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* sor roAN Hoc vIFT NAM

'BO GIAO DUC VA DAO TAO

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TAP CHi RA NGAY 15 }IANG TIIANG

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* vfi nAr roAN so sANH PHAN sd * cfctt t{}tiN MEr Df NG ro6N ouY ricH * * GIG sd nnusEY * l{i THr oL€rutPt( Tonu ouOt rd

Do?tn Viil Nam tai IMO - 96 : Tir trAi sang phAi : Phqm L6 Hitng, PGS Phan Duc Chinh, NgO Ddc

thity

I uan,

xndc Nguy6n Minh, Nguy)n Thei He, Trinh Thd Huynh, D5 Qu6c Anh, Ng6 Duc Duy.

ToAN Hoc vA TUOr rRE MATHEMATICS AND YOUTH

MUC LUC

Trqng

Ddnh cho cdc ban Trung hoc Co s0 For Lower Secondary School Leuel Friends Nguydn H{tu Bd.ng - Vd bai to6n so srinh phdn s6. 1 Gini bdi ki trrdc Solutions of Problems in Preuious Issue Cdc bdi c.iua s6 226.

NGO DAT TIJ

2 Tdng bi6n fip : NGUYEN CANH TOAN Ph6 tdng bidn tdp :

nOr oOruc arEN rAp :

I

B iographies of M athematicians. Nhd.n ki ni€m 170 nam ngity ra diti cila hinh hoc Lobasepski - Nguydn Cdnh Todn - MQt quA tnlng ving vi dai. 8 Db ra ki ndy Problem.s in this Issue T11230 ... TtOl230, LU230,L21230 Hoc sinh tim tbi

t2

HOANG CHUNG

13

o

16

Nguy6n CAnh Todn, Hoang Chring, NgO Dat Trl, LO Khic B&o, Nguy6n Huy Doan, Nguy6n Vi6t Hai, Dinh Quang HAo, Nguy6n XuAn Huy, Phan Huy KhAi, Vri Thanh Khi6t, Lo Hei Kh6i, Nguy6n Ven Mau, Hoing L6 Minh, Nguy6n Khic Minh, Trdn Van Nhung, Nguy6n Dang Phdt, Phan Thanh Quang, Ta Hdng QuAng, Dang Hung Thing, Vfl Dtrong ThUy, Trdn Thdnh Trai, LO Bri Kh6nh Trinh, Ngd Vi6t Trung, D4ng Quan Vi6n.

Bia 4 Bia 4

Young Friends Search in Maths Ngd Minh Nghia - S*y nghi v6 m6t bdi torin. 11 Ddnh cho cdc bun chudn bi thi bdo Dsi hoc For College and Uniuersity Entrance Exam Preparers. Hb Quang Vinh - C6ch nhin mOt dang todn qu! tich trong kh6ng gian. Hoitng Chrlng - C6.c s6 Ramsey. Pharu Dtc Chinh - Nguydn Khir Minh - Ki thi Olempic To6n Qu6c td ldn thn 37. Gidi tri todn hoc Fun with Mothematics GiAi ddp bdi : Vidt sd trong bin cd. Tudn Dang - Tim ctla vD.o vd dtrdng di.

Tru sd tda soan : 458 HDrng Chudi, Hn NQi 23f Nguy6n Ven Crr, TP Hd Chi Minh

DT: 8213786 Bi€n DT: 8356111

tQp ud. tri s4 : VU KIM THIfY Trinh bay; QU6C gbNC

a

S6ch girio khoa To6n 6, tdp 2, d phdn 6n t{p vd tinh chdt cria b6n ph6p tinh

trong N vi trong Q* cd bdi to6n :

So sanh bang nhi6u phuong phrip khric nhau xom trong hai phan sd f ,'a

T

z{o^

13., * thi phAn sd ndo I6n hon. Ddy lamOt bAi toan don giAn nhrrlgchfa drrngnhi6u ydn-dd trongchrrongtrinh Torin 6. Trong bii niy tOi xin trao ddi vdi cac ban vdi di63 nhrJ sau- ; Tlti6c hdt xin ndu tcim tSt nLrrng c6ch so uinh phAn s6 (trong tap hSp Q*) quen thuOc ld : 1. Quy ddng mdu cac phAn sd de cho rdi so s6nh cAc tit v6i nhau. Z.iiAt caclhan s6 di cho du6i dang cdc phdn sd cirng trl rdi so s6nh c6c

mdu vdi nhau.

5S ztrt 5Z I P z

,

$u."f,tf,,r:,i"d Y$ "nu,

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c

b

vdi don vi", theo tinh chdt : ndu

3. So sdnh phAn s6 theo tinh chdt : ndu ad, < bc th\t < 3 4. so ssnh ti s6 c.ic ol?t;: 5. Vidt c6c phdn sd drrdi dang s6 th{p ph5n rdi so s6nh cric sd th{p phnn 6. So s6nh sd nghlch dAo cria c:ic phdn sd, theo tinh chdt Choa,b,c, d* o, ndu ! . 1 thi,9 >, 7.. DUa vdo' tinh 'chdt b6c cdu cria quan he thf tu : ndu nL m .a c ( -thi-<- < -ud - ttndjd 8. So srinh "phdn bir cria c6.e phdn sd d6i

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Ti6p theo xin n6u th6m vdi c6ch giAi khic : 9. Ta ctj tinh chdt (d6 chrlng minh) N6uU

r@ Wf,

L3=5.2+3; 16 =7.2+2 .Tuong ttl ta cflng so sdnh duoc mQt crich nhanh ggn hai phAn sd ching hqn

N \ {' $ $

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89 89.10+5 5 "tac6 *< 4ndn 95< gs. 10 +4'a dooo gs' gsa .

(xem tidP trang 10)

$< \t N

a

(1)

Bni Tl I 225 : Tim sd add. trong hQ ddm thdp

phd.n th6a mdn :

Q)

- Nduy2 = I ta c6 4 = 2 +{199 -7= 2x 6y',-L99-x2-?* *x2+2r-lgd=o o(r-13)(x+15)=g +t; = 13 vi.r = -15 - Ndu yz =4 ta c6 16 = 2 + \[109=7 - 2x *196 : 799.- 12 - 2x +x2 +2x -3 = o o(x-1)(x+3):0 e+x=1vdu=-3 Thay cac gi6 triy = ! t, ! = + 2 vdcdc gl6ti c'&a x 6 (1) vh (2) vdo phrrong trinh ta thdy phrrong trinh duoc th6a mdn. YQy cdc nghiQm cria phrtong trinh ld : (x,y) = (L,2) , (L,-2\ , (-3,2), (-3,-2), (13,1), (13,-1), (-15,1), (-15, -1)

(2).Kdt hqp (Z) vdi (1) ta c6 :

(3)

o

cdng, Ddm Doi, Minh HAi. 4y2 = 2 *rtlsg -7 -% . o4y2 =.2 + r[r00= @ +IY Dd phrro.ng trinh cd nghiQm nguyOn thi rfm=@ +-IY = t[2 . lF:(x *1Y = = {F4W -1:@ +:IY phAi ln sd chinh phuong. Khi dd phAi cti : ho6c (r + 1)2 = 102 hoac (x + L)2 : 22 holc

(r+1;z =142

- Ndu (r + 1)2 = 102 + 4y2 = 2+ 10=+y2 = 3 phuong trinh kh6ng c

(L,-2), (7,2), (-3,-2), (-3,2), (13,-1), (13,1)

(-15,-1), (-15,1).

Hon nta, 0 < o < 9 (o € N) n6n chl c6 cic qp @, n) sau day l} th6a m6n (3) : (1,4), (2,5), (3,6), (6,8), (8,9). Va"ta cd c6csd tuong tlug sau d6y : 1996, 2828,2832,3996,6664,8164. Tht lai, ta thdy cA 6 sd ndy d6u ld sd cdn tim. . Nh?r, x6t. Cri 84 bai glLi, tdt c6 d6u gi6i {ring. Crig ban sau ddy crf ldl giai t6t: Trd.ntinh ?4ry $ To6n L6 Quy DOn NLa Trang - Kh6nh Hda), _Br)i Dtc Anh (7A Trong Didm THCS UdngBi - QuAng Nirrh), TritnThanV Son (g Torin NEngKhidu T* Ninh Binh), NeuvdnVi€t pQng (9ATo6n Nang Khidu Lane G"ia;e - Ha lec),_Ngrqydn Trdru Ngoc euans (9 T;6n LO Quy D6n LongKhrlnh: Ddne Nii).Vuonp Gia Vtl (6TC PTCS TrungVuone, HoanKigm"- Ha NQi), pd Qu6c Bd,o (gt frdir Dane Ninh Tn Nam Dinh - Nam Hii, Btri Anh fn (9f Tfi Li6m - Ha NOi), phqm Dinh phil (93 THCS t_ong Qin-h ChEu Thdnh - Ti6n Giang), Vit Tudn Anh (9 To6n THCS Nane Khid;'Th6i llqoyoq - B5c Thai), Nguy La"Giini 1CX Ighia-onn --Nghe An), yo Thanh itet ed hoc - Quy Nhon), Trdn Eic Son' (g ^Qlr6c Qhyyen_lhi trdn Ba D6n (eu6ng Trach - QuangBinh)'

oANc vr6N

EZIZB : Gid.i phuong trinh. nghiQm

Bari nguy€n

-

.41 = 2 a {igg -- rr- 2* - Ldi g1iri: Cd,ch 1. ciaNguyiln Thanh Tilng 8Ar, Hdng Bdng, Hei Phbng Di6u ki6n dd bdi torin crj nghia : -lb < r< 18. Tacd:

4y2 = 2+r[1$s424; = 2a{26s= @Try Viy ld sd nguy

. 2 +r[2oo

<4

I ,

1) Sd6A x 4 tQn citng bdng cd, 2) adAn -EZ x 4ld. sd chinh phuong. Ldi giAi : Tr) thidt ta c6 : ' a6ccl -Uc x 4 = rr2 x 100 (n e 7j) nx 100+6ex 10+d -frx4-n2 x 100=0 *1AO(n2 - 10o) = 666+d (1) Do 0 < b, c, d ( 9 n6n 0 < 668+d < 610 Cd.ch2 ciaPhamVan Tidn, gA, ; THCS bdn

