Tạp chí toán học tuổi trẻ tháng 6 năm 2012 số 420

r! ii , r i ,i t + E ETI xuflT sin rU tgoa rnp cxi nn xAruc rxArue tAru rsU+g oArun cHo rRUNG ttoc pHd rsOuc vA rRurue Hoc co sd Tru s6 187B GiAng Vo, Ha NOi oT ai6n tdp (04) {yzlaol; DT Fax PhZrt hdnh Tri su (04) 35121606 EmaliiiibTiiit,iain ocLuoitici@yahoo com vn Web httir //www nxbgd vn/toanhoctuoitre ,ffi f g CUOC THI uffuul*[frffii]E{ r*s*r* Dqt Z Ttdn hqc vd Tuili1rd sii 41 g, thdng 5 ndm 20 t 2 tld gitii thiQu ba bdi iI6 vui Dgt t Lan ndy chfing t6i xin gid'i thiQu fiAp bs bdi iti vui md[.]

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E:ETI.

xuflT sin rU tgoa rnp cxi nn xAruc rxArue -tAru rsU+g

oArun cHo rRUNG ttoc pHd rsOuc vA rRurue Hoc co sd

Tru s6: 187B GiAng Vo, Ha NOi.

oT ai6n tdp: (04) {yzlaol; DT - Fax PhZrt hdnh Tri su: (04) 35121606

EmaliiiibTiiit,iain-ocLuoitici@yahoo.com.vn Web: httir://www.nxbgd.vn/toanhoctuoitre

,ffi

f g

llirn ilo Pri rdPsa

Hdy tirn trong c6c hinh e, B, C, D, E b6n conh, hinhndo phi hop dddidn vdovi trisdl vd vi tri sd 2

nh0ng con sdchi khodng cdch (tfnh theo don vi m6t)c0o con

dudng gi0o c6c phdng trong nhd m6y TJ phang VvcA,b6c

b6o vQ phai di tudn d6m ddn hdt moi con dudng r6i sou cUng

tr6vd d0ng phang truc Theo bon, b6c bdo v9 phdi di theo lO

trinh ndo dd doqn dudng di cOo bac la ngtin nhdt?

GIA LINH

Bii6.PHAN GHIA IYIANH VUON

Bdn onh em duo c gioo vi6c don c6 6 m6t mdnh vudn hinh tom gi6c ABC, nhvng chuo th6o thudnxem oi ldm phdn ndo cOo m6i mdnh vudn Ngurdi 6ng ld gido vi6n To6n di ddn, lidn ro bdi to6n dd

sou: Hdy chio mdnh vudn thdnh bdn phdn bEng c6ch ldy bo didm D, E, Ftheo thO tu tr6n bo conh

BC, CA, AB soo cho sd do di6n tich cOo cdc mdnh vvdn DEF, AFE, CED, BDF ti 16 v6i tudi cOo c6c

ch91la 25,15,.l 2 vd 8 Bon hdytim c6ch phdn chio m6nh vudn gi0p bdn onh em d6

1

Tinh gia tri cua bi,2u thuc td mQt dang

todn thwdng gdp trong chrong trinh

THCS, trong do mQt s6 bdi todn tinh gid

tri cua bi€u thrc cd liAn quan d€n viQc

vQn dung dinh lf Viite dii hdi sqr- vdn

dang saig tqo vd linh hoat Bdi vi* ndy

xin-dra ra mAt sa Uai nip d€ bqn doc

tham khdo

t:t)a phu'o'ng trinh bdc hai

t:

, v8s ll

-tj-' ".r'i *',-0.

4t6

Kh6ng giai phtrong u"inh, hdy tinh giu tri t'ila

bieu thuc ri - -r:j

1

LN gidi Phuong trinh dd cho co A : I > 0,

16n6n lu6n c6 hai nghiQm phdn biOt .xi,r,

(x, > *r).Ap dirng dinh li Vidte, ta c6

(ira,i toin 2 Goi x1, xt-, x:, xa ld bon nghiim

cua phtro'ng trinh

(" + t)(x +3)(.r+s)("r +7) = 1.

Tinh gia tri tttu bit)u lhtrt' .{1-\,.\ q.y4

(Di thi try,€n sinh lnp t 0 THPT, tinh Hiti Duntg, ndm hgc 2005-2006)

Ldi gidl Phucmg kinh dE cho hrong ducmg vcri

frnal tofn 3" Gqi xt, x7 ld cdc: nghi1m ct)a

phtrong n"inh x2 +ZUM.r+l=0 r.'a x.,, xa ld cacnghiAru cua phtrrmg trinh x2 + 2005r + 1 = 0.Tinh gid tri ct)a hiAu th{rc'

(.r, + r )(x2 + -r, )(x, -ro )(.x2 - ,ro ).

(Di thi r:hgn hgc sinh gi6i ktp 9,

tinh Hdi Dtrcmg, ndm koc 2004 2005)

Ldi gidi De thay c6c PT dd cho lu6n c6 hainghiQm, n6n theo dinh li Vidte ta c6

\ * x2 = )004; x3 * xq : -2005; xtx2 : x3x4 =1 .

