Tạp chí toán học và tuổi trẻ tháng 8 năm 2015 số 458

xuflT siu rU r goa 2015 sd 459 rnp cni nn xAruc rsAruc NArvr rnU 52 oi rx cHo rRUNG Hoc pxd rHoruc vA rRuruc Hoc co s6 Tru s6 187B Gi6ng V6, Ha NOi DT Bi6n tAp (04) 35121607; DT Fax Ph6t hdnh, Tri su (04) 35121606 Email toanhoctuoitrevietnam@gmail com Website http //www nxbgd vn/toanhoctuoitre a a l ; ' !^ /' ** x &,& * &{ ffi&e 27' e, m*l; ffieei " ] *; HÃY DẶT MUA TC TẠI BƯU CỤC (facebook com/hoang heo 79) HÃY DẶT MUA TC TẠI BƯU CỤC (facebook com/hoang heo 79) a,( t' 'TIt U UG co sd / ///hi g[.]

xuflT siu rU r goa

2015

sd 459

rnp cni nn xAruc rsAruc - NArvr rnU 52

oi\nrx cHo rRUNG Hoc pxd rHoruc vA rRuruc Hoc co s6

Tru s6: 187B Gi6ng V6, Ha NOi.

DT Bi6n tAp: (04) 35121607; DT - Fax Ph6t hdnh, Tri su: (04) 35121606 Email: toanhoctuoitrevietnam@gmail.com Website: http://www.nxbgd.vn/toanhoctuoitre

,(

t'

/.///hi g{p mQt phuong trinh c6 d4ng

ell' u.r1P *r.iFQ=* (voi u, v, w, P, Q ld cdc

bi6u thirc chria An) md ta nhAm clugc c6c hing

i - , , ,.:.

so e,fvdcdc bi6u thuc Po, Qo chtadn th6a mdn:

lu.P^ +v.O^ =

{,luw(x)

le.(Po)' + f .(Qol' : e.P + f .Q

thi ta c6 th€ gi6i PT d6 nhu sau:

D* </F a; \[0: b suy ra a- = P; b' = 9

lu.a+v.b=w

'

lr.o^ +f b'=e.P+f .QGidi (x*) clO tim (a; b), tu cl6 viQc tim nghiQm

cia PT dd cho sE trd n6n tlcrn gi6n hcrn!

Lru j,: Tn (*) ta thdy (**) ludn c6 nghiQm

(a; b): (Po;Zo) Sau ddy ld c6c thi du:

Thi dU l Gicti pltrcortg trinh

lZi;- +i,5,: J"+7 =r*1.

PhAn fich Ta c6:

ft*-t.l+2=x+1

It* - tt' +D = 12 + 4x - x2) + (2x2 -6x +l)'

Nhuvdy e:f:1vd (Po; Qo): @-l;2).

Ldi gidi.|l51 ,{2,r[r-] =o; tr[Zf -6r.+7 =b

Suy ra a2 +b3 = x2 -2x+9 (1) Tir PT cl6 cho ta c6

a * b = x *7 ) o = I a | - b Q) Tl:my vito (1) ta tlu-o c:

Suy ra (4) ktdng x6y ra vd do d6 (3) kh6ng xhy ra.

rr0T rfiil0 pil.tl0il0 rniilil u0* ulr utr

VUHONGPIIONG(GY THPT Ti€n Du l, TiAn Du, Bdc Ninh)'VOi b:2,thay vdo (2) dugc a : x - l.

Vay PT d5 cho c6 nghiQm duy nh6t:, ='*,J' .

Thi du 2 Giai phunng trinh: LvF+20,t'-86 *

Tir PT tH cho Ia co' a * xb : 3x + 2 > a : 3x + 2 - bx.

Thay vdo (1) ta dugc:

- Yoi b - 5x2 t^4x -3 thi ta c6

x'+3 t;, , 2 5x r+x-t

' Voi b : 1, thay vdo (4) ta du-o c a = x - ! Suy ra

Nghre* (x,y) cira (I) vd @) ctng li nghiQm cira hQ

,lx2 +y2 +2

Do mz + 1 + 0 voi mgi m n)n PT thf nh6t cira

he lu6n c6 nghiQm Suy ra he PT il6 cho lu6n c6

nghiQm voi mgi rn.

' Gi6 sri (xo ;yo) ld mQt nghiQm cria hQ PT, ta c6

[*o -, =(yo -4)*

Lyo -t =(3-xr)m

(1)

(2)

Nh6n cA hai vi5 cua (1) voi 3 - xoi cia Q) va ys - 4

rOi trt theo vti ta dugc

(xo-2)(3 -xo) - Uo- l)00 - 4) = 0

2) Vi c5c tir gi6c BI<Iq vd ABiCt nQi tiiip n6n

tac6 i?i =flei =i6J

Suy ra tft gi6c CKIBtnQi ti6p

,' + u +| =(, - +)" + o + t + |> o + u +l> o.

^?lTuong W b' +a+i>a+b+)> 0 Suy ra

D Tn gih thitit suy ra ?-r * 2y - | : 3xy, do d6

i -*y+f :(*+yf -2(x+y)+t:(x+y-tf .

Mit kh6c, do CAFD ld hr gi6c nOi tiep vn SB ti6p

xtc (O) t4iBn€n SBD = BAC = BDF, suyra

SX ll FD Do 116 LIu/XS @ NIFD (c6 c5c cflp cpnh tucrng img song song).