}J:ry y2 = lhodcy2 = 4. _NhAn x6t : 1. Da sd c6c b4n gi6i theo cdch 1. Giai theo c6ch 2 cdn cci c6c ban : LA.Hod.ng Drtc Khd.nh, 9T, Chuydn Nguy6n Nghi6m, Drl"c Phd, QuAng Ndai ; to rnZn"n ViQt", 9As'Qr6c hgc, Quy Nhon. Binh Dinh. _ 2. Ce9 ban cci ldi giai t6t la : Phsm. Hd.i Trung,9T, NK Ti6n son ; Duong Quang KiAn, ?!rq*_ Trung Drtng, 9T, NK B6c Giang ; NguydnViQt Qttng, NgOVd,n ftic, 9T.NK L4ng Giang, Hn B6c. Deng ViQt Dung,92, Chuy6n Vinharrdng ; Anh'H"img'DAng,"9.0,, Sa Dric - ryU-ThS ; Dsng Thu Hubng, 8T, Chuy6n cdp II, Phri T\o ; Vu Mqnh Cuimg, 8A; Biri Dd,ng Quang,9T, Chuy6n cdp II, Tam DAo, Vinh Phri. N gu_ydn Thi N gqc Anh, 9T,L6 Hdng Phong Thi x6, Y6n Bdi, T?d.n Td.t Dqt, Ng| Qu! Duong, 8A,, Chu Van An ; Nguydn Tl^rdn Anh,8An Ngoc taffi, Gia Lam ; rrin"uuu Drlc, gC'i, Chuitjn

2

NhQn x6t : Bdi niy cti rdt nhi6u ban tham gia giei. Chi cti 6 ldi gidi sai : Ldi giai t6i ld cria :, Nguydn IIit. Duy 9 Hd Tdy, Nguydn Thi Minh Hod.ng 9 Hi Bic, Vo Chi Thd.nh 9 QuAng Ngdi, L€ Hod.ng Anh I Ha NOi, Phqm Thd Anh I H.it NQi, 7r&z Drlc.Son,8 QuAng Binh Pham Thi Vd.n Giang 8 Minh tlhi Nguydn Trung KiAru 8T Nam Hd, Trd.n Td.t Da, 8A Chu Van An Ha Noi, Nguydn Vd.n Thirnh 9 Ninh Binh, Nguydn Thi Thny Hqnh 8 Hba Binh, Le Thi Tdm I NghQ Arr.

DANG HI]NG TH-{NG

BldiT4l22S- Trong cd.c hinh thang cd.n c6 chu ui 2p, gdc kbdd.y l6nbd.ng a (o < 9?o),dqrug hinh thang c6 di|n tich ldru nhdt,

Ldi giai vfun tf;t : Ggi ABCD ld hinh thang (AB ll CD) c6 AB = 2a, CD = 2b, BC = c. O lit trung didm CD. Ta c6 2p = 2a * 2b + 2c

\

\ Hc(

<_-_

Ti Li6m, He NOi. V{r Vd.n Qu!,91., KimAnh, Kim M6n HAi Hrrng. Luong Si Tirng, 9T, chuy6n Kidn Xrrong Phan Huong Thu, Hoitng Thd Doanh, 8T, Chuy6n Thi x5, Thdi Binh Nguydn Trgng Ki€n, Hit. Thanh T\.r.d.n ; Dito Hodng Anh, 8T ; Nguydn Thi Hbng Dun g, I{ai Nggc Kha,9T, Trdn D6ngNinh ;Nguydn Nggc DiA.p,9B, Thanh Lrru, Thanh Li6m, Nam ltri. Dinh Httu Todn, 8T, NK Trrrong Hrin Si6u ; Trd.n Thitnh Soz, 8T, NK Th! X6, Ninh Binh. Dodn. COng Anh, 84, NK Ha Trung ; Philng Hd.i Anh,9A, Xi Mdng ; D6 Thi lloo, 7T, NK Bim Son, NguydnTrgng Phong,8C, NKThdnh ph6, Thanh H6a, Phan Thanh Trung, 8T, Qu6n Hinh, Nghi LOc ; Phan Thi Nghia,8A, NKY6n Thdnh ; Nguydn Tud.n Duong, 9ll L6 Mao, Vinh ; Nguydn Anh Til,9T, chuydn Phan BOi ChAu, NghQ An. Nguydn Thi Thtly Hqnh,8T, NK Thi x6; Trd.n NguyAn Thq, 8T,.NK Ha Tinh. Nguydn Minh Kien,6T'Vo Chi !hd.nh, 9T, Chuy6n LC Khi6t ; Nguydn Hdi Au, 8T, Chuy6n M0 Drlc, QuAng'Ngei. Bi,Li Tidn Dat, Trinh Duy Binh,9Ar, LO Lqi, Di Linh, LAm Ddng. Nguydn Ngqq-Minh, 9A, Ltrong Van Chrinh, Phri Y6n. Luong Trung Tud.n,9T, Bdi dudng gi6o duc, Bi6n Hda, Ddng Nai. Chung Nhdn Phil, 8"Ib Nguy6n An Khrrong, Hdc Mdn ; Pham Mifuh Hilng, 9T, Nguy6n Du, Gd Vdp, TP. Hd Chi Minh. Nguydn Chi Thdruh, 8T,n chuy6n Nguy6n Binh Khi6m, Vinh Long. PhATh ViQt San, 6 Ar, Phqm Th{ Vd,n Giang, 8A" chuy6n Bac LiOu, Minh HAi.

16 NcuyEN

Bei T3(226) : Tim td.t cd. a e N dd phuong

trinh

D

x2-a2x*a*7=0

=

+P=a*b*c Mitkhrich=csina. + S = (a +b)h = (a *b).csina Do dd S l6n nhdt*(a*b)c 16n nhdt. Vi o * 6vdcldhais6 dtrong cd tdng khOng ddi n6n tich crla chring l6n nhdt khi vd chi khi a*b = D " = f,. D6 thdy CK = c (K ln hinh ehidu ctraA xudng CD). Suy ra cAeh drrng : DUng LBHC o, BC : i.Yu dudng trbn vuOng d H e6 C (c, cB) c6t cH k6o ddi 6 K. K6 Bx ll CK vd Ky t CK cdt nhau t4iA. Tt dri suy ra didm D.

ki6n ld L, = aa.- 4a - 4ld sd chinh phrrong

Nh4rr x6t : 1. Bii niy nhi6u ban giAi rdt dii do chtta chri f gie thid.t cho p vi a trlc li dE kh6ng ddi.

2. Cecban giAi t6t bAi niy : Cao Bing : Ld.m Mqnh. Trudng,9A THCS

Hop Giang.

B6c Th6i : Vfi Tud,n Anh, 9CT THCS Ndng

khi6u Thrii Nguy6n.

phuonsv6io>3vi

Van Quy 9a HAi Hrrng)

c6 nghiQnt nguyAn. Ldi giai : Cach I (cria da s6 cric ban) Dd phrtong trinh cd nghiOm nguy6n di6u Ydia:0, lthiA<0 Ydia:2+L,=4th6amdn Y6ia > 3 ta cd L > (a2- 1)2 vl L > (az - l)2 +,o+ - 4a - 4 > a2 - Zaz - | = 2a2 - h - 5 > O *-2a(a - 2) > b dringvi2a>6,a-2>1. D6 thdy 6. (o2)2. V4y A kh6ng li sd chinh (a2_L)2

trinh. Theo dinh li Viet

'

Hd B6c : Tfd.n l{6ng Quang, 9T Chuy6n B6c Giang N guydn NhU Chudz,,8 NK Thudn Thdnh, Pham HAi Trung, 9CT Nang khiSu Ti6n Son. ' Vinh Ph:d : Trh.n Thi Tho 8, PTCS. Supe Phong ChAu, Bili Dd.ng Quang,9T Chuy6n Tam DAo, Pham Td.t Dat 88 Chuy6n YGn L4c, "Anh Hilng Dung,9A PTCS Sa D6c, Phri Tho, DQng Thu Huong 8T Chuy6n Phri Tho, __.YCq Bdi : Nguydn Thi NSoc Anh, 9"1 Ld H6ng Phong. Hd Tdv : D6 Hoirn* DiQp, CII BO t6ng,

Chuong Mi.

x1*xr: a2 (1) xrxr:a*l Tr) dd (xr- 1)(x2- 1) = - (a2 - a - 2) Yi xr, x2 € N n6u tt (1) suy ra x, 2 l, xr) l.Ydy at - a - 2 < 0 j0 < o < 2. Thtt trltc ti6p chi c6 a : 2 th6a

HAi Phirno : Nquydn Thanh Tir.ng, 84,

mfln dbi h6i bii torin.

Ha NOi : Ddo Phuong BilL,8A, BdVan Ddn, Vrl PhrrongNhi, TH, L€ Thi Hodng 8H TlrtngVrrong Tfr.n Phtic Long, 9APTCS Phan Chu lYinh. THCS Hdng Bnng "

tQ

Thrii Binh : Phan Huong Thu, 8T PTCS

cirt'rgxdy ra khi vd chi khiCM - CN, n6n ti (2) tac6:

Chuy6n Th! xa.

l. 2a,7

Nam Hi : Vir Trd.n Crtong, Nguydn Trqrlg Cuang, Dod.i Phuong, Hir Thanh Tud.n, Nguydn Van. Trung, Dd.o Hoitng Anh, 8T, Dd Qudc Bd,o,I{guy\ru l{bng Dung, 9T, Trdn Ddng Ninh, Nam Dinh.

Thanh Hcia : Doirn COng Anlz, BA Nang khidu Hd Trung, Nguydn Trqng Phong 8C NK Thanh Hcia.

Chaul Nguy 6n fhinh Quyn h, 8T NK'Vinh Hd Tinh :VO SiNam 9CT NKDrlc Tho, QuAng Binh : T?d.n Dilc Son 8 Chuy6n Ba D6n

2a ,iE > cM.cN 11ay 2(Z;m)- > scut, Suv ra S-^,^, dat giri tri ldn nhdt khi vd chi k:ni cM = Cfl'tt" la:khi DN : a * cN : o. - C4-: BM-rzAta cd AAND = LABM (rcgc) hat BAM = DAN :22a30'. Vay vi tri cria M, N dd diQn tich tam gi:ic CMN dqt gre tri l6n nhdt li: g6c BAM bing 22030'. Nh4n x6t : Cci 98 bdi giAi trong d6 cd 12 bdi giAi sai. Sai ldm phd bidnTa cho ring ndu (r; < sfil ve tai didm i^, flx^) : gk^) thi flr^) Ia gi6 tri ldn nhdt c&af(x). CEi cdn xdt vi du doq giin aav cung dfr'dd b:ic b6 di6u d

Thanh,"Tht Ddu IVIOt.