:iinro,.-ro:r, T?EH.HBI I

'x't3["lnd*i xl

phaong trinh x2-3'fzx-'[r=0 Tinh gia tri

cirs bi1u thti'c

BAr rAP ru Lr.iYEN

l Ggi x1,x2,x3,x4ld bdn nghiQm cua phucrng

trinh (x +2)(x+a)(x+6)(x+8) =1.

Tinh gi6 tri ctra bi6u thric x1x2x3x4.

2 Cho phuong trinh 12 -3ax-a:0 c6 hai

nghiQm phAn biQtxl, x2.

a) Tinh theo a gi6 tri cria bi6u thric

3ax1+ x] +3a a

b) Tim giiLtrinho nhdt cua bi6u thttc A.

(Di thi chgn hqc sinh gi6i lop 9

tinh Qudng Ninh, ndm hoc 2004 - 2005)

3 Gia str x1, xz ld har nghiQm ciia phucrng

trinh .r2 + ax + 1 = 0 vd x3, x4 ld hai nghiCmctra phrrong trinh x2 +bx+1=0 Tinh gi6tri

cira bi€u thric

, -,7vu'ot qua (Z + +J:) .

ffinai toSn 5 Gia sa phuottg trinh

at?'-rbr *c = 0 {a * 0) ca hai nghi€nt xt, x2

tho* trtdn ox1+bx2 *c=0 T{nh gid tri cua

biiu thuc M = a2c: + ac7 +b3 -3abc.

- (Di thi tuy1n sinh l6p I0 THPT chuy€n

Nguy€n Trdi, Hai Daong, ndm hoc 2005-2006)

Ldi gidi Theo clfnh li Vidte, ta c6

(d) cit (P) tpi hai di6m phdn biQt khi vi chi

khi A =a2 -4a>A oa<0hodca>4 (1)

Do x, +x2:a=xtx2 n6n (2) tro thdnh lol: o

* a) 0 fi5t hqp v6i (1), ta droc a> 4.

Ciu 3 a) Di0u kiQnx > -1 PT d5 cho trd thdnh

Dltt 2(p +1) = (2x)2 vit 2(p2 +1): (2y)z vfr

x,y nguy0n ducrng, 0 < x < y < p

Ta c6 2(y' - *') = p(p -1) nln 2(y * x)(y + x)

chia h6t cho p

Dop 16 vdr 1 < y - x < p, I < y + x < 2p n€nphdi c6 y + x: p (1) Suy ra 2$t -x): p-l (2)

a) ra c6 dDE =dZE :Oa *tB.

Suy ra bt gi6c BCDE n6i tit5p duoc trong mQt

dudng tron (Z)

,^ -^

b) De thdy 6Y=KE=TW,CLS:CEB:VTE.

Tt d6, ta co LVCS <.rs LTEV (g.g) = ry-=Y- TEW

hay SC.TE: VC.VE

ld Me ,SC = Rs,TE = Rr, VC =VE = rR7 n6n thu

VQy PT c6 nghiQm duy nhdt x:3.

b) Khi 2(p +I) ld s6 chinh phucrng thi d6

sO chinh phuong chin vi:, p li s6 nguy6n t6

t nro,.-rorr, '?S#EE 6

ffi&M ffiWm ffi68€ t * ffiffiQffi

C0u 1 1t,S diem11) Cho bi6u thuc

rJi - Jif ++'[un

Ji+Ji

v1ia>0vdLb-'0

a) Rrit ggn bi6u thfic A"

b) Tim gi|tri cria & d6 gi6 tri cua bi€u thuc I

rninh rdng ( p -l)(p + 1') chia h6t cho 24.

2) Tfun th ch circ cip s6 qu nhiCn (x ; y) thoa mdn

x2 -Zxy + 21t2 +4-y -13 = 0.

Cflu 3 (2 diem)

Cho phuirng trinh bAc hai (in x)

x2 -2(m-1)r+ m-3= 0 (1)1) Chtmg minh ring phucrng trinh (1) ludn c6

hai nghidm phAn bi6t v6i mgi gi|tri cua m.

2) Gqi x1, x2 ld hai nghi€m cira phucrng trinh (1)

Khix:y : z:3 thi r: -1 Gi6 tri nh6 nh6t

cl;ra t ld-1 dat tlugc khi x : y : z : 3.

b) Gia str t6n tai hinh vu6ng diQn tich bing

l) Tn mQt di6mA U6t t<i O b6n ngodi ducrng tron

tlm O kd c6c ti6p tuyOn AB vit ,AC v6i ducrng

trdn (B vd C ld c6c ti€p di€m) Ldy rli€m / thuQc

dopn th6ag BC (I khdc B, 1 khSc C"va 1 khactrung di6m cira dgan ,BC) Ducrng thang r.u6ngg6c v6i OI t1i I cdt duong thing AB t4i E va cl,:t

a) Chimg minh ring tam gi6c EOF cdn

b) Chrmg minh ring AEOF ld tu gi6c nQi ti€o

ducrng tron

2) Cho tam giitc ABC tudng tai A vd, cir dien

tich bdng a2 lvoi a ld s6 ducrng cho truoc).Chimg minh bdt d6ng thuc

BC +2a<irUB + AC)

Cflu 5 0 diem)Chimg minh ring trong t7 s5 tU rrhien bAt ki ta

ludn chon ra dugc 5 s6 c6 tdng chia het cho 5.

