TOÁN HỌC TUỔI TRẺ THÁNG 3 NĂM 2014

xuf,r eiu rUrgo+ 2014 s6 441 rnp cxi Ra xAruc rHAruc - NAM rx05l oArux cHo rRUNG xoc pHd rHOruc vA rRuruc Hoc co s6 Tru s6: 1B7B Gi6ng Vo, Ha NOi. DT Bi6n t6p: (04) 35121607: DT - Fax Ph6t hdnh. Tristr (04)35121606 Email: toanhoctuoitrevietnam@gmail.com Website: http:l/www.nxbgd.vnitoanhoctuoitre ee&& Glliit[fl]t il0.G'll[, rudl ilt xeww$@, ilP \ff 6 chdro mllng 50 ndm thrinh lap Tap chi Y Ed TH&TT (1s04-2014) vi h0 trE t6t hon cho A.l phong trdo dqy vdr hec Toin & c;ic truong b'n6 tf,Ong, trong ndm 2A#lrap chi sG t6 chuc hai euOc thi: 1. CuQc thi Giei todn chdo m*ng 50 ndm Tqp chi TH&TT - DAi tuqng dq thi: Hqc sinh b?c THCS va THPT. - Th6 te cuqc tt1i. CuQc thi gOm 5 vong, d6 thi tu Vong 1 d6n Vdng 5 l6n luot dwqc ding trdn TEp chi TH&TT tu th6ng 2 tsb 44q d6n thang 6 (s6 44q. De thi o m5i vong g0m 2 bdi toSn ddnh cho hoc sinh THCS vir 2 biri to6n dinh cho hQc sinh THPT" Hgc sinh THCS co th6 ldrm bdiTHPT nhung hgc sinh TIIPT lirm bdri THCS s6 khOng duqc tinh di6m. D6p 5n ctia c:ic d6 thi sG lin luEt dwgc ddng tu th6ng 6 ts6 444) d6n th6ng 10 (sO 44S). Kot qud cuQc thi s6 duqc cong b6 tr6n Tqp chi TH&TT thZrng 10 nhrn2014. - Thdi hqn gwi bdi dv fhl: MuOn nhit ta 2 thang kO tu ngdy cu6i thang ra s6 tqp chI co dang drO thi. - Quy cAch bdidu fhl: M6i bai gidi c&a m6t biitodrn phii viet tr6n mOt to,giSy (hoic mQt file) ri6ng. Phia tr6n m6i bii ph6i ghi 16: Ho t6n, dia chi truong, lcrp, x5 {phuo'ng), huy6n (quan), tinh (thanh ph6), i -,^ ., , i , s0 diQn thoai (n6u co). Tr0n phong bi hoic file ghi 16: 8di dtv thi GiAi todn chao mung 50 ndrn Tqp chi TII&TT. 2. CuQc thi ViCt chuydn dO Todn chdo mwng 50 ndm Tap chi TH&TT - Doi twqng dq thi Gi6ro vi6n d6 hodc dang dqy & THCS, THPT; gidng vi6n. sinh vi6n cv cdc truong DH, CD v;i ban dgc y6u thich To5n. - NQi dunE bdi tltt ttti: LA c6c chuy6n de bOi du0ng hqc sinh gidi Toin; cdc chuy6n dE on tQp, 6n thi cuOl c6p; nhirng tlm toi, s6ng tEo tronE viQc dqy, hoc To6n {bAc Ti{eS, THPT). - ThA E cuQc tlti:. M6i ngucri tham gia cuQc thi duEc g&i kh6ng qu5 3 bdi du thi kh6c nhau. Cdrc biri dir thi co th6 liir c6c s6ng kien, kinh nghiem di ddng ki, hoic doat gidi & Trucrng, Phong, Sd, nhung chua tirng duqc xuAt bdn thdrnh s6ch, biio vi c0ng chua tham gia cu6c thi ndo khiic. K6t qud cuQc thi s6 duqc cOng b6 tren Tqp chi TH&TT thiing '11 ndm 2014 (s6 a+g). Cdrc bdri hay co th6 duqc chen ding tren TEp chi, hoic in thirnh s6ch, ciic tiic gi6 duqc hudng nhuin b*t theo quy dinh c0a Tqp chi. Bdri vi6t tham dur cuOc thi thuOc l, bdn quy6n cua Tap chi TH&TT. - Ttldi han gtri b;ii dtv thi: Trvoc ngdry 1 511012014. - Quy cach bai dUr thi: Bdri du thi co the vi6t tr,On giAy hoEc ddnh mdry vi tinh bdng chucrng trinh soan th;io vin bdn word M6i bdi dLr thi phAi vi6t tren giSy (ho{c co file) ri6ng phia tr6n m6i nai pnal ghi 16: Hq tOn, dta chi