DANG VI EN

BniT6l228 t Chtng minh rd.ng udi msi sd

Khdnh Hda : Dinh Thi Phuong Thanh, Trd.n Tud.n Anh ; 8T LO Qr"ty DOn, Nha Trang, LAm Ddns : Phant Npuv2n Thane. 8T PTTH Chuyc-n Thanb LonS ila Lat. S6no 86 : Dodn Ngqc Minh, 8A, Hi6p Ddns Nai ; Pham Nsoc Huv- 8T L'i Tu Minh Nsec gTir ftdn Hrrng"Dao,

NEhO An : Neuvdn Aruh Tu.9T Phan B6i

ill."*f+ff"

TPHCM : Chung Nhdn Phrt 8"11 Nguy6n

An Khuong H

Chuy6n ThoAi Nggc Hdu, Long Xuy6n,

n-l > cr#._{) .c,,_o t=0 ' chia hdt cho 4tr,

Ldi giei : (cfia da sd cdc ban). Ta cri n-l n-l

'

An Giang : Hod.ng Thanh Ldm.,8T PTTH Vinh Lone ; Neuy,6n Ch; Thitnh. 8T1 Chuy€n NguyEn Bin'h lGiem ' Scic Tring : DQng Minh Thanfu, I PT CI-[ An L4c TAy,Xd Sdch Minh H&i : Pham Van Tidn. 9A1 THCS b6n c6ng Ddm Doi' Minh Hai'

tU nhiAn n >- 2 ta c6

t:0

vo *u rnuy

>cr*.-p ct -o=)1n -k)C"Ji,.-{) =>kc*, *, k:0 n-7

2k.(2n +.1)l ='"f\ry-:'

B,di T51226 : Cho hinh uu6ng ABCD, cdc djfuM, N ld.n luot bAn BC ud. CD sao cho MAN = 45o. Hdy tim ui tri cia M, N d.d dian tich tam gidc CMN dat gid tri lon nh(it.

e2.(2k)t(2n-2k+1)t Met kh5c, ta cci

n-l

B

k=0

k=o

Dodti

Tr) dd suy ra di6u phAi chrlng minh.

ir"*f, "r-r:"#*.{n = (2n*!;4n-r. (*) ^ Nhgn x6t : Da sd cdc b4n chrlng minh drroc dang thdc (*).

(1 + 1)2n - (t - 1)2" : Z>Cfi-r :4".

Nam Hd : Nguydn $nh Hoa, Nguydn Thu

Cdc b4n sau ddy cri ldi giAi t6t : TPHCM : La Quang Md.n. Ha NOi : Vu Td.t Thd.ng, Nguydn Qu6c Th&.ng, Trd.n Nguy€n Ngec, Phqm Minh Hod.ng, Nguydn COng Minh, Nguydn Hoitng Anh, Hod.ng Si Hilng, NguydnVu Hung.

Ldi giei : Dqt o la c4nh cria hinh vu6ng ABCD. TrOn tia ddi citatiaDC,ldy didm E sao cho DE = BM, ta c6 b,ADE : LABM (cgd'da d6 A.E-= AM vd. Wn= MAB-*$uy ,o4,a<- DAN : UIAE+ DAN 2 = 90o - MAN - 45o,' vi ta cri LEAN = LMAN (cgc), do d6 MN = EN = ED +DN = BM + DN -- 2a - CM - CN (1). Mdt kh5c, tt tam gi6c vu6ng CMN tal+i c6 MN :,[eWrCM. Thay vdo (1), drJoc : Za- CM _ CI/, {CM-CM * Za : (CM + Cloll + {eW +Tffi (2) Hcln n{{a, CIl{ + CN > 2,{C.tu1 3N va '{CY.-effi > {ACM. clv, va c6 hai ddu bing 4

Thr1y.

Hd Tay : Nguydn Quang NguYSn . F{Ai Hrrng : Trd.n Hoitng Vi€t, Phitng Dtlc

Ftrn Bic :Yu Duy Tu6.n. Hba Binh : L€ Van Manh. Thrii Binh : Tro.n COng Cui,n7. HAi Phdng : Hod.ng La Quang. Thanh IJda : Dinh Trudng Son' N guydn Van Quang, Nguydn. Minh Thanh, Vian. Nggc- Quanlg, t,ui'iruimg l{uy, Hd.nVd.n Thang, D6 Hbng Son.

Tud.n, Mac Dd.ng Nghi.

c) Ta cd f(-0,80) = -0,0083 f?A,79): 0,0064... f(-0,78) = 0,020... Vey x = -0,79. Nh6n x6t : Cric b3n sau dAy cri ldi giai t6t Trd.n N guyan N gqc 1 1 DHKE{T\T, L€ Qyy"S Nd.m la I{CM, Trdn Hui^r Lt"tc la Qu6.+gtsinh, Trd.n Nam Dung 10 NghQ An, Phan-Anh Huy 11 Da Ning, N{uydnfr.nh Chi 11, HaNQi, Hd Loi ThuAn i I Ddng Thrip, Trdn Drtc f!.q'Ary 19 Qirang ninr,, lrgu)dn riat Thq lp IgLq A", Eod.ru*Xud,nVinn ilHu6, Nguydn Qu6c Thd.rtg 10 DHSP, Nguydn Manh Tinh ll,IIAi-Hung, Phan Thanifiai 11 Lam D6ng, Nguydn Anh I{oa, LONam Hd, Cao ThdAnh lltIrt1,Nguydn Dang Tridn 1tr, D6ng Thrip.

QuAng Binh : Trd.n Dtlc ThuQn, Fhan DuY Hilig, Tiuong.Vinh Ld.n, D6 Dung, Trb,n Hilu Isrc. Trdn Chi Hba.

thoa ntan dibu hi|n

Thi'Atth.

Bai 18/226 : cho a,;i:;Ti[T?i'"ou" ab*bc*cd*d.e+e1:1.

Ch*ng minh rd.ng

QuAngNam - DnNing :NguydnVanDung. Einh Dinh : I{guydn Phuong Trinh, To

Pitttdc Sanh.

NghQ An : Duong Vdn Yen, La Thanh Binh, i DS.ni Dxc HS.nh, Le Vdn An, Nguydn Thd.i Thq, LA Hbng Hit, NguydnViQt Dung.

- Zcos1

?rd Vinh : Bili Minh. Thiln. Kh;inh Hda: Trd.n Tudn Anh.,

_

NGUYEN VAN MAU

Ldi gi6i. (cira da s6, cdc ban)' Vdi moi

BAi T'?i22G ; Cho phuong trinh

a; ) 0, i = 1,5,tac6

xt3-)e6+sx4-ax2+7=o

1-

a2 +bz +c2 +dz +ez +fz > L

a) Chang nzinh rd.ng phuong trinh c6 drtng

mlt nghiQm sd tht.tc b)Datxt=7uit

.)

' olo.* ot l_ ,"' ,2b" dzuz + zz ;la.cz+)a2>z"a a4d.2 *{"' , za" - I rz>zef.

osn'*-os,

ft+r= (x;78 +1)-3119 uoi ntei n G N*. Chtng minh riing day sd x^ co gidi hsn ud. hhi ddt xn : -limxn thi xn lii nghiQnt sd thttc n6i tr€n, ., c) Ditng mdy tinhb6 ffii hay t{nh g6.n ding nghiQm sd thutc n6i tr€n ddn hai chft sd th$p phdn.

Ldi giai : Dqtt f(x)

=^xr3

: x6 + 3rf4 - 3x2 + I . (a13-16)*3(r4- x2)+l >

\.

,L

'Ndur ilth\f(x)': 1>0

(0, l) thi f(x) = at3 - (r2 - 1)3 > 0

(1)

Ndu r € Vdir < 0thi f'(xl:13112 - 6xs +1?a3 - 6x:13*12 -6*1*2 - 1)z > 0 do d

I)O.

d1=--------1-rL:

tJt

111

b) Dat E&) : (v-7t3 a 1)-3113 Ta cri r, = 1, rn+l : g(rr.,) Vi hdm f (x > 0) vdi a < 0 te him siAm do d6 g(x) ld him tang. Mat kh6c r, : 2-jfit . , : xr *rr : g(xrl < g(r,) : x), ...ilay (x.) ldday giArh vati chan- dudi b&i 0 n6n tdn tai i,ldi han a : lirnr-. Qua gi6i han ta drloc

Q)

-2cosnl7.

Tt dd suv ra ^ :t ara2 + (", * ar)bt + 14* ar)ct + 1. ,1 ,1 * (4 +ao)d/ * \E+ar)d * orf. >2(ab *bc *cd *de *ef) =2. Dat sin(i+1)f . -= sin 7 Khi dd d6 dnng chfng minh drroc ring or: q*oz: q*or= q*oo: 11 --r -rl ,'u5-a.- 1J (1) vn (2) cho ta dpcm. Cric ban sau dAv cci ldi eiai t6t:. Dito Manh rhd;i. i0-A-PTfH HirneVuons. vinh Phn,Le ouan1 twdi.l0cr. DHfH TP.fICM, Neuvdn Anh YIoa, 10A L'6 Hdng Phong, Nam Hd,

a- 1313"- o-113 I | <+ I = (a-xl3 * l)qt3l3 - o,2 q r13l3 ---, (1 - a2)3 - o13 dat x,o = -a =-, (l - xf13 = - r'ou --**'ou (x! - tlz = 6 -flr.,) :0.

b2 >- zab

Nguydn Qu6c Thdng, l0A, PTCT-DHSP I He NQi,, Triiz Nguy€n Ngoc, 1 18, DHKHT\I, He I{Oi, ' ' DinhTluimg Son,7IT, Lam'Son, Thanh Hda.

NGUYEN VAN MAU

20) Hai bqn Nguy6n Thdnh Phttong, 10A, PTTH chuy6n Th6i Binh vi Pham Anh Drlc, 11T, PTNK HAi Hrrng da giei hodc d6 xudt bAi todn tdng qurit hon bing cdch thay trX di6n (chdp tam giric) D.ABC bdi ch

*t dd, BDT (*) drroc thay ddi BDT :

> nGM.

B,diTgl226: Goi G ld. trgng td.m met ABC c&a tt die.n ABCD ud, M ld. mQt didm bd,t ki thudc mibn tam gid.c ABC. Duitng thang qua M sotlg song udi DG cdt cac m,at phang DBC, DCA ud. DAB d A', B'ud, C'.

, ) i:r

"a', 3o) Ngoii hai ban tr6n, cdc ban sau dAy cci

ldi siai rdt :

DA' + DB' + DC' >. \GM (X) Ldi giAi : GiA sttM * G vddrrdng thir,gGM IJor grar : Lila sU /}4 + G va CIUdng tf)anq GM nio criao4dy ABc. Goi

He NOi : LA Tudn Anh, Ngutdn Vinh Chi, Nguydn Sy Phong, Nguydi Qildi Thang, Trdn NguyAn Nggc, Nguydn Vu Hung,

Hn TAy : Nguy6n Quang Nguy6n HAi Hrrng : Phqm Vsn Khdnh, Mac Dang

Ngh!, Nguydn Manh Tinh Thrli Binh : Quynh Hoa, Thanh Hda : Phq.m Nhu Nggc, Dinh Trxitng

Son, Vi€n Ngqc Quang)

fl:EY ""T-i"il* *uj;u,n Ar, Br vd C, li giao didm cria cdc drlbnE thing GM theo thfi tri- voi cdc dridng thine BC. CA vE A-a : Thd thi cric didm A.', B'.vd C' chinh ld Elao olem cua cac iludns ctudng thino thans DA.,DBr vd DC, u6lr f,__ drrdne th&ns di ouaO -* M sois sonivdi DG ' M sons sons v6i .D A (xem h'inh v"6 b6n).