INAX VAN HANH(GV Khoa Co Ban, DH Phqm Vdn Dinq @i"ng.Xggil

sru timvdgioithieu

Theo bdi rata c6(n+l)2 +(n+2)2 +(n+3)2 +ln+4)2 +(n+5)2(n+6)2 +(n+7)z +(n+8)2 + (n+9)2 = oz.Hay 9n2 +90n+285 = a) Do 9n2 +90n+285

ld s6 nguy6n chia cho 9 du 6.

M[t kh6c, s5 chinh phucrng a2 chia cho 9chic6th6du 1;4;7.

V{y kh6ng tdin tai hinh r,uOng ndo th6a mdn

1 Y nghia hinh hoc cria d4o hirn

Cho hdm s6 y = f (x) co dO thi (C), mQt

di€m Mx cO einn thuQc (C) c6 hodnh d0 ro

Khi d6 .f '@) bine he sO g6c cua ti6p tuy6n

cira (C) taidi6m Ms(xs;J@i).

2 Fhu'ong trinh ti6p tuJ'6n

Phuong trinh ti6p tuy6n v6i d6 thi (C) cna

hdm sO y = .f (x) tai diiim Ms(xo;f (x,;)))lit

y : .f 'Qo)(x - xo) + yo, (v6i yo : J @o)) (*)

- *.x .a

J rrlcu kren tlep xuc

Dudng thdng y=kx+bttdp xirc v6i duong

cong l' : f(x) khi vd chi khi hC PT

I f G):kx+b

)",

l.f '(r): kc6 nghi€m Khi d6 nghiQm cua hC ld hodnh dO

ti€p di6m cira hai duong tr0n

rr MQT SO DANG TOAN

D{IVG f Vi0t phu'crng trinh ti6p tuydn vfi

{I0 thi hirm sO

Crg}ri to6n l Vi€t S;huotrg trinh tiep tu1t€n

rhuoc tli thi

Crich gidi DC gi6i c6c bdi to6n loai nhy ta cAn

tim tga dQ ti6p di6m M (xo;yo) vd h0 sd

goc f '(xs),sau c16 sir dpng c6ng thfc (*).

* fhi OU t I,'ier phurmg trinh tidp tu.vAn v6'i

di thi(C) ct)a hitm so "r = -;rr -6-12 + 9x-2

tqi diim M thuoc (C)" hi& ring M c'ilng vr'ti

hoi didm t'trc tri c't?a (C') tao thdnh mot tcrm

CIuA oCI TH! HAM S0,

PHAM LE THANH DAT (Gy THPT Le LEi, KonTum)

l_

o ,Jz'+\4\2

*)nai lnhn 2 ViAt phurtng trinh tiep tuyOn

vo'i d() thi hdm s0 y= /'(.r) hiir hQ so gic kcho lrudr:

Ctich gidi Gi6 sir hodnh dQ ti6p ili6m ld xe.

Giai PT f '(til:k (voi An x6) tim duoc xs.

dci t:dt li\m cdn d*ttg, tient can ngang

luot tai A,.B sao cho tam giac IAB can,

ld giao diAm cua hai dudttg ti€m tan

Ldi.gidi Vi tam gi6c IAB cdn t4i 1 n6n

tuy6n ph6i song song v6i mQt trong

lanv6'i {

tlephai

() xo : -l hofc xs = -3 C6 hai ti6p tuy6n

th6a mdn dA bei Dty = v-tl; y = n + 5 fl

ftrai to6n 3 Vi& phrcng trinh ti€p ruydn

-; : ,

v'6i do thi ham so 1, - f (x), biil rdng ti€p

tuyAn d6 di qua di€m M(xs;ys)

Ctich gidi Luqc at6 chung dC giai bdi to6n

ndy nhu sau:

Str dung mQnh d6 ,rC ei6u kiOn ti6p xfc cua

hai cluong tld trinh bdy trong phAn ki6n thric

cdn nh6

9ry vC eai to6n 1 (tric ld qqy vC tim toa dO

ti6p diOm cira ti€p tuy6n v6i d6 thi)

*fni d,rtr 3 Vi€t phtrong trinh ti€p tm,€n vdi

di thi (C1 ct)a hdm so ! = xa -4x2 +4, bi€t

I ; ; ,, -.

rang tic;p ruyen do di quu diem hl(0:41.

LN gidi Duorng thing d di qua M v6i hQ sd

Dulng thing d tie:p xric v6i OO ttri (C) khi

vd chi khi h0 sau c6 nghiOm

l* -+* +4:la+4l

[4x: -&:k.