=9Oo - AEF, suy ra OA I EF OE ddng chimgminh MEF cn LABC; LAFY a LACD, sly

tit ch cdc cti6m nguy6n cta m6t phing

'Ntlu trong ba tlinh cira tam gi6c ABC, c6 hai

dinh, ching hqn A vi B ctng nim tr6n mdttludng ngang hodc mQt

hai lAn di6n tich tam gi6c

ABC lilm6t s6 nguy6n

tong qu6t, gia su cludng

ngang chira B nim gifia

c6c ducnrg ngang chtaAvit

C C6 hai khi ndng xiy ra:

- M|t ld: (h.l) Dudng doc chira B khdng bi kep

gifia hai tluong doc chta A, chfta C.

Khi d6 2S*, :2Sr**, *25** -25*o, -2Sr*

ld m6t s6 nguy6n

- Hai ld: (h.2) Ducrng dgc chria B nim gita hai

<lucrng dgc chr?a Avdchta C Khi d6

25or, =25*r* *25*rra-2Sr* -2Sro, -2Sqec

ld mot s6 nguY€n'

NGUYEN THANH H.NG

(GV trucrng THPT chuyAn DHSP Hd N|i) gidi thieu

OM: ON: 2J5, do d6 theo tlinh li Pythagore

ta c6 OA:4.Y+V M(2; -4), N(-2; -\.

Mit kh5c, do M, N thu6c parabol n€n -4 : a.22

) a- -1 vd (P): !: -x2:

2) DC dfip img ttugc chi6u cao, tru6c h6t xe tii

phii chon phucrng 6n cli vdo chinh gita c6ng

Tren parabol (P) x6t hai di6m H(9,-*) \s 2s) uu

-( 6 a[-i,- 36) ,

25 )cl6i xrlmgnhau qua OyvdHT:2,4

(img v6i chitiu rQng cira xe tii).

phuong Tuong \r a : (a-!+t\' =(o-q^_l\'

\z/\2.t

cf,ng ld s6 chinh phuong

*' b-g+l 22 -b-!-' =1 n6n a vit b ld hai s6

chinh phuorg li6n ti5p

Ciu 4.Yl BE L CA, CF L AB n€n BCEF ldttr gi6c

nditi6p= frE:frE =frF = LXFB cdntaiX

Dft THr rudru srNH vAo r,op ro cmurftu

1qUdruc D^t IIQC sU pH+M Tp, Hd cni mlrunNAiu usc zuts-2016

VONG I (120 phrit)

CAu 1(2 die4.

1 Cho phucrng trinh:

x' -21m-21x+m' -3m+3=O (m ld tham s6).

a) Gi6i phuong tdnh khi m: 1.

b) Tim z <l€ phuong trinh c6 hai nghidmx,vd

a) Cho m = J2 Vc (P) vd (D) trdn cing mot h0

truc toa dQ Oxy vd tim tga dQ giao dii5m cira chtng

bdng phep toiin

b) Tim m d,€ (P) vd (D) cdt nhau tpi hai <li€m ph6n

biet A(-rri )',) r'd B(x2; y) sao cho

nr5u kh6ng c6 gi thay AOi ttri se clu6i kip xe tai tgi B.

Nhung sau khi tli tlugc mQt nria quing ducrng lB,

xe khrich tdng v{n ti5c l6n 60 km/h n6n tl6n B s6m

hcrn xe t6i 16 phft Tinh qudng duimgAB

CAuu 4 @ di€m) Cho tarn giitc nhgn ABC (AB < AC) Ducrng trdn t6m O tluong kinh BC cit AB, AC ldn

luqr tai E vit D CA cbt AD @i H vd AH cEt AC tqi I<.

a) Chung minh tu gi6c BEHKn6i ti6p vd KAldtia

ph0n girlc cua goc EKD

b) Gqi AI, AJ lit c6c tii5p tuy6n cta ducrng trdn (O)

(1, J lit c5c ti6p di6m vd hai diOm D, r nim ctngmQt nua mdt phing bo ld tlucrng thing AIg

Chung minh ring IKE = DKJ

c) Chimg minh ba di6m -r, H, I thinghing

d) Ducrng thlng qua K vd song song vdi ED cit

AB vir CH ldn luqt t4i Q vit S Chrmg minh ring

b) Chnng mirrh ring A:2A124'+ 20134" + 20144'

+ 2}154'khdng phdi li s6 chinh phuong v6i moi

s6 nguy€n duong r.

CAu 3 {I diA@ Chorvd"l,ld c6c s6 thuc duong

thay d6i th6a mdn di€u kiQnx +I < 1 Hdy tim gi5

tri nho nhit cua bieu thi,r, p=( \ *' *'l[r' +'l +)/\ +x/

CAu 4 @ diem).

a) Cho tu.giSc ABCD n6i ti6p ducrng tron tdm O.

Duong th[ng r u6ng g6c v6i AD t1i A clt BC t1i

E Eulng thdng r,u6ng g6c vdi AB tqi A cdt CD

tai F Chimg minh O, E, f thing hdng.

b) Cho hinh thang ABCD vudng tai A vd B, M lit

trung di€m cua AB Ducrng thSng qua I vudng

g6c v6i MO c,it duong th6ng qua B vu6ng g6c v6i

MC tqiN Chrmg minh MN LCD

Cdu 5 Q di€nfi Sau khi di6m cianh xong, 16p

truong tuy€n b6: "Sd cac bqn c6 mqt ld m\t s6 cd

hai chii s6, s6 zoy hd hon 2 tdn fich hai chft s6

cira nd 9 don vi " H6i c6 bao nhi6u bpn c6 m{t?