Neh6 An : Neuydn Vi€t Dunp. Triin Nam n$oAns Dilc" I{anh,. N guydn Y hinh, Duong QuAng Binh : D6 Durug, Trung V[nh lin, Trdn Huu Lttc, Triin Chi Hba, Phan Duy Hilng, 'Thrla Thi6n - Hud : Cao Thd Anh, Doitn

Vi G vA M d6u nim trong tam gi6c

Xud,nVinh.l.

MA' ll GD ndn ta cd :

NGUYE,N DANG PHAT

MA,

-

MA'-

flU

B.di TlOl228 z Goi ho, h6, h" ud. lo, 16, l, ld.n luot lit d0 dili cdc duimg cao ud cdc duime phdn gidc in g u6i cdc canh d6 d0 ddi a, b, c cilit. m\t tam pid,c c6 ban kiruh cac duirnq trbn n6i uit ngoai tidp lit r ud. R. Ching min| riing : t tL t -]f; ro* ro*4" I o Ldi giai : Sir dung cd.c cOng thr?c tinh dO dai drldng eao vd drldngphAn gi6c cria mQt tam giric theo dQ ddi aic canh : o_

MA' - GD GD: 6n,GD: A'A s(MBCt * s(MBC) -: : A-GD:37CO, s(GBQ) s(ABC) vd hai diirg thtlc tuohg t(: Mn,:ffel=r#*n, e) ,h,:#on=r"ff;* (B)

s(ABQ

ho= o'lP@ -a)(P -b)(P -c), lo=

2{6; O +"rlp(p -a),

1

(4)

(1)

- {(P -bXP -c) vd hai ho thrlc

trong dcip =, (o * b * c), ta drJoc : ha b *c t, = or{b" tudng tu :

"rtob''l(P-a)@-b)

hb c*a 4= or[-*tl@-c)@-a)' hc a*b l,= Tl} dd, rip dung bdt ding thric C6si, ta duoc : ha hb hc T* ru*T'

(b +c) (c+aXa +b) (p --oXp -+Xp --r)

o262"2

+ s(MAB) = s(ABQ NM e mp A-BC), ta rhu d,rqc (+) vdi (5) vA do dd drioc BDT (*).

CQng.vd ddi vd (1){2) ,a q, ta dugc : MA', + MB', + MC' =3GD; _. Tg dci suy ,^, fr' + Dd' + Dn, = JGfu ;6\ Tit (5) ta drtoc bdt ding thric (*) cdn tim.

6

3m *7.1V

boin dQng lrrong suy ra u: eU u. Di6u kiQn dd va ch4m xAy ra ld ; u > f,vd f , > T, M5M nitra B

c, trlc ld tari giric la Cl6u.

Ddu dine thrlc xAy ra khi vd chi khi a : b = Nhdn xdt. 1o\ Bii ndv c6 rdt nhi6u crich giAi tr6n co sd srl duns c6c cdns thfc khdc nhau tinh d0 dAi dudnscdo'7a drrAneEhan siac cria m6t tam eihc khons fihtne chi the"ot6 daTeic canh i;b va E mh con iLeo ca"cric goc 4 B v\ C. Cliing han : ho= bsinC(= csinB), U = #"o"f;, r.r... Tuv nhi6n- cudi cirng d6u ph6i sit dgng

B.D..f Cosi ddi v6i 3 so

R vi c66 g6cA, B, C :

Chf l, th6m ring cri h0 thrlc sau dAy $ita r, r=4,BsinA B C LTsrnrs,nZ 20) Gqi A.trl: h, vd AD = lo , ta drrgc :

e.9

BIA|LZILhS z S, M, N, P ld car dinh cila m.Qt ht dian db u c6 td.m hin h cd.u ruEoai tidp lir. O. D bne dian'cudne d,0I chav theo au\ni USPM (hinh I) gay ra tai"O idnt thg tit Bo . Xd.c dinh cd.m ung tit B do m1t dine dien cuimg dO I cl14.y th.eo dudng' gd.p khnc MSPNI+{ gay ra fai O Oinh 2). n-

\v

i

- Cl, do dci :

hb C-A

B-C

/ At

FlA.r) :;l* ha t=cos 2 'ciing"ay,4=cos 2 hc

A-B

p Htnh 2

: cos 2 + cosT- * co5-

MAI ANH

A-B ,r"T:cos 2 Tfidd: ha hb Ftc B-C C-A T"6*l B.D.T. cdn tim

30) Drlng tidc cd nhi6u ban phat triOn dring

MOt s6 ban d5 srl dung kdt quA niy dd di ddn

,

Nh6 ringB

rangm6i r;?(i = a,6, c) d6u, {# , nhrrng trf dci suv ra B.D.T. cdn tim li khOnE dring ?;"3;T,"18;;m;;' ,,cr{< lvidod

>- 2r, 12,

le

NcuyEN oANc pnAr

at*tP Hudns d6n eiii : Coi tt trudns eAv bdi dbng di6n" / chay" theo dudng-MSPlfM Ia srr "chdng chdt" cia-2 tr) truBg : Bo EAy bdi dbng diQn ch4y theo MSPM vd .B1 Bdy bdi dbng di6n ch4y theo MPNM (tr) tnrdng do 2 dbng diQn chiv ar MP vdPM tridtti6unhau). Vi li do d6i xtliig 4 hu6ng theo oiV ve Br hu6ng'theo OS ftinh 3), D6 ddns thdv ON J- SKvd OS r KN. xet m4t phing,frar (ninn +). Chr1 y ringB, = Br Lrej *ry: t OS = ON ; tt dd v6cto B = Bo I Brcd phrrong Ko (oKL li trung truc cria sN]l_hrr6nqi': o d6n L vd cti dQ l6n B : 2I3ocosBOB.: l/3;

p

K

,#oo Iltnh 3 Nhan x6t: Cdc em cci ldi giAi dring: La ){ud.n QuyEn, 11.A, PTTH Ltrong D6c Bing Thanh lfcia, Hd.n Van Thd.ng, LLA*, THCB Dio Duv Tr). Thanh Hda. Phan Div Ililng,12CT Qs?ng Binh. M6t sd ein khong clii ra hrrdng cria B .

" Bai T !11228 ; MOt qud. cd,u hh6j luons m dang chuydn dAne ff€n rhOt dudns thd.np h.tm neane uoi-ui.n tdc"u thi ua iham udo mOt"oud cdunhdi luoni M dane chuydi doip cilnb ihibu tren d idie than e d6 u di i dru td c' u 13. Sau"u a c ham o u d. L/ ciiu ri chuvdn dilne udi udn tdc ui2. Coi uit clfam ldxuvan ffim. 86oil,arna sdt eiilahai oud.cd,ubdi dudis ndm neaie. Chilne tZ rdne sciu ua chant Qqi qltQc?ty.{Ap.fr" clyy{n 4ln7Yheo hudngba. t Lm d.rcu Hen d,e ua cnclm mv ra- tludne d6n Eiii : Gie ihi6t sau va cham. quA cdu m-bi bAtTrd lai : rip dune dinh luAt bAti tbin d6ng l,llt#$91[uoc'vai tdc i^t cria M sau va cham u : 6M u. Tinh tdngd6ngndng Trttrllc va ch4m vb. ?, sau'va cham, phAi cri T,4 T, mudn v{y 1 + # * 0 : v6 li. VAy sau va cham cA hai ouA cdu tiSn tuc chuvdn d6nE theo hrr6ng cu. Bdy grd 5p dlUng dlnti luat bA6

o rl5

MAI ANH

Nhfln ki niQm 170 ndm ngiy ra dtvi cria hinh hgc LO snspPSKI (r s26 - 1996)

m0rQuAmffiovAmoffiD$r

NcuvEN cANH roar.t

Nhdn dip dinh li ldn Fermat drroc chrlng minh, c6 tdc gie de viSt rhng qurl trinh 300 nhm tim tdi chrlng minh dinh li dd Ia m6t con gi d6 tnlng vi.ng. Trong lich stt toin hoc, cbn cd nhtng gd d6 tritng vdng kh6c md sau d6y t6i xin kd th6m m6t chuyQn. Ndu nhrr cAu chuy6n chrlng minh dinh li lon Fermat k6o dni 300 nam thi cau chuy€n chrlngminh ti6n dd Oclit cdn k6o dii ddn hon 2000 nnm. Ti thdi Oclit (a tnA H trd6c cOng nguy6n) ddn 1826, kh6ng bidt bao nhi6u gidy nrrc vi tdm tri da dd veo vi6c chfng minh cdi ti6n dd brrdng binh nay. Ndu ai h6i : "Deo dudi chrlng minh nhtr viy dd lam gi ?" thi chi ctf thd trA ldi ring : 'Dd la bidu hi6n tinh bdt khudt cta tri tu6 lodi ngrrdi ; tt th6h6 niy qua thd hC kh6c, ngubi ta khOng chiu ddu hdng trtr6c ti6n dd dd. Cd ngudi, nhrr nha torin hgc Ph6p ndi tidng Legendre (1752 - 1833) da tudng chtlng minh drtoc rdi'nhr.rng rdt cu6c c

mdt cdu) d6u kh6p kin vd trong thd gi6i ndy kh6ng cci dudng thing song song. Nhrr vay da manh nhay ki6n ring cd thd cci nhi6u hinh hoc kli6c nhau. Kd thrla y kidn niy vd chdp nhdn ringti6n d6 Oclit kh6ngph6i ld con d6l6gic cria c6c ti6n d6 khdc, L6basepski (1792 - 1856), mdt nhi torin hoc Nga, girio srr trirdng Dai hqc Kadan, mdi dtra ddn cho gia dinh cilc ti€n d6 ndy m6t dria con nu6i khric hin ti6n d6 Oclit ma sau niy thudngggi ln ti6n d6 L6basepski. Ti6n d6 nAy kh6ng chdp nhfln tinh duy nhdt cta dudng th&ng song song k6 tt m6t didm A trong mat phing (4 a) vdi dudng thinga. Tt dd 6ng kh6ng tr6c trd gi dd suy di6n ra c6c dinh li nhrrng d6u drroc nhtng dinh li ki qudc nhu cci nhirng tam gi6c md cA ba gcic d6u bang kh6ng tich c{c dai. vA dd tA nhtng tam gi6c cd di€n Ngudi ddi kh6ng ai hidu 6rtg vi 6ng sdng trong su dau khd vi todn bi chi trfch cho ddn khi ch6t (1856). Thdt ra thi cfrng c