Thay k tu PT thri hai vdo PT ddu c0a hQ vd

thu Ben, ta duoc 3xa *4x2 - 0 <+ x - 0

DNG 2 Tim c6c tli6m tr6n AO tni sao cho ti6p

tuy6n t4i aI6 thoa mfln tinh ch6t cho tru6c

*Thi drtr 4" Cho hdm so ' y =T x-2 co d6 thi

(C) Goi I ld giao di€m hai &tnng tiQm cQn

cila (C) Titn c:dc di6m M ftAn (C) de rcp

cQn ngang cia (t)) lin lrot tai ,4 va B sao

cho dadng trdn ngoqi tidp tarn giac IAB cd

diAn ilch nko nhat

Nhan thdy 'o !^*u : 2+ 2xo -2 - re, fl611

Ta c6 I(2;2) vh tam gi6c IAB vu6ng tai 1

n6n dulng trdn ngo4i ti6p tam grirc IAB codisn tich ld

*rni du s cho hdtn so y - x-l =+ co rti rhi

(C).Gpi I td giao die:m hai dtrottg tiim cdn

ci)a (Q Tim tr€n (C) diin M sao c'ho fidptu\,€n cr)a (C) tqi do wtong gdc' t,6i dudngthang IM

IM L Le h.k2=-1 <>(xo -1)o -1 <> ro = 0

ho{c xs =2 YAy c6 hai di6m cdn tim ld

M1Q;t) vd M2Q;3) Q

DANG J Bni tofn tri6n quan tli5n sii tiOp

tui en ciia tld th! him s6

*Thi dg 6 Cho hCm sd y=x3 -3x co dd thi

(C) Tinr nhirng di€m ftAn d.rdng thdng y = 2

ntd n'r dd kd dung ba ti€p ruyen din (C)

Ldi gidi Ta c6 !'=3x2 -3 Gia sir M(xs;2)

thuQc duong thing ! = 2.

Ducrng thing d di qua M voi hq s6 g6c k

YOu cAu bdi to6n tuong ducrng v6i PT (1) c6

ba nghiCm phAn biQt, tuc ld PT(2) c6 hai

nghiQm phdn bigt khSc -1.

Di6u ndy xfiy rakhi vd chi khi

{r: ox; -.t2xs 12:o , l'o " , (3)

I/t-tl=6(xo+l)*o .- l ro.-i

LJ

YQy M(xs;2) (v6i x6 thoa men (3)) ld c5c

di6m cAn tim tr6n duong thtng y =2 Q

DAI\G 4 Tim tti6u ki6n cfia tham sii OO trai

tluhrg ti6p xric nhau

*Thi dtl 7 Gqi (C,)ld do rhi cuo hdm s6

c6t tryc Ox, Oy 16n luqt tqi A, B phdn biQt sao

cho tam gi6c OAB cdntai O.

S ViCt phucrng trinh titip tuytin ,v6i dO thi

hdm sti ! : -4x3 + 3x, bi6t r[ng ti6p tuy6n d6

di qua diOm M(1;3)

4 Cho him s6 y=+ c6 d6 thi (C) Tim

x-1

didm M tr0n (C) sao cho ti6p tuytin cira dO

gi6c c6 chu vi nho nhdt

5 Cho hdm s5 ! = x3 -3x c6 OO ttri (C)

Tim tr6n duong.thing,x =,2 c6c di6m md tu

d6 k6 dring ba ti6p tuy6n d6n (C)

(m-l\x+m :

!=: (rzlathamso)

x-m

Chrmg minh reng voi mgi m*0, hq dO thi

(C,) 1u0n titip xric v<ri mQt duong thing

cO einn

*nro,.-rorr, '?[l#S Z

2) Tim m aC aO thi hdm sO 1t1c6 ba cliiim

cgc tri vi b6n kinh ducrng tron ngoai ti6p tam

gi6c t1o bcri c6c di6m ogc tri d6 dat gia tri nho

nhAt.

CAu II 12 diem)

1) Giai phucrng trinh

A Theo chuoxg trinh ChuAn

CAu VIa (2 diem)

Cflu III Q die@ Tinh tich ph6n

L

1dJ rl

'

J .orr.Jz + ri, z, '

C0u IV (1 diA@ Cho hinh ch6p S.ABC co dby

ABC liLtam giirc w6ng tai A, AB : a, -4C :2a.

I\4at b6n (SBC) ld tam gi6c cAn tai S vd

ndm trong m6t ph[ng vu6ng g6c voi cl6y.

Biet g6c giira hai rnlt ph[ng (SAB)

vd (ABC) bing 30" Tinh th6 tich khoi chop S.ABC vi kho6ng c6ch gifia hai

dudng thingSCvd AB theo a.

