THTT So 278 Thang 08 Nam 2000
Transcript of THTT So 278 Thang 08 Nam 2000
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
1/28
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7/21/2019 THTT So 278 Thang 08 Nam 2000
2/28
E
:
i-
'iw
c0NG
THUc
TiNH
rdxc
wffi
rewffiffiffi
wssffire ffi
Cdc ban
nh6
ch5c rAt
tUng
tfng khi
gap
nh0ng
ph6p
tinh
tdng
phtlc
tap, chdng
hAn
tinh A
=
1+2+3+4+5+6+7+B+9+tO. ruhfng
viOn
Oic6 thd
gifp
ich cho ban, gi6ng
nhucdc
nhA
nghi6n
cuu
toan
thoi
cd Hy
Lap.
1.
-i
i]
1-1
*
.
HAy
xi-{p
cdc
vi6n
bi
nhu
hinh
1
vd
l5y c6c
bdng
giAy
ngdn
cdch
ch0ng
tfri
oe dang
thAy
1+2+3+4+5*o
=
1616+11
=*
=r,
22
TU
d6
c6 thd tdng
qudt
h6a
thirnh
cOng
thtlc tinh
tdng
1
1+2+3+...+n='D(n*1).
2',
.
Hlnh
Anh
b6n
girip
cdc ban
d5 nhd
c6ng
thr1c,
d6ng
thdi
cUng
goi
y
c6ch
chfng
minh cOng
thtlc
Hinlt
l
DANH CHO
BAN
DOC
1) H)nh
2
goi
cho
ban c6ng
thrlc
gi
? Ban
c6 cdch
x6p
bi niro khdc
O6
tim duo.c
c6ng
thtlc
do
kh6ng
?
2)
H6i nhutr6n
otii voi
hlnh
3.
Ndm
phdn
thudng
danh cho
cdc
ban
g&i
v6 sdm
va c6
y
tLrdng hay.
rrataa
I
t
I
I
I
I
I
ttta
ii
i) i.r l{}
at
ot
at
{:i *
i-i
***
11 1.\. /*\
\_t
\-*/
i".,:
r)
{-1
ii
t-
*
t:
*
Tlr:,
1r
tl
i-
llli"l lf
Hinlt
2
ctAt
otp
Hinh
3
Tu
dinh
li Ole
dein
ndm
"c&o
hidm"
MOt
s6 ban dd
chfng
minh duoc dinh li Ole. Cdc ban dbu lh|y :
Ca.D
=
2C vd
Cm.M
=
2C.
TiJ
d6
se
cd,1*
1
-Ql
M
-c+2
=
1o1.
N6u
ca.
c->3rhi
-
-*1.1*1=l
rarthuAn.
CdCm2C2CC2-'CoCm442
maCa
Cn23,suyra Ct=3hoicCm=3.Khi Co=gthi
+=1-1nen3
x+y+z
Gie sfr:
r>
y>
z.Xt:
(x*y\(*-y\
+
1y-z)02-
z2)
+ (z-x)(22-
*)
>
0
e
2{rt+f
+;7
2
ryz+xz2+y*+yz2+z*+zy2
(l)
C$ng
hai vd
cria
(1)
vd,i
f+y3+23
vil
bidn
ddi
c6 :
3(f+y3+f)
>
(x+y+z)(f+y2+22)
>-
>
(r+y+e)
iW*
+
f+y3+23
>
x+y+Z.
DPcm.
D&ng
thric
xty ra 0vd
ab+bc+ca
=
abc.
.{ar+Zb, .{br+Zc, .{C+Zo,
r
+
,
+-/-YJ.
a0 0c ca
Gi.di. DLtl
=
r,
1
I
' a
U= i-=z'
Ta
c6
:
x,
,
z
> 0
vI
x+y+q,=
|
Ta
phf,i
chring
minh :
-lll
Xt
u'=
(x,
r'lZy
);
v'=
( ,
tl?z);
t
'=
(1,
"12
x)
lll
tht u'+
v'+
t
'=
(x+y+z;',12y
+
"l2z
+
l'l?x)
hay lx
f+
?+
/=
(t;
O)
vi lrl+flf
+trl
>
tt+f+rl
non
ta
c6
dpcm.
Ho[c
theo
bdt
df,ng
thric
Bunhiacdpxki
:
.lW6>
x+y+y:
'ltO4*+A
>
y+z+z:
{164r,
+xz)
>
z+x+x,
tr)
d6
cung c6 dpcm.
4.
DH
N0ng
nghiQp
1
Cho
a,
b, c
>
0vd
abc
=
l.
Tim minP
vdi
-bccaab
=ffi+
bh+b6+
cza+czb
Gi.d.i.
^tru6c
hdt
ta
c6
: N6u r,
y,
e
>
0 thi
:
*
*1
+I+
1+3+
1
y+z
z+x
x+y
| / 1 I 1.9
=
,(x+l+l+z+z+x)
(
r.,
*
yn*
,n
)4,
Hav:a*I*3->?.
'
y+z
z+x
x+y
'z
DEng
thrlc
xhy
ru
a lr.y=z
(day
lI
bdt
ding
thfc
Nasobit)
Datl= *r =y,L=2.
aDc
Ta
c6
:
x, ,2>
0;
ry4
=
1 vi
'
P=
* *L*L
y+z
z+x
x+y
xk+y+z\
yk+v+d
zk+y+zl
=r+-+--(r+y+i/
+z
z+x
x+y
=e+y+z)(
r
*
I
*'
-l)>
Y.
'\
)+z
z+x
x+y
/
2.
,w_l
'z
-2
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
9/28
Vfy MinP
=
6=S=6=l
g
a=y=4=l
I
5.
Dai
hoc
M6
-
Dia
chflt
MBCc6:0 tgx
+
cotgx
(1)
Gini.
0)
0.
Dft:
er+cotgx=
t>
2.
(1)
0.
DltP=u)4.
Ta
chfng
minh
:
flu)
=
u3
-
7uz
+
l4u-
8
>
o voi
u>
4.
L{p
b6ng bidn thiOn
c:0,a
flu)
ho[c
vidt
fla)
=
(u-l)(u-2)@a)
dd
suy
ra
dpcm.
7.
Dai
hoc
Ngoai
thuung
(TP
HCM)
Cho tarn
gidc
ABC.
Gid
stl
14
ld
m/t ilidm
thay
ddi ten
cenh BC.
Gpi
Rt, R2ldn
laqt
ld
bdn
kinh
vdng
trdn ngoqi
tidp cdc
tam
gidc
ABM
vd
ACM. Hdy
xdc dinh vi
trl crta
didm
M
sao
cho
ft,
+
R,
ld
nhI
nhdt.
Gini.
Ap
dung
dinh
li hlm
sd
sin cho
c6c
tam
gi6c
ABM
vl
ACM
ta c6 :
I
oat coo"
R,=
AB
:
R.=
'
ZsinZAMB'
L
Do
sin/AMB
=
sinZAMC
nln:
R,
+ R,
=
f,{ea*er,
#*r>
ltea*ee
Dtng
thrlc xby
ra
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
10/28
cAc
cu0c
rHl
roAnt
GOM
NHIEU
CAU
HOI,
TRA
LOI
NHANH
NcuvEN
vEr
HAt
thio
luan
vdl
10 bAi
tqp
nh6
trong
oo
qlrlt
1OA
tni
ding
trong
THTT
s6
257,
thdng
1 1/1998).
CuQc
thi todn
0-vttng
Ftanders
thuQc.nydc
Bi
c63'ubng,lbngtgbrirtSt?,u\(\\iirtstg\\e.m\{rn
trong
90
phrlt,
vdng
2
g6m 30 cau
h6itrdc
nghiQm
ldm
trong
60
ph0t,
vdng
3
gOm
4
bli
tap
kh6.