NCUYEN DTIC TAN

QP HO Ch{ Minh) Stu tim

aanur,r-roru, T?!I#E[

5

,riR NETDEP HIIIIH }IOCTIEM AITI

Ai'lrA'

a

,3f rong Di thi minh hea - Ki thi THPT QuiSc

t-F gia cia B9 GD&DT c6 m\t cdu tim gid tri

nhd nhiit cila bidu thtc khd hay vd kh6 md cdch

gidi di xutit td s* d4ng phuong phdp toa d0.

Bdi vi* ndy duqc xay drng theo liii tidp cdn

trt{c diQn vdi t*ng dqng todn, bdi todn cu thd

vbi phdn tich clinh hudng gidi vd loi gidi tudng

minh Hy vgng rdng sd gitip cdc bqn thi sinh sd

tu tin hon trong K) thi THPT Qu6c gia.

! (;l \l t, t Bt,'l vA IlF P]'

*"-l tri elr,r l tiiui pluxrng lrinh'

L.; L r[ - Jr i - r./;-;l

Phdn tich f tucrng khai th6c yiSu tii hintr hgc bQc lQ

tu v5 tnii PT dugc cho ducvi dary: x.J x + I + 1.,11 -i,

tu d6 ta li6n tu&ng <liin bi6u thirc tga ilO cria tich v6

huong cria hai vecto trong mit ph5ng tqad0 Ory,

Vfy PT dA cho c6 hai nghiQm ld x =1; x =l+4 J

*'l'hi dq 2 Gidi phrrrng trinh'"

r +,,r1-l+.,rf i:-1.

Phdn tfch Y tucrng khai th6c yiSu t5 tintr hqc bQc l0

tu v6 trai PT du-o c cho du6i dpng:

*'f hi rlq 3 Giai hi)t phrco'ng trinh:

', r-l +.r-3 >.[tr-:t' +2,-z (l)

Phdn tich.'f tuong khai th6c yi5u t6 hinh hgc An

chira trong bdi to6n 6 ch6: v6 trai BPT duoc cho

duoi d4ng: ,[A1+(r-3).1, tu <16 girip ta nh6

tt6n bi6u thric tga d0 ctra tich v6 hu6ng cria hai vecto trong m4t phing vdi hQ tga ilQ O4,

Di6u ndy xity ra khi vd chi khi a

hu6ng, hay i=k; &>O)

va y cung

- TOnN HOC

(, - 6fudig,i, sdasstg-zo]sl

VQy BPT dd cho c6 nghiQm duy nh6t x = 5 E

*Thi dq 4 Giai bdt phtrcrng trinh:

v5+1 +,,8.r-z+"'fitt-:r <tz (1)

Phdn tich O6 y, ring vC t ei ctra BPT (1) ld:

1.,1V+1+t.JN-z +1.J50-3r, ncn ta nghi ngay

tcri bi6u thric tqa <10 cira tich v6 hu6ng cria hai vecto

trong kh6ng gian vni hQ tga dQ Oryz.

3 50Ldi gidl DK:

Phdn tich 'f tucrng khai th6c y6u t6 hintr hqc bi€u l0

tu PT tliu cira hQ VC trai cria PT thri nh6t dugc cho

ducri dsng: x.,tnJ +JtI:P.f, * d6 ta li6n

tuong d6n bi6u thric tqa d0 cria tich v6 hu6ng cria

hai vecto trong mflt phing v6i hQ tga tlQ O.rry.

Ldi gidl DK: -2J3 < x<2J1; 2<y <12.

Suy ra ! =12- x2 = 3 (th6a mdn diAu kiQn).

Vfy nghiQm cria hO dd cho ld (x;y): (3;3) tr

*Thi dU 5 Giai h€ phtrong trinh:

(1

)4t +4.t'- +4:'+ l7:tt.L+4.v + l6:

l-lr-+l'+l2z=12

Phdn tich.

Voi PT thri nhdt cira hQ, ta li6n tuong d6n PT mQt

m{t cAu (S) trong kh6ng gian vcri hQ toa d$ Oxyz.

Voi PT thri hai cta hQ, ta nghi ngay ctt5n PT mQt

mflt phdng (P) trong ktr6ng gran voi hQ t9 a dQ O xyz.

Vi€c gi6i hQ PT duo c quy v6 bdi to6n hrong iluong:

X6t sU tuong giao cta (S) vd (P)

' Do khoing c6ch tu tdm lcta (S)den(P) bing

b6n kinh cria(S) nOn(P) ti6p xric v6i (S) Do d6,

he PT da cho c6 nghiQm duy nh6t.

C6ng viQc cdn lai ld tim tga itQ ti6p diiSm cria (S) vn (P).

Ldi gi,rti He PT c15 cho tuong cluong v6i

( r r\2 Itr-l)'+l y-* I *tr-2)'=l

I \ L)

l3x-4y+l2z-12=0

Trong kh6ng gian v6i hQ tga dQ Oxyz,ta c6

PT tht nh6t cira hQ x6c dinh mflt cAu (S) cO

Do d6 he PT tl6 cho c6 nghiQm duy nh0t

NghiQm cria hC PT le tea dO cita ti6pdi6mA gifia m[t ph[ng (P) vd mat cAu (S)

Gqi A ld tlucrng thing di qua tdm 1 vi vudng

lx =t +3rg6cvcri (P) thi A c6phuongtrinh: ) 1r=*-0, .

II CH(TNG MINH nAT DANG THLIC,

rinr crA rRI LoN NHAr vA GrA rRI

NHO NHAT CUA BIEU THIIC.

