NGUYfiN VAN THUAN
Rd )'-i -i I l )) z=y= I Pn= r2[&*.)= \r ) u2 z, -(4s*,\'lo I \4 / u,(ls*,) [n ) (nu )' l-+r I \,1 ) Yd*d&f;u { \+gt E{ S d {\Stt"Lidnf ' fir*enftc 56 3sB l4-2OO/\ 27
0ifri bii cHA| tttt2 2007 -Dot2
*nai F6. Cho tam gidc ABC vu6ng o A.Hdy rim ttit cd cdc b0 sdu di€m phdn biQt Hdy rim ttit cd cdc b0 sdu di€m phdn biQt
(M, N, P, Q, R, S) th6a mdn d6ng thdi cdc
di€u ki€n sau:
t) N, P ndm t\n canh AB; Q, R ndm ffAn
canh BC; 5, M niim tuAn canh CA;
4M^r = PQ = RE
3).L4c gidc MNPQRS ld m6t luc gidc tit nliti€p vd cd ba dudng chdo chinh MQ, NR, PS ti€p vd cd ba dudng chdo chinh MQ, NR, PS d6ng quy.
Ldi gidi. a) GiA sri ta dd tim duoc m6t b6 s6udi6m ph6n bir't (M, N, p, e, R, ,S) thoa mdn t6t di6m ph6n bir't (M, N, p, e, R, ,S) thoa mdn t6t
cit citc diAu kien n€u tr6n. Coi I Id diem tlong quy cria MQ, NR vd PS. Tir giA tt i6t .,iu Uai to6n, d6 th6y ring ciic tri gi5c nQi ti€,p MNpe,
PQRS vd RSMN dAu ln nhftng hinh ihang cdn
c6 cic dty MQ, PS vd ltN tuong r?ng song
song v6i c6c c4nh AB, BC vd CA cta tam gi6c
ABC (vu6ng tr A) tld cho. Tir d6 ta duo-c:
AMN =SPN =CSR (: p), ANM = BPQ=MSP
(: 1), trong do B vd y ldn luot ld d6 l6n c6c
goc i vd e cta tam giitc vu6ng ABC dd, cho.
Ti' d6 suy ra PQRS li m6t hinh chfr nh6t.Cfing vAy, AMLN ld hinh cht nh{t n€n AL : Cfing vAy, AMLN ld hinh cht nh{t n€n AL :
MN : PQ = LH. Suy ra L ld trung di€m cia dudng cao AH cila tam gidc ABC.
b) D6o Iai, qua trung cli6m L cua duong cao AH ta dung c6c dudng th1.ng MQllAB, NRIIAC vd PSllBC, trong d6 cbc cdp di6m N, P; Q, R
vd S, M tuong ri'ng nim tr6n c6c canh AB, BC
vd CA. De ddng chring minh dugc rang
MNPQRS ld mQt luc gidc rci n/i il6p, co ba duong ch6o chfnh song song voi c6c canh cta tam giirc ABC vd d6ng quy o I, d6ng tho'i
MN: PQ= RS,
c) T6m lai, lqtc gidc t6i MNPQRS nhdn daoc
ld duy nhtit, thoa mdn ddy di cdc diiu kiQn
dat ra cua bar toan. l l
(Nn6n x6t. 1) Bdi to6n tr€n ddy thuQc loqi todn
d{ng hinh trong m{t phing. Trong phAn a) cua loi
gidi chring ta d5 thuc hi)n budc phdn tich (cira bdi
to6n dgng hinh) de chi ra tinh chiii cua hinh (cu thi: lit cria b6 s6u di6m M, N, P, Q, R S) phdi dung. Tinh ch6t
tim tlugc ld: Ba doan th6.ng MQ. NR vd PS noi ba cap
di6m ph6i. tim song song v6.i ciic canh cua tam gi6c
ABC vit d6ng quy o trung di6m Z cua duong cao .)H. Ph6n b) chi cdch dtrng hinh phai tin (d6 ld b6 s6u
di€m) sau do chtrng minh hinh dvng duoc td hinh cdn tim.Phdn c) cu6i cirns ld phdn biQn ludn: biti torin c6 nghi€m dul nhAt.
2) Sd ban tham gia gidi bdi to6n ndy kh6ng nhi€u.
Nhi€u. ban trinh bdy loi giai con thi€u chEt ch0. Cu th€ Id da s6 chi thuc hiQn phAn phdn tich sau d6
.d6 vQi v6 ktlt luan bdi to6n c6 nghiCm duy nh6tl V6t
sd ban kh6ng trinh bdy c6ch dung ho{c c6 chirng
minh d6o nhung l4i kh6ng k6t ludn v0 s6 nghiCm cia
bdi to6n.