truong. xd (phucrng), huyen (quan), tinh (thdnh pn61 so di6n thoai (neu co). B-ri dr=i thi kh6ng qudr 15 trang viet tay hodc 10 trang d6rnh m6y. Tr6n phong bi hodc file ghi 16'. Bai dq thi Vi& cttuy\n dO Todn chdo mang 50 nam Tqp chf TH&TT. . Cdch grhi biti dqr thi: B2ri dU thi c0a hai cuOc thi tren gfri ve tap chi TH&TT bdng c6ch. + Girifile word theo d!a chi email: to a n h octu o itrev i etn a m @g m a i l -c o m + hoic g&i bai theo duro'ng bivu diOn. Phong bi co d€rn tem, giri v6 dia chi: Tap chi To6n hoc vi Tu6i tr6 1878 Giing V6, Di5ng Da, Hd NQi .: + hodc d6n Toa soan Tap chi gtli truc ti6p. . Alaf tnwdng; Gom gi6i c6 nh6n vd gi6itAp th6 cho c6c trucrng vd dcvn vi co nhi6u cd nhAn tham gia" NhCrng c6r nhAn vdr tip th6 doat gidi cao s6 duoc rncvi du vii trao gidi tronE L6 fi nienr 50 nim thdnh lQp Tap chiTH&TT t6 chuc vAo th6rng 1212A14. Rdt mong cac bqn nhiQt tinh hu&ng ung hai cuQc thi tr6n ! / .i"_.,.i ' 'r'rr1\G [{id r](, ${i LINH HEAT TFSNE effi reAru em&mrffi ffi@@ LE eA xoAt'tc (Phbng GD-DT th! xd Hdng Linh, Hd Tinh) f} am mO giai todn ld ti6n Ae *ri6t yeu OC tqc lJ tOt mon To6n. Long dam mC sE giirp cho ngucri lim to6n c6 dugc sU ki6n tri, ph6t huy tffi s6ng tpo vd tr0 n€n linh hopt khi gi6i to6n. Qua bdi vi0t ndy, t6i mu6n ctng voi bpn clgc nhan dien sU linh hopt trong gihitoinhinh hqc, drcctry cao Af{ vd phcn giit"c BD. Giti I |t'r ram chrdng trdn ruoi ti\p tctm giec AliC. biit 6il - 45o. Tim so tlo c:ilo 6k ,'o thO rc. ( tii 4 ali rhi frtitt Ot:,i, ri tan rhir 5i)) Ldi gidl Gqi O ld giao di6m cua AH vd BD,ta c6 Co ld phAn giiic cia triE -Id0=6d0. HaOK LAC (K e AC). Trudng hW l. Nriu K tdng voi D tbt ABC lit -;. tam gi6c dOu vi co O vira ld tnrc t6m rua 1d tAm dudng trdn nQi ti6p, suy ,u 6tre = 60o. Trudng hW 2. N6uKthu0c dopn khdc D (hinh la) DA, mdu thuln vcvi diAu ki6n d{t ra ld K e DC. Vi viy tnrong hqp 3 kh6ng xdy ra. * Khi ABC lil tam gi6c -l d6u hoac vu6ng cin t4i A ta cfrng d6u chfug minh dusc EDi : 45o. L Y6ry BAC = 60" hodtc 6k = 90o. H Hinh ]b NhQn xit Bdi toan ndy thwc chAt h kh6ng kh6, chi cdn ki€n th*c THCS. Vi€c dung OK vu6ng goc voi AC cfrng xuAt phdt tb y nghi n€u tam giac ABC diu thi 6Di = 45o, itdy chinh ld "chia khoa" d€ tim ra loi gidi don gidn cho bdi toan. fi Blri toan 2. Clto {am giac ABC t'ri phnn gidc AD; O lL't ciienr btir l;i tren AD (O khac A vd D). Tia tlo c'ot ccuth AC tai E, tia co ciit canh AB tai F. Chirns ntinh ,'ring tc,n giac AIIC t:dn tai :IIII .4ttatt-;-= + .]81 AE2 AC:2 AT;2 Ldi sini Qua O ke OM ll AB; ON ll AC (M e AC; N e AB - xem hinh 2). Do AD ld ph6n grilc, ta chimg minh ilugc AMONlir hinh thoi, suy ra OM: ON (: x). Ap dU"g dr$ li Thalds vio tam giSc ABE,tac6 OM ON OE OB BE _t -r-=-3I AB'AE BE,BE BE A (2) thi ta c6: Vi 1ld tam B duong tron nQi titip L4HC H Hinh la n6n.I e OC vit, OHI = CHI = 45o. Suy ra hai tam gi6c vu6ng OHC vit OKC bing nhau -5fu =dfu = 45o = 6fu =6Di = 45o s OKDIldtit glitc nQi tiOp =COD=CKI =45o > 2COD = 2(OBC + OCB) = ABC + ACB =90o = EZe =90o. Trudng hW 3. N6u KthuQc clo4n DC,ldtirc D (hinh lb) thi tuong tu nhu trrdng hqp 2, ta cfrng c6 ODKIliLtT giitc nQi titip, suy ra DOC = CKI = 45o > BAC :90o. Tuy nhi6n, khi d6 Eic ,67d = 9oo 11 + _!