*Thi dB 6 (DH Xtriii A - 2003) Cho x,y,z ld

ba s6 daong vd x + y + z < l Chbng minh ring

t, l /-, r t, r

-,l*"+ ,+^l!'+1+1lz'+ Yx-Yy-\z' 2 >VEz

PhAn fich, Bing sU "va chpm" vd khoanh vtng c6c

dpng toSn BDT, ta ph6t hien c6c biilu thric d4ng

q'+ F' 6n chta dudi c6c d5u c[n b{c hai, vcri d6u

hiQu ndy y6u tii trintr hoc dAn 10 diQn DiAu cin quan

tAm ld viQc lga chgn toa d6 c6c vecto i, i vi, i

thich hqp, khi d6 voi viQc sri dpng BDT Cauchy cho

tune b0 ba s6 duong( x;y;z),[f - \x f t) Y z) thi gi6 thi6t

Ding thric xiry rakhi vi chi khi * = y =, =! 3'" n

*Thi dg 7 Cho hai sij thqc x vd y kh6ng dm

- -.,1

thoa mdn di€u ki€n x + y : l Tim gid tri nh6

nhh ctia bidu thtrc p =3.[izf +2"{+g+9f .

Phdn tich Nh{n th6y bi6u thric dd cho ld t6ng cria

hai bitiu thric Bing phdp bir5n AOi acrn gi6n, ta vii5t

duoc c6c bi6u thric ndy duoi d4ng @ +ff +y',

aEs6r

(Thdi gian ldm bdi: 180 philt)

Bni 1 (2 di€m) Cho hdm s6

y=x3 -3(m+l)x2 +9x-m (C).

a) KhAo s6tvdvE 0O tfU (q cuahdm s6 khi ru = 0.

b) Tim m d.6nim s6 c6 hai di6m cuc tri x,x, th6a

Bni 3 (0.5 die@ Giniphuong trinh:

1og* ("r - 2) + log, (x - 2) = logzQ - 2x)

Bni6(l diim).Chohinhch6p S.ABC c6 AB=AC=a,

Ga =300, SA L(ABC), g6c gita hai mdt phing

NhQn xdt Hinh thanh dugc tga dd c6c vecto lir chia

kh6a cria cbch giiri ViQc t6ch hai s6 h4ng ctra t6ng

chria trong cdc ddu cdn thdnh ba s6 h4ng girip vi6c

x6c dinh tga dQ c6c vecto duoc d6 ddng.

Binh lu$n Th6ng qua mQt s6 thi du d ren, ta

th6y r6ng v6i m5i bii to6n c6 th6 c6 nhi6u c6ch ti6p

c6n kh6c nhau, ching han nhu: khai th6c tinh chdt

hinh hgc, phuong ph6p hdm s5, vfln dung BDT trung

gian, Tuy nhi6n, v6i c6c bdi to6n niry, cilch giiri

khai thac tinh chiit hinh hqc cho nhirng loi gi6i gon

gdng vd trong s6ng.

(SBC) vd (ABC) la 600 Tinh theo a the tictr klOich6p S.ABC vd kho6ng c6ch tu trgng t6m G cta

tamgifuc ABC d,6nmdtphing (SBC).

Bii 7 (1 dii!m) Trcng mqt phing voi hQ tga tlQ

Oxycho hinh ru6ng ABCD c6 tdm 1(1;4), tlinh

A nim tr6n tlu&ng thing c6 phuong trinh

2x+y-l=0, ttinh C nim tr6n ducrng thing c6

phuong trinh x-y+2=0 Tim tga tlQ c6c tlinh

A,B,C,D cta hinh rudng dd cho.

Bni S (l di€m) Trong lfudnggan voi h€ tqc tAa dQ O xy z cho tli&n I(l;2;3) vd 4Atph5ng (P):4x+y-z-l=O.

Vi6t phucrng trinh m{t ciu t6m 1 titip xric v6i m{tphing (P) vd tim tsa ttQ ti6p rlirim.

Bni 9 (0.5 die@ LQp s5 t.u ntri6n c6 4 cht s6 tt6i

m6t ph0n bi6t tu c5c cht sd 0, 1, 4, 6 Tit:hx5c su6t

dC s6 l6p dusc ld s6 qu nhi6n kh6ng chia h6t cho 4.

Bni 10 Q diA@ Cho 3 s6 thpc ducrng x,y,z thba

-111

m6n - + * + - (3 Tim gi5 tri."nh6 nhdt cira bi6u

^!z -20t5 r ,2015 ,20r5 r -20t5 -2015 t -2015rr.i,n,E-^ T! ,! -t Lz rAtrruu' 1

xtrrt +fw - ,rno4rt*l t ,nw **teot'

NGUYEN VAN XA

(GV THPT YAn Phong 36 2, Bac Nnh)

GDine thric x6y ra khi vd chi khi i vd;'** """;;i;i;"'

1 Giai c6c phucrng trinh sau

hu6ng Di6u kiQn cAn de u vd v cr)ng hudng ld / ^

j jr ?r | ) a)Jx-2-J4-x =x2 - 6x_ 11:

12 4 6y - J 3

2 Giai c6c briLr phucrng trinh sau

ti6ptath6y p=s./iT khi x=l ui, y=? a)JP +2x+iri-tJiPTax.+t;

4 Cho x,y,z ld ba s6 thqc d6i m6t kh6c nhau.

Chimg minh ring

H ld hinh chidu cfia G rAn BC; HE,HF,HG

lin fuqt cdt dudng thdng qua A song song BCtei X,Y,Q thi O ld trung di€m W vdHX= HY .