CuQc
thi
Olympic
todn
d
ntdc
Bi
g61n
3
vdng
:
vdno thi
so
tuvdn
q6m
30
cAu
h6i
trdc
nghi6m,
von[
z
gom
eb
ca-u
nOi
tric
nghiOm
vd
bdi
t&p
nh6
ldm
trong
90
phtit,
vdng
3 thi
chgn
hQc
sinh
gi6i
qu6c gia
gom
4
bAi
tAP kh6.
CuOc
thitodn
Canguru (t6n lodi th0 c6
trii
s6.ng
d
Oxtralia)
d
Oxtralia
c6
tir
nhfing
nim
80
gOm
cdc
cAu
h6i trSc
nghiQm,
d6n nay
c6
tdi
nfia
triQu
hqc
sinh
tham
gia.
Cdc
nhA
todn
hqc
PhAp
vd
Oxtralia
dd bdn
bac
td chrlc
cuQc
thi
Canguru
khOng
bi6n
gidt
(Le
Kangourou
sans
lrontidres)
tir
ndm
1991,
d6n
ndrn
1993
da
lan
sang
qdc
nudc
chAu Au
vd
du-d. c
sq,
0ng
hQ
cria
Uy
ban
ch6u
Au
vd
UNESCO
trong
D6
dn 2000.
.SO
trrgng
thi
sinh
tham
gia
cuQc
thi
nAy
tdng
dAn :
1991
1993
1995
1996
6
Ph6p
1 20
000
420
000
500
000 600
000
d ngodi
40 000
200
000
500
000
nudc
Phdp
D5n
ndm
1996
da c6
hoc
sinh
c0a
26
nulc
tham
gia cuQc
thi Canguru
kh6ng
bi6n
gidt:
Phdp,
Drlc,
Anh,
T6Y
Ban
Nha,
ltalia,
Lucxembua,
HA
Lan,
Nga,
B6larut,
Ba
Lan,
S6c,
Sl6v6nia,
EstOnia,
Hunggari,
Max6d0nia,
M6ndavi,
Rumani
(chdu
Au),
Oxtralia
(chAu
Uc),
Bu6ckina
Phas6,
COt
Divoa,
Mar6c,
Madagasca,
MOritani,
SOnOgan,
Tuynidi,
(" ''i11.Phi), Libing
(chAu
A). Cdc
bqn mu6n
bi6t chi
ti6t
hdy gt]i
thu
d6n
dia
chi
:
Kangourou
Sans
Frontidres
50
Flue
des Ecoles,
75005
PARIS-FRANCE
ho{c
tr6n
lnternet
:
http
;
//Aaruw.mathkang'org'
Xin
gidt
thiQu
mQt
s6
cAu
h6i
trong
cu6c
thi
cZigiri
xi6is
oi6i
iia
ii^
rsgz.
iuoc
ini
td
Cac
cuQc
thi
todn
duo.
c
td
chfc
bdng
nhi6u
hlnh
thrlc
khdc
nhau.
C6
hinh
thtlc
thi
dua
ra
cdc
bdi
ioan
tro,rrg
dOi
kh6,
ldl
gi6i
ld
mOt
chu6i
suy
luQn
phtc
tap,
dbr hbr
tirnh
0Q
phhn
Yrch
\6ng
hgp
kh5
cao.
C6c
bdi
todn
loqi
ndy
dugc
ddng
trdn
bdo
chi
dd
c6 thdi
gian
dAi
suy
nghl
gi&i
d6p
hoic
dirng
trong
cdc
LuQc
lhi
chgn.hqc
srnh
grdi'
Hinh
thrrc-khe;la
bai
thi
gOm
nhiou
cdu
h6i,
bdi
tQp
khOng
qud
kh6
vdl
y6u
cAu
tri
ldi
nhanh'
Loai
nav
doi hOi
nqc
sinnbiot
dp
dqng
dung
va
nhanh
cai;
fien
th(c
todn
pnd
tfrong
vdo
cdc
trudng
hgp
khdc
nhau,
cdc
tinh
hu6ng
thtlc
t6, tap
dugt
suy
luAn lOoic.
Mdt
khdc
bai thi nhiiiu
cdu
h6i
n6n
c6
thd
ba;
g6m'cdc
phAn
khdc
nhau
cta
chLrcmg
irinh,
kid"m
ira
sLr'nidu
oi6t
rQng
v6
ki5n
thrlc
todn.
Trong
cdc
cuQc
thi
ndy
thf
sinh
khOng
duoc
s0
dqng
cdc
loai
sdch
vd,
mdy
tfnh,
thudtc
t(nh,
gi6y
v6-dO
thi
mA
chi
duo.c
ddng
thudc
k6-thdng,
6o,irpa.
Bai
thi
cho
trong
dqng
cAu
h6i
tric
nqhiOm
(thi
sinh
dtrgc
chgn
mQt
trong
cdc
ddp dn
c6o
s5nj
hofc
cdc
bAi
tQp nh6
(thi
sinh
phAi giAi
bdi
tap
nhurng
chi
ghi
ddp
sO mA
khOng
trinh
bAy
ldi
qidi),
di6u
ndy
girlp
cho
viQc
chAm
thi
nhanh
ch6ng
(c6
thd ch6m
bdng
mdy
tinh),
dOng
thdi
vugt
qua
duo. c
hAng
rdo
ngon
ngfr
gitla
cdc nudc
khdc nhau. Vdi nh0ng dqc di6m tr6n hinh
thtlc
thi
todn
gOm
nniou
car]
h6i,
tra
ldi
nhanh
c6
thd
ddnh
cho
d1ng
dAo
hqg
sinh
&
nhiu
vdng,
nhl6u
ntdc,
dflp
r4g
nhu
cAu
giao
lur
todn-.hgc
gi0a
cdc
virng mi6n
khdc
nhau.
Ngudi
ta
t6 chfc
thi
cUng
mQt
dO
bai vAo
cirng
mQt
thdi
di6m,
sau
d6
gAt
bAi
gidi
ngay
qua
p&t
drpn
(trong
tudng
lai
c6
thd
qua
thu
di6n
tit)
v6
cho
Ban
t6 cht?c
cuQc
thi,
Hinh
th0c
thi
nAy
d5 duo.c
td
chrlc
d My,
Oxtralia,
Bi,
Phep,
cdc
nudc
D6ng
Nam
A
vi
mQt
s0
nudc
chAu Au.
d nhi6u
nudc,
cuQc
thi
ndy
duo.
c coi
ld vdng
so
tuydn
cho
cuQc
thi
Olympic
to6n
qu6c gia,
Sau
dAy
lA mQt
s0
cuQc
thi
:
Trung tAm
gi6o dr;c
khoa
hq-c
vA
todn
hqc
DOng
Nam
A
(RESCAM)
da
td
chrlc
thi
todn
nhi6u
c6u
h6i,
trd ldl
nhanh
d
Malaixia
(7/1996)
vd
cho
9 nudc
EOng
Nam
A
1sltsoa1.
CuOc
thi
nAy
c6
3 vdng
:
v0ng
1
ddnh
cho
c6
nhdn
gdm
20
ciu
h6i
ngin
trong
90
phtlt'
vdng
2
ddnh
6ho
cd
nhdn
gOm
5
bdi
tap
nhd
trong
180
phtit,
vdng
3
cho
ph6p
cA 4
tht
sinh
trong
dQi
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
11/28
chfc
ngay 2113/1997
cho 2 trinh
dQ
:
cdc
tdp dQ trt,
dQ tam
(tuong
duong
ldp
I, ldp
9 THCS)
vd cdc
tdp
lyc6es
(tuong
duong
cAp
3 THPT)
Thdi
gian:
75
phtit.