_ - AB, AE 1t = oM =; (1) Tuong tr; ta c6 111 _f AC,AF_X ss aar ta-zorar T?3I#88 r Hinh 2 ( t.l (B-E) rutCV o VN)V :tttEgV a yWtN 9c eL .D JV gg gv osildstuqffilNn gc ugc Eueql qqq q g,qNI[ er fng .lN : gW u?u QIy= qry gc et 191ll9rB oeql ',(pu uo.4 Euo.np w^ )v'gY e+c !eq, nql tuglp oet8 cgc PI }onl wl dr 'g loc 'NWV cgrE urel dgq reoEu uo.4 8uo.np pn @'q w$ tat 14€)I*11:flQ - ,,)l ,;-,' 'ti _\ ut I 1i '. i"'1' ", 1:-l\' .'!-l W\,i ():t! ) r'i,J)\ ir i1 rilt1'. itltl tn! ,)ff ilLli:) ;i;).: ! \rrt-' -:)1')t ,', ti.) ':- iltl(U II:f{ q:l '(o nm !g^ ?q uonb nru oa gs lbp Suop rcnp u?ot !?q uztlu uttlu Suiu ptpt u?p u? (q ngc 'quts coq onc coq qulq u?ot 7p13 Suoa ryqt u? n?tp ?l @g .ryd Buenp o"t oot qcpc 3uo,r7 ruoq qull Bupu pt1t1 p1 7 t7to3 :coq ryury 3uo.4 elq pp tt ttulp tpc Eunp u?^ I qrD) 'Lpry (, n?u rup ruq Buo4 ,nnt1u )?W !?!3 Wry n?lqu oc (o nn3 yQx ugt11r1 (o<z*,.,61 :te6:74 9c 3r @ ocrher og"ra F#uQu w):trryprN ,3 g I'z-' I+-//\= PY N t- z1 l" ffi=#= z4€0.#:? rpcr(Q '(cgrE ugqd ]gqc qup e^ soler{J n rfuip) QY = wx, = ^o: : ,y: = lg e.r.(n5 )g )s os Hs Hg 'H) : IHZ er,(ns 'IJ : IH u?u )hl : VW pW ,HV IIIWWU)HT WI^?HT TI!iH'ZqC?) JV JH -_= )g sH *itiou-iTr?r eB-ts&Wrus , ffi }sH ruH#.r" .JY =OV * H) )g og Hg er dns (cgl8 ugqd Wqc LIUtr) #= t l.lulH _go .vyt .)H '-oy rc Hg gc e1e^e3 II r{uip oor{J 'p tu1,(nb Sugp INg'e) ,ttyE 4W Buqnp eq 9c )gycgl? tuel 'I Wq) @ GtD tug tqt )t' ,1,': "'ti tltt I l'1 !r / :_l i!!!tt:! .'t,,,l / {r, / i(i ,)g 'y7 li.:l .t,ili i'1ia:7t qi 1 :t,ttil iit.t:;tr !).\ ,'l,i/ ri,,i i;u i.'it;:.,: ! !'y,. it:t.': jj'ir.r,.ti,,, ,r., , ,, , ,;;,.'i,ii,'. iill ]i. ;, ':,;,i ,,,,. 1 '";;. iri.r.iJ ,n - 'cpt17 tg{nb ru8 Suonq )o) D.t tpp ruoq quu ug) ,uonb oqy Bugqy n7? ,tV,t nqu ntld Suenp at Suanq )?) .Liy :Cn '(qV :gy) cg8 ?uona. quoc doc )D) rat 3u9nd cotB ruor cpc u)1q runx ultl hqd Bupnp )D) or otlt ?p lq8u [ns Sugnq ?g 4t ,Bugna cg73 woy Dnr oD) Suonp ?^ quor ?^ cqtw Aq u?P ?q u?!t qu u?ot !?q rutB rcn&u oLp rupl ?p z.4Y . zJV ,W ,gY _ ___ *_= . t?!\tplD7?xugtlN I I I I i" 'yptryc agy cgl8vml e.r ,(ns 'CV : gy',iy : gV u?u Cy > gy lL '0 : d + /S - { Wn Suonqd er.rc urgq8u e1 JV ',iV igy 'gy gc e1 eler1 4t quip oeqf '(,5:).iV + )y: Zy + gy e Q:) trcv: zY{v € ,IY ,JV ,IV ,gV (1-+-: :-+: - lglqlgF op) I I I l 1' j "' TV'JV - lV'gV - TY , JY gY 8Y T-:- (,T,IIII er.