Y AQ X

BHDHinh IChtimg minh (h.l) Gii su EFnBC =R ta c6hdng ditim di6u hda co b6n (BC,DR)=-1

chi6u xuy€n tdm A l6n tludng thing EF ta co

(EF,GR)=-l n6n chirm I1(EF,GR)=-1, lai

c6 HG L HR, n6n tu tinh chAt chr)m phdn gi6c

suy ra HG liL ph6n giSc oia fFF V{y trongtam gi6c HW c6 HG = HQ ld duong cao vit

ph6n gi6c, do cl6 tam gi6c HW cdn tpi 11, suy

ra Q ldtrung dii5m W vd HX = HY .

gO AC sau dugc dat tCn ld E.R.I.Q boi kiiin tnic

su Hy Lqp Kostas Vittas ld vi}t tit c6c cht c6i

dAu cria cpm tu titing Anh "Equal Ratio InQuadrilateral" tqm dich ld rj' s5 bing nhau

theo cilng mAt ry so sd thdng hang.

Tdm tit Bdi vi6t ndy tlua ra tdng qu6t cho mQt

bdi to6n hay clugc nhidiu b4n dgc quan t6m tr€n

Tpp chi To6n hqc vd Tu6i trd v6i phdp chimg

minh th6ng qura tinh chdt chilm diiu hda vd img

dl;rngbd ai e.nl.Q,th6m vdo il6ld mQt vdi img

, i

dgng cua bar tong quat.

Trdn T4p chi TH&TT sO +OZ thhng 12 ndm

2010 trong myc Di ra k) ndy c6 bdi toSn hay

nhu sau cria TS Nguydn Minh Hd

QBii to6n 1 (Bai TL)1402" TH&TT thring l2

nanr 2010) Cho lam giut: nhon ABC dtu)ng c:ao

AD M ld ntot di1m thuQc dcten AD Cic' du'd'ng

thdng BM, C'M rheo thtlr tu cfu AC, AB tcti

E,F:DE, EF theo thu' tu' cdt c'dc dtrit'ng trdn

dtrd'ng kinh AB, AC tin lu'ot tqi K, L" Chu'ng

:-.

nrirth rarrg dtro'ng lhdn,q noi trung tliAm t't)a EF

KL di quo diOru A.

Loi giii bdi to6n tr€n itd <l6ng tr6n T4p chi

TH&TT sO +OO thing 4 ndm 2011 Sau mQt vdi

tim tdi nh6, chring tdi nhfln th6y ring bii to6n

tr6n ld trudng hqp ri6ng cira bdi to6n sau:

ilBii tofn 2 {'ho lant giar: ABC' t,c\ P ld ,liint

,L",,_

hit k' tt'ong nlat pltarry PA, PB, PC theo thi'tu

c'dr BC,CA,AB rqi D.E,F.PA citt EF tai G tI

lit hinh c:hiiu t't)a G trOn BC; HE, HF tin tupt

c:at dru)'ng lrdn ngoai riip c:oc' latn giat'

I.AE HAC tai M, lti khac H; HG cdr &rd'ng

thring qua A song song ttri'i BC' tcti Q Chilng

minh ,orrg du'dng thang nrii trung Jic'nt

M N EF tli cJttu diim Q

DC gi6i bdi to5n ta su dr,rng hai b6 dC sau.

g6 Oa 2J Cho tam gidc ABC vd P ld di€m

biit k) trong mfit phdng PA,PB,PC tdn luqt

cdt BC,CA,AB tqi D,E,F; PA ciit EF tqi G

g5 OC ta tiSt qua co bdn, bpn dgc c6 th6 tham

kh6o nhi€u c6ch chimg minh trong nhi6u tdi

li€u kh6c nhau Trd lai bni b6n2.

Ldi gidl (h.2) cia st AQ cit <luong trdn ngo4i

ti5p c5c tam gi6c HAB,HAC lin luqt tai S,I.

HE,HF lAn luqt cit ,l,g tai X,Y Ta th6y

A,S,H,M thuQc ducrng trdn (K) ngoai ti6p

tam gi6c AHB vd A,T,H,N thuQc <lucrng trdn

(L) ngoai ti6p tam gi6c AHC n€n

Y'S.ru = YJ,I .fiI vit W.Ve=Ytt.Yn, suy ra

Vi tu gi5c ATCH ld hinh thang cdn vd c6 HQ

ld clucrng cao n6n fA-CH =zQA Tti <16:

E-=:,* XI )A+CH =:-4: ){A+TA-2QA

Tri tl6, cfing theo UO 0A 2.1 vd UO Ae 2.2 thi

trung dii5m Q cin XY vd c6c trung clitim ctra

<ldy chung tdi xin ddn ra mQt vdi vi dg.

f,iBii toSn 3 Cho tam giirc ABC vd P la ,lia'nt

btir tq, PA,PB,PC lin ltrer c:dt BC,CA,AB toiD.E,F' GOi H td hinh chi€tt c:uu D lin EF

HB,HC ldn lmll <:dt cdc dudng tritn ngoai tiitrt

tan.t giac: AttE,AHF tai M, li; Q td hinh chiitr

t'rio A lOn HD Chirng ntinh riing dtrinrg n6itt'tt)lg Lli['nt t tra BC vu MIV cli qrta dienr Q.