OC
tni
gOm
30
cdu
h6i
:
Cdc
cAu
h6itit
l d6n
10
duo.
c
3 didm
m5i
cdu
;
cdc cdu
h6i tir
11 d6n
20
Ouqc
+ didm m6i
cAu
i
cdc
cAu
h6i rU
21
d6;
ao
duo. c 5
Oidm
m6i cdu. Thf
sinh
duo. c chon
1
cdu
tri
loi
d0ng nhdt
trong 5
ddp
dn A, B,
C,
D,
E.
cAc
Lop rucs
CAu
6. Cho
ba didm
A,
B,
C khOng thEng
hdng.
Th6m
didm thf
tutren
mat
ph&ng
chia
A, B-,
C sa-o
cho
4
didm
d6 lQp
thAnh
rnQt
ninn
Uinn hdnh.
C6
bao nhi6u
didm v6
thOm th6a
m6n di6u tr6n ?
A:1 B:2
C:3
D:4
E:6
Ciu 10.
Budi
sdng. LOra
dang trang
didm,
lloeng
nhin
thdy tron
guoJrg
cni66
o6ndno
qud
lSctre.o
phia sau lung.
M{tdOng
hO khOng
ghi
bing
chrf
s0 ArQp.
C0 b6 k6u
l6n
"DOng
ho.ch6t
r6i
Bdy
gid
4
gid
k6m
5
phrit
A ?'. Lora
da nhAm.
H6i
dOng
h0
l0c
d6
chi mdy
gid
?
A:8h05
B:4h55
C:7h55
C6u 12.
M6i
c4nh c0a
hinh
ch0
nhit
duoc chia
thAnh
3
doqn
bdng nhau
vA cdc
didm
chia
duo. c
nOi vdi nhau
di
qua
tdm
nhu
hinh ben.
H6i di6n
tfch
phdn
Od trdng
bdng
bio
nhi6u
phan
c6
cdc'.,dAu.
chAm ?
nqoAi hdng
rAo
md huou
c6 thd an
duo. c
(tfnh
theo
m')?
A:96
8:99,14
C:1O2,28
D:105,42
E:108,56
CAC LOP
.
C6u
3j Sdu
nga
duOtng
dAn d6n
m6t
ddo
trdn
(bUng
binh),
trong
dd
c6
2
dudmg
mQt
chi6u
(chi
duqc
di
vdo).
Quanh
dio
trdn
phii
di nguoc chi6u
kim
dOng h6.
H6i
c6
bao
nhi6u
cdch vdo
-
ra khdc nhau mA
kh6ng di
qu6
1
vdng
quanh
dAo trdn
?
A:12 B:48
C:24 D:30
E:28
CAu 9.
Xem cdu
13 THCS
Ciu 1 1.
Xem cAu 1
2 THCS
GAu
20.
(c0ng
ld
cdu22
THCS)
Dem chia
st5
1O...O,.trong
cdch
viOt cO
tggZ
ch0
sti
0 vi5t sau
chf
s5
1,
c-ho
15
thi
duo.
c
du
ld
bao
nhi6u ?
A:1
8:6 C:9
D:10 E:12
CAu 21.
Xem cAu 30 THCS
THPT
,=NlP\*,',
::i.::::i::.::\
1...:::::::.:.:.1:
,,.,.,r,:,.,:,:,:,:{
QD
A
}l.l,,,l.i.l,l.l,l,,
.:::.:::::.:,
/ l||||1.
|{A---o#+
Z
4llll:t:l:::l:l:l::tl
l::::::l::::::::::::::::::::::::::
.t::.1
/::|::::::t :
D:8h55
E:4h05
.A:1
B:1/2
C:1/3
D:1/4
E:2/3.
CAu 26. Nau
r(f(x))
nhi6u
?
A:
(x)
=
-2x+3
C:
(x)=+fi-s
E: f(x)
=
-4x+
1
=
4x-3
thi
(,
bdng
bao
B:
f(x)
=
2^'lx
-
3
D:f(x)
=2x-3
D:ar/5
E:
phu
C6u 13.
Trong dudng
trdn
bdn kfnh
3cm
vE mOt
hinh
ch0 nhAt
ABCD.
Ggi
l,
J,
K,
H
ld
trung
didm
cdc
canh
cia
hinh chrl
nhAt
dd.
H6i chu
vi
c0a
hinh
thoi
IJKH
bdng
bao
nhi6u
(cm)
?
A:6
B:9
C:12
-
.t
Vua
qua,
Bg
Gi6o
duc
vi
Dlo
tao
dl td chrrc
trong thd
L0
kr niOm
90
nlm ngly
sinh cfra
Gg
T4
Quang
Bfru
(23.7,1910).
Bd
tru0ng Nguy0n Minh
Hidn
Aa Adc Uii
Oien
vtrn
ddnh
gi6
cdng lao
tg lon
cfla
CS
Ta
Quarg
Buu
trong
c6c
hnh
vuc
kiroa hoc,
quOc
phdng,
ngo4r
giao
vi
dic
biQt li
rong
su
nghi0p.gi6o
dUc. Nhdn
dlp
nhy,
THTT
rrqn trgng
girii
thi6u
bhi
vi0t
crla
GS.
TSKH
NgO ViCr
Trung
v6
nhlng
k niirn
vcn
CS. Ta
Quang
Buu
(trang
2y.
*
_Chiiu
ngiy
1.8.2000,
Dar
hqc
Qu6c
gia
He NOi
da rd
chric
Lf
tuy6n duoug
khen
thuong
c6c
hoc
sinh dat
gihi
cao
trong
c6c
lii
thi
Qlyftpic
quiSc
t6,-Ciao su
-
Vign
ii
ilguy6n
VIn
DAo,
gi6m
d6c DHQG
Ha NQi
da
nhiOt li0t
hoan nghOnh
vi
chric m-ing q6c
.em
h}t
sinh
gilnh
thdng lE uong
iac
H
thi Olympic
qu6c
t0
u&
v0,
D{c
bitt, diy
li
16n
diu
tin
ci,3
em
d4t
huy
chwng Ving
m1n Toin
cfia do?tn ViQt
Nan du
lA
h7c
sinh
khdi
chuyn Toin
-
Tin
hgc
crta
twng
DHK\TN
thuQc
DHQG
HA
NAi,
Cir6o su
dI
d6nh
gi6
cao ss
c0
glng
cila
c6c em
hoc
sinh cr)ng sU
d6ng
g6p
cfra c6c
nhi
gilo,
sq
chdm s6c
cfra
cdc
vi
phu
huynh.
NhAn dip niy
DHQG
Hi
NQ
di
tuy0n durrng khen
thudng
cdc emhoc
sinh
dat
giii
vd
nhiOu
nhl
gi6o
c6 thhnh tich
lroug
vi6c
boi duoug
hoc
si,h
gidi'
THTT
thuoc
h)nh
chO nhAt
CAu
26.
Nguoi
ta
gdp
doi
m6t
td
giAy
hinh
chff
nhat sao
cho
hai chi6u
rQng
trUng.nhau.
Trong
m6i
?n
ggp
doi
ti6p
theo
cAn
phdi
gap
vu61g
g6c
vdi
I_an
gap
trudc.
Sau
5 lAn
gAp
ngudi
ta
cit
mOt
ch0t
o4
dinh hinh
ch0
nhQt
d.5
gip.
H6i
c6
bao
nhi6u
t6
thtng
d
b6n
trong td
giAy
duoc
m0 ra
nhu ltic
ban
dAu
?
A:4 B:9
C:18
D:20
E:21
CAu 30.
MOt
con huou cao
cd
dtrong
mOt mAnh
dAt
hinh tam
gidc
vdi d0
ddi ba
c.anh lA
20m,
16m
vA 1
2m. MAnh dat
c6 hAng
rAo thdp
nhung
huou c6
cd
dai nen
c6 thd dn
c6 xahh ngon.b6n
ngoai
manh
Odt
cacn
hAng rdo
2m.