(ns (d pt (t) +f MB AB MC /17 - # = ffi;'fr =fr + MB.NB = AB.EB; MC.NC : AC.FC - 9Y = 12'!3= el MC-NC AC.FC Tt (1) ve (2) suy ra dpcm. NhAn xdt W MAB = NAC vd ddng thitc cdn chyn7 minh c6 dqng ti s6 cila tici cdc doqn thdng n€n ta vd th€m dadng trdn ngoqi tidp tam gidc AMN. Khi gidi todn hinh hoc, hoc sinh cdn daqc luyQn tQp nhi€u cdch vd dudng phu khdc nhau, qua d6 s€ tich lu! &tqc kinh nghiQm trong gidi todn hinh h7c. fi ga; tohn 5. {-'l:r., ri:tst 1litir: ,11}{" t:i 1;iiriil ,2i1i, tt'()t .g.lD i{S ,,, {id-1. 7 r't't:.JI) 1lir !t,ti tli, t. \t ti tttt) tlt(t .iiti . t A r 1! tt,trt, qiii.r : 1r' \i. Ch{rng minlt .4{'l\,1 1lL.rv". Ldi gidl Cdch t. (h.5a) Ap dung Bdi todn 4 vi,o A,ABD ta co Mptkh6c BAD = CAD , r (gla thret) suy ra LABK aLAMC (c.g.c) * BKN = ACM. Trong t& gi6c nQi titip BKCN cingc6 BKN = BCNsuy ra feM =6dfi (dpcm). NhAn xdt Lni gidi o cdch 1 c6 duqc nhd vQn &tng Bdi todn 4. Loi gidi 6 cdch 2 thAt bt hai khi vd thdm &rong trdn ngoqi ti€p LBNC. Hinh 5b ran rep tsni tip t. Duong trdn t6m l nQi ti6p tam gi(c ABC ti€p xirc vfi c6c cpnh BC, AB, AC Mn luqt tai D, E, l.Qua E ke ducrng thing song song voi BC cdt AD, DF lin lugt tai M vit N. Chtmg minh M ldtrung diiSm ci:r- NE. Bni t4p 2. Cho hinh thoi ABCD canh a. Gqi4 vd r lin luqt ld b6n kinh rlucrng tron ngo4i ti6p cdc tam gi6c ABD vd, ABC. r1l4 Uhtmgmmnrang .*.: .t(ra llAi t$p 3. Dudng tron nQi ti€p tam giitc ABC ti6p xirc voi AB, AC hn D vd CM cdt DE tai I. ,IMDM Lntmg ffunn rang -;-: - rL CE' Bni tip 4. Tam gi6c ABC ngoai ti6p duotrg tron t6m O, cqnh BC ti€p xric v6i (O) t1i D. Ggi Mldtrung.diOm cin BC. Chrmg minh OM di qua trung tliOm cinAD. Dinh chinh . Biri T3 1440 - TH&TT thilng 212014 c6 in thir5u mQt do4n. Xin d'qc lai nhu sau: Chung minh rang na -5n3 -2n2 -lOn+ 4 kh6ng chia hi5t cho 49 voi m5i sO tu ottl6n n kh6ng c6 du 3 trong ph6p chia cho 7 vdr chia h6t cho 49 trong truong hcr-p cdn 14i. AM.AN (,qA\, DM.DN \DB ) Gqi M ld cli6m trln BD sao cho IeM :Eei'. Tuong n; MCDtac6 B g#:( {^\'q DM.DN', IDC ) Ta co AD ld phAn gi6c trong cta tam giitc ABC ^ AB AC .: nen - = ft, ket hqp v6i (1) vd (2) suy ra # = # Do N vd N' ctng thuQc AD ndn N=N', suyra ACM = BCN (dpcm). Cdch2. (h.5b) Ve duong frdn ngoai ti6p LBNC, cit ducnrg thirng AD tai K (k]16c N). Tri gi6c n6i ti6p BKCN c(t Gfi = ffi, md ABM = CBN Gin thi6t) =d?fr =78il > LABM cn LAKC (g.s) AB AM +_ - AK- AC' ,1 :x.L,r-rr4, T?gI#gE B nft nloutonn toP s rixrr HArfrxn NAvr rrQc zotz - 2ot3 Bii l. a) Ap dung UiCn eOi (o- b)t : a3 - b3 -3ab1a- b)tac6 x3 : 4J1'3x= x3 +3x=4J2; (1) f : z+J1-3Y> Y'+3Y=2+f,' Q) Tru theo tung vC (1) vd (2) ta dugc x, - yt +3(x-y) =-20J2. Suy raM:-20J2. b) Ta thAy I + x + l; * - x * | lu6n ducrng vd x': 0 kh6ng h nghiQm cta phuong trinh dd cho' Do d6 var t=x+L, phuong hinh de cho tr0 x 215 tnetnfr-h=;e5t2 -3t-t4=0 (r;u t 1) ft=2 Q (t -2)(5t+7)=0o1, =_1. Ta tim tluo.c L5 nghiQm cua phuong hinh de cho ld x : 1. nai Z. a) Nhin cdhai v6 cira phucrng trinh thri nhAt ctra hQ phuong trinh dA cho v6i 3 ta dugc 9 : l2x - 3i - lf , uc vdro phucrng trinh tht hai