./5cos_r,+sin,r: Ciu V Q diAm) Chrmg minh r5ng voi cac s6

thsc duong x,)',Zthod mdn ('t^!.:')' <4ry2,

\ 2ot2 )

ra ac6 vx n {/-, l' <2012.

x+rly y+'la r+Jry

(Thi sinh cht duoc ldm milt trong hai phdn A hodc B)

B Theo chuo'ng trinh Nflng cao

Cflu VIb (2.,die@ 1) Trong m6t phing top

1) Trong mflt phing to4 dQ Ox1,,cho di6m dQOxy,chodiemM(-3;4) vd hai dudngthdng

M(3;-1) vd tludrngtrdn Q):l+72*Zx*{,-11=0 dt'.x-b -?=0; dz:x-y-0 Duong thing vi6t phumg trinh duong thing d qua M , cat d q.u M:

-".at !:* (: PTs img o A vit B

(z') iheo *9t oay.u,g i6 do dei nho nhai " s.ag thang cho r/, W:,3M8-.'.vi6t biet rlng cli6m,4 c6 phuong tung dQ trinh ducrng.ctucrng

2)Trongkh6ng gjanOrvz, chohaiduongthang 2) Trong kh6ng gian Oxyz, cho di6m

A

'I-3=.^ =-;" ,1., '' j .11= 3- u; M tl;l;t) ciucrng thing d.:-2:l-y ='

l2-51-ltlll

I cat dlvad,ltuong img o AvirB, d0ng thoi grao di6m cua d va (P) ViCt phuong hinh

khoang o6ch tu A ilen rn{t phing (a) bing duong thing A chua M, cit a valf\tuong img

VO Vi6t phuong trinh dudng th[ng A, bi€r o BvitC sao cha ABC 1]r tam gsic cdnt4i B.r6ng,Jidm ,a c6 hidnh dd duong - C6u VIIb { diA@ Voi m5i s6 thirc a , gQt

",)ur,i*!T,r"Y3,,"J",#Truflr#11

Tim m6dun cira sd phac v +z

GV THpr D6ng Hwng Hd, Thai Binh)

TORN -clirdiU@ HOC

pEqyfl$r,e"# mAx\T

Cflu I 1) Ban cloc tu gi6i

2)Ta c61(1;0); M(O;2)thf t.u ld cliOm udn vd

di6m cgc dai cria dO thi (q.

PT hodnh d6 giao tliiSm ctra (4 vd (C) ld

I -3x2 +2=k(x-l) e (x-1)(x2 *2x-2-k)=Q.

(4 cit (C) tai hai drdm A, B l<hdc Mkhi PT

x2 -2x-2-k=0 c6 hai nghiQm phAn biQt

kh6c 1 e k>-3.

Gi6 su A(*r; yr), B(xz; yr), chir f ring 1 ld tAm

clOi xrmg cria (Q nOn d6 am gs6c MAB vu6ngt4i

M *n AB : 2MI : 2'li C6 ba <lucrng thing thoa

mdn dO bei h y= 2(x- t); l=f+(x-1); "2

t _ -11

v-' '-(x-1).

,2

Cflu II 1) DK sinZx * 0 PT tucrng ducrng v6i

cos2 3x = cos2x o I + cos 6x = cos 2x

2

<> (cos 2x -\$ cosz 2x + 4 cos 2x -l) = Q

Eao s6., = *!arccosf +€) + knlk ez).

2) Gie sri l(x;y;z) ld di6m th6a mdn

( s s 13)Ddo s6 Ml : : l.

Ta chi cAn xdt mi6n nghiQm mh sirx > 0, cosr > 0.

1.

Him sd .f (t)=t.2012J' ddng bi6n tren (0; 1),

ddn dtin PT (*) <+ sin-t : cosn.

Dap ,4 s6., = 1* kn (k e Z\

C0u VIb 1) L: y: -1 ho[c L: Y: 5.

2) PT mat phing (P): x -2Y + z + 4 =0

CAu V gien d6i

I MO DAU

Pierre de Fermat di trmg dua ra bdi to6n sau:

*fii tor{n I (L3di to6n Fennat) Tim di€nt

trang mdt turn gidc 9o gric lon nhdt khong"vur"n

qwa 120' sao cho t6ng khoang cdch t*'di€m ifui

ddn lta dinh cua tam gidc la ngdn nhdt.

Nhd to6n hgc, vft lihoc Evangelista Torricelli

dd chimg minh clugc bdi to5n ndy vh ph6t

bi6u thdnh mQt dinh li, thulng goi ld dinh li

Torricelli du6i ddy

d.nlnn H 1 (Dinh li Torriceili) " N€u trong tam

gidc ABC cho trwdc cd gdc l6n nhdt kh6ng

vuqt qud 120' dgng daqc didm F* thda mdn

tinh chdt AF+B: BF*C = CF*A=120, thi F*

chinh td di6m th6a mdn bdi toan Fermat ddt

ra Ta g4i diAm ndy td di€m Fertnat

-Torricelli, hay gpi gpn ld di€m Fermat

th* nhdt

Dlnh li ndy kh6 quen thuQc vdi bpn dgc, xin

kh6ng chimg minh t4i dAy C6ch dimg ali6m

Ton'icelli hay diOm Fermat thri nhdt c6 li6n

quan v6i bii to6n Napoleon sau d6y

flgai toSn 2 (Bdi to6n Napoleon rhu nhdt)

Cho tant giac ABC Vd phia ngodi tam giat'

dyng ba tam giac dAu BCA', CAIS', AB{:' Khi

do tdm ba tunt gidc ddu BCA', CAB', ABC'

thdng AA', Bq', CC' d6ng quv tai dieu

!er*wt thir nl'tdt.

