chuyCn LC Quf D6n; B4c Li6n: Ly Hodng ThiAn, l1T, THPT chuy6n Bac Li6u; Ddng Thip: Dd Hodng I/ieL lOT, TIIPT chuy6n Nguy6n Quang Di6u; Long An: Pham Hodng NguyAn,1lAl, TIIPT Hdu Ngtria, Dirc Hda, LA Tr{ Phil, lOTl, Pham Quiic
Thdng, l 1T1, TIIPT chuy6n Long An.
HO QUANG VINH
Bii T91463. Gidi h€ phaong trinh:
(*-2)'+(22+l)2
Ldi gi,rti (ctra nhiOu ban). Trong mgt phhng Oxy, dift i = (x; y - 2), i = (x-2; 2z +l),| = r6t +1; 5z +l) .
HQ phuong trinh trO thAnh
.NCu a=0 thi x : 0, y : 2.Thay vdo (3) ta
e 922 -62 -10=o e z =1t Jil
.3 3
. N6u i + d thi tu hai phucmg trinh dAu cria hQ
(I) suy ra 6 vd Z cr)ng phucrng. KCt hqp voi phucrng trinh thf ba cria (I) suy ,u i:Zi tto4c
i=-26. G2JN G2JN v- J 1+Vll ) ? I' (1) (2). (3) ^r ly+l=2(x-2) lz=l - Voi c =2b=4 c>i [52 + I = 2(22 +1) ly =2* -5 la.b=O t-- la.c =0 (I). li4 =,14 _a . [x=3
Thay viro (1) ta duoc: I +4x-21=0<)l
lx=J
Voi x : 3 th\ y: l; v6i x: -7 thi y: -19.
^: lv+1=1(x-2) lr=-l
- Voi c =-2b>1" <>l- 3
l5z+1=-2(22+l)
J=l_Z*
Thay vdo (1) ta duoc:
3x2 -8x+l=00r:41ff =3
Thu lai, ta th6y c6c b0 ,t, ( O, Z,
(/-\ /-\ I 1 -/1 1\ I o; z: + l. (: ; t ; l). (-7; -le: l), (3) /-\/\ | 4+r/13 .l-2,113. I I I 4-J13. l+2V13. I l [ 3 ' 3 '-].,i'[ , ' , -1)
d6u ld nghiQm cria hQ tld cho.
YNhQn xdt ,Crtngdtng phuong ph6p vecto <16 gi6i h6
nhrmg mQt s6 ban tl6 tl6 m6t nghiQm. Lli do ld khi c6
hQ O, c5c ban ild khing dinh ngay ld c6c vecto 6
vd J cung phuong, diAu ndy da d6n di5n b6 s6t
truong hqp khi i =6 , c6c vecto 6 vd i c6 th|
kh6ng cr)ng phuong. M6t s6 ban gi6i hQ ndy bing bitin tlOi tt4i s6, tuy nhi6n loi gi6i cdn dai. C6c ban sau
c6 loi gi6i ngin ggn:. Phrfi Thg: Hodng D*c ThuQn, Ducrng Gia Huy, H6 Quang Huy, l0 Toan, THPT
chuy6n Htng Vuong, ViQt Tri; Trdn Quiic LQp,9A3, THCS L6m Thao. Bfc Ninh: Zd Huy Cadng, 12
To6n, TIIPT chuy6n Bic Ninh; Mdn Drhc Binh Minh, 1141, THPT Y6n Phong sO y tfghiem Chi, llLl-
K10, THPT YOn Phong s6 Z. UA Nii: Nguydn Vdn
Diing, Hodng Thi Thu Hodi, 10A14, THPT Ngqc
T6o, Phric. Thg. Hu:ig YAn: Pham TiAn DnAL llAl,
THPT Trdn Quang Khdi, Kho6i Chdu; Nguy€n Th!
