a) Ndu m = 3 thi PT (3) vh (4) dcu c5 dang
"f = 0. Suy ra v6i m =3 thi PT (3) tuong duong
v6i PT (4).
b) Ndu m = 4 thi FrI (3) vn PÍ(4) đu c6 dang
I + 2x = 0. Suy ra vdi rr =3 thi PT (3) tuong duong v6i PT (4).
K|'r ludn. F f (3) tuong duong vdi PT (4) khi vn
chi khi rr =3 hoac m = 4.4
Qua c6c bdi to6n tren chac cdc ban da thay duoc
phdn nho tdm quan trong cfra viec iip dLrng didu
ki0n cdn vd dri đ giai bni toiin tim didu kien.
Dd luy6n tAp xin mdi caic ban ldm mot so bii
tap 6p dung sau dAỵ
BAi 1. Tlm a đ cdc phuong trinh vi h0 phuong
trinh sau c5 nghiOm duy nhat:
a) .6= *Jg-* =o;
b) J3rr+G-r-J3 {-\r)(6-r) =,1'
,l i[Cr+rY +\@x-rf +:[4x') +á1 :{; '
ft gai to6n 3. Tim a s:r,o choctia b hA phLtong trinh sau c6 ttgh ctia b hA phLtong trinh sau c6 ttgh
Jlr-tl.t'fys:lIt+ta*t\b.ry| -a) It+ta*t\b.ry| -a)
lvr=1
1'
[bx =0.
b) v6i a = -thc (IV) c6 dang {-'^t-F'l :]
" t l=l ", {J**t*Jii=oIx+y=3r. Ix+y=3r. [.r-v+..+ v=ư2 d) {r'r*rr 2 =a+t i
BAi 2. Tim a đ vdi moi gi6 tri cta b ho phuong
trinh sau c6 nghiOm:
[a{f t }r )+r+y=1-2ly- x=b. ly- x=b.
Bei 3. Tim rr Ad nai phuong trinh sau tuong duong:
(1+m2)x2 -Z(nP -l)x+nt2 -3=0
vi x2 + (m -1)x + m' -3m +l = Ọ
I
GfufirL(
Aire rrt&ru
-'',ta-
8 '-Qr,i thdtlain hr Aluilitrldo gidnh drrot tinh
etim crio ngrroi y6u lorin TrGn luin hoc A Tuiii trb
lubn cd nhring bti vi6t, nhrlng chuy6n muc nhrr'[hudn
bi thi vdo Doi hoc| 'Bon doc fim tdí] 'Dd ro ki ndy]
'[6u loc b6':- [hring do gidp do mor cdch rhi6rhrr.c
cho chdng em-"
NGU,EN MANH IUAN
(Th6n 5 xo lhrii Y6n, Dfc Tho, l.ld Tinh)
*K '- Nhd' cri bdo lqin hoc a Tuii trb, em do rim
drrọc r6t nhidu bdi todn hoy vrfi nhring kii gitii sdng
too khi6n miii ldn doc ld em cdm thdy rdt thdn phuc.
Quo bdo, em cfing hoc drrọc cdch trinh bdy m6t bdi
lotin suo cho logic, chdt ch6 md ngdn gon. (6 gitio
khen em rdr nhidu vd su fi6n bo do va m6i ldn nhrr
vdy em diiu nghi đnluin hoc &Tuiiirri, rhdm crim
on bdo do girip cho em y6u rhich vd sdng too hon
lrong vi6c hoc vd gitii Totin-"
IRINH IHI IHUY LINI|
(500, Hrjo Binh, Hd Trung, Thonh Hoti)
* ".- Didu đu ti6n Tuin hu aTudi fib doy r6i ld srr
khi6m t6n, srr hoc hrii vd vuon l6n kh6ng ngdng trong
hoc top, nghi6n cdu chf khdng duọc tg đc vrfi nhring
gi minh dong crị Td mdt con ngurii cd phdn ki6u cdng,
lu mfr'n, l6i do nhdn ro rdng ki6n thdc crio minh cdn
ndng con ldm, bdn ftfin minh crin kdm crii ldm, minh
cdn phdi hoc hrii thdt nhidu nvọf uin hoc aTuiii trb
do mong d6n cho t6i srr ki6n ri, cdn r'h0n, girjp r6i
phdt huy ric sdng too, doy t6i cdch hoc, ctich nghi* 0O
ld nhring diiiu quy gid md trudc đy tdichuo crị
Irikhi ldm quen vrfi luin hqc aluiii trb,tlidO ri6n
bO rdt nhidụ Nhring phrrong phdp hoc totin, gitii rodn
cioTuin hu &Tuiii trbdlt mong Ioi cho rdi nhidu
didu lirhrj vd bd ich, girip rdi co k6i quti 16r hon rrong
hoc t0p, nhiir ld ri mdn Todn-"
_
DIIONC VAN AN
(llAl, THPI (hdu Thdnh, TX. Bd Rio,
Bti Rio - V0ng Tdu)
* "- 5r, 2 ndm kd ili ngdy đr bdo, em ctim thdy
minh nhu rhoy đi nhidu vii m6n lodn. TrGn bdo lti
nhring ki6n thfc horin todn mtii md diii vdi em,
lhrrc srr đ ld nhring ki6n thfr quy bdu vri đng
lron lrong..Top chf dong drrọc cdc bon hoc sinh đn
doc rdt ndng nhi6t, during nhrr kh6ng cd srr phdn
bi6t ltro rudi, hotin cdnh..l'
NHT VAN UNH
(Tdn Ydn, Bdc Giong)
TOft'C HOC
I\NÁ \NÁ =N Na vis xEp z ra didm M phhi
pnAr cHlA nNn vuOrc
thanh cac lay giac,c6 dien ticb bdng nfau c6 dien ticb bdng nfau