Cr)ng vdi bdi to6n ndy d6n d6n bdi torin

Napoleon thri hai

*gai to6n 3 (Bdi todn Napoleon ttrrri hai)

Ckrs tam gitrc ABC (A phia trong tam gidc

drmg ha tam giac ddu BCA", CAB", ABC""

Khi do ba tswt gieic diu BCA", CAB", ABC"

thdng AA", BB", CC" dong qtry tai di€m

Fermat tkw lzai, ki hi€u ld I; .

Cdhaibdi to6n Napoleon d€u Lh6 quen thu6c

vor cdc bryr hgc sinh, xin h.ong chimg minh

@i diry DO thu4n tiQn cho ph6n sau, ta ki hiqu

t6m cua ba tam gi6c d6u BCA', CAB', ABC'

lAn luqt ld T.,T6,T, vd t6m ba tam gi6c dOuBCA", CAB", ABC" lAn luqt 1A ,S,,,Sb,,S

II MOT SO TiNT{ CHA.'I'

*Tfnh chit t" F* ndm ftAn dadng trdn

, ,:

ngoqi titip cdc tam gidc BCA', CAB', ABC'

Chftng minh Thlt v4y, giA su iludng trdn

(ABC) clt ducrng tri:n (CAB) tai F' (h 1).

A'

{{inh I Khi d6 dF)=180" - IYe =120o Tucmg tu

IFD =120" Khi d6 ta cb 6?C:120o suy

ra F' e(BCA') vit F'= F* la

Chimg minh tuong tu ta thu duoc

:']

ddng phuong cua ba dudng trdn (ABC'),

(BCA',), (CAB',)

*'finn cfrit l Di€m Fermat thtr nhiit niim

tr\n dadng trdn ngogi fidp ktm gidc S,S65,.TOAN HQ(

10 -qidikre

Chirng minh (:h.2).

A

Hinh 2

Su dpng tinh ch6t ctra g6c n6i ti6p cho c6c tir

gi6c AF*S6C vd F*BCS,, tac6

*finn cn6t S Tdm cila hai tam giac diu

T,T6T" vd 5"565" trilng vdi trong tdm G c{ra

tctm gidc ABC

Ch*ng minh Gqi Mldtrung cli6m BC (h.4)

J^

Ta c6 { ndm tren AA', theo tinh chat 1 thi

AF* ld trgc dlng phucrng ctra hai dulng trdn

|ABC) vit (ACB), n€n T6T, L AA' Theo tinh

ch6t cria trgng tdm tam gi6c ta c6 (chf y

trong tAm cua tam gi6c BCA)

Tir cric tinh chAt ftOn ddy ta di d6n mQt dinh li

nOi U6ng trong hinh hgc Euclid, do nhd hinhhoc June A.Lester n6u ra vdo nim 1996

*ninn Ii 2."(Dlxh li Lester) Trong tam gidc

ABC, hai di€m .,. Fermat F* vd F-, tdm dadngtron ngoqi tidp O, tam dadng trdn Euler N citngndm ffAn mQt dadng trdn

C4,b"g minh (Theo Nikolai Ivanov.Beluhov).D0 chimg minh dinh lita cinmOt sd k6t qui sau:

S0 dA 2.1 Tdm N &rcrng trdn Euler cua tam

gidc ABC chia doqn OG theo ti l€NO:NG-2:|.

BO de nay kh6 quen thuQc, ban dqc t.u chimg minh

g6 OO 2.2 Cho tam gidc ABC Gei T ld giao

Chftng minh Ta thdy LTAC <n ATBA, vi vfly

m{i ei6 23 Cho tam gidc AqC F ld di€m d6i

xtntg da B qua.AC, Q ld di€m d6i nrng cila C

qua AB ,Kd ti€p tuy1n tqi A.crta ,dadng trdn

(APq cdr PQ tqi T Khi d6 di€m d6i x{rng vcri T

Liy B', C'brn luqt d6i ximg vdi B, C qua

A-Ta c6 tam gi6c AB'P cAn t1i A vlr

Eip =180" - Eid'-FZa = 18oo *2EZa.

Do d6 AB'P : APB' : BAC, nln B'P ll AC

Tucrng ti C'Qll AB, suy ra B'P vit C'Q lu6n

cit nhau Goi giao di6m cua chring ld A', dA

dang chimg minh cluo c AA'B'C': LABC Ngodi

ra LAB'PcnAAC'Q OC tnay B'elA'P),C elA'Ql

.- J

vd f nlm ngodi doqn Pp.