ID

Hinh 3Ldi gini G.3) Gqi 7, S hn luqt ld giao di6m cua

HM,HN vuAQ Ap dUrrg bdi torin 2 vio trudrng

hqp P b6t ki cua tam giSc AEF vor EB, FC vd AD

tl6ng quy t4t P tathu duo c tti6u phii chimg minh

Ho{c c6 th6 chrmg minh quc ti6p theo c6ch khSc

nhu sau:

Ta th6y theo tinh ch6t ctrtm ilidu hoa tbt HQ lit

phdn gi6c ci:a SHT md HQ 157 suy ratarngiircHST ciln tai H Do d6 Q lil tung ttitim 57 vdHST = IIIS Vi EF ll STn€n lN^t:180'-ANH

=l30" -frfu =fu Do il6 ASAN a MBA

=,, nur,r-roru, t?8ilrHEE

1 1

Chimg minh tuong tu c6 SA.TA=TM.SC suy ra BC,CA,AB ldn ltqt tgi D,E,F; H la hinh

chi€u ctia D l€n EF; HB, HC theo tha tq clit cac

./

dtro'ng tron ngogi tiep tam giac HAE, HAF tqi

M, N khac H Chung rninh rdng trung didm cua

hinh cht nh6t, suy ra AS li phin gi6c ngoii dinh A ctra tam gi6c ABC n€n ,4S ll EF Gqi

K,L ldn luqt li giao di6m AS v6i HM,HN.

VAn su dgng md hinh bdi to6n 3 ta suy ra.FI,S ld

phAn gi6c cua KHL , md DH I KL suy ra tamgi6c HKL c6n tai g Vdy S ld trung tli6m

KL Ap dpng bii tofun 3, ta c6 S,T,P thinghdng vd d tr6n c6 S,T,R,Q thing hdng Tn d6

suy ra P,R,Q thing hdng.

OBii to6n 6 Cho tam giac ABC vdi trung

tu,v€n AM Cac didm E, F ldn lu'qt thuQc CA, AB

sao cho EFll BC H td hinh chi€u cuq M bn

EF HB, HC ldn lwqt cdt daong tron ngogi tidp

tam giac HAE, HAF tai K, L Chu'ng minh

rdng HK = HL.

TM.SC: SN.TB

ai Z.nl.lta c6 tli6u phii chimg minh

NhQn xet ViQc 6p dung bdi toiln2, bii torin 3 vd

m6 hinh gi6i cira bdi to6n 2, bii tobn 3 vdo c5c

tam gi6c kh6c nhau t4o ra nhi6u bdi to6n moi ho[c

U6 AA moi kh6 thil vi Ta tltin mQt vi dir kh6c 6p

dgng bdi todn 3 nhu sau:

QBii to6n 4 Cho tam giac ABC voi cac dudng

cao AD,BE,CF K ld hinh r:hi€u ctiq D bn EF;

KB,KC lin lactt cdt cac dtrrng trdn ngogi ti€p

tam giac KAE,KAF tqli M,l{ khac K Gqi P, Q

tdn taqt ld trung diAm MN^ BC' Chrtng minh

rting PQ vd DK cdt nhau tAn fi€p tnydn tui A

cita dtrdng trdn ngoqi fii\p um gidc ABC

Hinh 4

Ldi gidi @.a) Gei S,R'lAn luqt ld giao ili6m

KM,KN vdi tit5p tuyi5n tpi A ci,r- tludng trdn

ngopi tii5p tam gi6c ABC Chtmg minh tu.rng t.u

bii to6n 3 suy ra DK ld phdn gi6c 6Ei Lqi

, , |,,

c6 ket que quen thuQc ld EF ll RS, n6n

DK IRS md DK lai ld ph6n gi6c ffi suy ra

tam gi6c KRS cAn tai K Gqi I ld giao diiSm

DK vd R^S thi 7ld trung di€m RS D6n tl6y ta

th6ry ngay mO hinh cira bdi to6n sl5 3 ld trung

dii5m ctra c6c do4n thing MN,BC vit di6m 7

,.:

tnang nang.

Sau cldy ld mQt img dr,rng tlgp kh6c

OBdri to6n 5 Cho tam gidc ABC, drd'ng trdn

wsx&wm mAqr

Drfgc sg d6ng f' cria Cgc 86o chi - BO Th6ng rin vd rruy6n thdng vir

ndm zors, m5i sd rqp chi To6n hgc vh rudi rre r6ng tir sz ftang hiQn nay

l6n ao trang rudt vdi gi6 bia la r2.Eoo C6ngfrs6 vdi 8 rrang t6ng ndy,

nhidu chuy6n mgc nhtrz Tim hidu sdu th€m Toiin hpc phd th6ng, Toiin hgc

vd ddi sdng, Sai tdm d ddu, Cdu lqc bO T-oiin hec, s6 duec xudt hiQn

thrldng xuy6n tr6n Tap chi, nQi dung c6c chuyOn mgc sC ddy dfi vh phong

phri hon.

ndt mong c6c thdy c6 gi6o, c6c em hoc sinh vir bqn doc y6u to6n fing

hd doc, vidt bdi cho c6c chuydn ffiVC, dd rqp chi ngdy cirng ph6t tridn.

ra AM,BE,CF rl6ng quy Chrmg minh tucrng tu

bdi to6n 3 co HM ld phAn giirc friL vit HM

di qua trung di6m KL, suy ra tam gi6c HKL

cAn tai H Tt d6 HK = HL (dpcm)

C6c bpn hdy citc ldm bdi tAp sau 6p dpng bdi

toin2,3 ve UA di E.R.I.Qdd luyQn tfp

OBlri to:ln 7 Cho tant giat ABC nii tiip drxtng

n'r)n (Ol Dtrd'ng trdn ngoai tiip tttru giac BO{' t ir

AO rqi K khdc'0 Ltil; r'tit'diOm E, F l,-in ltrrtt /!tru)t,

C:A,,18.suo c'!'Nt KA la gthdn gi,i, Lil' KE:^ Kt';

!in ltrrvt cdl t't!,-' dut)'ng trdn ngooi /iep iuui giti<-'

