Toán học và tuổi trẻ Số 150 (4/1986)

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Mời các bạn tham khảo tài liệu Toán học và tuổi trẻ Số 150 (4/1986) sau đây để nắm bắt được những nội dung về câu chuyện học các con số; đếm được và không đếm được; chung quanh đường trung tuyến; bài toán J. Garfunkel và một số nội dung khác.

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Nội dung Text: Toán học và tuổi trẻ Số 150 (4/1986)

BAo ITA EAI 'rrrANc. m0r fit Tdng bi/.n tQp : Nguy6n Cinh ToAn Phd tbng bidn tgp: Trul s,ll : 78 TrIrr liung Brc "- IIA NOi Ng6 Dat Tri OAy n6i : 52825 Ndi cfrrc-r'dn ahi c6c ba;n tr6 vAu to6n c&w fiffiuy$m B'BB ffi$&{: CAf, f,$il ss n6 negc I}IEP (r d-6n ioAn herc lA nguoi la ndi nga.v f .noiviec d nu&c ta, trong nhiing nIm gAn {l{; c6 d[n siog iao: mC qul thr.rc IAm toAu, rgt trAo luu luyOn toAn phb th6ng khA rAnr 16.. hge teAn, Ap dung loAn, cu nrA-y rudc lA Khong'ir thev da nbi ridng, khonfir rrd dd 116 - hdng liBn I Ilgt: to{n, vf,n rlurrg to6n iuOn lu6s thAnb diSu luy0n rdi nghg rhuAt glli ctc bni dDi,h6i ta phAi s6dg tao. to{n hh6. \IQt e6 b4n df, lhuc grr thAnh tlat: ilE Vd-r' riri la pl:ii sAn5 rao toin nhrr lh6 nAo ? giAnh duge nbir.ng giii thr&ng cao uhfit, ill trri C6 bnn cho rlng ra kh(,ng thE sABg t4o terAn thAnh nhiing nh{ toAn hqc thrre srt, df, rhAnh ho r cluoc vdi vdn ki6n thirc plri tir6nq it 6i. cdng trong v rt\c ilra tc6n lor: vdo phqc vr.l Dd chi li nhfrng ban quA bi qurn vE rrrinb. Lieh nhirng vigc ich nufc k:i nhA... T,uy nhitr: nr{t s* toA n hoe t15 chj ra rfrng rf t nhilu nhA toAn siJ c*c ban khfie lgi tr& thlnh { lho > lam LiAi {-t hgc n?,i t;{rg dI lAnr ra c6c phrlt. nrinh toAn l;ip r0i seu dd ehlrrg durTc gi hsn ntla. elri t L hqe drlng b&'ng v6n it (.)j c5e ki6n thrlc 6 trndng phb thOrg. NhA lodn hoe phip Fcrrnat dn phet cdn ng&rn ruii luydn tiEc e{i n thdi vAng non } d[ tr0i qua. "l'lreo rC_rr nghi, c6i chinh li cAc ban hl,t'l .,',, qiA th rryEr nbi ridng rAng phuong trinh d6 ehua cd rn6t phuang ph6p hoc terAn vA rln r t y-' :: :", n ) ? khOng c6 nghiQm ngu_y6n. luvgn tu duy toin hoc nr$t edch th&t dirug d6n. l't ch6 ilio s{u suy nghi ve vrf c giAl phuoirg L{nr toAa kh6ng hin ehi lA gi&l cic bii tofn trinh bic n{m tr& l*n bing c[n thtre {mgt dti kh6. Gi&i tlugo bAi todn k h6 ban chl c6 dutre flof n rnA Abel truric 60 cirng dl nghi ldi khoing 50% sing tao. Biri vi. lt rrh{t lA b0Zo de ) E Galols dfl dE ru$r cir mdt li. ihu5 tir toAn hoc thu0c vB ngudi d{t ra b$i to6n. Cho nfn cai ma ngAy na]. mang tOn 6ng.. Bt qu-v$t lf & tIt, tlitu quan trpng rnA chring ta mudn thrio lufn hp ehiu khtr r1*o sAu ruy nghi, c6 piuong ph6p ll iAm aae dlb c6 ttuoe s6t c6i iox kir; tue lA tu duy dring, phuong phAB sdng t6o dtug. v$a dE.rAng tao todn hge lA th6 nio,
etrg i,to c{e ngAnh hhoa hge kh{c toAn hge = o,r.l6lt + ... + dt1 0 + rra + ddi b6i eu slug t4o tr6n trlng budc di nho nh6. * {or to*l + ..-F bn.ro*k}.s Toln hoc l:ri rrrang flfis lhr) cirn sq tli duy 1r'rin v6i 5 = t 1- r0*'t -1-- riit:k i-... --.. I + r0*k.S tugng. Eiri vd1' lrude lre t ln phfti n{m bfit ilrtrrr: tu turlng trriu t*ong dC,. Snrr d6 {a vin drrng 1.. a{ing !qo n6 dt di d6n lhiLng f ! rrllng mrii la 'i'rtc l{ S :: U !0, .!}, ni'i cirr:h ilric moi sd thAp & trong hin th&n todu hr.rc I{0t nrit ilir: i,hri phin v0 hnn ,,,,in i.,nin li nhirng s+ phiin. nfrrr crin t()il n hoc 1i liirlr Idqic ch{t cht:. \l;i Iliii tird (ir s(5 htirr li iirrirc -rtm nhir iA cA ud diBu n(ri, t'i6t ru s:ru pLii t:r hQ qu& cirrr rtrirng thip phin I:iiu l\ao hr,ir:: vir Iran rihung tuiin diEu rlA xo;. viit ', rl ttd duc,rc miuh chrlrr:g tinh [rotrn. ]l!il rir'irh trr nhit'n t:r s0 hfii pbiri chdng riring dfn rira'nd. itrirr v*y hirng ngdv ia phii rdn rAs sti thlpr;ihin vd ha* kh0ng turirr hoin cirn,8 luySn eieh tu rlu-v i:hr-rl ch€, ddng thoi r€n rto vai trd c1u:rn tiong nhu cA,'sd hilu ti. Trot! lu.' Qn mOt tu dr:r' phfrng khoirng s{ng lao. DrLu gi:r dilh cii:r cAtr con sd drrsc tttra thOm vdo quan lrong hcvn ci li c5{ra11 ta ph&li r0n Irrl'Qn ciie s6 ti:i1 p phin "iO han kh6,rg tuIn hoirn b:'it .