BO dC 2.3 dugc chimg minh n6u h chi ra T,

B', C'thing hdng Sri"dung dinh li Menelaus

cho tam grdc A'PQtathAy

T bhn kirTh T,B ; theo tinh ch6t 3, F* nim

tr6n (S,.9r.9") n6n f,S, ld truc rling phucmgcua hai du<rng tron (5"565.) vd (7,;T"B) Y\

G li tdm duong trdn(,S,SrS.) (tinh chAt S; nOn

F* vd S" d6i ximg nhau qua GTo Tucrng tg

ta c6 F- vit f, d}iximg nhau qua GS,

Xdt tam gi6c GS,T, co F* vit F l6n luort d6i

ximg vdi S,,To qu1 GT,,GS, Gqi Q ldr giao

ili6m cria tiOp tuy6n @i q cua dudng trdn

(GF.F-) voi F*F- vd O'dOi ximg voi Q qua

G Tt UO AC Z.: tathdy O' eToSo

Tucrng t.u ta c6 O'eT6S6, O'e 7,5, OC thay

T,5',T656,T,5" ld c6c ducrng trung tryc cita

BC, CA, AB n)n O' chinh ld t6m dudng trdn

2-Suy ra F*,F-,O,N cirng nim tr6n mQt ducrng

tron Dinh li 2 dugc chimg minh hoin tohn il

TOAN HQC

tg - qirOi$e-S"fggJe-39l2*

fli kltuo.,vat c'ltinh thirc k& qua hgc tap cuu h7c sinh lop 1I d6i vo'i c'ac mdn Totin, I'lgti'vdtt

l\rrr riting Anh rtiiitr ltlti' 2()tt-:0t: t'tro Bo Giio dut' t'tt Doo tao tlii tltruti' td t'ltti<' tlr ]*\)

tlen 30/12,/2011 t'tii cpfl,tttd 340 trud'ng T'HPT.tqi 63 tinh - thanh pho, m6i fi'trd'ng c'hon ngiu

nhi0n -t0 hoc sinh iJdivi0t nay xin gio'i thi€u v0 Noi dung khr)o sdt mon'fodn

r- MUC OiCm

Kh6o s6t nhim d6nh gi6 thgc tr4ng chdt luqng, hiQu qui day hqc m6n To6n lop 11 THPT sau

khi hoc xong n6i dung tinh clOn thoi di6m tu6n 4 thfug 12, v€ ci b€ rdng vd chi6u s6u cua ki6nthric vd ndng lgc tu duy; nh.am cung cAp th6ng tin cho cdc co quan quAn li gi6o dpc trong x6ydlmg c6c giiliphdp hqp li d6i v6i ph6t tri€n gi6o duc trung hoc Tu d6 g6p phAn.vho vi6c chinh

li, cli0u chinh chucrng trinh- s6ch giSo khoa To6n vdr tpo c<y sd cho vi€c thi6t kC chuong

trinh-s6ch gi6o khoa THPT cho giai doan ti6p theo

rr- Ngr DLlr{G

NQi dung duoc lua chon thi6t te OC ki6m tra g6m 5 chuong v1i 2l chtr d0 vd 33 chu diOmchinh sau clAy.

Hdm si lrro.'ng giric vd phrrong trinh lutrng giric

Hirm s6 lugng gi6c (Hdm s6 lucrng gi6c, chiOu bi6n thi6n, d0 thr.

Phucrng trinh lugng giSc co bin (sin x : m; cos ,r: m; tan x:m; cot x:m).

Phuong trinh lucrng gi6c thucrng gip @t+b:O, ay' +bt+c:0, (/: si11r, cosx, ) asinx+bcos x:c).

*; " :

I o lr?'|, - xilc silot

Quy tic d6m (Quy tic cQng, quy tic nhAn)

Ho6n vi - Chinh hqp - fO hqp (Ho6n vi, chinh hqp, t6 hqp)

Nhi thirc Newton (C6ng thric nhi thric Newton, Tam gi6c Pascal)

Ph6p thu vd biOn c6 (Phep thu, khong gian m6u, Uien cO).

X6c su6t cua biiin c6

Da1' si - Cai, sd cgng vd cdB si nhrtru

Phuong ph5p quy n4p to6n hgc

Day sO lOinn nghTa, c6ch cho ddy s5, bi€u diSn diy s6; day s6 tdng, giitm,bi ch{n).

C5p s6 cQng (Dfnh nghia, sO hang t6ng qu6t; tinh chdt, t6ng cira n s6 hpng dAu).

C6p s6 nhAn (Dinh nghia, s5 hang t6ng qu6t; tinh chAt, t6ng cria ru sO hAng dAu).

Fhip ttdi ltinh vd phdgt tling dqng trong m{tt phdng

Phdp bi6n hinh Phdp ddi hinh

Phdp tinh ti6n Phdp vi tU.