Tich luy bi6t bao nhi6u thdt bai nhrr vdy cudi ctng da drra ddn m6t srr nghi ngd : "Hay li ti6n d6 Oclit kh6ng phAi ln h6 qu6l6gic cria cric ti6n d6 khSc ?'. PhAi ncii ring hdi dd tu tudng coi rdng h.inh hoc Oclit li chdn li duy nhdt, tuy6t d6i dring drrdng ngg tri trong ddu ric rnoi ngUdi ; nhungcring d6 c

(xem tidp trang 15 )

8

BAi Tgi230 : Goi E, r ldn lrrot ld bdn kinh eic dudnE trbn nsoai tidp vd nOi ti6p cua Lam Ai;Me. ehrlnf minh 'rane ridu c^ci hc thtlc Er = R(cos2A* c"os2B * cos2B * cos2C * lziiiAiiidi ziinnlinc i-z1irucsinA1 thi tam gieicABC ln d6u'

cAc l6p rHCS. Bai Tf/230 : Tim nghiOm nguydn drrong

ctia phrrong trinh :

tr + 2..f' + tr = 4500ydix < y < z oO rrnNu UAN (Minh Hdi).

sau :

a) Hay.xdc-dlnh r viy theo & dd ba didm M,

Bai rlo/23o: Gia ,o rfi:';{lf#i,l ,u* tr6n dttdns thane chtla dudng chdoA-B' cria mat ban ABB' A' c(ra ft ot hinh hOp 4BCD . A'B'C' D', M * A,M * B' vdMA : kMB' (k + o\. Goi N vd P ld c6c didm ldn lugt nim tr6n c6c a.rang thing BC' vi CD' sao cho -ffi = rilD, Ft = yfr'. N va P th&nq hdns. .uf |fr i,EqlH;.??i",?1:E'#f*['ffi"":fl]F tlidi4Ai-chfr nathFiharihrnolhatfl

Bidi TZI23O : Cho biSt m6i phtlong trinh x) - ntx * D :0vdx2 - nx *q= 0d6u c

Bit$F,.1^.,

(Hd N0i).

F{oANG Ncec cANs (ld Trnh)

Bei T3/230 : GiAi Phrrong trinh :

. x! - 4r[i.x - 5 = 0

cAc ob vAr r,i Bhi Ll/230 : Ba quA cdu - kh6i lgqng nlo , trLl ud. ntrbing nhau cri thd trudt khdng ma s6t trOn soi dAv xuv6n tAm vd cang lgang, m, vd m, dtro'c ndi vdi-lb xo tudng tdng h, k2 nhu hlnh-vc. O vi tri cdn bing mr eAch ntrmdt

BiiariT4lz}O: Chq mOt hinh cht nhQt cri chu 'ri ktrOncnii ananZt[2 vd mOt trl gi6c cti'c.ic dinh nim tr6"n cric canh khdc nhdu cria hinh chrr nh4t dci. Chrlng minh rdng chu vi cria tri giric khOng nh6 hon 2'

TRAN DUYHINH

5*s?t*al'r*t#iry"?"rhBT\'sr.i"Alt*il"#d ntr. Ban ddu, gh6p m.o vdi m, r6i n6n lai m6t do4n o thA tay ra, khi mo vd m, tdi d tri cin bing thi mrtdch ra chuydn dQng theo dny t6i va cham vd,o m2.

Coi va cham, gh6P, t6ch c:'ua mo ddi v6i

m, ud. nzr li hoin toin din hdi'

-

B-qi'tFtz;sp :. che r*ug uff!:;';!l;(o, S) vi didmA c6 dinh tr6n dci..M6t didm M chuydn d6ns tr6n (O).'tidn tuv6n v6i'drldne trbn tdi M cit iig" tuvdn t;i A A P. Mot Eudns trbn (O, , tr,) Uign ttrion di qua M vd ti6p x-uc vdi AP tqi P. Chrlng minh rhng (Or) lu6n tidp xric vdi mQt drrdng trbn c6 dinh.

NGUY6,N KHANH NGUYEN (IIdi Phbng).

cAc r,6p THCB BaiT6/230:X6ts6a:1- W + t4. a) Tim da tht?c bic ba c

1) ?irn-m6iquanhQgtitaK, , K2 , lvdadd sau khi va ch4m vdi in2 , mo qvay trd v6 va cham dring tai didm cdn bing khi m, dao dOng drroc 1 chu ki'. 2) Suv ludn tido' ch6 ou6 trihh tidn theb. 86 oia kich thrr6c bria cilc quA cdu,

s6 a lA nghiQm.

uar ru6 urEN (Thdi Binh).

b) Chting minh rang kh6ng tdn tai cdc da thrlc b6c nhdt ho6c bfc hai c

@a Nai).

a b+c ,n='{n -r

'

rnAN xr-rAN oAwc

R th6a m6n di6u ki6n :

Bni 18/230: lim tdtca"ufiiirJJlk:f : R---+ lr@) - r@)l =uE*o,?u'*Y,[f1YtF#

' (I

Bl}LiL2l23O: Cho mach di6n nhu hinh vC trons dd ngudn cci sudt di6n dOng E, di6n trd tioilE i = 2?2 cicdiQn trd Ri = 80Q, Rz ='40Q ; Rs = 26,67Q (ldy 26,67 = ff\ R.t = 8t). 86 oua di6n trd c6c ampe k5, cric dAv ndi vd cac-khda-K,Kr. Ifti K, d6ng, R, ng6tiv6n kd chi 30v. l

TRINH uANc crn^rc.

b) Frove that there is no polynomial of degree 1 or 2, with integer-coefficients, such "that a is one of its roots.

PROBLEMS INTHIS ISSIIE For Lower Secondary Schools T1/230. Find integer-solutions of the

T71230. Let be given three positive numbers

equation

a, b, c and an integer n > 2. Prove that :

tr+2.9*52=4500

satisfyingx

satisfying:

has two positive roots.

Prove that the inequalities tu I ft, m.n < < p * e, nlq < ip do not hold simultaneously.

T3/230. Solve the equation. xa-E{ix-5=0.

{5,#\n-r T8/230. Find all functions / : R - Il, lf(x) - f@)l < 5(x - q)2, vr € R, vq e Q. Tgl23O. Let R and r be respectively the radii of the circum-circle and the incircle of triangle ABC. Prove that if 6r = R(cos2A * cos2B * cos2C * 2sinAsinB * * 2sinBsinC + 2sinCsinA) then the triangle ABC is equilateral.

T10/230. Let M be a point on the line containing the diagonal A-B' of the face ABB'A' of a paralellggined ABCD.A'B'C'D', M * A, M * B'andMA: kMB'(h * @.

T4l23O. Let be given a rectangle, the perinreter of which is not less than 2t[2 and a quadrilateral, the vertices bf which lie on distinct sides of the rectangle. Prove that the perimeter of the quadrilateral is not less than 2. T5l23O. Let be given a circle (O, ,R), a fixed point A on it, and let the point M move on the circle. The tangents to the circle at A and at M intersect at P. A variable circle (O, E1), passing ttirough M, touches the line Ab at'P. Prove that the circle (O1) touches a fixed circle.

Let N and P be two r:oints resr:ectivelv on the lines BC'andCD',Nn = rrt','Fb = ;FB'. a) Determine r and y in terms of ft so that

these three points M, N and P are collinear.

For Upper Secondary Schools

b) Deduce from it that thdre exists and infinite number of lines, cutting the lines AB', BC', CD' arrd DA'at four points, forming and harmonic quadruplet.

T6t230.Let(]: 1- 11, + 114. a) Find a polynomial of degree 3, with integer-coefficients, such that a is one of its roots.

\

Rd rdng mOt bdi to6n don giAn nhrrng ctng

VE BAI TOAN...

chrla dung nhi6u vdn d6 li thf .

(Tidp theo trang 1)

Cu6i ctngxin n6u vdi bd.i tAp vdn dung

ta cci m6t cdch chrlng minh tinh chdt sau :

Choa, b,c,d, e N,nduo < b < c < d,vh.at

38

d:b*cthiad,

4zB ; 6sE

. 3B3 625 3871 va va tu, )

1. So s6nh c6e ph6n s6 theo cdch hgp li nhdt : -49 u' 1oo' 4s vd-

7345 bDb I

47 96 /J 65

b-a d-c b'd!:>

Thdt vAy tr) gie thidt suy ra fb-a=d-c t- I0

2. Chtlng minh ring

ad

,-Z,r-3=;.;+

+_

199 10d 100

2J 3J

11* s'z

Nghia ld : Cho hai sd t{ nhi6n bidn thi6n cd tdng kh6ng ddi, thdthi tich cria chring cd.ng l6n ne'u hiQu cria chring cing nh6. Vi dq : 95.98 > 94.99

t9952 > 1992.1998

a2>(o-1)(o+1)

=

txQ;TTt'

11 ") r, '* "' 11b) -+ ^*... Hudng ddn : ta c6 n3 > (n - l)n(n + 1) + 221 n)- @=fnQt+t

{n -*-

Thdt ra tinh chdt niy crlng dring v6i d, b, c, d e Q+. Khi dd tax6t c6c ti sd tudng ring thay cho phdn s6.

n = 2;3; ...

d. MOt di6u thri v! lA nhd so s6nh ph6n sd mi

10

SUVffiffiffitrWfr ffiOwmlilummm

o

sau :

- Phrrong trinh de cho cd dang :

Nhtr moi ldn, vrla mua duoc tap chi "To6n hoc vi tudi tr6" la t6i b6t tay vio giAi ngay nhirng bdi mi b6o ra. Ldn niy, tdi da g4p bai toin sau :

f(x) + s(x) = f(x) .. 8@ + g(O . ef@)

- C6cbidu thitc (x2 - 1) vitxcdthd thaybdi

12 +x-l

(*).

. 3-r + @2 - t)d

NcO ulNH rvcnia Lhp 1272, chuyAn cdp 2 - 3 i-e Khidt Qudng Ngai. Xem lai dd bei vi ldi gifri tr6n tOi cci nhdn x6t

-, TOi da c

- 56 e c

(**)

xtf-t

- il + (x2' 1)[d - lJ =o

Ta x6t uic trrldng hop :

'=

.

f(x) + 8(x) ld mOt sd drrong kh:ic 1.

<+

l\x.kz - l) :Oc=+l .

fx:o lrz-7:0

Bei to6n I : Giai phuong trinh : f(x) . as@ + s(4 . af@) v6i a Ldi giai bii to6n ndy ttlong tu nhu tr6n.

L

:

NghiQm cria pt ta nghi6m a^,lf,(')

!.