-..4 rr:6t "t.r rnln!l ci .1 thr.r,' su. r-r lii Chring rluor'6-,,;i l* r:Ac sd vd ti, nEhia li 4 kh,1ng lilb vint lhrir:ir i'i -:rj r"ir* s6 nl4r)\'0n. \'6i --t i.hring iar lrii i nrir r6ch gilo Iihoa ilai s6 lLrp l0r l? r'a. Chu,rrrg trinli xr::r ra thA.t giirn rIon. khii ni0nr sri n-ril r.ij)g 1l r,'15 tir0 Lhlri cln bic tt_v r- cac sJ drr*ng ttr'(-. C6 ii cii*;1 cl tl*iin ni{n: nhr vA-v n€rr kh$ng it ban flir. r oi thrr ,*g hqc l- - thu_v6t vtr chi lr,o Ndu ren, xCt ici lai t{L , i i,r th{y ring 'rong 11i srti tHnr, th6oh r"ld uh:iu nhOng bit tr,itn bibu d,!0n $6 thep phan. tait cir c*(: ch& sri rlFrr r l:rlaa bb D. 'f hr-1 rlt)rrlg nSir ba* hoe thril k-t, vi ix c*.r:.6 lilr0ug qriL 9, tiri: li\:6 rlrr ctir lAc d!t chrr,rng.l r.:\ [l lhi .:irlrg si ].hiis ra nhibtr ;rh6p riria riro l0 lhur: hl$n nhii,u lin. {. ir rnit eiiErr dlng su-1 il::lli.' t{i}u rhua tir*r ha.y lirrrr lA r6rr ta ldv rrr0t s6 Phln 3 eirir truung I n6i vb sd v0 ti Cht'rng chia . [ro lfl (lu :l ni:ln \'_6i rriQt si'r e hit cho l0 tr l-ridi- nhrtrg 'r;d rlon iiiin ruh[t lil nhiLng s6 tu drr 5 ia c! trnc si, cl:ia trdr r ho 1U, '.f c li dLr 0. 'lr;.lr rlar:3 riir .ii ahi0n 0, '.1}, :]... Vrri r'{te sd n:i-v 1:r co tht c{ng, ;..1"r ltr rl,o la l;it sri ngogt nhAn rnQt tirch khir tu nlriSn.'frrv nhi€n ctng l0 klr; tlrur h qri ..'*c ph(,p tinh nhin, chia cho phii luu 1. c6* b+.r r*.ng .db qrian n.iQnr duqr: cic r6 loai nni tr(rn. ia rrrtrr.rn khie phuc tinh trang nrlv thi ,hi i:(r cAcl: la trt thay s6 ttt blng rd.ng sd 6 eir*g ql!an trgxg nhu. thim tlri cda rnOt s6 nELr_rr:n t6 bet ki p. Iihi J6 r'Ac chir sd c6 vai trd rlii,r b;Qt hon, t:[c sd t.2,3... Ioili ao. at'. . ., b , bA, . ". chi nhin ci.c giirlri0, I, X,..., ngwiri rii phAi tr[i qira r{t nhiEu n5.nr trong }ich rr phiit tribn v[n rninh. lio nhu cIu cICri sdng p,- 1 M,,i dieu,- n6i 6 trSn v6.n ed J nghia vOi loAi nguli phii lru, ph&i nq {ghi nhd) nrA p* thly tho 1ii*, .S:0,cIca...(k v$i ci = p = I thr_y e ho .\ a I09 ..I fi tr0n. \'ay lA Lan dn di ngtrrli !a ilir sirn.q too thEni c{c eon s6 - l, '- 2. - 3.." tilc'lrt cf;c sd ngu_r'6n. Am. Ctng do dda ii thrr!6t cac :ij 7r * pha n, rrrdi chucrng tlua n nhu r iu dr:i s{iug phii chia Is:. mfr ngrdi ta dE trorrg cri* i,i' ,iru"r Ji sii tr;4n clai. Ngrrdi ta cll slng t:ro th€rn cAc ,i6 m(ri IA e6e sd phArr, hr-v dring t'Ac srj p'- gririln dl r:tri ra ring phuong n6i r{ch kh6c li e{c s6 lriru ti. Cing trong lt inh lrert:ral 'k.ha!(ig iLF giii tf uoc bing phrrong phln t'lIu si,"+]: l(rp i0l,] vB FIinh ]roc bau ilucrc p hl.p si: cl-tp.. bi4t rn0t lfrth ctrr!ng rninh rlcrn gi&n r&.ug s6 il,-, Chtin& la tr'& r F ';rii phr''Fr kliti ciln. \'i la dlinh rlr.rclr,g eh4o eira hirrh vud,:. dt.yn l,i ktr6ng i.hts nghitr cirn ,itt' /i ,,:iia nr0l sr) lfi rlrQt .6 nrA liry lA .s11 lriru ti. i.v ia ndu clri :iet li sir g, frr cAt: rhu:r bir: r't llrl h."L*g sd itrn dlu, r'ho DOR cIn s6 ngrt.r'f,n tn lii;ing thb brhu diCin dQ d&r ciur hrir: 2 ,: iilr l'ttr'rl sil int [itriril , ,', ni lrilr Nor c{eh dudng i:hdo hinit vrr0ng iltrn vi. khA, i:\ Trl;u
dinh c&e s6 ta thtm rnQt < sd kj. la r i, !l = bit kb hiQr sd A tnr ha-v duong hay'blng kh0ng. - l, lf thuy6t s6 dai e6 triln tlgi Nhu- vAy, trudtg cd ph&c ld m& rgng cria trudng s6-thgc b&1 NEu .{ ( {i rlri nhtr