Trong-cdc
sO siu
d6y, s6
ndo
bi6u thi
gan
dUng
nhdt
di6n
tich
phAn
c6 moc
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
12/28
THI
TOAN
Ki
tniTodn
qutSc
to
1tuo1
lAn
th0
41 duo. c
td chfc
tqi
Taejon
(HAn
Qu
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
13/28
Bni 3.
Cho
tnltrc
s6
nguyOn duong
n>
2.Ll6c
dtu
c6
n
cnr,
bq
ch6t &
tren
mQt
dr:Ong
thtng
nim
ngang sao cho
khOng
phti
tdt
c6 chring
ddP
0
tai cing
mQl.
didm.
Vdri
m6i
sd
thuc duong
1,, mOt bwfic chuydn
c6c con
bq
ch6t
duqc
dinh
nghia
nhu
sau
:
Chqn
hai
con
bg
ch6t
nlm 0
hai
didmA
vl B
trong
d6
A
nXm
bOn
tr6i B.
Sau
d6
dd con
bq ch6t
O a-nney ffi
di6m C
nim
trOn
ihdng
thtng
nly
vir
0 bOn
phtri
B vfi
BCIAB
=X.
X6c
dinh
tdt cL
cdc
gi6
t4 dtrcrng
l.
sao
cho
vdi
mgi dim
M n[m
ren
du0ng thtng
tren
v]
vdi
mgi
vi
tri
ban diu
ctra
n
con
bq
ch6t,
ta d0u
c6 thd thuc
hign
dugc
mOt sd hrru
han c6c budc
chuydn
dd Crn
to}n b0
c6c
con
bq
ch6t
tdi
cdc vi
tri
nlmbOn
phAi
dimM.
NGAYTHUHAI
Thdi
gian
ldm
bdi
:
4
gid
j0
phrlt
Bei
4.
MQt
nha
[o thu0t
c6
mOt
tr[m
tdm
thO
d6nh sd
rir
1 tdi
100.
Ong
ta
s{p
xdp
tdt
ct
chring
vtro ba
chidc
hQp, mQt
hQp
son
mdu
d6,
mQt hQp
son
mlu
tr{ng
vtr
mQt hQp son
mtru xanh sao cho
mdi hOp ddu c6
chria
it
nhdt mQt tdm th6.
MQt
kh6n
gil
dugc
d0
nghi
chgn
hai hQp tty
f
trong ba
hQp
n6i
trOn
rdi tir
mdi
hQp
dugc
chqn
rut
ra
mQt
tdm
th6,
cQng
hai
sd
tren
hai
tdm th6
dx
rut
ra
vt
thOng
b6o
k6t
quA.
Chi ctn bidt
duqc
tdng
nly,
nh}
6o thuflt c6
thd
n6i
chinh
x6c
chi6c
hQp mh kh6n
gifl
d6 dl
khOng
chqn.
H6i
c6
tdt
ce
bao
nhiOu c6ch
s{p xdp
100
tdm
thO
nly vtro
ba
chidc
hQp trOn
dd
6o
thuflt
nly
luOn
thtrnh cOng
?
(Hai
c6ch sfp
xdp
thO
duqc
coi
ld
kh6c
nhau
ndu trong
hai
cdch
xdp nly
c6
it
nhdt mQt
t{m
thO
duqc
d{t
0
hai hQp kh4c nhau).
Bni
5.
X6c dinh
xem
c6 tdn
t4i
hay kh0ng
sd
nguyen
duong
n
thba man ddng thrli hai
didu
kiQn sau :
n
cO
dring
2000
udc
nguy0n
td
ph0n
biQt
vl X
+
I
ctuahdt
cho n.
Bei
6. Gil srlAIll,
BHz,
CH3ltr ba du0ng cao
ctra
mQt tam
gi6c
nhgn
ABC.
Dtrlng
tron
noi
tidp
tam gi6c
ABC
trp
xric
vdi
c6c
canh
BC,
CA,
AB
tuong
r-mg
14i
71, 72,
73. Gqi
h
ll
dr$ng
thlng
ddi
xtmg
v|i
H2H3
qua
7273,
/2
lI
tludng
thflng
cldi
xung vdi
H3Hl
qua
I3I1, 13
lI
sin(cosx).
Solution,
Because
of
periodicity,
we need
consider
only
values
of
.r
belonging to some
intdrval
of
Iength 22. Thus,
assume
that
-nl2
3
x
0.
Hence
cos(sinr)
>
sin(cosr)
for
nlT < x
n.
xn_rlxn+x)
xn(xix2)
T41278. T\e
similar ftiangles ABC
and
APtCr
satisfy the
conditions
:
A1
lies
on the
ray
CB,
B,
lies
on the ny
AC,
C,
lies
on
the
ny
BA.
Prove that
the orthoccnlcr
of tliangle
APP.
coincides with
the
circumcenlcr
ol
triangle
ABC.
TSl278.
The vertices
of a convex
hexagon
are
labelled clockwise
by
6
consecutive
even
numbers.
We
make
a
change of
labelling
by
choosing
a side
of
the
hexagon
and
adding an
integer
to
each
number
written
at the
extremities of this
side.
Is
this
possible
that
after
a finite
nurnber
of
such changes,
the
numbers
written
at the vertices
ol
the
hexagon
becomc
all
equal
?
FOR
UPPER
SECONDARY
SCHOOLS
T61278.
The
sequence
of
numbers
(u)
(n =
0,
1,2,...)
is
defined
by :
uo
=.1,
ut
=
l,
iln+z
=
I999un*1-
un
for
every n
=
0,
7,2,
...
Find all natural
numbers
n such
that
a,
is
primc.
'l7l27t].
Prove
thal
h/t
sirul'+sin2l+sin3r-2(en+1)+
+.jf,l
+5
L
nam
tim
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
19/28
ctAr
eAr
ri rnuoc
*
Xdt
htrm
seiflx)
=
2x.arctgx
-
ln(l
+
x2)
c6f'(x)=2.arctgx.
Vdi.r
>
0
thi/'(-r)
>
0;
vdi
x 1+ln(1+-r2)
=)rr*rn(l
+
xz)27=ffiffi
.irrr.
Tn'( 1)
vh
(2)
suyradidu
cdnphtri
chring
minh.
Nhin
x6L 1)
MOt
sdb4n
chrlng minh
bdt ding thrlc
(1)
qu6 phfc
tap.
2)
Tdt
cd c6c
ban ddu
gili
ddng
vtr rdt
d6ng
khen.
Clm
on c6c ban.
lErr6NcNsAr
Bei
T71278.
Tim
nghi\m
duottg
cia
phuury
tritth
,,1
I
rrn(r*1)'*i-
itn( t*1)'?
=t-x
x)
\-
f)
Lcri
giii. (cia
Trdn
Tudn
Anh,
tdp
12
To6n,
tnrdng t
Qu
DOn, Nha
Trang, I(hdnh
Hdu
Trdn
Dinh Nguyn,
PTNK
DHQG
TP H6
Chi Minh
vt
mor
sd ban
kh6c).
Vd'i.t
>
0,
dua
phtrung
trinh da
cho
vd
d4ng
tuong
duorlg
sau
:
(x+r)rr(,
.1
)
-
(x3+x)hr(
t
+
i)
=,
-,
e(.r+l)ln(r+1)-r=
=
r[
(.r2+1)
rn(
r
+
i
l_
,
]
o)
Dar/(.x)
=
x[(r+1)rn(r
+*)-,
],
tr,i
rrl
c6
d4ng
11x1
=
f(*)
Q)
'tacof
'(x)=
(2x+1)ln(
,11-z
=(z,r+1)[rn(r+*l
=]
r:i
2
D{t
s(x)
=
*( t
.i
)-
l,
voi.r>
0.
