)
z
:
"or9
(
+(
"o"'A-B
-r).in9
2[
\
z
)
2
-'(""'#-r)
C( A_B .\
cos-l
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Vdn Trung,
t lA2,
THPT Binh

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Dan
Pl,oqng,
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Bao
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chuy6n
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