7' sU m0u thuin. Rd rAug lfr chirng ta iil klidng 0-*et-'**P $*"*-'+ 5 ... tl6rn mA v6n bi6t t*p hqp e6e hgt e {t li. hitu han / ,v ,/ v Ii$t quf lrinh cl{m bao gid cirng cho ta m(rt . ,// I /r' / sd phlrr t* erl thb. t'rcng dinh. nghia t$p hqp *1 *3 * 5... hiru han & tr&n, n ehlnh ld s6 phln tit eira ,I. II& r$ng kh6i ni$rn ed ph&l tir' ngndi ta gqi I /,v i / ./ ,/ hai t{p hqp lI crir cring,'lqrc lugng u6u e 6 ni$t su tuurg ung 1-1 giiia edc phEn tir erla ching lr/ 2d 3,/ {_L P,6 rdng td, hai t{p hSp hiru han c6 cring lLrr , ,q NA ,t ,) luong hhi i,A' chj khi chirng c6 edng s6 phrtn lit, ./v Vi vA-v t,6t t{p hgp hitu h4n khdng bao gi6 c6 !/ cring h.rc lugng vdi rnQt lrip cou th{t sr"r cfra r,/.)./ 34 5 n6. DiErr ndy kh0ng edn iltng n&* al6i vdi nrdt ,) ),, 2 ; thp v6 ltan Vi du nhu cAe sd tr; nhi€n vA cAc tz t/ s6 duorg e bin c0 cdng luc lugng th6ng guri sh' [/lr tu.lng ur,g n dtdn ?n. 'l'bng qu{t hon, m6i mOt 1 ./ ',) 345 tip vO han bao gio cirng chua m6t lip hEp cr:xr '..- ed cring luc lrgng vdi lAp e6c sd lu nteiOn" Thflt 33 3:1 3 r,Ay, cho tt'u&c n phln tir khAc nhau a1,., t Qi CAc ban cirug cd thb h6i t4p hqp t{t ci c{c etia. m6t tAp '.'d han f , ngLrdi ts ludn luOn c6 tir} tim thilv ri6t phXn tri oa41 cia Y lihAc vdi st5 thnc c6 phii ld rn0t tip hqp ddur duoc her: klr0ng. Cil.u triL ldi e€ tA khCrng. 1'hAt viv, gii. i:ar,r phhn i& trcn. l)iEu nir^r cho phdp ia xAy drrng urilt dfry vd hilu .c6e phin tir irh/i* nhau rit ta cd thb sdp r{p l{t c& cdc cd thqc thAnh rnQt dny vd han u|, uZ, us.... tlnl' biA,r dian cec ar, oz, r.rg,... cira I. D[y nfry c6 cdng luc lugng sd thuc dr:6i dar;g cac pL;l* -rd thrlp pi;An r'6 han : v#i tS;, c6c sf Lu nhiFn thOng qua sU tuong ftng n ddn an. \hrr v$y, ta c6 thB coi c&c sd ul ,'nl * 0, 1::".,. tr nhirln co lrrc ltr,:ng nh6 lhAt trong cile tAp v6 han. UZ = n? + [i' o21 c22 c2g uS = ng * 0, ca1 c32 cts Il6t t&p hop c6 cr\ag lrre lugng vOi t{p irqp cAc sd tu rihi6n duge goi lA /Qp hqp ddm dugc. l-.i di: li l.l&ng m0l pb6p d6m ihOng thudng ngu
'3i+ "d pttt p3,,... pl" $ \ r ": st $fit vrii ai, Iri, ci Khi d6 ta ngu-r0n vi kli0ng tm *$ &r#c r,6 o'] b5f [a, a] = ,1"u'lnr' . omat['rl" . Bdi I/1,1,?. Voi mdt s(t titrgt t to kg lridu lr]1ri td ngugAn ldn nh(tl klfin11 ttu'ql rlud :r. Hdy tim chtr .. (,r, oi p|'rialor' r;rJ .. ,rnirr[ao' t'n] ' sd t6ncdng {hdng rlon ui) cri. td tt/s-:+ r/1'oe0 = -"r \'r . Bdi qirii: (Nguyin L8 l)t1n9,1 gCt. PTCS Tr0n '. v. r' .. r'Jr i , Qr6c '!'r,.1 n Nha Trrng r 5 Fhnm Qui.lnh IArr llCT. t'r' tr} ... pln"*[';n' bo c;'] PlT'llLa Qrrt t)dn Nhn 'Irang, Ld Iltru 1-ft)ng I IB' la,b,,:1.,,, plnax{or; PTTII Lam Son Thnnh l'i6s, #{im.Yudn Son llBC]' bn' erl br, cr] . Vinh Phri ). (s,b. ci - pl"irll,i , ,mi*:lon . i/s+/l]'seo = ;+4y'6 )ees Do dd ta ehl cBn el,rrlng ,r,l ,tt.r ur; nrdi i, sd x4r a :5 i'r-v/6-vt b = s * r/ti D6 Ii trai mt cira pt & hai r'6 b&ng rrhau ha-v ih ogh it nr cric phrroog tr iuh - 2nra: ..- { /r,l ' ,na- l1,l ,'il * ,: - tor * I = o, loi, 1,i,,":l ma [,,i - max t r're ld ,,?*ioa*1=o lci, ni] '= b2-tfrbfl:o :2ini:r l,ri, bi, ci| '* min lor, r,i] * rnin lbi. cif * T't d6, vrii mgi s6 tu nhi6n n ta ilEu c6 min lci oil oo*2 - l0ontl * an:0 Do vai trd crla LIt. Iri, ci d6i rung n0n ta co lhb b"+z * i0b'+t i bo:0 gii thi6i ai>. hi) cr. Khi dr: ding thuc t16 thinh Kf hi$u sa = (lo * b" ta c6 h iln nhitn : Sa*2 -* I0sortl * sn : 0 -o1 * bt - ai e Zci -- 0i".' ci -- ci 2o1 l)o vd.y Sn*z -{- 'So chia hdt eho l0 vd Nidn ;rit: Fhin l&n c{e bqn grii btri d€n sn*4 - .Tn = (.f614 * ,ru*z) -- {so+e ;l-,rnl ehia h6t 16 lqi gili ilirrg. IiiOng ban tlai Phu&c Tu{n'tBuAp- cho 10. Do sl = 10 r,fy.91, 55, 59,..,$gsg tlbu ehia dqng.br}a il&ng thire al6 j v6i BCNI{ vi tiCl,i$ efra h6t eho l0 hai s6 eho ba s6. Nhuugdoo(i-:{o-(rne, D B.K o( (s - r{iltess ( r ,.