,*;
DE
thdv
s'(x)
=
I
I
uw
vtqJ
6
\^,,,
-
,(r*D
*
_r___+
4
4x(x+l)
(l+x+l)
< 0 v->0
VQy
S(x)
l)
hdm
gi6m
khi,r>0, mtr
rims(.r)
=
rmI
m( r
+
i
)
-
*
]
=
o
r-)l{
x-+rob'
\
^'
**;'
+
g(r)
>
g(0)
Vx
>
0
Do
2x+I
>
0
Vx
>
0,
nOn
tt
(3)
suy ra
f
'(x)
> 0
V->0,
tric
llfl.r)
lh
him
t6ng
khi
>0.
Do
vfly
Q)
a
*=
12
y
>
0 thi
ln
|
,2'-1"
'yx+y
2)
Trong
sd 21
ban
gili
sai,
thi sai
ldm
chri
ydu
lb
:
-
NgQ
nhfln
mQt
htrm
sd
ll
d6ng bidn.
-
Tinh
todn
dgo
htrm
cria
c6c
him
siu
viQt
sai.
Dic
biet
cd mQr vli
ban bidn
ddi
;
,l
1
tnIt+i)=tnt.tni(tt)
\
^x-
/
x'
3) C6c ban
sau d0y
c6
ldi
gitri
tdt.
Hi
Ndi:
NgryEn
Hodng Thgch,
10T,
Ha
NOi
-
Arnsterdaq
NguyZn Tutn Duong,
l0T,
Trdn
Tdt Dqt.
l2B
to6n,
DHKHTN-DHQG
Hr
Noi;
Tp Hd
Chi
Minh
: Hodng
Thanh Ldm,12 I'of,n,
Luong Thd Nltdn,
1l
To6n, DHQG Tp Hd
Chi
Minh;
Thrii
Nguyn:
Vd
Quang
Dftc
vd
Trdn Dfirc
Mgnh,
11
To6n
NK
Th6i
Nguy0n
Vinh
Phric: Nguy4n Ti6t
Thinh, llAl,
Nguy4n
Trmg
ljp,l2A,
chuyOn Vinh Phric
;
Bdc Ninlr: E6 Arth Dfir, Nguy1n Ngpc
Cudng,
H6a 11,
n
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
20/28
crAr
BAI
r'irnuoc
Todn
11,
Trudng
TI{PT
NK
B6c
Ninh;
tlli
Dtrong:
Nguy1n
Thanh
Hdi,
11 To6n,
THPT
Nguy6n
Trai,
Htri
Duong;
Thanh
H6al
Phan Vdn
Tidn,
l2T,
TIIPT Lam
Son,
Thanh
H6a;
Nghg
Ant
DinhThanhThudng'
11A,
THPT
Hermann Gmeiner,
NguyZn
Thanh Son,
11
Tin,
Khdi
PTCT
-
Tin DHSP
Vinh; L0m
D6ngt
Phan
Thi
ThanhVdn,
ll
To6n,
THPT
chuy0n
L[mDdng;
Ninh
Binh:
Er)ift
Htu
Tiep,
11
'l'ohn,
THPT
Luong
Vtrn
T\ry;
Vfing
Tiru:
Trdn
Quang
Vinh, ll
T2,
THPT
t
e
Quli
Dtin.
PHANHUY
KHAI
Biti T81274.
Cho
tam
Sidc
ABC. Chtmg
minh
rdng
ABC
1
+cof
I
+cosf
1
+cos;
-
^'.-G-t.
,-
>
3\5
trong
dd
cdc gdc
A,
B,
C
do bdng radian.
Loi
gi6i.
Cdch
1.
@rta
TO
Minh
Hodng,
11T, THPT
chuy0n
Hf,i
Duung
vtr
nhidu ban
kh6c).
X6t hlm
sd
f(t)
=
tgt+
sinr
-
2t,
t
.lO,t),,{O)
=
O
Ta
c6
f
'(t)
=
*,.cos,
-
2,f
'(0)=
0,
,
[O,l)
os-,
f
"(t)
=*.-
sin
>
o,
I
e
[0,
|
)
cos-,
Suy
ra
f
'(t)
>
"f
'(0)
=
0,
/
e
ddng
bidn trong
I
o,;]
Suy
raflr)
>flO)
=
0,
M[t
khic,
trong
MBC ta
luOn c6
ABC
T
=
cotl+
cotg,
+
cotS
ABC
=
cot\cotgrcotg,
,r;@=3{7,
nOn
Z> 3{t. Thay
vio
(1)
ta
dugc
dpcm.
Cdch
2.
(crla
da
sd cric
bgn).
vdi x
e
(0,
;
)thi
sinxcot|
DOi
vdri
clc
g6c
B,
C
ta c{lng c6 bdt
ding
thric
tuong
tU
(*).
Do
vfy
ABC
I
+co5
I
+co5
I
+cos,
+-
--
ABC
ABC
>
cotf+
cotE+
cots;.
(l)
ta
c6
(*)
18
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
21/28
ctAr
eAr
ri rnuroc
Vdn
Tidn,
1147,
TTIPT
Kinh
MOn, T6 Minh Hodng,
Phqm
Thdnh Trung,
Ya
Xudn
Nam,
Chu
Ngpc Hung,
NgA Xudn Bdch,
NguyAn Tidn
ViQt
Hwng,
l0T,
Brti
Drry Thlnh, 11T, TIIPT
chuyOn Nguy6n Trai;
Be Rla
-
Vring
Tiu: Nguy4n
Vdn
Thdnh, Phan
Hodng
Va,
12T,
TIIPT
chuyOn
LC
Qu
DOn;
Hud:
Nguy4n Du
Thdi,
1lT, DHKI{
Hud;
Hn
N6i:
Ddo
Tudn
Son,
NguyAn
Hodng
Thgclt, lOT,
D(ng
Nggc
Minh,
llT,
THPT Amstedarn"
Phan
Nhdt
Thdng, 11A1,
PmL
TOn
Dric Thdng, L
Chi
KiAn,11T,
Hodng
Tidn
Mqilh,
Nguy1n
Kim
Thanh,
l0B, Hodng Ttng,
L2A, Ngi
Qudc
Anh,
11A, Nguy4n
Quang
Hdi, 10AT,
DHKHTN-DHS,
Ngry1n
Trung KiAn, 10A2, Hdn
Thd Anh, 11A1, DHS;
Hi Tinh:
Nguy4n
Thta
Thdng,
1lT, TIIPT
chuyn; NghO
An:
NguyAn
Dinh
Trung,
NguyAn Thanh
Son,
lLT,
Vfr NguyAn
Bdc,
lt Tdt
Thdng,
11A,
DHSP Vinh;
Ninh
Thugn: LA Tidn
Trung, L2A,
Ddo
Dqi
Duong,11A2,
THPT
Chu Vdn
An;
Lam
Ddngt Phan
Dinh Hdi
Soz, 11A10; Khtinh
IJin:
12
Th Khdnh
Hiln,llT,
THPT
I.
Quf
Don;
Gia
Lai: D(ng
Tudn
HiQp,llC3,
THPT Hing
Vuong;
'
Quing
Ngdi:
Trdn
Thdi
An
Nghia;
Quing
Tri:
.
