chuyOn Vinh Phric; Ngh€ Ant Nguy4nVdn Minh, llAl,
THPT chuy€n Phan BQi ChAu; Quang Namz Nguydn H6ng
Stn,l}fl,"IWl chuy€n Nguy6n Binh Khi€m; Phri YGn: Vd
Vdn Huy,l 1A1, THPT L-e Hdng Phorg, Tey Hdq B€in Tre:
Khdns Hitu HiQp, llT, Huynh COng Bdng, l7f, THYI
chuy6n Bdn Trẹ
CUYEN MINH DUC
*Bei TnB87. Trong mQt phdng cho haitam giac ABC, ÁB'C' v6i r, r' thtr tr ld bdn tam giac ABC, ÁB'C' v6i r, r' thtr tr ld bdn
kinhicic dudng trdn nQi tiilp chilng, ct)n R' lit bdn k{nh dudng trdn ngocli ti€p tam ,gidc
ÁB'C'. Cht ng minh rdng ,v6ị ntoi diint P
trong m(t phdng đ ta c6 bat đng thtTc
[,,in4 *.iné)", * [r,n I *rin4) pa
( 2 2) [ 2 2)
+(rin!*rin4)pc u6rr' .
\ 2 2) .R'
Ldi gitiịTrudc hdt xin ph6t bidu kh6ng chtmg
minh ba bd đ don giin.
Bd dA /. Ndu r li b6n kfnh duong trdn nOi tidp cfia tam gi6c ABC thi min{.Bc,CA,AB\ > 2r.
Bd đ 2. Ndu R, r thrl tu ld b6n kfnh ductng
trdn ngoai ti0p vi bdn kfnh ductng trdn nOi
tidp cfra tam gi6c ABC thi
r = Rsin!..in4.rin9.
222
Bd dA 3. Vdi mgi tam gi6c ABC,tac6
.A B C-I
sln-.sln-.sln- 2 2 28s -.
Ding thfc xiy ra khi vI chi khi tam gi6c ABC
đụ
Tr& lai vi€c giii bii to6n Tl21387 .
Theo BDT tam gi6c, tU c6c bd đ 1,2 vd 3 , ta c6
_ -u2 +3u-1 _ u(3-u)-l ,O
-uz +2ưl -u2 +2ưl
nOu I < u <2 (v\khi d6 3 - r> l).
'Lln+l-2=
-2u2, +5un
-u3 +2u, +1
_ -Zutr + 5u, -2(-u2, +2u, +l)
-u|+2u,, +l,, _.) ,, _.) un =- -u| +2u, +l | -ul *2u,*l I * ..._ -r, L- //?\
2-un*t Z-un " Z-un
Đng k6t qui (3), bdng tinh to6n truc tidp ta
c6 2<un 13, Vn= 1,2,3,4,5 vi I 1uo12.
Dilng kdt qui (2) vi (3), chfng minh bang quy
nap theo n = 6,7, 8, ... ta nhAn duqc I 1u, 12 vdimoin)6.
HO qui ld, u1 + uz I ...* u, ) n, Vn e N'.
Suy ra
,,l{1(r' * u2 * ...i u,) = *oọ E
(Nnan x6t. l) Ket hSp (2), (3) vdi (l) ta suy ra
2,1 = url thlutluq< I +..E < zs vi
-2
[=in4*rine),o,a*frinC * rin {)ru[""'z -'2) \ 2 2) [""'z -'2) \ 2 2)
*( t ,inL* , ,in 4) )1rc
=(PB+ PC)sin {*(Pc + ea\sinl
'*gei Lll387. I)tit m()r ctiAp tilt xouy chiitt ,'i
giti tri hiAu thrng U ' 60Y vin hat dtiu rtttit
'clocrn'mut'h
Rt-í mic' n,li riip thi cm)'ng đ
đns cti(n quu tfutun nruc'h li it :/ n\ / n\
.1,,ctrsl l00rr +i I (A). ,Vir nii tirt hui hun
. 4)
ctrc'c'irtr ttt cliAn C br)ng mit d(ịt'rlrin riiin thi
c'trd'ng dQ đng ctiAn quu cfutqn ntut'h lir I -\
| lL
ị: l,.osl l(t0nl -- ltn).'l"in hitltr Iltfít'trrtr
i [i
tliin dp o htti clLiu do(tn nl(tL'h
Ldi gi(ií Theo bhi ra thi /61 = 1oz ren tdn-e tro
Zr=Zz'
Do d6 n2 +(2,, - z, )' = R2 + Zr.2 ,
suy ra 2,. - Z,' - +Zr.
m (l ) cho 2 nghie m 2,. = 0 (loai) va Z, = )/,,
Dien 6p hai driu doan mach c6 dan-e
u = 6oJicos(l oorr + pu ) Q) ir tre pha hon r.r lh g,,vdi
Z,-2, Zt -zzt zt
lantPl=
R =- R
Z,
i" tr€ pha hon rr II g,.voi lanrlt, =
-- ff' . - ff' . 2 -.),2R' >!!-s.ilu -6rr' 2R' R'
va, Irin4*rinC)r.a '=,[""'2 - 2) \ 2 *[ rir9*rir4)ru2)
*( ,inL*rin4l ,r' r6n'' . o
[ 2 2) R'
{Nnan x6t. C6 it ban tham gia girii biri to6n nirý Cric ban c6 ldi girii dring li:
Vinh Phric: N,grr.vlrr Munh Qutitr, I lAl' THPI chuyCn
Vinh Phtic; Nghd An: Ngttvtitt Vittt Hodttg, lOTl .