Thay vio phrrong trinh (**) ta cci

x=0 X:7 x=-l x=0 I:1 x=-7

Nhin bni todn md rQng d ,"u, [u(x? ,.ur'r, ,u, "cAm thdy" chrfa th6a man, bai tod.n chrta.tdng qu6t l6m khidn t6ilai mOt ldn nira ph6i suy nghi vi cudi cirng thi t6i ph6t hiQn ra li : "Minh chrra

(Doe>1)

md firig [f(x.y + g(r)J thdnh ) f,O>

cdc bidu thirc f(x) ud, g(x) bdt ki vio dd.

Nhung md khi md r6ng 'ir';l *s(r) thdnh

ld nghiGm crla phrrong trinh. Dx.. (* - 1) < o: , [r.o [*-1o-Il-t_1>o l(*r-l)td-11

phuong trlnh (**) v6 nghi6m.

\ f,<.> thi cach giAi tr6n cci cdn gli tri kh6ng ? i=l Thdla t6i giai vdi tnrdng hop n = 3.

[*-1>o lA--t_t<0

+i + , -

Bni todn 2. Giei phrrong trinh : fi(x) +fr(r1 +fs@) = f1@) .;fr(x) +1r1x). oh@) + fs@) . aft@) v6io ld s6 drrong kh6c 1.

l*ro b)1,={,,+ r _ y"-r.o lGz-Dlt-u

Do dd, trong trudng hgp ndy, phrrong trinh

dd cho vd nghiQm

GiAi: Ta vidt lai phtrong trinh drr6i dang : fi @)taf ,O> - 1 I + f .(fltof t o7 - 1 J + 1 r1xy taf {4 - t l.

B).r . (/ - l) > o

Ta chia 3 trtrdng hop : t * fi(x) . f(x) . fs$) : 0 : glhinhrr b6n. * fr(*) , fz(4 , fs@) ctng ddu : giAi nhtr

tr6n

.. * [, < o D)*{.

:}

,

-+{

lxz-1

lt - t . o ll-i-1

x /(r) khric ddu vdi fz@) v?t fj(x) : (Vi vai trd ft , fz, /, binh d&ng) : thi ta roi vdo ng6 cut. Nhtr vdy, bdi torin niy kh6e v6i bii to6n 1 ld khOng c

Cho n6n d6' giei dudc bai torln vdi n = 3, tdi

dua ra bii tod,n sau :

( ^ lkz-tv*-tt>o l)rl-i -rl, o + phuongtrinh (**)v6nghiQm' , Qo dri, tror,rg trudng hgp niy, phuong trinh de clo v6 nghiQm. V4y, pt ita cd cB B ngtiiqm ta x:0rx=-lrx=!.

(Xem tidp trang 15)

11

:+ Dhudn}trinh (**) vonghiQm

0r0l| r{ilr]{ Ihfi ut{$ ruffi ollr llcl| Inoil$ tll0lra 0lAfl

HO QUANG VINTI (NghQ An)

A.

Trong bhi vidt ndy, chtng t6i xin trinh bdy vdi c6c ban m6t crich nhin mQt d4ng to6n qui tich trong kh6ng gian, vdi hi voirg ring n

m4t phing IBCD] c6 dinh, n6n ta d1t dorin ngay, ring (theo 1i hrdng cria nhan x6t 3) K pE chay tr6n drrdng trbn (ho4c m6t phdn cria drrdng trbn)

Ddu ti6n ta hdy x6t hai qui tich cd ben sau

ddy, mi h6n ld c6c ban khdng thd ndo qu6n :

;;;"s --tfh i;B

(trong dd AAt L tBCDI). Ta cri ngay ldi giAi bii to6n.

Ldi gid.i : Phd.n thuQn : Drlr.gAAt L {BCDI,

l. Qui tich co bdn 1 : Qui tich nhitng didm ll4 (trongmat ph&ng) nhin doanA-B cddinh drr6i m6t gcic vuOng la dudng trdn dtidng kinh,4B. 2. Qui tich co bdn 2 : Trong kh6ng gian, qui tich.nhtng didm M nhin doan AB c6 dfnh du6i mdt gcic vu6ng li mat cdu dtrdng kinh A3.

I

khi dci do BK tAK BK L AAt (iAAt L IBCDI) l+BK

L IAAFI + BK t AlK.Didm K lu6ri nhinA,,B cd dinh du6i 1 gcic vu6ng n6n K chay tr6n dtidng trbn dudng kinh ArK (dudng trdn ndy nim trong mat phing/BCDl cd dinh)

Gi6ihqn: KhiM= CthiK=KrvdiAK, t BC I

7{chuydn d6ngft6n KrK,

Phd.n dd.o; Ldy didm Kbdt kj'tr6n K1K2,

n6i BK cdt CD t4i M. Ta chrlng minh : AK t BM. Thuc vAy, do K e KtK2+

BK LKAt,DgMt LIBCDI + AA, tBK YQy BK t [AA,K] + BK t AK inay

BM L AK (dpcm)

3. Nhq.n ref ; Ndu didm M vta nhin AB cd dinh du6i m6t gric r,'udng, vrla nam tr6n mat phing cd dinh tPl di qua B th\ M thu6c dudng trdn dttdng kinh AP. (Trong dci A, In hinh chiSu vuOng gcic cta didrn A l6n mAt phirie [P]). ' Thrrc viy : Do A c6 dinh ; [P] ld m6t ph&ng c5 dinh -didmA, c6 dinh. Mat khdc, do : AA, r [P] lA, t MB) AM t MBI

Kdt ti,n : Qui tich cria K ld cun g @K, (MAt

phdn cria dudng trbn drrdng kinh ArB)

ndn : | + MB L lAAtMl, do dd : MB t MAl. DidmM nhin doanArB c6 dinh dudi 1 gcic vu6ng, d6ng thdi thu6c mat phing c6 dinh [P] n6n nri thudc drrdng trdn drrdng kinh ArB, dtrdng trdn ndy nim trong mat ph6.ng [P] cd dinh.

Vi du 2 : Cho hinh l{p phrrong ABCD . Apppl tdm O c4nh a. M ld mQi didm chuydn ddngtrongdo4nAB. Tim qui tich didri }/, chAn drrdng vu6ng gdc ha tt didm C xudng drrdng thhngMO ?

Bay gid chring ta sE v6.n dqng y trrdng cria nhAn x6t III vio vi6c nhln vh d{ do6n 1 dang torin qui tich trong khOng gian.

(Trich d6 sd 103 cAu V - BO d6 thi tuydn

sinh) 1990.

vi dq -I ; Cho trl di6n ABCD vh M ld m6t didrn di d6ng trong doan CD. Drrng AJ( vu6ng gcic vdi BM. Tim qui tich cric didm K ?

Dga vio nh{n x6t fl ta thdy didm cdn tim qui tiehH ludn nhin do4n"OC e6 ainn drrdi mQt gric vu6ng, d6ng thdi nci nlm tr6n m4t phing IABCpi c6 dinh n6n ta du dorin : ]1sd ch4y

O bai toan niry, nhin vdo hinh v6 chdng ta nhAn thdy ring didm K luOn nhin doah AB c6 dinh du6i mQt gric ur6ng d6ng thdi n&m tr6n

(Xetn tidp trang 11)

t2

CAC SO RAMSEY

HOANG Ci-IUNG tTP Hb Cht Minh)

Tax6t cricbii toanbidu khddon giAnsau ddy. Biti todn 1 - Hai uo chbng mdi bqn d+t nt|t

bua fiAc ; ffong sd khdch miti,

th6a (xem hinh 3o) ; cbn ndu 3 ngUdi niy ttng d6i kh6ng quen nhau thi (b). drroc th6a (xem hinh 3b).-(trudng hgp A chi quen vdi nhi6u nhdt Id 2 ngudi thiA nhong quen udi it nh(it lit clttdng nZt-li6n trdn c6c hinh i neudi "ai,hung 3o,-b dtroc thay baag drrdng n6t drit. nh1118 du'dng n6t drlt drtoc tlray bing drrdng n6t li6n vd (b)hoac (o) duoc th6a). Bii iorin 1 di dttgc giei. Sei phAi tim li r = 6' Bdi trun 2'Nh.ttbdi trun l, chi thay (bl boi h') : (b') bit uo mudn cd it nhat 4 ngudi tilng dii khOng quennhau. Ho phdi mitiitnhdt ntdy ban dd (a) hoQc (b') duoc the ?

Ta s6 chrlng minh rdLng s6 phAi tim la 9,

nghia ld :

(i) ndu chi 8 ngrrdi thi cti thd chi ra tnrdng

(a) \ng chbng m.udn c6 it nhdt 3 nguiti tilng

hgp md cA (o) l6n-(6 ) khOng dudc th6a ;

(ii) ndu cti 9 ngudi thi luOn th6a dugc (o)

Gidi GiA sii c6c b4n (A,-8, C, D, E, ...) drloc mdi

hoac (b').

.1

dOi quen nhau ; b) bit, uo ntudn c6 it nhdt 3 ngudi tilng dOi kh6ng quen nhau. Ho phdi miti it nhd.t bao nhiau ban dd nlong mudn cia chbng (a) hoQc mong m,udn c{ra uo (b) duoc the ?

,f

L Ilitth I

oQ

Thuc viy : (i) Ndu chi cd 8 ngudi thi c

(ii) Trong tnrdng hq'p c<, 9 nggdi, ta x6t hai

Iltnh I

Ta s6 chr3ng minh ring s6 bau drrgc mdi it

- C6 m.Qt ngudi (A) queru udi;t nhdt 4 ngudi

ngdi quanh mQt bin trbn (hinh 1) ; ndu hai ngrlbi (A va C ch&ng han) quen nhau thi ta n6i,4 vdi C bing mOt dui,ng n1t libn ; ndu hai ngudi (A vit D) hhdng quen nhau thi ta n6i A uoi D bdng rtadt dudng net dit. TrOn hinh 1, A quen Iren nlnn r, r'r quen vdi B, C, E vi khOng quen v6i D, -F'. nhdt phAi li 6, nghia lA : hqp ci (o) ldn (b) khong dugc th6a;

(i) Ndu chi mdi 5 ban thi cii thd xdy ra tnldng ' (ii) Ndu mdi 6 ban thi ludn th6a drroc (o)

c

A ,

hoac (b).

khA nang cd thd xdy ra :

tttlh s

D

ThAt vdy: (i) Gia st chi c

hhdc (8, C, D, E,hinh 5). LOc dci ndu trong4 ngrrbi ndy cri hai ngridi quen nhau thi (o) ' dudc 'th6a, cbn ndu khOng ai quen ai thi (b') drroc th6a, dpcm. - M6i nguiti (trong 9 ngudi) chi quen udi nhibu nhd.t lit S nguiti khdc. Thd thi Phii c

(ii) Xet khi 6 ban (A, B, C, D, E, F) duoc mdi. GiA srl A quen udi it nhdt ltr.3 nguiti (8, D, E) ; kie dci, ndu c

I

B

t1

.F

,6

.a

Htnh 3

Theo biti tod.n 7, trong 6 ngudi niy c

.Hr"

Htnh 6

13

3.9

Bli torin 2dddttoc giAi, sd nh6 nhdt phAi tim

ldr:9.

n) phtic t4p hon rdt nhi6u. Cho d6n nay, ngudi ta chi mdi tim drroc rdt it s6 Ramsey : r(3, 3) = 6 r(3, 4) : I r(3, 5) = 14 r(3, 6) : 18 r (3,7) = 23 r (3, 8) = 28 r (3,9) = 36 r (4, 4) =18

Tr6n ddy li tnrdng hop don gi6n cria l6p cdc bii to6n do Frank Ramsqt nghiOn crlu nim 1930, vd nay drtoc ph6t tridn thdnh m6t dd tdi l6n trong li thuydt Graph.