Qu)nh Hoa, l0Al, Tri€u Ninh Ngdn, 11A9, THPT DugnC Quing Hdm, Vdn Giang; Vfi Tiin Khang, Vfr Tudn Dqt,l0 To6n 1, THPT chuy6n Hrmg YOn. YGn
Bii: Hodng Ngoc Dting, 10 To6n, THPT chuy6n
NCuygn Tat Thenh, TP. YOn B6i. Thanh H6a: N4"yii" Khdi Hwng,l0A5, THPT Lucrng D6c B[ng, Ho5ng Hoa; D6 Thiry Anh, 11 To6n, THPT chuy6n Lam Son. NghQ An: Cao Hiru Dqt, l1Al, THPT chly6n Phan BQi Ch6u. Hi Tinh: Phan Vdn Duc NhAL 10T1, TI{PT chuyCn He finh. Quing Tr!: Nguydn Th! Phaong Linh, 10 To6n, THPT
"t rryen I.C Qu;f D6n, D6ng Hd. Thira Thi6n Hu6,: Phan Trdn 4 + (22 +l)2
t, n.r,r-ror., T?[I#S
Hadng,l0Tl, THPT chuyOn Qu6c hgc Hui5. t<tr6ntr Hitaz Nguy€n Quang Dc1t, l0 To5n, THPT chuy6n LC
Quli D6n, Nha Trang. Quing Nam:. Ng4ftn HuyHAi, l)ll, TI{PT chuyOn NguySn Binh Khi6m. Phr[ HAi, l)ll, TI{PT chuyOn NguySn Binh Khi6m. Phr[
YOn: 1/96 LA Phuong Trinh, DSng Bdo Vinh,11 To6n
1, THPT chuy6n.Luong Vdn Ch6nh, Tuy Hda. Long An: Phgm Qudc Thdng, 11T1, .Nguy€n Binh An,
11T2, THPT chuy6n Long An. D6ng Thfp: Ngtqt€_n
L€ Minh, Dd noimg VieL lOT, TIIPT chuyCn NCuyqn Quang Di6u. Vinh Longz NguyAn Minh Long, Trdn Minh Qudn, 11T1, THPT chuy0n Nguy6n Binh Khi6m, TP. Vinh Long.
TRAN H TU NAM
Bni T10/463. Tim hdng sd m l6n nhiit dA niitddng thac sau dung voi moi s6 thryc kh6ng dm a, ddng thac sau dung voi moi s6 thryc kh6ng dm a,
b, c, d'.
(ab + cd)z + (ac + bd)2 + (acl + bc)2
> ma(b + c)(b + d)(c + d). Ldi gidi.Xdt a=l,b =O,c = fl = Ji thi ta c6
22 + 2 + 2 > *.Jl.JT.zJT = Ji >- *.
Ta sE chimg minh b6t ding thric (ab + cd)2 + (ac + bcl)2 + (ad + bc)2
> JT.a(b + c)(b + d)(c+ d) (1) dring voi mgi s6 thlrc kh6ng dm a,b,c,d.
Thpt vQy, theo nguy6n lli Dr.ichle! trong ba s6
a-b, a-c,a-d t6nt4i hai sd c6 tich khdng 6m.
Kh6ng mat tinh t6ng qurdt, ta gia sir
(a-b)(a-c)>0.
Suy ra d(a-b)(a-c)>O
=d(az +bc)> d(ab+ac)
> a' d + bcd + ad2 + abc > abd + acd + ad2 + abc
= (a+ d)(ad + bc)2 a(b + d)(c + d)
Do d6, 6p dtmg bAt dang thric Cauchy, ta c6 (ab + cd)z + (ac + bd)2 + (ad + bc)2
I
> )@b + rd + ac + bd)2 + @d + bc)2
Z= =
|{o + a)' (b + c)2 + (ad + bc)2
> A @ + A(b + c)(ad + bc) > J-2n(b + c)(b + A@ + A. Nhu vfly m = J1 ld sd thuc lcrn nh6t thoa mdn bdi to6n. tr
YNhQn xdt. Chc bpn sau da chi ra: nAt Aang thric (1) ld bdi to6n T61452 (voi x =b ,, =L,, =4 sau khi
aaa
chia c6 hai v6 cho oo n€u o> 0, n6u a : 0 rdring (l)
tlung). C6c hgc sinh sau c6 lcri gi6i tlitng: Vinh Phtic: Phirng Vdn Nam, 10A1, TI{PT chuy6n Wnh Phic; Hir NQi: Trdn Ba Khdi, 11T2, THPT chuy6n DHSP He Ndi, Trdn Nhdt Quang, I i To6n, tlti'T chuycn
KHTN, DHQG Ha NOi, Nguydn Vdn Dilng,10A14, THPT Ngqc T6o, Phfc Thq;Hung YEn: Nguy€n Th!
Thanh Lan, 11A8, THPT Duong Quing Hirm, Vdn Giang; Thanh H6az Nguy€n Khdi Hwtg, Nguy€n
Danh Thdng, N.guy€n Bd Tudn,10A6, TI{PT !o*g
DIc B5ng, Ho5ng H6a, D6 Thily Anh, 11T, TIIPT chuy6n Lam Son; Hi finh: Phqm Vdn Dwc Nhdt, 10T1, THPT chuy6n Hd finh; Quing Blnh: Hodng NhQt Tudn, 10T, THPT chuydn V6 Nguyen Gi6p; Quing Nam: Bll Ngpc Giao,K04-01, THPT chuyCn
NguySn Binh Khi6m, TP. Tam Kj'; Long Anz Biti Khdnh Phong,10T1, THPT chuyCn Long An, Phqm Hodng NguyAn,llAl, THPT H4u Nghia, Dric Hda.
NGUYEN MINH DIIC
Bii T11/463. Cho sd nguyAn du'ong m > 2.
Chtrng minh rdng vcti m7i s6 tw nhiAn n > 3 thi
ll
: lll -l
so