+y l(r r:/o lsq31 : sre3 - I phli ll s6 ehia cho I(t du 9, tuc ld chi! s$ hAng don vi cfia no lA g Bdi 3/147 : Hdu rlic dinh a d? b(it phrong rVft{n rdl. Bdi nA-v e6 th! giii bing cAch khai trinh saa ddg c6 nghiQm dug nhdl" tr?r rIi thuc Niulsn cho (St2y'f1eoa un logoll * rog*y'[:-2r,3 Y . (5 * 2y'6'-;oos nhrr hdu h6t e6c ban khfle tIE linn. x toe"({;z:;E- +1) < o. N'8.7 LdE ilAi: ieia c6c bgn Oodn anfr l"rung (Ngh$ Tiuh); Etnh I'tin Hring (Qu&ng Nam *,DA ltiS"ng))' Bdi 2/147: /{g liifu (a, f ,... e-l uri [a, b,,.. e)ld *'2, * tlO" ASCaN ud AS,-Nlf crta cac s(t" Hdg ehung minh DiEu kign: c)>0, a*t;or2 rdng attt a, b, c Ld ede sd tq nhiln thl Yoi 6lBu kiQn iI6, dlt {;z:2r;; * t, t>s t, ta e6 tht h.[t phuong tlinb dil cho du&i [a, c1" fu, b,cjz "i[1 dgug rau: [a, DI . [0, cl . [c, a] (a, b] . (a, c). (c, a) log"?.logz 1l (t+1i. rogzVil+l- *(? ( log" Ldt gidi: Fh{n tleh ra thr}a sd nguy0n t6 1) N6u a ) I thi f '.tt =' log"{1-f- t; ' togzVts*2 q=pl1 pi, ... pt,, b = p?, pls .:. plo, la ham dbng bi6n khi l>0 rn /(3) = lega4 X x tog2y'[ = ]o8a? X 1og211
I)o vAv 11)+i,)>::t"y,rz-2, *t)S g[t t986 1986 . phuong trinh nAS,kh6ng thB c6 ughi$m duy, x sin{i -}- l/2 x : llsinat} n lrA t. iel i=l . f '-Xtir 0 { rr ( I. lihi d6 f, l) ti 'bidn hAm nghich khi tl0. Oo vAy (l) e t(l ,t.t:z-2r*l}a ha1. ,r1/ 1 i-1 986 I 980 I{ or.," *:t^ S; -I (:r - (2) .',o' i=1 L Tt (li r."i i2) c6 : I 986 I ,r*2-rr,,*2(o '" -- 'u'?-2"+lVo r2:< '2 lfoirr:i ' liit Lut!n..Kh0ng tSn t:ri a db bAr phuong . i*1 i=1 lrinh cd nghigrrr du-v nlrit (loi s6 e5* siny"i h{ng l/2 la 1r, srl cac sinii v.\{ hlng I !d q t:i c6 \lii 4l t47 : Tim tfr.t c& ta() hdm s6 Ntri^: dlah Ir,*o=1986 lrong 10. *l tnd,t n,fin dtcu A.:ch: {"ep-g=:9C3 Y t "" lplz+q*(2piqt2 /(r* s = /(,). f yl y .r, u> 0 viy gi{ trj I0n nb(t
DIu bdng xiy ra kh, ccs!.r : slrttr +> a) nr6i doan nflm tr'6n nr6t canb.
4ectga = i4 |rt)z - ,uNl)ir:. J/^V (3) Tu rliEu kign dt tr&i su y ra diQn tir:h toiln Lln luol rct cAc laru gi6c vuirng pAb vd COD philn tr di$n ASCD ntrri nhir khi YA chi khi la rlrroc: difn tieh xung quanh (kt hi$u Ii S.\(t ,) rrja tir dl,D? *,t(tit;z -t.r2): 4.x? + ltct-tz *,12) dign d r nbo nh6t.. Ta cd : Do tCb = ilIN *21i nt:n 4c D2 xM.,t2 1- 4lt2 + 4MI/ Il, 2g-x{, ,rl?+h, +by oz*;, *.1 -,r,i Vi li-v: Suy ra 4r2.+- 411x,. 1l t 4(1tz - 42) + MNz . *= t,tN1*t/x. lt + tr2*{n? ApDz, (..2 n hz) + bz (uz r nz) * ,'2,;2 * h?) +- * 4sir ,r2 4pD2 * a2) (,l) 't'u (S) vA (1i ta th{y ndu chon , + 2db'l (* . hJ )( yz * dz ) *'lo cf (r2 -r h2,(:2 r n2) + = .,1 ,tz - tlz thi si cr * zl. { ty'+ r,'"rzJl,l (t) eorgcr d2 * n' :: c6Dst. : till ^ip dqng bf,t tI&ng lhrc B,rnhi:, crlpr tii cho cAe Tri cAe l*p lu,irr tlAn su-y ra dilrn ^p n*nr lr€n Irr] s6 (r. y, h, h); (r', ;, Jl, h); r.e (,1, z, h, h) OC va cAeh 0 rrrOt hhoing lt| = { ,p * Iit* lir f i) ta duoc i lh6a min tlit e 5. cSc tIiEu kiQn .ria dir bAi, vd aslq.2 o!(rz +nz)4^*(u2+hz)*,'(:2-t h21 +. _t--;-;: +2or,(r'g+ht,f tac( r:*h2) IZt,c{q;ar,ty rl} khi d6 r};,' . a: ar.,.!* ,,'" ji n" . r)ibnr p' I)Au r = , xiv ra khi vi chi khi .r.ih-gl1= - 1, har .r= U =:. SaLr khi khai tr';Bn r.A rrit gon r6 cl6i rir'rng rrii P qua tI cu,rg eo tinh chtit tucrng phii cirn (2r ta c6: lr.r n iru rlib ln P. (2) ++ ,tsiq > (n * 6 1-c i.,2 + r.-l lr,) t r.r 2 ,_, '2 )lhin ril: 'l'rang tdt ea r.ii;,: barr grii t,ii gla; .* lslq ) {o + n * c}2. trz * r.si*. t6i tda soan. chi cr ban l.,d ,\rrTnyr'rn Dinft r\oirydn c6 loi giii drrng. os,t)- \/V-,it+h1c\'ztl + ^2 \,\l,C -corist. N. it It Tn' d6 srr! ra 5",, nhdn gi{ tri rrhir ir h{i khi vA Bdi 9/117 : Cho ta diln AtjCt) r:6 AB = c, ehi khi i'= y =:. \tij' diQn tir:h ioirn phtsn tri ilC=a, CA:b cr-, dinh, cdn dinh D chughil tlqlnq diQn ABCD nh6 nhSt khi vA ehi khi r = U = -- sro clro kholrtg crich tt'r l) din ddA (Altt,:) IuAr .tric khi vA chi khi hinh chi6u ctia D rudng mIt lufin bun11 h &fitrn.r tlbi. 7 inr t'i tri tlinh D tt\ di(n phfrng (,1I3C) trrlng vdi tlm i!udrrg tron'n;ri tidp tlch todn pltdn /ri dt€n ABC,D ld nhit nhdl, Arrc. Loi oidt t Nhqtt rd.l : I)a s6 cAe ban e6 