Sd r (3, 8) = 28 drroc Mc Kay vi Min tim thdy nam 1992. Ngudi ta chi xric dinh drrgc cQn tr6n clia moi s6 Ramsey v6i dinh li Erdoas - Szekeres sau dAy:

Ta x6t bii to:in tdng qu6t c&a hai bdi tcr,5n tr6n. Biri todn - Tim s6 nguiti {t ruhtit d.6 trong hg lu6n c6 m nguiri tilng d1i quen hhau, hoQc n tlgudi tilng dOi hhdng quen nhau.

Sd phii tim drroc ggi la sii Ro nlsey r (m, n). d nai toa, 1 thi m : n = 3, cbn d bdi to6n 2 th\ m. : 3,'n = 4. Ta dd chrlng minh drrgc ring r (3, 3) : 6 vd r (3, 4) : 9. Tuy nhi6n v6i nhtng gi6 tri l6n hon c&a m. vi z thi vi6c x6c dir,h r (m,

Vdi mqi sd nguyAn m 2 2 ud. n >- 2, ta c6 : r (m, n) < r (m - 1, n) * r (m, n - 1). Ching h4n : r (4, 5) < r (3; 5) + r (4, 4) : 32 r (4, 6) ( r (3, 6) +r (4,5) < 18 +32 = 50 .1.

cAcn NHIN wrOr DANG. . .

Liti gi&i; Tim qui t{ch E (ViQc tim qui tich udng tU). DUng

(Tidp theo tgng 12)

tiwL' I t

\/__

tr6n drrdng trbn (hoac mQt phdn) dudng kinh OC, (voi CC, t ABCpi)t; cti ldi gi6i bAi to6n.

i I,t- i A+--\ -

o

\

Liti gid.i; Goi C, ra glao olem cua CB, ud C,B.

OH t [SAB], d6 chrlng minh .Ff ld didm cd dinh. Do OH t ISABI + OH -l- S,E. TheogiA thi6t OE r SE. VQy SE t IOEII| + SE tEH.Enhindo4n Slf cd dinh drrdi 1 gdc vu6ng, ddng thdi nim tr6n m4t phing

/V

CHt L ACi

drtdng kinh S.F/ (trt hai didm S vd.Ff), drrdng trbn ndy nim tr6n tS ; (d)l c6 dinh. Phd.n dd.o; C6c ban tu lim. Kdt luQn; Qui tich E li drrdng trbn drrdng kinh S.F/, trt S vd f/. Dd kdt thric bdi b6o, xin (drra vio mdi cric ban thrl dirng crich nhin tr€n y tudng crla nhin x6t 3) dd $hi ac bai tQp sau : Biti tQp I : Cho tarn g76c d6u ABC vi S ld didm chuydn dQng tr6n drrdng thhngAx vuOng grfc vdi m4t phing IABC\ DUng CD t SB, BE t SC a) Tim qui tich c6c didm D vd E I

Didm-Ff ludn nhin OC, ed dinh drrdi m6t g

dudng trdn drrdng kinh OCr).

b) Tim qui tich I/, truc tdm tam gi6c SBC Bdi tQp 2 : Cho tam giSc ddu/.BC c4nh o c6 dinh vd cdc nrla dudngthingBr vd Cy vudng gdc v6i m{t tABCl vd d v6 cirng rnQt phia d6i vdi mat phing dy. Hai didm M, N chuydn d6ng tr6n Br viCysao choBM *CN:2k(k- chotnt6c)

a. Chrlng minh mat \AMM luOn di qua 1

drrdng thing cd dinh

lu/\ /z'\ / rt F---___j/( DoAB trcccrnll+AB Lcclkacc, t tsc 1 ttt d6 : cc' t IABCF I + CCt L OH,M4L l

Tim qui tich didm If.

t4

b. Goi O li.trungdi6mBC. DUng OH r MN. Phd.n dd,o; Mdi cdc ban trr lim. Vi d4r J ; Trong m6t phing [P], cho m6t didm O cd dinh, m6t drrdng thin&(d) c6 dinh kh6ng di qua O, mOt gcic vu6ng rojr quay quanh didm O, cric canh Ox, Oy cilt (d) theo thf tu tai A vd B. Tr6n drrdng thing vu6ng gdc v6i m6t phing [P]'va di qua O, ldy mQt didm S cd dinh. Drrng O,E I SA ; Otr' I SB2{!m qui tich circ didm,E vdF khi gdc vu6ng xOy quayquanh O ? (Trich d6 s6 136 - BO dd thi tuydn sinh 1990)

srrY NcHi vb ...

' (...tidp theo trang 11)

mQtt ud .ta phd.i c6 'sr/ ddi xing giila ft(D bfi{*) - il uit f1@)t/l,@ - 1l ff6i i * i). Cudi cirng, fii da tim duoc biri toan tdng qud.t nhu mong mudn ld.:

Blri to6n 4: Gid.i phuong trinh :

tt

BAi todn 3 : Gidi phuong trinh .: ztft (x) + fz(x) l fj(x)) : 111x1.J2@) + f1(x).qh@) + f2(4. ah@ +f2(4,J3@) +fj 1x1. af1@) +1j 1x).: afral

(ru-1)

lr'r.l = ft(x)

t-

I

2 rJ,Et - Ji@l + i= I

Vdi a lit 1 sd duong khd.c 1. Cdt liti gid.i hoir.n tod.n tuorug h.t rlhu ffeft. . Nhu uQy, dd gid,i d.uocbdi tod.n khi md rQng

+frrr)>d@)-6fz'i)1 +...+ f,(x)p@ - ot' @)1

i=1

Vdi a lit m|t sd duong khdc 1.

f(r) + g(x) thdnh ) no thi khi bidn d.di ub

i= I /

mOr euA rRUNG vANc. . . (Tidp theo trang B)

n$rdi ta crings6m nhQn ra ringchinh tinh chdt cdc m6i quan h6 gita cac sd vi cdc phdp tinh mdi d

NCT

giA, ctng gi6ng nhu dudng th&ng tr6n mlt cdu thdt kh6ng cri hinh Anh li nhtng sqi gid.y mdnh carug thdng trong hhdng gian md c6ng thing tr6n mQt m4t (cdu grA hay cdu thAt). Tt dd, ngUdi ta mdi nhdn ra ring trong khi suy di6n to6n hoc, nhtng tinh chdt nhrr "mAnh", "thing" chA dirng ddn d dAu c6 mi chi nhfrng tinh chdt ,,hai v6 quan h6 girra didm vi dudng thing nhU : didm xric dinh m6t drrdng thing duy nhdt', m6i dtng ddn ; da thd thi thidu gi nhting lo4i ddi tuong cd m6i quan h€" hai ddi tudng loai nly xric dinh duy nhdt mQt ddi tudng loai kia" nhu : "hai m{t cdu brin kinh R x6c dinh duy nhdt m6t m6t tru trdn xoay b6n kinh r? ngoai tiSp hai m6t cdu dd" hoac nhrl "Hai nghi6m cria m6t phrrong trinh bdc hai xdc dinh duy nhdt phtiong trinh dd, li lai mQt h6 sd ti 16 kh6c khong". Cdng di s6u vio hrrdng suy nghi mdi ndy ngudi ta cling tim ra nhrrng thd hi6n eu thd kh6c nhau ctra hinh hoc LObasepski nhrr thd hi6n cta Felix Klein trong dri m6i dudng thing thd hi6n thdnh doan thing, thd hi6n cria Henri Poincar6 trong dd m6i dudng thing thd hi6n thdnh mQt cung trbn. Trr tudng niy cflng trdn sang dai sd vd

15

Ki THI ot EwIPlc roAN eudc rff (IMo) LAN THU 37

PHAN DIJc cHiNu - NcuvEN rHAc rr,ttNrt

(b) Chfng t6 ring nhi6m vu dci thrrc hi6n

dttgcndur=73.

:97?

IMO ldn thrl 37 drroc td chrlc tir ngiy 517 ddn ngiry 171711996 tai Mumbai (Bombay) - thir phri bang Maharashtra cria nu6c CH An DQ DU thi ki ndy c6 426 hoc sinh cliua 75 mi6c. DQi tuydn Vi6t Nam, g6m 6 hoc sinh (4 hoc sinh l6p 12 vd.2 hoc sirih ldp 11) : D6 Qudc Anh, Nguydn Thdi Ha, Phq.m La Hi.rng, Ngo Dar Tu(in (l

Biti 2 (Hinh hlc phang, Cq.nada): GiA stt P ld m6t didm trong cta tam glec ABC sao cho : trE- frE = fic- Gc. GiA srl D, E ldn luot lir t6m cac hinh trbn nOi tidp cta c5c tam gTdcAPB vdAPC . Chrlng minh ringAP, BD, .CE ddng qui.

Bd.i 3 (Dqi s6, Rumania) : Gqi S : {0, 7,2, 3, ...] la t4p hgp tdt cdcirc sd nguy6n kh6ngAm. Tim tdt ch cdchim x5c diuh tr6n S vd cci gre tri thuOc S sao cho :

f(m +f(n)) = f(f(m)) +f(n) vdi mgi ne, ra thuOc S.

So v6i ki thi IMO nam 1995, thd thrlc td chrlc cria ki thi ldn mdy cci mdt srla ddi nh6 : Sau ngiy thi thd n]ndt (101711996), t?it cit c6c phci dodn duoc drla tdi noi d vi ldm viQc cria cric trrrOng dodn. Kd tr) dd, cung nhu c6c tnrdng dodn, tdt cd cdc phci doin bi c6ch li tuyQt d6i v6i hoc sinh cho tdi khi Ban td chr?c (BTC) ki thi da quydt dinh xong giAi thudng.