(Di dans tun TIITT s6' i53 thdng 11.2006)
A- A, At Az Ạ\ A, B- A,
l)','I),, D, D. D1
Hinlr I
AiDi vdAi tDi
(i = 1, 2, ..., n) ta duclc ]tt tam giric blng nhaụ 2) C6 b6n dinh hinh vuong thu6c ba tam
gidc nOn phii c5
Do C.,D, l) Chia canh l) Chia canh AB vd canh CD cira hlnh vuong ABCD thinh rr doan bang
nhau bdi citc
diOm theo thir
t{ Ar, = A, At, Ạ, ..,, A, = B
r,) Drr = D, Dt,
Dz. ..., D,, = C
(hinh I ring vdi n = 5). Ndi
B
Hinlt 2
3?7 N
Dd chio mimg xuan mdi 2007 ngudi ta du dinh
sap xdp 2007 chrdc đn theo m6t vbng trdn l6n
& m6t-quing trudng, m6i đn c6 mdt trong ba
mdu Tim, D6, Lam sao cho miu Tim li it nhdt'
Hai cdch sap xdp đn coi ld nhu nhau neu qua
m6t ph6p quay quanh tAm vbng trdn mot g6c ld
360'
boi niro d5 cua -"" thi cich xip n)ry gi6ng
' 2001
hot cdch xep kiạ
Dinh cho ban doc
1) C5 bao nhieu c6ch sap xep dBn mhu neu c6
kh6ng quii ba đn cirng miru lidn nhaủ
2) C6 bao nhieu cdch sap xep dbn m)u ndu c6
::::: i:':.'::. : :1 : ::i :.":.':':. : :1Ỵ . . . . .
it nhat hai dinh cirng thu6c m0t tam gi6c, chang
han 1)r C, D. Gih su tam gidc cZin tim la MCD' . ED.DC Sur,r, 1
AD.DC Sntúr, 3
,1
thuoc doan EF th6a miLn ED = ió vir
EFllAB. Dd phan chia da gi6c MDABC thhnh
hai tam gi6c thi hodc M thuoc dudng ch6'o BD
(hay AC),hodc M trtrng vdi F (haY E).
. Neu M thuoc BD th'iS,,,r, = ryrry
o Ndu M tring v6i F thi khon-e thd ke BD (theo
trOn) n€n chi ke duoc AF, nhung
^ AD.EF S.t,r, r, S1H,,'
VAy khdng th€ phan chia mQt hinh vuong thinh
ba tam gi6c c5 di6n tich bang nhaụ
Nguiri ta da chfrng minh rang kho1.S- thd phAn
ctia m6t hinh vu6ng thhnh m6t so le cdc tam
gi6c cd <li6n tich bang nhaụ
Nhtin xit. Cdc ban gini bli đu c6 ket luAn
dring nhung phdn ldn lAp IuAn khOng chat chÉ
Ctic ban sau c6 ldi giai tot:
l) Dd Thi TluL Tluio, l1T1' THPT chuy€nNguy0n Trdi, TP. Hii Duong. Nguy0n Trdi, TP. Hii Duong.
2) Dttong Van Att, 11A1, 'l'FIIrf ChAu Thinh'
TX. Be Ria, Bi Ria - V[ng Thụ
3) Lúong Xtdn Htry, l0A1' TFIPT Ti0n Lir'
Hung Yen.
PHi PHI
\1\t\1\t\
HELLỌ.. (!'iip trang 4)
I-7. Let be given a real nurrber k in the interval
(-_ll 2) attd three pairrvise distinct real nulnbers
ạ h. c. Prove that (,rr , hr -c: - k(uh t /,c r ca)) > ( r r r )e(:-k) 'l- - -. f {,r-h)r (/,-. )r (r'-.r)r ) 1
When does eqtrality occur'?
I-8. Does there exist l positivc intr'ger a such
that irr thc scqucnce of uumbers (rr,,) defined by
,r,,: ,,t -t rrr for all n : l. 2. 3, ... evcry two
corrsccrrtive ternrs are coprirne integers?
I-9. Does there cxist a positive integer n such
that one can assign to each veltex A1, A2, ...,.1,,
o[- a couver l-poltgon an integer (these n
integers arc no1 necessarily,' clistinct) so that
i) the surn of these n integers is equal to 2007, and
ii) for every i:1,2,.... r, the nur.nber assigned
to l, is equal to the absolute value of the
difference of the numbers assigned to 11.1 and
A, - t. (with the conventiorr Atr , | = At and
A,.t=Az)?
I-10. Suppose that in the coordinate plane
every integral point (ị ẹ point rvitlr integral
coordinates) was colored in one of trvo gir,'en
colors, Prove that there exists a inflnite set of
integral points of the sante color. fbrrnirrg a
figure admitting a center of symrnetrỵ
(Di thi dcing rAn cdc s6 -3ii, Ji6, 357 vd đp
cin diing trAn cdc ,yA SSl, 358, 359. Danh sdch
ctic hon doot giai cliing trOn tO SOO 6.2007)
PROBLIIMS ... 1t'iofu n.ang t7)