lui giiri khirng rl -holn hiio do b6 qua tru.ong hop Ii nim nge,ei tarrr gi&c :{BC. Cdc bqn godn Anh Trang, r.a Chdrr Ittnh Tri c6 ldi gili dring nhui:g dAi. llan 1.C Mru llrtng dilng phuong phAp toa cld dE giii ring cho loi gi&i ilung. N Ii. 1I Bai lO/147 : Cho tit diin ABCtt co cae oe lh ld ilt, a2, al, a4,rr5, 06,. g9l rtt, n| p, q ld, r10 ddl ctic ttaqn thdng adi cic tlinh cfra I{r rii€n udi treng taot mdt 16l dt6n udi di nh tlg. cl*ng mtnh he ihric: ,& i{A DTT I. \AIJCI, IIAI L BC, HRI LCA Vd It(:r L.48. Khi d6 ra cirng cb DAtLuC, DhLCA tt tt:1 )_ trg DAt 1i^dr =lLl, ilRr = lul ra //C1 = l--l 'la qui udc r)0 Ahi r'A jf nirrn eriug phia "{ al6i vdi dudng thfing .BC I 5 < 0 rrong trudng i, hnp nguoc lail o = 0 klri ti nlm tr0n BC. (.)ui rrdc lu*ng tU d6i vdi y vA : hi d6 ta rE c6 : .!i . d.t * ty * c: :, ?$ass I
FGI I bdng hilu binh phaon.o hot s6 uta q6o. H6! ca E"? ntw Ltdg co Lic ndo trd,nbing abm todn it 0 kh\ng? ^2+u? "p'1-q2-llgx i=1 Ldt gtii: (.Tbeo ib4n NgaL,0n thi Ngug€t 16p ' Loi : Goi 41, Dt, Ct, D! l:i trong -t&m cira ,iiai S1' truting PfeS lioing Phuc, iloing li6a, ehc milt d6i diq:p c{c rlinh A, Ij, C, D. Goi M va 'l'hanh ilca). X l5n lrrcrt lA dibrn giira cirs CI) tir IlAt. Nlrfin x6t r[ng bi"rh phuong ciie m{t c6 ngur'€n chia cho 3 h itc du I hotrc du 0 (tlrr I khi s6 ei6 -\cr 16n lrrot r.Ac A A,tM uA Allrln rJrrcrc: '1..3, au 0 khi sd il/, 2AA| '- AVz + Atiz - KM'I ) : 3) Tti d6 srl]'ra hiQu binh * Att!-+ r/l x (L82+.1.i? -, 817,':\ ,- KM?l') plruong l sd ngur€n khr)ng chie h6t cho 3 khi 'zxvz vA chi khi m0t tronq 'J s6'd6 lih0ng chia h6t ..+6,{d? = 1A}12 * 2ABz _ lt,\ L- t q .'j I cho 3. Vi. vA-r,seu rndi llo ri:a 2 s6 r'i. th:r1'vrir: dtl biOu hrnh phuong ctia ehung. sd cic sd kh6ng F_ * = .1.4,y2 .{ ?.1.,3 (1. ,,,; - r( ! e.,\' \.r / chia hdt cho 3 ho[,: khOng tha-v clbi bo{c gi6m tli :1. MA trcng i00 s(' ltr nhi0n ddu tita c6 4 = 4AlI2 - ?.{g" Ltt' . (l) lco - 33 = 67 s6 lih0ng chia h€t cho 3. ilo 3 d6 trdn bflng lii6n ludn co nr6t s6 16 c6c sd l-in lrror xer c{e [,, X:A khOng ehia hdt e ho 3 VA-y khdng lhb e6 lul niro vn I Bf. I) ta c6: lldn bing gDr:r toAn sd 0. Auz: l/rx ( A()2+ JDz -cD212\ (t) Nndn .i//: (.6e bau Lrlar ltbng T'lmnh, ligLyin vd BV? = tl) Bcr+RD? -cDzl,t) (31 Quong :tftnh, Phan Hbng Vi4 . Ngd Duc Cht(n, 'fhn1' l) (3) vio (l ta riuo,: tjhu ludn Yinn (lrtp ?A tlrrong P TCS Hirng, Bing "A llii Phong) cfrng cd ldi giai t6t. 6,4al = t(Ar:!'*.tttz cDdlil*2at2* .. B. B" I( t2/li*7 : Cna A ArlC ciG '= 1")" ud '(ttcz+Rt)z .'rr2\ 3\ t l -\BAi C = t32e. So.srinA dO ddi phdn gidc ngoii cac ,.1oc I) ua C. + 3.{ {? ", tRz-lAcz I ADL - t,Ls(nt:z *'gnz i (:Dz, . {4, Ldt r_oiiti: /-\Ta erl //\- ll .tc- '30".-;ct' = 24".-.{NC-: l}60 - 2lt : 12'' -/-\\ Tr:ong lu la e6 : il rnh C q ]JC : (lN 3Ii8f=6.1e1 BCz + Bn2-tllx t.2ycoz +-na,2l (b\ Suy - ra [\ nC.l'cAn /,/-- //_\ zct.! = c # + c 82 +c D2- t l.tx (a n2 a- a D2 + o.qz ) \6t Nlrt khac CB,tr:810 vd BCII.:180'' l3?o:l,Sa \DD!= psz + DB? t Dcz'*r I3lARz vRLz + cazl (t) + naC: ;8oo - 8la - {84 : l8o, C0rrg r'€!. vdi r'6 cic dLi ng ttr ric (4). (5 , (6), (i) S*y ra Lnltc cAn dinh B ') BC e Btrl ta ,lucrr': YAr 1tM : C.',V (dpcm). Nlidn rit: \6i e.hung eic bau giri bii tl6n atrEu AAI + Brizt * ccl . ioi .7 giAi direg. Rdt troan ngh0nh cle ban & P |CS dd kip thdi htr6ng t4g mgc db re ni-r'. = * (rr, + Acr + eoz *Bcr+cn2 + naz) s N 0 **'\zYY' -1 hry m2+nl*p,-l- r'-* : o12 (tpcm) $e1 i=l Nlrdn sel: CAc ban Phem Qainh Thu, \,6 Tdn PhLil, Cltfiu tlinh Tri, 1'd Dnc Ddng, &ao Xttfln San dtng phu ng ptrAp r.6cto itE gili cirog cho ldi gi&i ng{n gr2n r.A chinh xAc. n , N.K.M I t^t r i BAi ll/l4z : c6 g{do etti lan b&nq 100 s6 rv l"p Y nhicn ddu t,en. rbi qol mlt ioc stnh lAn rde 2 il{ sd bdl ki1 uAghi lln b&.nq sd binq ht€u binh phtronq lroi sr) uia xda. .Snu d,i, nfit bqn khric lql t6a dl - p,r 2 sd bdt k! tct l!r? cd sd uia ut(.!t ut lei utlt l€q