Kt\C, KA.B ioi ,1.,', ,V iiitric' K {'hti'ng mirih rrhtg

r-iia'lt3{ thiug nii lnlt,q ilient r.'rtct 1:1.'" Mii tli t1tru l.

ffi$3iri to*m 8" {.r)t,, iitiii gi'':t 4}}i-'t,ttt f}, J l,i

!;,.ti ilielm dini; gir-it tr"iit 1;irr-'itr gitir"gtit' ftA{' E"

il' j:t itinlt, 'hiitt ttitr f) lcn {'itr ,18 fi lit hiqli

rhiirr t't.rtt {) t0n BC !! lir hitrh t'lticu c'u,i [) l0n

r'r,t /18" tlC cut r-'r.ic ,ltritt,q trort nguai tidlt 71;1111

t:icit [{AE.Ht\l;' tui l,l I'l kltdt' f{ Ckti'ng minhrirtg trung clienr uict ,'d,- cilton thong Bq', ,VIN

vo ,4!{ thdng hitng

OBii tcrdn 9 {jho Nont sitir' 18C t,i'i tt'trc tdut

H P li tJiAru bi)t k.i'trcyi Alt DtN)'ng trdn rtgotri

rit ;t t"'ric tirnt 51icit.' i\Pt] APC lin tu'rt crit C,1^

,4ll tui E, l' tt{}, tlC lin luo,r t'dt tlur)'ng tronngooi iirys tctm gidc AHt',AHE tqi M,l,l kh{tt

tt {'hrlrng minh ring tlud'rry thang n6i cliant

I-t'nroint' t'rttt htri tttnr gir.it' AElr,t\MN di quu

tllr'Ill A"

gOrtrr-rqrt, ryAIr

GIAI PAP:

lot crnt MANG DAM cHAT xV rHudrr

1oi aAng fiAn TH&TT sd 455, thdng 5 ndm 201fl

@ di gi6i cira Long dd sai 0 ch6: Voi BDT

*-/ d(A;(a))< 9J3 , d6u 3'-') xay ra khi

a = b =c K6t hqp voi dlng thric 2a+ b+2c =O

suy ra a: b : c : 0, mdu thuSn v6i giA thi6t

a'+bz +c'>0.

Ldi gi,rti drtng hz

Ta c6 vecto chi phucrng cua d ld i=(Z;t;Z).

Ldy M0;0;2) e dvdgsi fi=(a;b;c)

(a' +b' +c' >0)h vecto ph6p tuy6n cira (cr)

Do mp (ct) chria ttuong thing d, n6n ta c6

NhQn xitz C6 hai ban c6 dilp in tlirng ld: Hir NQi:

Dinh Vdn Anh,l2Al, THPT Hd Xu6n Huong, Sli I

Nguy6n Quf Eric, qufln Thanh Xudn NghQ An: Hi

Thanh Tilng,l2C1, THPT Kim Li6n, Nam Ddn.

HOANG CHr(HdN1i)

KET QUAE.4 DUNG CTIIIA?

flr) 1i6n tuc tr6n dopn [-t;t] n€n dpt gi6 fi lon

rhit, gi|tri nh6 nh6t trCn tlopn ndy oO thi cria

hdm lr) ld parabol quay bA l6m xu6ng phiadu6i, c6 hodnh d0 dinh

2 _ -1 [-t'tl .

x =- z\-D-i't "'l'

vd minA = t{-lrllTir}.,f(r) =minU(-ll;/(t))

= min{-3; 1} = -4.

Ban hgc sinh tr6n tl6ldm dfng chua? CLch gihi

cira bpn nhu th6 ndo?

DAu "-" xity ra Y,hi a : c, md a2 + b2 + c' > O

n€n a = , = -X+ 0 Khi d6 chia hai vti PT (1)

Q nt"g ta dang s6ng trong mQt th6 gi6i th{t

lU phric tap Tu thi trudng chimg kho6n cho

di5n c6c thim hga thi6n nhi6n, moi thir duong

nhu ld kiit que cta nhirng th5 lpc phirc t4p,

kh6ng th€ lucrng tru6c tlugc Nhmg li6u chring

c6 thi5 bit ngu6n tu chi mdt vdi quy tic dcrn

giin kh6ng? Hay n6i.c6ch kh6c, li6u mQt hQ

th6ng tlon giin c6 thO sinh ra nhirng hdnh vi

h6n <lQn, kh6 ti6n doSn duoc kh6ng?

(Lop 12 Tban, THPT chuyAn Hd NQi - Amsterdam) - Phdng dlch)

chri,ng ta nghi sg h6n ttQn ndy se tdo ddi mdi

Chdng phii khi con ki6n di chuy€n trong ludi

h5n dQn, thi n6 chi c6 th6 vE ra nhtng hinh cdn

h6n dQn hon hay sao? Th6 fr*g, sau hcrn

10000 bu6c, <IQt nhi6n mQt quy luQt hi6n ra: con

kit5n vE ra mQt hinh g6m 104 budc <luqc lAp di

l{p lai, t4o thdnh mQt "xa lQ" tr6n ludi 6l.u6ng(h.3) That ld ki di0u!

Hirth 1

Mgt hC th6ng di6n hinh ld mO hinh con ki6n cria

Langton, dugc s6ng t?o bdi Christopher

Langton vho n6m 1984 Trong md hinh (h.1),

r, i.

mQt con ki€n sdng trong lu6i v6 hpn cdc 6

T6rg.Ban dAu t6t cA cilc 6 ildu mdu tr6ng M6i

ldn, con ki6n il6i mdu 6 mi n6 <lang dimg: n6u

ld mdu trlng thi OOI tnann miru den, n6u ld miu

clen thi OOi tnann mdu tring Sau d6, n6u 6 <16

i ,.:

mdu trdng, con ki6n sE quay ph6i 90" vd bu6c

titip, n6u 6 d6 mdu den, ion kirSn quay trSi 90"

vd ti6p tgc hdnh trinh Cflng nhu 1u6i v6 han c6c

6lu6ng, s6 luort di cira con kitin ld v6 han.