vd thu ggn ta tlugc *' i Yt : 3(*' - f) e (x +y)( i' xy + f -3x + 3Y) : 0. * V6i x-r !: 0 <> x: -!, thti vdo phuong trinh thri nhAt ctra hQ ta tlu-o. c Zi - +x + 3 : 0 e 2(x - l)' + 1 : 0, phu<rng trinh vO nghiQm. * V6i * - xy + f - 3x + 3y: 0, tni theo timg vi5 voi phuong hinh thf nh6t cta hQ ta dugc ry fx=3 -x-3y * 3 :0 <+ (x- 3Xy- 1): O oli=r. Ta tim du-o. c hai nghiQm (x; y) cila hQ phuong trinh de cho ld (3; 0) vd (2; l). b) Ta co P=abc-(a+b+c1+!+L.:-# Do a, b, c ld chc sO tU ntri6n n6n P ld sti ngtryOn o M =L+l*f -* ld sO nguyen. aDcaDC Do a, b, c c6 vaitrd nhu nhau, kh6ng m6t tinh t6ng qu6t ta giit thi}t a < b <c, suy ru a ) L; b>2; c 2 3, suy ra TONN HOC 4 I cflrdiSe, S0 eat ta-zotO I I I I I t l-1 1,1- U(-*=*- 11+-+-<-+-+-<2 a D c aDc a'b'c-l'2'3- = 0 < M<2= M: I (vlMlds6 nguY6n) 1111 + -f -J = | - a'b'c abc ) a+ b + c: (a-l)(b- 1X, -l)+2. (*) Ntlu (a - lxb - 1) > 4 thi voi a < b < c ta c6 3c> a + b + c > 3c > (a - l)(b- lXt - l) + 2 + 3c> a@ - l) + 2 > 3c > 4c -2 + c 1 2, tr6i voi di6u kiQn c > 3. Do a * b + c26 n6n tu (*) suy ta a - 1' 0, suyra b-t> 1,suy ra(a-l)(b- 1)chic6th6 nhQn gi5 frliLZ hoflc 3. Ta tim du-o. c mQt b0 sd (a: b; c) thod mdn ld (2; 3; 5). Vqy c6c b0 sd tU nhi6n phdn bid1 @; b; c) thoit mln bdi to6n giim c6c ho6n vf ctra (2;3; 5)' Khi d6 P nhfln gi6 tri nguY6n ld 21. Bdi 3. Ycl. a, b, c duong a + b * c: I ta c6 abc3abc3 -=-Q'1-t =- l-a- l-b' l-r- 2- b+c c+a a+b 2 o2(a*o*'{}-* I *l')=o v!\u,","/\b+c c+a a+b) .(t I r\ o(x+ v+z\l' a' +l l=9 \ir y z) (voix : b + c> 0,Y : c ! a) 0, z : a + b> 0) o( t*v-z\*( v*1-z\*(z*t-z) = o [vx)\zYl\xz/ oG-y)'*(:t-z)2 *(z-x)2 -o<+ x=y=z xy yz zx € a: b: c e ABC li tam gi6c dOu. Bii 4. Tu gii thli:t ta c6 AD ld trung tr.uc cria BC vitla phdn gi6c ci.r. 6k. rt do ta cflng nhQn thAy MNAP ld hinh vu6ng. M[t kh6c do fr@ *frFF =90" + 9oo = 18oo nan ANHP ld tu gi6c nQi tiOp duong trdn ttucmg kinh NP' , } Suy ra ZruP =VNP = 45o (ctng chiln AP) -m=1ED (:45o) > ABDH ld tu gi6c noi ti6p >ZruA = ADB =90' hay AH L BH' C il m 0$ Hec $ilt oil mil T0Ar rOt I n HAr DumG NAM HQC 2ot2 - 2013 (Vdng I) (Thdi gian ldm bdi: 150 phtit) Bdi 1. 1Z die4 Cho bi6u thric , Ji+F*_++J;_iF4 f-E 16 ,Il +- 1l- x' x2 b) Tim c6c s6 ngly)nx,y tho6 mdn ^\,.:L-,*_ *:-L /1Dnn nn_ nn3 , BE CE 2 y +2xy-3x-2:0. c)chtmgnrinhCEDF'EF:cDr"itffi=#' BAi 4. g,S aiAml O).ui? tam gi6c BEF c6 mQt hinh vu6ng BMKN a) Cho a., b, c lirbasO tiru ti thod mdn abc : I nQi,tiep (K e EF, M e BEvdN e BF) sao cho . a b c az U cz . ti s6 gifra canh hinh vudng vcri b6n kinh duong vit *+-+ ,r=;*;+7. chtmg minh rdng ?ig. rinh c6c it ,jat io, ior* ba s6 a, b, c ritbinh phucrng trdn nQi ti6p tam gi6c BEF ld 2 ' - "-' -*- :r:?.r:J.:e.l*J1 9::.