Bei 6/150. A. B, C ld ba g6c ,:iia A,{BC (do bing Itaclianj. X'im gi"L tr'! nhi, Ehdt cira bieu rlu'c M -.(A * Ilil"tBzr.3 lihi A.4,BC rhrrr- dbi. ,\'grryin Ffin tr{(iu BAi 7/150 Giii Yn bi-on Iuiin phrrong trinh : V sinl. + 1v4t + V;;1;+ * '.tr) = -V *.-fr- + V .*'rri -I gur;drt Ydrt MAu Bdi l/ iSE. Iln1. sd rhue l,,.l,I* , xic rlinh r- ,- -s+ 'nhtr Bni 8/l5O Cho tanr gilic ,lItri. Dunng thlng sarr l d qua 'l eit BC iai 11, lr6n Jl1 lii 1' O sao cho ri1 : l, rrn+l = 1 * otaz.,. ao, Yn)l. ,lO: 113X.{f1. Ndi fi v1i O cit AC-tai.R. Ndi n C' v6l 0 dl AB tai S, { Ar diFrrr R', S' Ihn lrrgt D{t 5. = : llat. Iinh lim 5r. ld d6i xitng ai"ra 1i vi S qua O. k=l n*- Qua R'(S') k6 clutrng thi,,g song song vdi Ed Lt*c Thdt AC (AB) c*t .,tB tai rv, rit Bi. tt.ti N (cit Bt) tai P, r:[t ,{C tai 0). Xf ,: c1!nh vi tli riia d db tbng diin tich bai, ' Be! 2/15O. DBy s6 nguy0n cluong lr"ltO tarn gi/rc tlir lir nh6 nl:dl. th6a mii n rlibu liiQa sau : { riln Thiip 116 ( at'-( oo * :0, ot 4 ar ( rt * !t], n:{cs{or*:0.. Bei 9il5O. Clio tanr giic .lIl{,'roi che iludng Chung minh rrin6l : s6 thip phAn b = 0' a0l1a2"^ eao ha tri' hai dinJr:1 r.d B cit nhau tai U. trong 116 e6c chir sr5 erin xo, a|, az .. drroc vi6t Chring nrinh r&ng hai clLrdng lrung tuv6n ili qu:i Iien tidp theo thri' tq d6. lir trhong tuln hoi.n' hai dinh d v{ B vu0ng goc "-tii nhau khi vA. chi Bd Hang ARh khi hinh ehi6u M'cria i.I xtrr)lg ducrng trung tuy6n ili qua dinh C chia trong ciuong trung tuy€u dialdrn hai doan theo ti It 1:8. BAi 3t l5o. GiA thi6r o ( ar d ,a { ,. ( ,, Rii Vain Thanh vA o,r''[(n.f r),:l{ I Chung rrrinlr li ng Bdi 107150. Qua mdt .libm t/ bf t ltj' thuQe nrQt ranh eria m6t hinh lnp phu'ong co thB dqng ii i/, t, n,i! = V ! ,.-- \ I n lltr-i a;) - cluae bro nhi0u rndt phreLng, r ir"t hinh l{p phuong ,\ " "rlb i=i / i=t, rld sao cho thi6t difn tltu
\- Ttm hibu sfiu. ttu6m to&n hqc phb thdng l\ CffiUNG QUAffi${ BrytrNS TRUNG trUYffiN NGUYf,N TLPTING L clrrrlng trttrtg ttt'l tin thi cdn e6 r6n dE 'gi 2. Duong tldi trung xtr6t pltrit lt! rl6t tlinh nira ilb n6i v(ri e6,: tran'/ l)trrlng trung cira tanr gi(c cii qrit giao
t/' dr) cria hai c4nh kE, tuc lI SB/SC = rz lb2'Chung i KCo dAiRli'ra doan "liQ :: P6-' minkr tlnh cbf,t nAy dua tr'0n ntrdn x6t: Do rinh chdt 1 : KRf ,lB v KQls{) hay KQ'/J13 = IiQl AC va 01iQ' - -l' srrv ra hai ItB dign tich ABS tam giAc KQQ vd-;tBC dbng ,lanq vtri nbau. .\ {.' d i0n tic h .{SC Do tl6 g6e^KQQ'* C ha-v KP/lQQ', nhrr vd.v I(P I I't' di qna ditnr gitra cria 1iQ hay i(,Ii nirm tr0n 4. 'Ia co - _*==:--;-- Chr'Lng rniirh clrirng trung lu-v€n .xudt l)hit tri dinh P crla b" -f co dAnh cho ban. i.'l'bng hinh phrrong c{c liholng ctch tt mQt dibrn l.l{t k-i n{rn lr0n rn0t cqnh cfra tam gl{e r16n hai cagh kirr rE drrt r:rre trbu lihi dibrn dlri trtlng v0i chAn i!rrrlng rldi trung tuong itng vdi canh il6. lrinft Goi lr Id mft rlitnr b{t kj' nirn Chtl'ng trBn,elnh tr]C civt tam giAc ,IEC. * vi 17 ld khoing c6ch tri P ddn e anh A1J r'il ,4C, rt) ring ta c6 i. i) c;.*bu = 2.s (s ta aien tich tarn g;Ac .4BC)' Ap J1; $N J dung bflt iliing thuc tiunhiacdprlii ta tlugc ': tam'gi6e I'jQl1, Iuong lr.r chLlng rninir cldi v6i (c:t: * brJ)z = 4s2 ( lc? * b2 t(*2 * g''t, I(Q vn f(fi' Yey Ii lil trong tAm cira trrrn gi6c PQII. _a r c.l Tu lr2 * ,f ) = t*' dlAr su\- ra nrin , ,; 9. l'a m gi6c PQll c6 diQn tich bing*- r,\' c'* b- a'+ b'* c't' t ilat tai rli?m P cri tinh ch{t .r/c = u/0, tit'c li trong iki s ld dif n tich eira lanr giie AgC i Ir := S (chan il rrdrrg tl6i trung AS). e, b, c IA d0'dd.i citc canh. 0. Ba dtrdng d6i trung cia m$t tam giAc 10. Trong t{i eE. c{e hinh larn giirr: ndi tidp dBng tlui t4i m6t d.ibm. DE ding chfing minh trong tam gitc ABC thl tann giAc PQfi la tem giAc iinh eh:it nd.