Ph6p quay Phdp d6ng dpng

Dw)'ng thdng vd mdt phitng tromg kh\ng gian" Quun hQ sorcg sr.u.p4

Dai cuong vO ducrng thing Hai dudng thlng chdo nhau

Dulng thdng vd m[t ph6ng song song

", nro,.-Zqrr, '?[l#8! 15

{ri- e}EiiN sc}4N un xmao sAr

Thoi gian ldm bdi 120 phrit, trong d6 phAn trbc nghiQm khSch quan (TNKQ) duqc HS ldmtrong 40 phirt, ph0n tU luQn (TL) duqc HS ldm trong 80 phrit

Ma tr$m diiMri,c tIQ nh?n thfc

DN, s6 h4ng t6ng

2 di6m Tinh ch6t, t6ng cria

zl s6 hans ddu Cdtt24YOffiN I-NQE

2 cdyr;

6 di6m 8 cdu

2./ Atem 22%

Hai ET ch6o nhau CAU 19 IY.2

4 TNKQ:4 di6m

6TL: 36 di6rTOng: 40 tli6n

40%

6 TNKQ:6 di6n

4 TL: 24 di6n

Tdng: 30 di6n 30%

30o/,

12TL:70 di€n

7\ot, TOne: 100 di6n

MO te ma trAn

r Vd c6u trric

T6ng s6 42 cdu,trong d6 c6 30 cAu TI'II(Q vit 12 cduTL.

30 cAu TNKQ:30 di6m (m5i cau 1 di6m)

12 cduTL:70 di6m (m6i cdl o mric nhdn bi6t 5 cli6m, m6i cau 6 mfc th6ng hiOu 6 di6m vdx.

moi cdu o mftc vdn dung 6 diOm)

c V€ mf'c ttr6 nhfln thrlc

LTS Do khu6n kh) td tqp ch{ c6 hqn, phdn gi6i thi€u d€ chi fiA sd daqc ddng trong EQc sancua Tap chi Mdi ctic bqn ddn doc

Biet Hgc sinh nhAn ra, nhd lpi, x6c dinh clugc, t6i hi6n ctuoc c6c dir 1i0u, sg kiOn, kh6i

ni€m dinh li quv t[c tinh chat dd duoc hoc

Hi6u

Hoc sinh bir5t duoc ki6n.thfc dl hgc vd f nghia cira n6, sir dqng ki6n thric d6nhrmg chua c6 fU 1i6n k6t cAn thi6t v6i c6c ki6n thric kh6c ho{c c-hua thAy duqccbc imgfimg dAy dri cira n6 O mric dQ niry, hgc sinh c6 th6 dnng ng6n ngt cira

minh d6 giii thich dugc., minh hga dugc, chimg minh dugc c6c dir liQu, sg kiQn,kh6i ni6m, clinh li, quy t[c, tinh ch6t, ttd hqc

v0n

dung

Hgc sinh sri dpng c6c ki6n thirc d6 hoc vdo m6t hodn cinh cu the dC gini quy6tnhirng v6n d0, bdi to6n trong tinh hu6ng quen thuQc, ho[c tucrng t.u nhu ntrirngtinh hu6ne da bi€t ho[c tinh hu6ns m6i khOns quen thu6c

ts

T]AC I-fiP TH[,S

'\.*#-BiliTll4z0 (Lop 6) Tim c6c gr5 tri nguy6n

BdiTZl4z0 (Lop 7) Cho tam gi6c ABC nhsn,

kh6ng cdn tpi ,4 Duong trung tr.uc cila cbc

cu*t AB, AC theo thf t.u cit t u.rg friy€n AM

t1i E, F Goi giao di6m cira BE vd CF ld K

Chtmg minh r[ng AKB : AKC,MAB = KAC.

(GV THPT chuyAn DHSP Hd N6i)

BiitiT3l420 Tim t6't cir c6c b0 ba s6 nguy0n

(x; y; z) th6a m5n ding thic

Zxy + 612 +3a-l x -2y - z | = f + 4y2 +922 -L

(H7c viAn Cao hpc Todn, Kh6a 2010-201 2,

phdn gi6c trong cua g6c BAC cat AC hi D

Gqi 4 F thir tU li hinh chit5u r,u6ng g6c ctra

D tr€n AB vd AC, K ld giao di6m cta CE vir

BF, H ld giao cli6m cira BF voi dudng tron

ngo4i tit5p tam gi6c AEK Chtmg minh ring

(GV THPT Phil M!, Tdn Thitnh, Bd Ria - Vitng Tdu)

BitiT7l42O Cho a, b, c ld c6c s5 thuc khdng6m c6 t6ng bing 1 Chrmg minh ring

(t + a2 xt + bzxr + c2)

= [f)'

rnAN ruAtr aNu

QP H6 Chi Minh)Bni T8/420 Cho tam gi6c ABC c6 di6n tichS Gqi x, !, z lAn luqt ld khoing c6ch tir mdtdi€m M trong tam gi6c d6n ba dinh A, B, C.

Chimg minh ring (* + y + z)2 > +"'6S

Ding thric xby rakhi ndo?

(GV THPT Nguydn HuQ, Qudng Tri)

s6 nguy6n avdb).

rnAN VU LINH(SV l6p K45.1 1.06, Hpc viQn Tdi chinh)

B^iT101420 Tim sO L tcrn nh6t sao cho

Ja + 2b +k + Jb + 2c +ja + Jc + 2a +3b

>k6[i+Ju+r,f,) dungvoimgi sii duonga, b, c.

TAHOANGTHONG(GV TT Thing Long, TP UA Cni UinttlTC6ru [{Q(

3ffi - q*iue- sti_a:9sf912._