Hodrtg Minh Phyng,
llT, Bqch
Nggc
CAng
D*c, l2T,
THPT
chuyn; Thanh H6a:
Trinh Khdc Tudn,
M,
THPT
LC Hodn, Hodng
Minh
Tidn,
1lB,
THPT Bim
Son,
Bri
Ngpc Hdn,
l0T,
Phan
Vdn
Tidn,
LZT, Lanr
San,
N
guydn
Vdn Trun
g,
I 1A1, TIIPT Hflu
LQc
1
;
Phri
Tho:
Hodng
Tudn
Anh,
llH, TIIPT
COng
nghiQp,
Nguydn
Dinh
H)a,1041,
chuy0n
Hing
Vuong;
Vinh
Phict
Nguy4n Hodi
Va,
l0Al,
Nguy1n Ddc Tnng,
Dd
Mqnh Tnng,
11A3,
12 Mqnh
Hilng,
llA, NguyZn
Mqnh
Hd, NguyZn
Trung
l$p,
l2A,
I2 Khdnh
Hing,
12A3,
TIIPT
chuyn, Phnng
Anh
Dilx,
I}At,
Trdn
Trung
Hidu,11A,
THPT
NgO
Gia TV, Nguy4n
Trudng
Giang,
1141, THPT
L0
Xoay; Hii
Phingt
Luong
Minh
Hdi,
l0T,
Dodn
Duy
Trung,
l0T, Phqm
Dfir
HiQp,
l0T,
Trin
Phri,
Trinh
Quang
Hidu,
lla,THPT
NgO
Quy0n;
Bdc
Ninh:
NguyZn
Vdn Thdnh,
L2A,
TI{PT
Thufln
Thanh
l,
Nguy7n
Ngpc
Crfrng,
ttT,
Ngtrydn
Vdn
Tidn,
1241,
THPT
chuyOn; Y0n
B6i:
Trdn
Vipt Yn,
llA2,
Ngq,dn
ViQt Hdng,
l1A,
TIIPT
chuyOn
NGUYENVANMAU
Biti'191274.
Cho tam
gidc ABC.
G7i
p
ld
n*a
chu vi
tam
gidc
ABC
vd
R
ld btu
kinh
dadng
trdn ngoqi
tidp tam
gidc
d6. Ggi
D,
E,
F
theo
thdt
tW
ld
tdm
dudng
trdn
bdng
tidp
cdc
gdc
A, B,
C.
Chrtng
minh rdng
Dd+nf+roz>afgpn
Ddng
thrtc
xAy
ra
khi
ndo
?
Ldi
gifli.
@Ea
theo Trdn
Va
Huy,l6p
9/3, Tnrdng
Nguy6n Binh Khi0rq
BiOn Hda, D6ng
Nai)
E
'Vt
D,
E,
F
le
ffim
dudng trdn
bing
tidp
c6c
glc
A,
B,
C cta
tam
giec
ABC
nn
EF,
FD,
DE
lin lugt ll
phin
gi6c
ngotri cdc
g6c
A,
B,
C;
d6ng
th0i AD, BE,
CF
ld
phAn
gi6c
trong c6c
g6c
d6 vl
ddng
quy
0
tflm
1
dnong
tron
nQi
tidp
tam
gi6c
ABC,
nhung
ddi
vdi
tam
gi6c
DEF
ti
AD, BE,
CF ld c6c
du0ng cao ve
l
le
rgc
mm.
Ldy
didm
,/
ddi
xtmg
vdi
/
qua
ttm O
dndng tr0n ngopi
np
MBC
thi
J la
t6m duOmg
trOn
ngo4i
tidp tam
gidc
DEF
(vi
durdng
trOn
t0m
O
ltr
duong
ffdn
Ole cta
tam
gi6c
DEF).
C0ng
dE
thdy
ring cAc
trung didm
Ao,
Bo
vi
Co
ci;a
ID,
IE
vd
IF
lin
lust la
truns
didm
cdc
cung
bi ctra
cdc.urg
ft,
ffi ui.un
g
fri
cfra
drdng
trOn
tam
O,
do
dd
: JD ll
OAo
vd,
JD
=
2OAo,
tt) dd suy
ra:
JD
=
JE
=
JF
=
R'
=
2R,
vi
JDIBC,
JEICA,
JFJAB.
Chri
f
ring
di
tam
gi6,c
ABC
ll nhgn,
wOng hay
tri
thi
DEF
bao
gid
c0ng
la
tam
gi6c
nhgn
vA
do
d6,
tim
"I
cria dudng
trdn
ngo4i
Up
LDEF nbm
trong
tam
gi6c
DEF.
Gqi
,5'
li
dign
tich
tam
giic
DEF
ta
c6
:
S'= S(JBDa
+
SQCEA) +,S(,rAFB)
=
1
='(BC.JD
+
CA.JE
+
AB.Jfl
=
pR'
Z
hay li
:
S'
=
ZpR
(1)
Ngotri ra,
ddi vdi tam
gi6c
DEF
ta c6
mQt
bft
dtng thric
quen
thuOc
:
Dd +
s,f
+ FDz
>4J'i5.
e)
19
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
22/28
crAr
BAr
xi rnudc
Tu
(1)
vd
Q)
ta
dugc
BDT
cfuL
tim
:
od+nf+FDT>a{:epn.
Eing
thric
dat
dugc
khi
vtr
chi
ktri tam
gi6c
DEF
ld dOu,
vl do
d6,
khi
vd
chi
ktri tam
gi6c
ABC
ltr ddu.
Chrt
ttzich.
C6 tdt
nhidu c6ch chring minh
BDT
(2)
quen
thuQc.
Sau
d0y
ltr
mOt
cdch
chirng
minh
kh5
dcrn
gi6n
(cong
cria
bvn
Tr&t
Vo
Hny).
Tacd:D*+DFz+Ef
=
=2ADz
+ Ad
+
AFa
+
Pf
(3)
Ap
dung
BDT
BunhiacdPxki
vI
COsi,'ta
dugc:
Art+AFz>l4r*AO2=iuf
2AD2+nf+ed+Af>
>2ADz
.|nn'>uAD.EF{,
=
+rl3,s'
(+).
Tt
(3) (4),tathu
dugc
B.D.T
cin
tim.
Nhfn
x6t.
1)
Sd
ban
tham
gia gi6i
bhi
to6n
trOn
khtr
dOng,
c6
ddn
hsn 380
bai
g&i
d6n
Tda
soan.
PhAn dOng
cac
f,an sfi
dgng
nhidu
cOng
thric
vh
BDT
luqng
gi6c v
cac
gOc
trong
tim
gi6c, vi vfly
ldi
giAi
n6i
chungddi
h6i
tinh-to6n
phric
tap,
cOng
kdnh.
tni
giti neu EOn
II
ngdn
gqn
nhdq
khong^nhrrnf
th6, chi
d0i
h6i v0n
dulg
kidn
Ii,,lc
Oon
giln
thuQc
chuong
uinh
}Iinh
hoc
THCS.
2)DC
y
thm
ring,
nu a,
b,
c
vtr S
ltr d0
dli
c6c
canh
v) dign
tich
crja
mOt tam
gi6c ntro d6
thi
ngodi
BDT
(2)
quen thuQc
:
az
+
bz
+
,'>
+r/3
,s,
chring
ta cttng
c0n
c6
mQt
BDT
manh
hon
:
u2+b1+c2
>
+{:s
+
(a-b)2
+
(b-c)z
+
(c-a)2
vtr
t}
d6,
thu
duqc
BDT sau
:
ab
+
bc
+
ca>
qfis
(5)
Sau
khi
nOu
ra
vh chrlng
minh
BDT
(5)
niy'
ban
Bili
Viil
Dang,|)A2,PTCT-DHSP
Hn
NOi
vd,Nguy1n
Hodi
Vfr,11A1,
trudng
THPT
chuyOn
Vinh
Phdc
dx
dd
xudt
BDT
sau
:
DE.DF
+ DE.EF
+ DF.ZF
>8fPft,
mnnh
hon
BDT cAn
chring
minh.