3) C6c ban sau d6y c6 loi gi6i t6t hon c6 (trinh bdy
chAt chc vd ddy du c6c budc crja bdi to6n dung hinh):
Phri Thg: Nguydn Ngoc Trtlng, 9A1, THCS Ldm
Thao, Ldm Thao; Hi NQi: Dri Vdn E*c, l0A2 To6n,
Dinh Hodng Long, 10Ar kh6i PTCT, DHKHTN
DHQG Hd N6i; Th6i Binh: Nguydn Ti6n Huong,
10Ar, THPT Quj'nh C6i, Qu!,nh Phq; Phri y6n:
Pham Quang Thinh,8H, THCS Hing Vucrng, Tuy Hoa.
NGUYEN DANG PHAT
BITEX
CONG TY CO PHAN XIUK BINH TAY IBITEX]
Tp.HCM: 11O -1 12 HQu Giang PB 86; Ha NOi: 1 28 Bpch Mai G.Hai Ba Trr-n-rg
NhA phSn phdi ehinh rhrlc mdy rinh GASIGI, rai Vi6r Nam
A\T \
HAN HANH TA TRO CUOC THI NAY
rctr{ }pc
56 358 (4-2007) & qirmgA
'*gai V7. Cho k td mlt sii th?rc thuQc khodng
(- 1;2) vd cho a, b, c ld ba sii thryc d6i mQtkhdc nhau. Chung minh riing ta c6 biit ddng khdc nhau. Chung minh riing ta c6 biit ddng thirc sau: (a2 +b2 +cz +k(ab+bc+ca))x ( t 1 l )qrz-ri - l - r -r- l\---:---------a [(a-b)2 (b-c)' (c-a)') 4
Hoi ddng th*c xdy ra khi vd chi khi nao ?
Ldi gidi. Ap dung bAt ding thrtc
lt8_+_>_: . (x+0. y +0),tath6i, _+_>_: . (x+0. y +0),tath6i, x2 y2 (x+y)2 111 (a-b)' (b-c)' (c-a)' I >-+ (l)
(a-b)' ((b-c)+(c-a))2 (a-b)'
Chring ta chi cAn chirng minh b6t tling thric
a2 + b2 + c2 + klab + bc + ca), '4(2- k)(a - b)2
Hay ([ + 2)(a + b)2 + 4kc(t + b) + +c2 > O 121 Xdt tam thftc b{c hai, An \d (a + b):
/(a + b): (k + 2)(a + b12 + 4kc(a + b) + 4c2,
ta c6 biQt thric
L' : 4Pc2 - 4c21k + 2) : 4(k - 2)(k + 1)c2.
Theo gii thiet k e (-1 ; 2), suy ra A' ( 0. K6thgp v6'i k + 2 > 0, ta di d6nfla + b) >0, v6i hgp v6'i k + 2 > 0, ta di d6nfla + b) >0, v6i moi fr e (-1 ; 2). NghTa ld bAt ding thirc (2) rlugc chirng minh.
Tri (1) vd (2) tathdy bat ding thri'c 6 dAu bdi dugc chfng minh.
Ding thirc xhy r7 khi vd chi.khi trong ba s6 a, b, c c6 mQt sd bdng 0, hai s6 cdn lqi khdc 0 vit
-:.
dot nhau. lJ
(Nf,6, x6t. MQt sO bqn su dr,.rng c6c k6t qu6 trong
', ,,-,.: ,^ , , :
bin,"Thiet ldp cdc ba.t ddng thirc ti mOt ddng thuc cd
di€u kiQn" trong. cu6n "TuyAn clton theo chuy)n d€
Todn hoc vd Tu6ftrd" Quydn 1, NXBGD, ndm 2005,
lrang 12, cfrng cho loi giai dring. Sau dAy ld nhitng
ban c6 ldi giai t6t:
Hii Duong: Triin _Vdn Hqnh, 10A5, THPT Ninh Giang; Hir\Ql Dd vdn D*,:, 7cA2 Todn, Nguydn
Lru Bdch, Trlnh Ngpc Duong, l1A1 To6n kh6iPTCT, DHKHTN - DHQG Hd NQi; Phri Thg.: PTCT, DHKHTN - DHQG Hd NQi; Phri Thg.:
NguyAn Ngpc Trung,9A1, THCS L6m Thao; TP. H6
Chi Minh: Vd Vdn Tuiin, 1,0 Torin, PTNK - DHQG
TP. H6 Chi Minh.
ud eueNc vINu
*nari I-8. Hoi cd tin tsi hay kh6ng s6
nguyAn daong a sao cho trong ddy sii (a,), xdc
dinh bhi a, = n3 * a3, vdi mqi n : 1,2,3, ..., thi biit ki hai si5 hqng li€n tiiip ndo cilng ld hai
sd nguyAn t6 cilng nhau ?