THFrf Do Luong I; Ben Tre: Khting Hint Hiip'll1',
H ttinh Cittg Brittg, l2T, THPT chuyen Ben Tre; QuringNgAi: Trl D,inlt Dtuttg,l2A3. THm s6 I Dfc Phd' NgAi: Trl D,inlt Dtuttg,l2A3. THm s6 I Dfc Phd'
(-'+( \/2P.,q+ PB)sin: +( \/2P.,q+ PB)sin: > stn -.sln -.stn- x (l) 7r'^ > _-.lr x 4R' '[r-.;t;4.qh] (.Á B' c') Isrn-srn-.srn-l\ ! 2 -/ rit (4) Tt (3) vd (4) suy ra Qr = -Qz
tanrP, = -lang, vAY
(s) ( l0) Do d6 ir = locos(l00zr/ + eo - Ar) = 1,,.or(toonr * 1) t6r " \ 4) i: - = locoslroonr \ -]l t2) (1) TU (6) vi (7), chf Y din (5) ta c6 Tc .. It {Po-Qt=; tSl. 9,-h =Qrt*q'}' =-; (t)) Til (8) vi (9) suy ra Q,, r,. = + 12 Thd ( 10) vho (2) ta du-ọ c u = 6oJicor[,oon, " --"-. . i] l2 ) o NGUYEN MINH HA seeltr-roror TffinHffi zS
(Nnan x6t. Nhfing bz.ur sau cliy c6 ldi gihi t6t:
Ngh6 An: l-? Vrin tlr6ttg, l24l, THft Quj,nh Luu 4.;
Quring Ng[i: N,grr.vlrt Tdir Ddn,g, l2 Li, THn'chuyOn Le Khici; Ci Mau: Drong Tltii Drottg, 12T2, THff
chuydn Phan Ngoc Hien.
NGUYEN VAN THLJAN
*tt,il L,21381. 'l'ir n()r khi t'itt
t'ittlt mit ,li,t mit khottng l5rrrtltrttg hq thtilt t'ríi t'tirt tit' t'1 tltrttg hq thtilt t'ríi t'tirt tit' t'1
2ntl.s n,qrrr)i trt 7th6ng rtr mit t'til
the o lthurmg lhin,q .clil'ng,
htr6'ng l)n, tran v6i t,dn l6c r,.:
lSrn/s clrii t,tíi tttrit tldt. 'l'int
klyt,rn,q t'ticlt llttt nlkir ,giit,t khi
t'tirt t'it t'til trong tyrti tr)nlt rrti ('fut,q = l0r,/sr. 'l t | ''' I lo l,
Ldi giriị Chon truc O-r hudng
thing dfrng lOn tr6n, goc O lh vi trf n6m vat.
Phuong tr)nh chuydn d6ng cLra khi cdu vh v0t
ldn luort l):
.f,, = Ill = *2t : xz = r,/ '2-4t =l}t -5t2
Khoing ciich gifra vdt v) khf cdu cr thdi didm r li
l=xt-xr=201 -5t2
DC thay / dat cuc dai khi r = 2s (tru6c khi khi
cliu cham dat), vd gid tri cuc dai bang 20m. D (Nnan x6t. Ciic birn ctl livi girii d(rrg vir ngin gon:
Thanh H6a: N,grn'lrr Tlti 'l-lttr ltd. l2Al , THPT HIiu
L6c ll; Ngh0 r\n: Ngtn'itr Tltutrlt Sott, l0Al, THff
Di6rr Chiu 3, Li Vcitt l!tr&tg. l2A l. THt'l' Quj,nh Luu 4;
Quring NgAi: N,grrrht lloii lltio, l0 Toirn 2, 'IHPT
Chuydn I-e Khidt; Binh Dinh: N,gd N,r;ọ D6il,q, lOT,
THPT Chuyerr Le Quyi D6rr; Kon Turn: Cutt'l'lrtttrlt
lli, l0Al. THn' chuy0n Korr Turn; Vrnh [,ong: Ngtrvitr'f lriin Qrrutrg, lOT2.'l'rLitt Hotitrg. I2't, THIT
Chuy€rr Nguyerr Binh Khi€rn; Vfing Tiu: Pltittg'iltutrlt
Hu,- 12 TA7 THPI Viurg'l'irụ
NGUYI]N XUAN QUANG
Doc lai cho dfng
1)'ong'l'l{'lT s6 390 (thirng l2p1y)tl1trang [7, d€'l't t/:tltt
c6 su r-rhirn lin, nay xin cloc lai nhu sau:
-&-n L=kr+kt.
.l''@) l'b)
Thirnh thAt xin l6i tic gizi vi ban doc.
]'H'Tf
í
THE LE (;UI RAI
CH(} TAP CHi T(IIN HOC & TU0I TRE
o llii cirr clfnh rniry htuc viet tal' sach sị
khong clap x<iạ trcn ntot rniit giaỵ llrnh
vC rO rlng. Kh6ng gLti bdn photocopv.
Ncu biri cla chc blin nOn giri kin Íilc.PhOng chir nOn lh unicodc. PhOng chir nOn lh unicodc.
. Mtli bari dii khOng quri 2(X)0 chrlr hoac khOng qlrir 4 trang Ạ:l chO hlin vi tinlr
. tiai dich clin gLri kcrn l-xtn phtttocol'rt' biri goc.
o M6i đ ra cldu c6 kirn lt)i gir.ii viikhong ghi 2 de tre n cirng I td giriỵ khong ghi 2 de tre n cirng I td giriỵ