3)
Ngoli
c6c
ban
da
nOu
ten,
cac
bqn
sau
day'cung
c6
ldi
giti
tuong
ddi
gQn
hon
ct:
He
NQi:
NSl
Qudc
Anh,
11A
To6n,
PTCT-DHKI{TN,
DHQG;
Hda
Binh:
NgryAn
Mm
Ttq\n,
lOT,
PTNK
Holng
VAn
Thq;
Vinh
Phric:
Nguydn
Hodng
Trung,llAl0,
THPT
NgO
Gia
TU, Lflp
Thach;
Phri
Thq:
Hodng
Ngpc
Minh,gc,
THCS
Viet
Tri;
YGn
Fl6ir-
Nguydn
ViQt
Hdng,11A1,
THPT
chuy0n
YOn
B6i;
[Iii
Dutrng:
D(ng
Dt?c
Ilry,
1045,
TIIPT
20
Nam
S6ch;
Nam
Dinh:
D(ng
Hftu
N/ro,
10A'
THPT
Trdn
Hung
Dao,
Dd
Thi
Hdi
Ydn,9B,
THCS
Htri
H0u;
Thdi
Binh:
Luu
Hodi
Nam,
l0A,
THPT
Phu
Duc'
Qulnh
Phu;
Thanh
H6az
Mai
Vdn
Hd,
10T,
TI{PT
Lam
Son;
NghQ
An:
Nguy1n
Thanh
Sort,
11
Tin'
PTCTT-DHSP
Vinh;
Quing
Ngdi:
BDI
Quang
Minh,
12
To6n,
TIIPT
Lc
Khi6q
Khdnh Hda:
Trd"n
Trung
Dry, llT,TIIPT
chuyn
IJ
Quy
DOn,
Nha
Trang'
NGUYENDANG
PHAT
. BdiT1,Ol274.
Ba
mfi
cdu
(Ot, R),(02,
R2),
(Ot,
Rt)
tidp
xitc
vhi
nhau
t*ng
dAi
mQt
vd
c*ig
tldp
xfic
vhi
m(t
phdng chfia
tam
gidc
lnC
4i
ba
dinh
A,
B, C.
Gpi
S
ld
diQn
tich
tam
gidc ABC
vd R
ld
bdn
ktnh
dwdng
trdn
ngoqi
"tidp
tam
gidc
dd.
Chilmg
minh
rdng
:
2S
R,+Rr+Rr>n
Ddng
ththc
xdY
ra
khi
ndo
?
Ldi
gif,i.
o,
(Duong
Mgnh
Hdtg,
12A,,
THPT
Nf,Ng
khi6u
NgO
Si
Lin,
Bdc
Giang)
Theo
gib
thidt
c6c
fii
gi6c
BOyO3C,
co3orA,
AOTO2B
lI c6c
hinh
thang
t
luOng.
Suy
ra
:
fR,
+ n, 2
oror>
BC
=
a
]nr*n,
=otot>cA=b
[n,
+
n,
=
otoz>
AB
=
c
a+ +c
__s
=4
ru,
R>zr)
=R,+R2+Rt2-2
r
R,
vdi
r lh
b6n
kinh
dulng
tJdn
nQi
tip
MBC'
lRr+
R3
=
a
lR,
*R,
=
D
Dang
thtic
xhy
ta *
l*,
*
R2=
6
[n=2,
+
MBC
l[
ddu.
NhSn
x6t.
1)
Day
ld btri
todn
c6
rdt
nhidu
ban
tham
gia
gili.,Tuy
nhiOn,
nhiOu
ban
gili
qu6
dhi'
Dlc
biQt'
c6
mot
ban
gi6i
sai.
2)
Khi
gili
btri
to6n
nhy
c6
nhi6u
bqn
da
cdg{ng
ve
ch
ba
hinh
cdu
(O1,
R1),
(Oz, R2)'
{q,
\)'
C6c
ban
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
23/28
crAr
BAr
xi
rntloc
nOn nhd rlng
khi
gili
bli toan hinh hQc,
dic
biet
ta
hinh khong gian,
khong nhdt
thidt
phAi
ve mgi
ydu
td
hinh hgc
c6 trong dd bAi.
3) Cdc ban sau
ddy c6 ldi
giAi
tdt.
Scm La
:
Nguydn Btch Vdn, 12T3,
TIIPT
NK
Son
La;
Quring
Triz Hd Khdc
Hidu,10T,
THPT
chuyOn tf
Quf
DOn;
Hi
Nam:
11D,
THPT
Duy
Ti0n
B;
Vinh
Phric:
NgayZn Ddc Tnng,
11A3, THPT
chuyOn Vinh Phfc;
Phf
Tho: Vrt
Cht CAng,
l0AI,
THPT
chuyen Hr)ng
Vuong;
Tp Hd Chi
Minh:
Trdn
Quang,
10T,
TIIPTNK,
DHQG.
NGUYENMN{HHA
Bdi
Lll274.
TrAn
mQt
mdt
bdn ndm
ngang,
nhdn,
dgc
theo
m1t dadng
thdng, ngadi
ta d(t
ba
qud
cdu cd
cilng ktch
thadc, khdi luong
cila
chrtng
hn
lwqt
ld
m,
M
vd
2M
(hinh
ve).
Qud
cdu
m
bay
ddn va
chqm ddn
hdi
trryc diAn vdo
qud
cdu M.
Hdi
vdi
tt
sd
ndo
cr)a
thi
trong
M
hA cdn
xdy ra
v*a
drtng mQt
va
chqmntra
?
mrrM2M
eoo
.
Htrtrng
d6n
gifri.
Ki
hiQu
vy v2ldn
luot
lh
vdn
tdc
cfia
qu6
cdu m vi
M
sau va
cham
lin
i;
6p drJng
c6c dinh
luflt
b6o totrn
dQng luong
vI
btro
toln
dOng
n[ng
(thuc
ra Ii
co
ndng)
ta
c6 :
ffilo=
mV,
+ MV,
-
m3
Suy
ra
M=a.
Nhgn x6t.