Ldi gidi. Gi6 sir t6n tai s6 nguyOn duong athoa mdn di6u kien bdi to5n. Xet n:3a2 + 2a, thoa mdn di6u kien bdi to5n. Xet n:3a2 + 2a,
ta c6:
* a,,-t = a3 +(n+1)3
: (a+n+l)la2 +(n+1)2 -a(n+l)l i (a+n+l)
hay an*1i3a2 +3a+l (1)
* a,=n3 *ot =(3az +Zal +(a+1)3 -(a+7)3 +a3
= 1(3a2 +2a)3 + (a+1)3 I - (3a2 +3a +7) .
Md l(3a2 +2a)3 +(a+1)31 i (3a2 +2a)+(a+1)
hay l(3a2 +2a)3 +(a+1)31 i 3a2 +3a+1 n€n ani.3a2 +3a+1 (2)
Tt (1) va (2) suy ra (an,an*t)23a2 +3a+1>1 .
MAu thu6n. Di6u ndy chi'ng t6 kh6ng c6s6 nguy6n duong a ndo th6a mdn di6u kien s6 nguy6n duong a ndo th6a mdn di6u kien
ac ual. o
{Nn3n x6t. l) Kh6ng co nhidu b4n gui loi giai cua
bdi ndy.
2) C6cban c6 lcri gi6i dlng vir ggn:
Phr[ Thg: Nguydn Ngoc Trung,9A1, THCS Ldm
Thao; Hda B\nh: Vil ViQt Dfing, 12 To6n, THPT
Hodng Vin Thu; Hn NQi: Nguy1n Lrru Bdch, 17A7
Torin, EHKHTN - DHQG Hd N6i; NghQ Anr Hodng Dac Hdi,1lA1, THPT Phan BQi Ch6u.
VU KIM THUY
*gai I-g. Hdi c6 t6n tqi hay kh6ng si|nguyAn duong n sao cho ta c6 th€ ghi lAn nguyAn duong n sao cho ta c6 th€ ghi lAn n dinh At, Az, ... , An cia n-gidc l6i AAz. '.4, n s6 nguyAn fthdng nhiit thiil dii m.Ot khac
nhau) d€ cdc di|u ki€n sau duqc d6ng thdi thoa mdn:
IOfiN * qudiw ffi
1) Tdng cr)u n s6 ,fuqc ghi bang 2007;
2) I,'6i *6i i = , ,: | , 2, ... , n, stl duoc ghi tai dinlt
/,l,irtg gi,i tri tuyet doi cua hieu hai so dtrcrc
ghi tai hdi dinh ,41 1 vd A, o z. (Quy artc An,, , = l,
t'it 1,,. z= A) ?
Ldi girti. (Theo b4n Nguydn Khdc Tuyin 10A-1,
THPT Hoing Hoa 4, Thanh H6a vd Nguydn
Luu Bdch, 1141 To6n, DHKHTN - DHQG
Hn Noi).
Ki hi6u ar ld s5 ghi t4i dinh Ai (i = 1,2, ...,
n + 2) trong cl6 coi an*1 : a1, ana2 = a2.
Ch[ ]i ring v6i s6 a tiry f thi lal - a bing 0
hodc bing -2a,tuc ld lal = a (mod 2), do d6
nnn
Lo,: Llo,-r,*rl = I(o, -a1*1) (mod 2),r=1 ,=1 i=l r=1 ,=1 i=l
n
hay \a;: 0 (mod 2) nhung theo gid thii5t thi
l=l n
Lo, : 2007 ld sO le. Di6u m6u thu6n ndy
chring t6 kh6ng tdn tai sti nguy6n duong n diS ).,
nthoa m6n y€u cdu tl€ bdi. D
,
<(Nhin x6t. l) Nhi€u b4n giai theo hudng sau: nn^nn
\al =\ai -aial' =z\fi -z2o,o,*r=0 (mod 2), suy ra
ia11a1-t1+io,=i4=o(mod 2) n€n fa;=O (mod 2).