}, lilng cAch sri' dung m$.t trong cAc c6 tbng blnh pbuong cAc canh ln b6 nh{t. tinh ch{t kb trbn cira tludng ddi trung. Chri'ng minh tlnh ch{t ndv dqa trtn nhin ?. 'l bng binh plruong cic kho&ng c6ch tir m$t x6t: Tbng binh phuong c{c khcing elch tir rngt dilm bi{t kv trong nrdt phing tam giAc ddn ba dibm b{t k;:'trEn rri$t tluong th.1ng cho trudc canh.cr"ra n6 sE dat 6uc tiEu khi alibm d6 trirng ddn hai dibm c6 dinh cho lrudc sd dat cuc tiBu vrli giao dibrn crja ba rlrr&ng cldi tnrng. khi diEm d6,trrlng vdi ehAri iiudng vuOng g6c Chrng miah tinh ch{t nAy tuong tu nhu chLi'ng ha tu diEm gifl'3 eria hai rlibm c6 dinh xudng mioh tinh ch{i 5. duong thing eho trudc. .1. Ggi giao di&m cira ba dudrng tl6i tlung cr3a So bO qrra 10 diEu thri vi rni ldj dd ns6 ctng tarn giAc ,lAC ln I( Tt 1i ba bl duong vudn;1 c6r ban trong eAu ciruyOn ( Chung t1r-ranh tlrrdng g6c ,(P. KQ, {11 xLr6ng La canh ciia tanr giirc truug tuvdn ), cAu chul,Qn sE cdn dr'ri, ild ughi- ARC. Dibrn 1i chinh lA trong tAnr eia tam c6e ban hEy chring minh ti r.r.il cAc tinh ch{t giac PQR. l, 3. l, 6, 7, 9,'10 xem d6 ln nh&ng bei iA p phAi h.rtn thinh & nhA chric cAc ban hr:e gi6i Chirng mlnh. (xern hinh 3) mOn ToAn. eA* ffiF* T'ffi.&,N ffir* T,&'P Ldrp cudi cdp phb thdng crr sa l. Don giAn bilu t l: rie 2. Chtng minh r&ng {a2*lr2-,,? * 2Dcj a*b*c a* h- c (I.b-c" *J-- , c*a, " ) a-D\ Tinh gi{ ir! hrtu thrlc vdl a = 8,6; b = 8i;; I , x (\ Ir c ,i+i_) rg =3 , c-(t a bI n6uo*b*c=0 12
3.Chtn6 minh rtrng n5 - 5n$ * 1n chia h6t aetg (dl2) * e otg (lllr) * eotg (T/2) = cho t90 vdi n lA s6 lgu;'8n !i{t ki-. cotg (o/2) cetg {F/e) cote (T/2) = {. 4 Ptrin rir.:h ra ilrus. sd Gili. phuong rinh : (a*o*.)t*' ,s-63*.3 ,og"/u n2 * litugru, *3 + 4$togo* d i = 0 5. Cho tarn giAc d.llC. Tr'€n l8 vA ,{C, vE phia ngr-,ii trrrn giAe ta d\rng 2 hinh vuGng 5 Cho mQt rn{t p}r&ng (P) va 2 diBm c6 dinh ABDti vi ,,1C,11,V" ChLrng rninh ring truag lu"tdn J, B nim ngoiri rn[t ph&ng d6. qulr..'1 eria tam giAc dtrN, keo dAi clrtnh lA Tim qu! tiob nhEng dibnr &f nXm trong m{t dudng c:ro cfrii tem giic -'1 8C' pbiug (Pr sso eho Phqm 'l'hu ilti a) .{ldlJ - gt,u n) aez " Mtla = /t! oho trurlc +\ LUP l0 Di, Eirt thdi ' * t. C.h,r bz - 4uc )t); a$0: l,'l) l"l +lal Lop 12 ' Chi'rng rninh rl'rng phrong trinh ar2 + b.r * e :0 cd 2 nghigm ,L rz vi. l.xrl ) t. lr2l ) t. L Gi&i b{t phucrng trinh : t Cher hinh th;:ng ABCD (a1llCD) c6 AIi=l: ' (r2-y.ll***z;-o. BC = 4 i CD *'5 ; DA:2 'f iab AL : BI) f. Ch', hai palabcrl 3. Cho ,')a 1 ; q > 1. Ch{rne rninh rirrg I g' t2+a-2 ra -r :llr2 - c-:.1 ' tlF +v 1;-t --< ,c. ,IIlv viEt phuong ti inh cAc dudng ti6p iu1 6n :chung crja hai p'.rral-ro1. , 4. Giiri irh.uong trinh l? - ix - I = y'.'+J. 3. Cho hArn s6 5 Cifii phuonp trinh t- - t- 9: e'il *,r'2 (a - 1) + r (r * a) * i 1l ryUY ,tzl;L = l/,-1il i clinh ll x-!'!c-t . cit trBc hcanh XAc c6c gi6. tri e ia a d'& db thi hAm s6 tqi 3 dibm phAn biQt. ' ff gugin I'd.n .ll Qu {. Cher hinh chdp tam giS.c alEu S. ARC. Goi K lA hinh chi6u vudng gdc cia B l6n Sd, gdc gliia Ld p l1 mlt ph:ing {KBC) ve rn[t ptring (eBC) iA rp. Cho thb tieh hinh ch6p Ii. ABC lt l', lr6.v tinh ctQ l. 'Iirn tdt cii eie sd thuc :r sao cho eAe sd tlti dudng cao cta htnh ch6p S. ",lRC 0 ; logz ( -,.os 2:') , k,gt (.srz.i') 5. Cho h$ b5t phuong trinh :. ttreo thu tn dC tao thanh rn$t cAp s6,:dng 2. (lho a, p le 2 g6c nnqn lli6a rrrlu: l,le*n +Vl-J