Cfc
em
c6
ldi
gitri
dring vh
ggn
:
Hi NQi:
NAng Hdng Dwcmg, 1042,
THPT Trdn
Phf,
Bqch
Vdn
Son,
12C,
TH
chuyn
ngn,
DHNN-DHQG
HI
NQi;
NguyAn
Ngpc
Dudn,
t)B
To6n,
PT
chuyOn To6n, DHKHTN-DHQG
Hi
NOi;
tl:ii
Phdng:
NguyZn Minh
Quang,10
Li, THPT
Trdn
Pht PhqmThdnh
C1ng,10
Li,
THPT
TrAn Phri;
Nam
Dinh:
Drtng
Cao Son,
10
Toin
3,
THPT
L0
Hdng
Phong;
Hrii Ducrng:
Va
Xudn
Nam, 10T,
TI{PT
Nguy6n
Trai, Trdn
Vdn
Binh,
10L, THPT
Nguy6n
Trai; Bdc
Ninh:
vfi
xudn Tidn,
llAl, THPT Thuan
Thhnh l;
Thdi
Nguyn:
Nguy4n
Hodi
Nam,
1181,
THKT
S0ng COng;NguyAnNgpc Anh,11
To6n, THPT
chuy6n Th6i
NguyOn;
Yn
Brii: Hodng Tidn
Dfrng,
11A1,
Nguy4n Thanh
Binh,11A2, THPT
chuy0n
YOn
BAi;
Hi
Nam: Ngrydn
Nggc Tdn,118, THPT
A
Dung
Tin;
Vinlr Phicz
Trdn
Ng;pc
Dang,10A,
THPT
NgO
Gia
Tg, Lflp Thakch;
Hd Duy
Hung, LlAl, THPT
MC
Linh;
NguyAn
drtlt,
l0l)2,
I2 Khdnh Hting, Nguy1n
Duy
Hmg, 11A3, NgryAn
Mgnh
Hd,
1241;
Ti6n
Giang:
Trdn
Tdn
LQc,
11
Li,
THPT
chuyen
Ti6n
Giang;
Tp Vring
Tiu:
Hd Thanh
Som,
10T1,
THPT t,0
Qui
DOn;
Thria Thi0n -IIu&
Nguy4n DuThdi,llCT,
PTCT,
DHKH
Hud; Di
Ning:
D(ng
Quang
Huy,
10A1,
TIIPT
te
Quy
DOn;
Quing
Namz
Phan
Nguyn
Nhu,
11/7,
THPT Trdn
Cao
VAn,
Tam K ;
Tlranh
H6a:
Trdn
Trpng,
1lB,
THPT
Bim
Son,
Nguy4n
Vdn HiQp,
10E, TIIPT
QuAng
Xuong
I;
Nghc
Anz
Nguy1n
Chi
Thdnh,
10A6, Nguy1n
Eftc
Giang,
NguyAn
Dinh
Thdi,
10A5,
12
Quang
Phuong,
l0A3;
12 Trpng
Gidp,
l}E,
THPT
Nghia Din;
Quring
Tri:
LC Vfr
IIAi,
1245,
TIIPT
Vinh Linh; Khrinh
Hda:
Nguy4nThdnhSon,
ll
Li, THPT
chuyn L
Quf
D0n,
Nha Trang; Hi
Tinh:
I2
Hdng
Qudc
Tigp,10A,
THPT
Hdng
Linh
MAIANH
B*'i
LZl274,
MQt
msch
din
c6
so dd nha
hinh
v :
Tffi
ttAB
=
u
=
2\7fi'sint
1Ont(V)
Lo
ld
mQt cuQn
ddy thudn
cdm
cd
cdm
khdng
Zy.=
30Q;
Cold
tlt din cd
dung
khdng
Zg"
=
50e);
X ld
doqn
mqch cd chdt 2
trong
3
phdn
t*
R,
L,
C
mdc
ndi tidp
nhau; ampe
kd
(1)
(2)
^
(M-m)vo
2mvo
JUVfaiVr=--:Vr=-
M+m
L
m+M
,
Ki
hiQu
v',
ltr
v0n
tOc
cria
M
sau
va
cham
v1i
ZM
(va
ch4m ldrr
2),6p
dqng
cdc
Gnh
1u0t
b6o
toln
n6i
ften
ta
dugc
:
Zmvo
v
2--
3(M+m)
mtl
mrrt
MtB
2-
2
2
Ddu
tni chring
t6 sau
va
ch?m lan
cdu
M
chuydn
dOng
theo
chi6u
ngtrqc
khOng
xly
ra
va
cham
nAo
n[a ta
ph6i
di6u
kiOn
:
*vrm:
Zmv^
(M-m\v^
*
lv,l>
lv'"|
=
u
(-
t.
-
'
'' 3(M+m)
-
M+m
2,
qua
lai. Dd
cd
c^c
21
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
24/28
crAr
BAr
xi
rnudc
nhiAt trd
I
=
0,BA;
hP
sd c1ng sudt cfia
doqn
mqch
AB ld k
=
0,6.
1) Xdc dinh
cdc
phdn
t*
crta
X
vd
d0
lhn
cila chilng
2) Vidt bidu
ththc
crta u7,6
=
uy.
Hud'ng ddn
gifli.
1)
Tdng
tr0
cria
m4chZ=Y
=*
=
2500.
/,0,9
R
Vi
k
=
cosg
=
'r*
0
nOn m4ch
phf,i
c6
R
+ X
chriaR,
vdi
R
=
k.Z=
l5OQ
a)
Trudng hqp m4ch
c6 tinh
ctm
kh6ng
:
Xg6m
RntL,tac6:
*
=
Rz +
(Zy+
27,-
26,)2
)
Zr=
"IV:P
+
(Zc"-24)
=
200
+
20
=
22oe+r=*(n)
)7t
b)
Trudng
hqp
m4ch
c6
tinh
dung
kh6ng
:
X
g6mR
ntC,taco
Zr=E-P
-(zc.-4)=r1oa
* c
=-1 -'
..^
18r
'-
''
2)
G6c
lQch
pha
gita
u
vtr
i
ll :
g
=
tarccos(0,6)
t
0,921(rad)
D0
lQch
pha
gitra
u,vd
u
le
(g,
+
0,927)
a)
Khi X
gdm
R
nt L:
Z*=@2,
x
266,3{)
.?aat?'i'
THE
THUC
GUI
BAI
DU THI GIAI
BAI
KI TRUOC
.
Bii
ddnh
m5y
vi
tinh
hoic
vi6t tay
s4ch
sOtrOn
mQt
m4t
gidy'
.
C6c
tron
U,Ol.
nfr"a.i
d6
hq
tOn,
l
-
7/21/2019 THTT So 278 Thang 08 Nam 2000
25/28
4-6'.7
MOT SO
NGHICH LI
CAA
XAC
SUAT
I{GrIICffi'
r,i
r}E[rEfifr
NGUYENDUY
n6N
(EHKHTN-DHUG
HdN\i)
Nghich
lf:
Khi
tung
4
ldn
mQt
con s:(,c
sic
can
ddi
thi x6c
sudt
dd it ntrdt
mOt
ldn
m6t I
xuft
hien
ldn
hon
Il2. Trong
hic
d6 khi tung
24ldn hai
con
sric sdc
cOn
cldi
thi x6c
sudt
dd
it
nhft
mQt
lin
hai
con ddu xudt
hiOn
m6t
1
b
hon
ll2. Di6u
nly
thflt
d6ng nggc
nhiOn,
bdi vi
khA
nlng
nhfln
dugc
mQt
sd
1
gdp
6 l6n
khl
ndng
xuft
hign
hai
sd 1,
cdn
sd
24
thi
gdp
6
ldn
sd 4.
Giii thich:
Ndu
tung
mQt
con
sric
sdc c0n
ddi
k
lin
thi
sd
kh6
ning
c6
tlrd
c6
bing
6ft.
Trong
sd
d6
c6
5ft
sd
kh6
n[ng
khOng
xudt
hiqn
mflt 1,
vi
do d6 xdc
sudt
0d xuft
hiqn
ft
nhdt
mQt
ldn
mat
I
blng
:
pk
=
r-f
I)*.
\6/
D6
kidm
tra
lai rlng
P1
> tl2
nu
k
>
4.
Trrrng
tg x6c
sudt
xuft
hiQn
it
nhdt
mOt
lin
hai
m[t
1
khi tung
,t ldn
hai
con
s:f,c sdc
b]ng
:
e*=t-f-)-
\36l
8*
ll2
v1t k
> 25.
yi
viy
"gi6
tri
gidi
hpn" cria
sd
lin
tung
dd
x6y
ra
didu
trOn
ddi
vdi
m$t
con
sric sdc lI4
'rl
ddi vdi
hai
con sric
sdc
ltr
25.
Chti
thich.
*
Nghich
li
tr0n
lI
thSc
mdc
cta
Dd M0r0
(De
Mdrd)
khi
rrao
ddi v0i Pascan
(pascal),
nhi hqc
gi6
vi
dai
cria
thd ki
17.
N[m
1654
Pascal
vi Phecma
(Fermat)
cting
tim
ra
clclrr
gili
thfch
trOn.
Tuy vQy,
eich
gifu
thich
nly
khOng
lim
De M6rd
th6a
mtn
vi
6ng cho
ring
"kh6ng
p_hn
hqp
vdi nguy0n
li
d
le
cfia
gi6
tri
ti0u
chudn".
Nlm 1718
Moavro
(Moivre)
da
chi
ra
rlng
"Nguy0n
li
ti
lg
cta
cdc
gi6
tri
ti0u
chudn" kh6 gin
vdi
chin li,
b0i
vi,
ndu
p
ld,
xic
sudt
cria sU kiQn nlo
d6,
thi
gi6
rri
riu
chudn
k
c6
thd
tim
tr)
phr:ong
trinh
:
(1-
p)*
=
ll2
(phuong
trinh ntry c6 nghiQm
ndu
0