t=l r=l t=l r=l
Nhung theo gia thiet thi iai = 2OO7ld sO le. M6u thu6n.
l=l
2) MOt s6 b4n cdn x6t s6 16n nh6t a trong day s6 (a)
(i = 1,2, ...,,n) d€ suy ra d5y s5 tr6n duoc chia thdnh ttng bQ ba s6 d4ng (q, a,01, ddn d€n mdu thu6n. Chri f
rdng n€u n = 3kthi t6n tai ddy s6 (a;) (i = 1,2, ...,3k) th6a m6n ydu cAu thri hai crla d€ bdi vir ia1 : 2ka,
i=1
trong d6 a: max{ail i = 7,2, ...,3k1 .
3) Ngodi hai b4n n€u tr€n, c6c b4n sau c6 ldi gi6i dringvd gon: vd gon:
Phri Thg: Nguydn Ngsc Trung, gAl, THCS LdmThao;
HA NQi: Dd Vdn E*c,70A2 To6n, PTCTT, DHKHTN
- DHQG Hd NQi; Hda Binh: Vil ViCt Dfrng, 12 Todn,THPT chuy6n Hodng V6n Thu, TX. Hda Binh; Bic THPT chuy6n Hodng V6n Thu, TX. Hda Binh; Bic
Giang: LA Trung KiAn, 12 To6n, THPT chuy€n Bdc
Giang; Thii Binh: Iri Tdt Dqt, 1141, THPT E0ng
50 s6 3s8 (4.2oonrffiiffi
Th\ry Anh, Th6i ThUy, Nguy€n Ti€n Huong, 1041, THPT Qu!'nh Cdi, Quj,nh Phg; NghQ Au DQng Tudn
Anh, llA1, THPT chuy€n Phan B6i Ch6u, TP. Vinh;
DIk LIk Trdn Nhu Ngpc,.l0A4, THPT Nguydn Binh
Khi€m, Kr6ng P6[ TP. H6 Chi Minh: Vd Van Tudn, l0 To6n, PTNK DHQG TP. H6 Chi Minh; Bn Rla -
Yfing Tiu: Dinh Ngoc Thdi, ll To6n, THPT chuy€n LC Quf D6n, TP. V[ng Tdu.
VIET HAI >kBai I*10. Trong i,i.it ,nhing toa do, mdi
di€m c6 toa d0 ngu.tiii i.iirctc' to bo'i mQt trong hai mdu cho lruoc. Ci::i,;-g ntinh rdng tdn tai
,:
mot tc.ip hcrp gc)nt vi, .;,'t. -l;1tt: cri toa.dQ nguy)y
h mAt hinh c6 tam ciii:.).':i;:? ,,.) d6ng thrli tdt
ca cac diAm thu6c tri:- .: -:- .i,', ,iLrrtc td boi
citng nt(tt miu.
Ldi gidi. Ta quy uoc, trong phin trinh bdydudi dAy, khi n6i toi di6m ta hi6u d6 ld di6m dudi dAy, khi n6i toi di6m ta hi6u d6 ld di6m
c6 tga d9 nguy6n.
Hi6n nhi6n, y€u cAu chimg minh cria bdi to6n
tuoxg tlucrng v6i y€u cAu chimg minh t6n t4i
mQt phdp aOi xrmg t6m bii5n v6 hpn c6c diiim
cirng mdu thinh chinh c6c elitim d6. Ta sE chring
minh <tidu vira n6u bing phucrng ph6p phin
chimg.
That vay, gii su ngugc lai, b6t ki ph6p d6i
xirng tAm ndo cfrng ldm d6i mdu t6t ch cilc
di6m, c6 thti trir ra m6t s5 hiru han tli6m md ta
sE goi ld diiim dec biet.
X6t phdp d5i xfrng D1 tdm Ot vit phdp dOi
ximg D2 tdm 02. Xdt hai duong thing \ vd 12
md 11 ll lz, D{l) : Dz(l),= 12 vd tit cit circ
diOm <l6c biqt cria Dt, Dz d6u ndm trong d6i Z
gi6i han bdi /1 vd /2. Khi cl6, phdp bi6n hinh
tich D.,"p, ld m6t phdp tinh tii5n bi6n m6i
di6m nim ngodi d6i I thdnh ditim cirng mdu
v6i n6. Tt d6 suy ra ph6p d6i xring tdm X, voi
X ld mQt dii5m tiy 1f nim ngodi dii Z, sE bi6nm5l aicm I mi vecto xy :zk.qo, , k e z,