Nsdy thi thti hui. 1117.11996

Biti 4 (Sd hqc, Nga) : a, b ld' nhtng sd nguy6n drrong sao cho c5c s6 15a * 16b vd 16a - 15b d6u ld binh phuong cria nhtng sd nguy6n drrong. Tim girn tri nh6 nhdt cta sd nh6 nhdt trong hai binh phrrong dy.

IMO ldn thf 37 duoc chinh thrlc khai m4c virc 917 tai H6i trudng cria ViQn nghiOn crlu cd bin Tata. Tt ngay 6/7 ddn ngiy 8/? li kho6ng thdi gian ddnh cho cric trrJdng dodn c'fa BTC so{n thio dd thi chinh thrlc cria ki thi. Phrrong thfc ldm dd cria ki tlri nam nay li su kd thta phrrong thrlc lim d6 cria ki thi ndm ngo6i vd cci cAi bi6n d6i chut nham giAm b6t su ceng thing cho c6c tnldng doirn md v6n dAm b6o dugc tinh kh6ch quan trong viQc llta chon vd drlnh gi6 mrlc dO d6, khd c:ioacdcbdi torin dU tuydn. Dudi day li Db thi cila IMO liin thit 37 :

ABCDEF ld mQt hinh luc giric l6i vdi AB song songv6i ED, BC songsong vdiFE viCD song song v6iAL Ggi Rn Ro l?u ldn lrrot li bdn kinh cira cde drrbng trbn ngo+i tidp cActarn gSdc FAB, BCD, DEF vd gSip 14 chu vi crla hinh h1e giric. Chfng minh :

Ngdy thi thti nhdt. 101v11996

(c) NhiQm vu dci thrrc hi€n dttoc chingn6u r

Bdi 1 (Td hqp, Phd.n Lan') : Chci hinh chtt lohdt AtsCD vdi lABl = 2O,lBCl = 12. Hinh chit nhAt dri drrgc chia thinh 20 x 12 hinh vudng don vi. GiA sil r ld m6t sd nguyOn drrong cho tru6c. M6t budc chuydn ddng xu tt m6t hinh vu6ng.niy sang m6t hinh vu6ng kh6c ld drrqc ph6p khi vi chi khi khoAng c6ch gSita cde tdm cria cric hinh vu6ng dy bing flr. NtriQm lru cria ta ln tim mQt d6y nhirng brr6c chuydn li6n tidp dd chuydn dugc do-ng xu tr) hinh vudng cd dinh A ddn hinh vu6ng c

(a)xo=xn=0 (b) Vdi m6i i vdi I < i < n, La c6 hoQc xi- xi-t - p ho{c xi- xi-t = -g..Chrlpg t6 ring tdn t4i mQt cqp chi sii (i, j) v6i i < j vit (i, j) * (0,n)saochox,= x,.

p RA.+Rc+Rr >i. Bdt 6 (Td hqpt, Phd.p); GiA sfi D, P, Q lir ba s6 nguy6n drrong v1in > p +g. GiA sil N.o, xp ..., xn li nhtng s6 nguy6n th6a m6n cdc di6u kiQn sau ddv:

Theo qui dfnh cria cac ki thi IMO, thdi gian lirm bdi ctra m6i ngiry thi Id 4 tiSng rudi vh didrn tdi da cho m6i bei lA 7.

* Stl phari loai ld ctra ban tti ch0c

16

(a) Chrlng minh ring nhiQm vU d

sinh bi didm 0 6bdi2, ctng nhrr 6bdi 4. X6t v6 todn cuc, kdt qu6 bni ldm cria cdct}lli sinh d IMO ldn tht137 k6m hon k6t quA ldm bii ctra ca'cthi sinh d IMO ldn tri6c. Ndm nay, trong tdng s6 426 thi sinh chi c

- 35 huy chuong ving cho c1'c thi sinh dat

tt 28 ddn 42 didm.

- 66 huy chrlong bac cho cac thi sinh dat tr)

20 ddn 27 didm.

- 99 huy chrtong d6ng cho cac thi sinh dat

td 12 ddn 19 didm.

Nhu v6.y, day ln ki thi co mrlc didm tdi thidu dd trao huy chrrong vdng (HC\D thdp nhdt tr) tru6c t6i nay.

K6t quA cu thd cria DOi tuydn Vi6t Nam nhu

sau :

TrSi v6i th6ng l0 t?i c6c IMO trudc ddy, bdi 1 ctra D6 thi nnm nay lai kh6ng phAi ld bAi drroc HOi d6ng chon dd (HDCD) d:inh giri la bai d6 nhdt trong 6 bdi to6n thi. DAy ld mQt di6u bdt ngd cria ki thi ldn ndy, Di6u bdt ngd thrl hai, cdn 16n hon nhi6u , cira ki thi nam nay ld bii 5 vd bdi 6. Theo ddnh gia cria HDCD thi bei 5 thuQc loai bdi trung binh (x6t v6 d6 khci, d6), nhung tr6n thric t6bdi to6n ndy dd "h4 guc nhanh" hdu h6t crlc thi sinh. Chi cd 6 thi sinh (2 cria D6i tuydn Rumania, 2 oia D6i tuydn Acm6nia, 1 cira DQi tuydn Hln Qudc vd 1 ctra D6i tuydn Hi Lap) dat didm t6i da d bni 5 vd cci tdi 313 thi sinh bi di6"m 0 6 bni ndy. Trong khi dd, Bai 6, mdc dtr duoc HDCD drinh gi5 ld bdi kh

Giei

Ho vd t6n

Theo ddnh gi5 chung cria cdc dodn thi Dd thi ndm nay gai gric hon so v6i dd thi cria nim 1995, vd d6 thi crla ndm 1994. Tuy nhi6n, cd m6t di6u may min li cA 6 bdi to6n thi n6u tr6n d6u thuQc c5c dangbii khri quen thu6c ddi vdi c6c hoc sinh cria DQi tuydn ntl6c ta.

B2

B5

B6

B3

B4

Tdng s6

o

B1

Huy chrrong ddng

D6 Qudc Anh

n

I

2 7 6 0 0 18

33

Ng6 Ddc Duy

Huy chuong vdng

,.7 t

2 7 5 5

7

Nguy6n Th6i Ha

Huy chrrong vdng

1

1

4 6 0 7

Trinh ThdHuynh

7

2 2 3 0 31 I

7

27

Pham LO Hing

Huy chrrong bac

n

n I

4 I

q4 dl

7 2 0 4

7

Ng6 D6c Tudn

Huy chriong vdng

doat 3HC), Anh, Hin Qudc, Nga (m6i d6i doat 2HC); Belarus, Bungari, Grudia, An DQ,Iran, Ixraen, Nh{t BAn, Singapo vi Ukraina (m6i DOi doat 1 HC).

TrongBAngxdp hang khdngchinh thrlc cria BTC ki thi, vdi tdngs6 155 didm. D6i tuydn Vi6t Nam drlng d vi tri thrl 7 sau cric D6i tuydn : Rumania (187 didm), Hoa Ki (1S5 didm), Hungari (167 didm), Nga (162 didm), Anh (161 didm) vd Trung Qu6c (160 didm).

L6 bd mac IMO ldn t]rrn 37 da drrqc td chrlc vdo chi6u t6i nghy 16lT.Tqibudi 16 ndy, Trudng BTC IMO ldn thrl 37 d6 chuydn giao 16 cd ctra IMO cho Tnidng dodn cria Dodn Achentina - nrl6c chri nhi cira IMO ldn thf 38.

X6t ri6ng vd HCV thi c6c D6i tuydn Rumania vd Hoa Ki dat nhi6u HCV nhdt (m6i d6i doat 4HC), tidp theo ld c6c D6i tuydn : Hungari, Drlc, Trung Qu6c, Vi6t Nam (m6i dQi

7 2

ciAi tlzfp bii

vrEr sd rnoruG BAN co

Ldi gid.i ctia Nguydn Thi Nhung, 7D, Phi

Thtl, Kim MOn, Hd.i Hung.

Ta dd. bidt cd.ch dibn cd,c s6 tit 1 ddn 16 uir.o hinh uuOng c6 16 6 sao cho tdng cdc s6 crta m6i hd,ng ngang, mdi hd'ng dgc uit. m6i duimg cheo dbu bd.ng nhau (xem TC THlTTf s6 7 (229)119s6). Thi du

13

3 2 16

10

i1

Nhfn x6t. Bqn Vd Thd DuY, 10A7 Trd'n Qudc Tud.n, Qudng Ninh cfi.ng c6liti gidi dilng, nhung tdng cd.c s6 tuanm|i du?mg dbubd'ng 66' Bqn Nguydn Httu Thud.n, 77A3, Ddng Hit', Qud.ng Tri, c6 tdng cd'c sd ffan m6i dudttg chio dbu b?in g 48.

Ini gid.i cia Bili Thu Hd,4A, Minh Td.n, Kint Mdn, Hdi Hurtg (hinh 1).

8 5

t2

t

I

siNH pHtjoNc

1

15

t4

nI

riTU CUI VAO VA DUONG DI MAt khu tridn td'm so db nhu hinh dudi.

Bd,n cit Vua c6 64 O (32 6 den, 32 A trd.ng). Bdy giit chiabdn cd uua thdnh 16 hinhuuOng nh6. (Mdi hinh uu6ng c6 2 6 d.en uit 2 O trang xen ke nhau) ud. ghi cdc sd trong cd.c 6 cia hirth 7 uito 4 6 c&a hinh uu1rtg nh6 tuong ung cia cit ta sE duoc cdch uidt sd trong bd,n cit th6a md.n yau cd.u crta bd.i ra ld. :

4

bang 4

2) T6ng cdc sd uidt trong cd.c 6 trd.ng ciam6i hitrug ngang hoQc nt6i hd.ng dgc dbu 34 uir. cttng bd.ng tdng cdc sd uidt trong cd'c O den cia mdi h?rng ngang hoq.c m6i himg dqc.

3) Tdng cd.c sd uidt trong cd.c 6 nd.m fiArt m6i

duimg chdo ciabitn cd dbu bd.ng 68.

Phdng c6 mui ftn td.l6i ra. Phibn cd'c ban tim girtp xem khu tridn tam t6i da c6 mdy l6i ud.o dd c6 thd c6 d,udng di qua tdt cd cd'c phbng m6i phbng mOt ld'rt ud. tdi l6i ra.

riJ.,.

13

1) Sd cd.c 0 d.uoc uidt cd.c sd tit L ddn 16 dbu

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ci6 2000d Hai nghin tlbng

TTVT NhA XBGD in Di6n Hdng nQp lrru chidu thSng 411'996

Sip chfr tai In tai Nhd In xong vd

ISSN:8066-8035 Chi sd : 12884 M6 s6 : 8BT32M6

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