*:i.:*3.i".T.913:.!r_!:,. C- b) Theo phAn a) thl M, N, A, P,ll cung thuQc Bni 5. Ta nhpn th6y dudng trdn duong l<nhAMvdtlucrng kinhNP. .4 4 4 F= x * ! * ' Suyra AHM =90o = AHB ) H,M,Bthdng ' -€+yrwy)-6*;yr*4- €+;\lr+x) - cdc gi|tri nguyOn criax di5 A c6 giltri nguy6n. Bdi 2. (2 die@ Gihicdcphucrng toinh { J? 4x+z+",8T: =Jx1+J? +2x-l; b) (4x+DJx+8 =3x2 +7x+8. Bni 3. (1,5 diAm) a) Cho f (*) : (x3 +l2x -3llzotz . Tinh f(a) vli a =il6 -8G * il6* 8".6. hang > BHN = MHN = MAN = 45o (1) , , _ (cnng chdn MN). Y\BI I/AD n€nBI IBC = ABI =45". Cnng v6i gi6 thiet cria bdi tobntanhfn th6y tamgi6c IAB vudngcdnt4il + BAI = 45o. Ta c6 AIB+AHB=9ff+90o=180" n€n AIBH ^. .i ldtugi6cnQi ti6p, suyra BHI =BAI =45o (2) Tt (1) vd (2) suy ra EEi =m. MIt kh6c, do cSch dpg ta c6 N vd 1 ctng nim vO mQt phia cua BH suy ra H, N,lthing hdng. b) Cho a, b, c ld ba s6 ducrng c6 t6ng bing 3. Rrit gon A, tim Chrmg minhrlng in.#.iA.} Bii 5. Q diAm) Cho tluong trdn (O; R) vd hai duong kith AB, CD sao cho ti6p tuy}n tqi A cta (O;R) c6t c6c duong thtng BC, BD tqihai di6m hrcmg img ld E, F. G1i P, Q ldn luqt ld trung di6m oia cfuc doqn thing AE, AF. a) Chrmg minh rEng tr.uc tdm H oiua tarn gi6c BPQliltrung <li6m cria dopn thing OA. b) Hai dudng kitlhAB, CD thoilmdn diAu kiQn gi thi tam gi6c BPQ c6 diQn tich nh6 nh6t. yro*o =_+_+_ € +f[x+y) rs? +iys,+21 1? +i\z+xy .( 4 4 4 4 4 4 ) =:l **y * !*, - ,** , 2[{r'+rr'Xr+ y) G +z2ysr+21' 122 +x2yr+x1)' Ap dung bstdingthkc i +t.ry (d[ng thric C xiry ru khi vd chi khi a: b) ta c6 or!(_Jtly'l_* (y2+22)z - (22+x2)z \ - 4\1x2 + rz )(x+y)' ( y2 + z2 \( y + z)' (22 + x2 )(z + g ) =L(dtY'*Y'*" *,:lr\ 4\ x+y y+z z+x I 's\ ,- - y+z - z+x )= o\x+l+z)=4' VW F d1r s64 nh6 nhft b5ns ] Gni, :l:, : \). t. nnr,r-rorn, t?[I#ff E MQr sO pmudn{e_emAe qfu pillmtu Tflltffi m un PHUBilIE mlflH u0mur (Trong nhimg ndm gdn d6y, m5i de thi t6t 'L rgttigp fftPf vd tuy0n sinh DH, CD thunng co mQt ciu gi6i PT mf, hoic PT l6garit' r.lfrirn"gitp "a. uai hqc sinh thi tot, bii vitit ndy xin gioi tt i9,, mgt s0 phuong ph6p gi6i thucrng gap de gidi hai loai PT tr6n. 1. Phudng ph6p dda vG cung co s6 Cdch gidl Su dung c6c ph6p ti0n dOi tuong duongv6i0<a+l,tac6 o qf(x) - os?) e f @): g(x). . logo f@)=1og,g(x) ^ {f t*l: B(x) - i,r(rl > o (hoac g(x) > o). *Thi dq l, Giai Tthucrng irinh _ I I V-r ,.i-1.10,51r' -t - Jr-'t. LN gidl DK 0 < x + 1.PT dd cho tuong duong v6i 1-rzJi112 1EJ ,E; - r,ffi ^ ,EJ-E; = t Ei L'" z', -L v L 1 Ji-r 1 - Ji-t 2 r/x+1 =Ji+t=:(Ji-1) o Ji :2 e x = 4 (thoirmdn DK). u *Thi dtt 2 (KhdiD - 2011). Giai plu.rary trinh 1og, (8 "') * Iogr (.'/f+, . J -;) -2 = o. Ldi gidl DK:-1<x31.PT dd cho tuong duong voi 1og2(8 -x2) -losz(fi ., + fi -r) -togrl = o e log2(8 - ,2) =r"g, [+(Jr -, -r Jr+,)] e8- f =4(J1a11*d-r) lE n6 ouY (GV THPT Le Lqi, Kon Tum) <> (8 -*2\2 =p(t.J=) (l) -\ l Ddt t=f-7 (r>o)=1- x2 =t2. Khid6, PT (1) tr6 thdnh 1l +t212 =32(l+t) o t4 +14t2 -32t +17 =0 e (t -l)2 (t2 + 2t +I7)= 0 e / = 1 (dot2 +2t +17 > 0,V/ e IR ). Voi t :1, tac6 I- x2 = t <> x = 0 (tho6 mdn DK). . 