"\!-z 7 F bi cflt bi vi rdc thidt diQn tao bdi e{r: m[t a I il6 irhfing 116 vdi F tiOi mirt kh6ng cd dlibrn chung. thu'a s6 ngur,6n t6- ld nl = p1 .. {)* n6 lhiy Bi6t lr'rng F c6 m cq.nh vI n ilinh. r hri'ng d6n diBnr nz - f ,r,1)= I * ulpr * ..1 r6fi6. r6i iniirh rilng li6u n )' imi 1'12 thi iiin tai ri 6! nrirl l* dibrn n2 nhll'd0i i!i?.nr n3 /(nr) vir cu phring ciit khnng rli qtra ili?rn ndr., t:ia ii(). li6p tuc mli nhLr th6. I hirng mirrh ri.ng r r'.'i bdt r* dlilnr rL,{t l.ihirt n1 )> ti nAo thi rl6n rr,rt ltit' nAu Ll6 ci dibnr li11r\11 thn hu, : 03-5- r 986 nh{y d€n cria thAu cL{u sC i{ 7,8,7.8... '1'hdt gtart: lStl plrrit tsdt 4: (iho di1' sr! I u" | , {r: df nh n hLr s:itt l DAi'{$ SACH I}OI TUYftN TiOt, SINfl ul !2.rrngi= un 1-i; r !.(I1 t}lf,l \rfT r{asl DIJ THI ToAI$ gud" Td un{t -Vr l* l lhn rnte g? T4tr yACKAYA, [iA LA:'{ n) t.'l'irrr sd La*g tlru 1980. IIAM I986 Brii 5 : Ch,: mrit trrr:r giic ADr) b{r k1:. 1'1;,,,0 l lrri 1!rth 1-fi, Dai lto ' srr i.,hq nr liA Ntii l. thrr0c vl\ corrpr hI1, durg tr6n tanh [,',1 c*c rlibrn t !'ht'r 4 IIi Itd:, [)si huc l0rrg rio1, Ilt \Qi. 'tr. .12. .la, .4e ilit tir L') vA trOn cauh Ct] ch,c Dtti ltoc su pi,anl [lA "\'r,rrydn Plugng ?"ttrirr, ? diBnr lii, llt, 83. tia (liB tri C)sao cho tlign tie h cde lom giirc CAll)1.:l rllrrtr, BtAzflz,..t2Bjr13, B2A3gs, N0, I -'trBs4a 8;1,1 84. ArBaA B4Ali biLng nhau, 4.N, rrgdn ?uiin 7'rung. L)ai hoir lbog hcp BAi 6: {16 nr0i cr;n ehrlu ehfiu nhAy Ir.dn cAc ltd N0i dilm n*,uyOn cii:r tis sii I0, " ccl ttreo qui lLr.At sau: 5. v guL1{n [Iing Sttn, Quing \a rit - .D] \ii,ug' Iir rl i6rrr xrllt plr[t r,1 c6 darrg ph:lrr rich ra b. Sodrt Al IIrii, NgtrQ'tinh, EAI TOAffi J. G&&FUffiffiEt HOAUC Bffl TAN \/Aii rrfirrr iS6l, rhi tosn hqc 11] la J Gerfunkel minh duge r5.ng khfrng tliuh e [ra (i arfrrnkr:l lA t di, tt,ir id.m rnot rhi ngtriQm thu vi s"u ctrdy : driug ilAn.ilsir thd nira, tl6n6 thoj r oi Garrjner Ons r:ligrr nrOt cAch nglu niieo 50ii tdm giAc vA thr l-o vi lrng rirng tli thu r1uor, c6e kdt qui. gur'6t d!rrh di;rg rn6.r iinh di$n tfr krbm tra thfr tuong tu sau dAv: j r()ng tain riai: b{t t.i te xem It} iii() lsrn gr5.c dd lif1 co rtt ru duoc nrQt .tupu r'6: qu-{ Iu6t chung uaogiiir tfe _rt1ui6 cria rarn ;/?
ehung mirft: D[t o:u+a, D= u*c. [Yi rff.ng, r* (5) vn bij dE tr€n ta nh*.n rluoc: t: = 2u lrong d6 I u vA u d ir -d:lu Iihirilj: fiu * 1116* t.( ma-t- hn * /" { mo*1r+*"( (t) tl = ,t, ll "l{ ' r'/ t1 0i)?l * { (y'fl ) (o . b +- c) vri ito chintr }d ftiBu rna *,r12 -- c{, j -,,2 ,r2l (,,'- .* i,'vr ir:irr,: tliri.c ta-ciin 9;hrii r:lrirr:;3 i:rinh. l)o','r.) t.ai toAn drroc x&y ra khi vit tlr! klrr .r --- 0 (tr'rc,i: 6,. giii qr,r'6tl '[a e6: lrno : rliir,tl-.'--: ] Kh0ng giinr rbng quir ru c(r r Irl = h: I. i,hi tlrl tu si ti, ;/:i.(,,.( l: i jt si&i thi6t k,.( l: * y'1.3ra-.-..;;+ ur---;ultnt'l) f :t,t vrr (16 2a{t +c;2c(ir{i;0 t c_>l.Ni:Lrta 1f{,1 : 2ar; =/'i '*.i: . \i rri,ril f '(t:r (31?) (i. *" o217Jr1,,2 -i-" rc2 + rllrt! ' tl.r-v lA:" (t -- l/)s i? * ry} i- 6 i)ilu llr. Ii -Thay c biri (t + or 2 v,,
\;7-- Goi ,""1, c[2, d.g - it hh6i lucrng d&u l&e, dlu f - alplrt * az\ztz * esps."-r (l) rna drit vi than cin tlri6t uA tbda nr;rn cAc diEu kif n : i1E dun sOi m0t lit nrrue (cu thE q : 0,11 lit, ?"2: i Dun sOi dir kh6i luong nu6c rnong mudn. ={},1 I llt. zg = r},3 kg) rl * rr * :r'3 '' '\r (Zl Fr, [3r, I3r - giA tiBn mQt lit diu lira, . . Fnt hop vrii thcri gir; cho ph6p l rl&u ma dtr, vI 1 kg than. tr(et) "-( 7' : tztlil ( 7' ; 1s(r':) ( l' (J) {cu thb IA p1 = 6 cl0ng . Pht hqp vdi khdi lrrtrng nhi6n liQu r:o thb e6 . At, At, As I * Fr-4rl6ngiFs:2dbng) khdi luong dtru l&a, dB,u zlxl - 0, :r2 ) 0. rs > 0 (St.,- ds = {r,9 kg) Nghi6n er'ru tr6n tung loai bE p ta rfrc d jni , 7' - I(ho&ng thrii gian cAn dun dugc eAc hdrn s6 tuong r'rng : ,r, ii,z, r:3 sdi nrdc. _:l,u;,ffif, Hi; ;iiri,; l1(r'1): ltOtg(0.314t1) -F 1 : tz(rz) = :l0tgi0,lill2xz) * -i- 1,5; tR(xs) : 6019 0,t90x3) + tt. l'ir (f) rtt e3 = if rL --12 rbi rhtv vAo (l , (3), (4), I1(r1\, ts(x*, t3(r3):.thdi gian chn thi{t clL b6p (5 ta rli d6n bAi to{n crre lri 2 b,rtn i)r}ng I, bdp 2 vA bdp 3 ilua sdi phuong 1rh{p hirh hoc (xem bi