Ban dIu con ki6n vE ra nhirng hinh kh6i kh6 d6i

xrmg (h.2)

Hinh 2

Nhung cdmg ngdy, hinh vE cfurg tro nen rOi rirrl

dudng nhu ld ng6u nhi€n Vd theo lE thucrng tinh,

hy*g di, vE l€n mdt ma tr{n h6n loan, nhrmg

r6i thft ki lp, ld "xa lQ" 104 budc ludn hi6n ra,

dd li sau mQt nghin hay .mQt_ triOu lucrt

di-Dudng nhu "xa 10" dy d5 "h0p d6n" con ki€n <16

con kitin dAn ve n6n, b6t ki5 trpng th6i ban rtAu

cua cdc o.

Su diQu ki ctra con kii5n Langton c15 s6m dugcc6c nhd to6n hgc ph6t hi6n, nh.mg thft rl6ngtii5c ld d6n b0y gid, hi6u bi6t ctia chring ta vO

mO hinh 6y vdncdn khd h4n hgp Cohen - Kong

d5 chimg minh tlugc mi6n di chuytin cta con

ki6n kh6ng bf ch[n, tuc ld con ki6n lu6n cli xa

vo han so vdi tli6m xu6t phdt ban ilAu Tuynhi6n cdu h6i v€ tinh t|,t y3, cira "xa 16" v6irchua c6 ai tr6ldi duqc

Tri con ki6n cria Langton, phii chdng ta,s6p d6n

gdn v6i mOt md hinh mi6u ti duoc th6 gi6i ta

<lang s6ng? Vd hon nfia, phii chdng mgi s1r

ki6n, hdnh clQng di6n ra dAu tu6n theo nhirngquy tic don gi6n nh6t cO thiS, md ta chua kh5m

phb ra? CA.u tri loi c6 lE sE ddnh cho c5c ban

clgc vd nhirng th6 hQ mai sau.

*r nur,r-roro T?[I#E[

BitiTZl4Sg (Lop n.Biiit a vd bldcfuc sli thlrc thay

c16i sao cho <ta thtrc A(x)=x2 -2m+2a2 +b2 -5

c6 nghiQm Hdy tim gi6 fi nh6 nhAt cira bii5u thric

p=(a+1)(b+1).

NGUYEN TAN NG9C

(GV THCS Nhon Mi, An Nhon, Binh Dlnh)

Bni T3/459 Gi6 sir a1,a2, ,an ld c6c sd

nguyOn ducrng ph6n biQt th6a mdn

Biti T41459 Cho tam gi6c vu6ng c6,,n ABC

Q4P: Aq, O D trung di6mAC Qua Ck6 dy*g

thdttg d vludng g6c vbi BC Gqi Cx ld tia <l6i cria

tia CB, Mlitdilm bat kj thuQc Cr Goi E ld giao

diiim ctra d vd AM,Ili giao didm ci.n BE vit OM

Chtmg minh r6ng khi Mchuyr5n clQng tr6n Cx thi

1lu6n thuQc mQt cludng cO Ainn.

W Quoc DUNG (GV DHSP Thdi Nguy€n)

Bni T5/459 Gi6i phucrng trinh:

*'-z*=zJrx-r.

TAMINHHIEU

{GV THCS Y€n Lqc, Y€n Lqc, Wnh Philc)

CAC LOP THPTBni T61459 Giai bat phuong trinh:

2x3 +3x

rJz-*.

7 -2x

NGUYENVANNHO

(GV THPT Nguydn Duy Trinh, NghQ An)

BiliT71459 Cho tam giSc ABC nho.n, khdng c6n

cb chc <lucrng cao AH, BE, CF (H e BC,E e AC,

F e AB ) GSi 1, R l6n luqt ld t6m tlucrng trdn nQi

ti6p, bdn kinh clucrng fdn ngo4i tiep tam gi6s ABC;

M, N, P l6n luqt ld trung di€m BC, CA, AB; K, J, Ltheo thri t.u ld grao <li6m cua c5c dudng thing MI

vd AH, NI vit BE, PI vd, CF.Chtmg minh r[ng

1113 -+-+-

>

-.

HKEJFLI

DAO QUOC DUNG

(GV THPT Le Viiit Thudt, TP Vinh, NghQ An)Bni T8/459 Cho a, b, c ld d0 ddi ba canh cira

mdt tam gi6c c6 chu vi bing 3 Tim gi6 tri nh6nh6t cira bitiu thuc: T = a3 + b3 + c3 +

'l5abc

NGUYEN VAN THIEN

(Gl/ THPT TAn Phti, Ding Nai)

ufx ror oLYMPrc roAN

Bni T9/459 Chyrg mmh r5ng

"g -si sl5 nguy6n duong n thi c6c s6 sau ddy d6u ld sd chinh phuongro([r+ * dst'] r)([r+ + JE)'*r] + r)- oo ,

o([r++ JBI']+ r)([r+ + fi)n*r]+r)- oo,

a) Tim s5 c5c nghiQm thgc kh6c nhau cira

phucrng trinhlffix)): Q.

b) Ggi cr ld nghiQm thr;c du<rng l<vn nh6t cira da

thttcflx) Chimg minh ring [o''ol chia h6t cho