2. Phddng PhaP d{t dn Phu u) Phwong trinlt dgng *.o2f (x) * n.of G) * p :0. Cach gidi. Ddt t = of @) u61di6u kien I > 0' DSndt5nPT mtz +nt+ p=0, tim/suyrax' *Thi dV3. Git)i ythwtng trinh .r -l ,., ,.a 5.1;i-r *-1.2T- l7-o, Ldi girtL DK: x > 0. PT dd cho tuong iluong voi Ji-t 5.zJ;-t +12.27 -17 :0. Ji-t Ddt t =2 2 (t >0), taduocPT: 5t2 +t2t -17 :0 e r = I hoic, =-!(loai). Ji-t c Vdi t =1, ta c6 2t; = 1<> '*;' =o o x = 1 (tho6 m6n DK). n b'; Phtong trinh dgng mlogf, f @) + nlogo f @) + P = 0 Cdch gidl D4t t-logof(1. Ddn d€n PT mtz +nt+P =0, tim/suyra'r' *Thi dv 4. Gioi phy-gj::L 2loga(-r2 -x) + :Jlogo1x - 1)2 - 2bga x = 4' Cftueil [i cho [i $ti mrruni0ptttPt ui thi uio Oai hoc 1 Jx+1 3 G; TORN HQC 6 r qUOtU" se aat ta-zoul _ Loigi,rti DK: x >-2.PT cl6chotuongduongvoi 2logalx(x -llt + :.,0 fogot, + - 2loga x - 4 = 0 e Zloga@- r; +:JZrogo1, - r; - 4 = 0. Ddt t = Jrt"tu@ -n (r > o), ta duoc pr: t2+3t-4=Oet=1 ho6c t=*4 (loai). vcri r : I, ra c6 Jz togo(, - t) : t I <> loga(x -1) :, e x - 3 (tho6 mdn DK). *Thi dr1 5. Gidi phtrrmg trinh Iogrr_,*7(z$r2 + i?"r+9)r-log2"-:((rr: +23x+ 2i) = -{. a 1 Ldi gidL OK: -1< x + -l.PT da cho tuong duong voi 7og3r*1(2x+3)2 + log2*41(2x+3)(3x+ 7)l= 4 e 2log3,*r(2x + 3)+ log2,*3(3x + 7 ) - 3 = 0 (2) Do log3,*7(2x + 3).log z*+t(3x + 7) = I, (t \ Vxe l-;;+* l\{-l} n6nn6uddt \2./ I e x = -; (thoa min DK) ho{c x : -2 (loai). 4' VQy PT dfr cho co mQt nghi€m , = -l o 4 NhQn x/a N6u trong PT co cdc s6 hang log.t@)g(x) vd logr(,) /(x) thi ta c6 di6u ki6n tuong tmg ld 0 < f (x) *1,0 < g(x) +1. Lfc d6, n6u dAt t :logy61g@) thi 1 Iogg1,) JG)=7. t\ Pfuwrrng trinh tlgng *.o2f (,) + n.(ab1.f Q) a 0.62f (x) = g fto{c aftf$) *oo?f@)d$) +p.{@6zf@ +qif@) =gy Ctich gidl Chia hai v6 cho b2f(*) @oac r - "f(x) 63[txt ). r6idat t =[+ I (, , o). \D) *Thi du 6 (Kh6i A - 2006). Gidi pliuo'ng trinlt 3.8' + 4.12''' - I 8'' * 2.2'l' = 0. Ldi gidl Chia hai v6 cira PT cho 27* , taduoc / r \3' t' ,t \2r ( 't\x ,[;,J .ol;) -t;] -2=o /'r \x Eat r =i* | (r>0), tadugcPT: \rl 3t3 + 4t2 - t -2 = o <+ 1t +t121zt -2) = o ) €t- = tdo t>0).Voi J <>.x = l. tr d'1 Phutrttg tritth dung m.a-f G) 1r.6-f lx) * p:0, voi a.b =l Cdch gidl Gia su a > l, ta df;t 1 - of G) (r > o), khi d6 6.f G) =!. *Thi rlg 7. Giai lthtto"ng trinlt (i - vB)'+ tr,.(:-'/5)' = 2'*r. Ldi gi,fiLChia hai v6 cria PT cho 2*, ta dugc [+)'.,u['-f)'=, (3) ^^ . . 3+.,6 g- l; o6 i, ring + + = l, ndn n6u tlat , =(t*5" l.,,,0) thi [#l' =+ [z ) [/ )t Khi d6, PT (3) tr0 thdnh , *!-9- 8 e t2 -gt+16 : o e t = 4. t voi r = + rntf*-fl' :o <+ x - logr*us 4. \,/2 "a nnr,r-rrrn, T?[HrHt[, V t:! tni(?). =? t =1og3,+7(2x+ 3) (l * 0) thi log2,*3 ( 3x +\ =! . t Do d6, PT (2) trd thdnh zt+!-3 = o e 2t2-3r+l = o <+ [':1 ' L'= 1' c t :7 <> logrr*, (2x + 3) = 1 e 2x+3 =3x+7 e x = -4 (loai). lr ., = r<> logrr*, (2x + 3) = - e (2x+3)2 = 3x+7 e 4x2 +9x+2=0 3. Phudng phep l6garit hoa, m0 hoa a) Phtong trinh dgng of @)6s@) = ", voi a,brc > 0. Cdch gidL Ta c6 th6 lOgarit hai v6 voi co sii a itua vd PT dpng "f (*) + g(x).log,b = logo c. *Thi dV&. Gidi pharmg trinh 3r_25x_17x = 245. Ldi girti Do hai v6 cira PT.da cho d6u duong n6n l0y lOgarit co sO 3 hai v6, ta du-o. c logr(3* -2 5*-r 7 x ) = log.r(72 .5) e x -2 + (x-l)log3 5 + xlog, 7 : 2logt 7 + log, 5 e (;r- 2)log105 = 0 e x - 2.a b\ Phwong trinh dgng otog6@+c) - *. Cdch gi,rtL Ddt t =1o96(x + c) = x = bt - c. * Thi drt9. Gidi phuong trinh 3log2x *x:2. Ldi gidi. DK: r > 0. D?t t :lo1z 1s 2 v, - )t , ta dugc PT mfl theo An t: 3t +2t = 2. Ddt f (t)=3t +2t > -f '(t)= 3r ln 3 + 2t ln2 > 0,vt e R = /(r) ld hdm d6ng bi€n. MAt khec, ac6 f (O) = 2. Do d6, PT f(t)=2 c6 nghiQm duy nhAt r = 0. Suyra log2x =0=x=1. tr 4. Phfidng phdp dda vd phttong trinh tfch Cdch gi,fiL Khi gap PT d4ng m.f (x) + n.g(x) = mn + f (*).5@), ta thudng bi€n d6i dua vd PT dang lf @) - n)ls@) - m)= 0. *Thi drl 10 1t<h6i D - 2010). Giai phaong trinh ,12x, Jii , "r3 - nz-,tGi , 1x3 +4x-4 .+ -tL -'+ TL Ldi girti DK:x >-2.PT dd cho tuong duong voi Oz+"lii 12t -4 _ 1) _ 2r3 12ar-a _ 1) = 0 / - r\ o 124x-a _ ,.,[ 4z+Vx+z _ 2x" | = e. \/ . *'a =la4x-4=Q4ev:[ (thoi mdn DK). o 42+Jx+2 - rx' € 2'17+z = *3 -4 (4) o6 5, rang * > 114. X6t hdm s6 -f(x)=2.,17i-f +a t'c" [V4;**). rac6 I .f '(x) =+ -3x2 <o,vx e [il4;**) lx+2 = f(x) nghichbi6ntr6n [V4;**). Mat kh6c, ta c6 f (2) = 0, do d6 PT ( ) c6 nghiQm duy nh6t x = 2 (thoilmdn DK). Vfly PT tld cho c6 hai nghiQm x :l; x = 2. a *Thi dr; 11. Gidi phaong trinh zlog?ex = log: r.toer(J, + 1 - t). Ldi gi,rtiDK: ;r > 0. PT dd cho htong duong voi log?3 x = zlolz *.tog.1Jii -t1 <> 1og3 ,[tog, * -zlogr(Jz* +t- 1)] = o log: x = 0 <> x = 1 (thoimdnDK). . logs * = ztoet(J2* +1 -t) <> 1og3 * = log (J2* *t -I)2 o x = Glr:+l-t)' e 2Jzx n : x *2 e *2 -4x =o e x =4 (do:r > o). Vfly PT dd cho c6 hai nghiQm x =l; x = 4. a 5. Phtldng ph6p srl dqng tinh ddn diQu cfra him s6 a) Phwong trinh ilu'a itwqc vi phnorug trinh dqns f(u)= f(v). Cdch gi,fii. Str dpng tinh ch6t: Cho hdm s6 y = f (x) don diqu tr6n t4p D. Khi d6 .f (u) = f (r) o Lt = v, vcri moi u,v e D. *Thi dB 12. Gidi phuong trinh Zr'-, +g3-2x * x2 + 6 - 42x-3 +3r-r2 + 5x. Ldrt gidl PT de cho tucrng ducrng voi 2*2-* +36-4x * x2 + 6 - 24x-6 +3*-r2 +5x of +i -*-t-* -24x4 +4x-6-3H'751 X6thdm sO f(t)=2t +t-3-t,tac6 f'(t) = 2t ln2 + r + 3-t ln 3 > o,vr e lR. nh f (t) ld hdm sO eOng bitin tr6n lR. Tt (5), tac6 f(xz -x)= f@x-A e *2 -x=4x-6 e *2 -5x+6:0 <> x=2hodc x=3. E TONN HOC B -clirdiU@ [...]... NGUYEN XUAN EiNH Bdi T31 437 Tim cac so nguyAn x, )) z miin diiu kien 6(i,t- 1) +3( ;r2 +y'r')+zG' -9x):o thoct (r) N6u Do 3( x ndn2z2 -2)i3;6y2 i3; i3 = Tri (*) ve (**) 3y222 i3; e) (*) 33 i3 22i31**', suy ra * h€t cbc ban tl6u tim thing k6t qug Chc cbch gi6i ln kh6 phong phri C6c b4n c6 loi gi6i ngdn ggn ld: Phr[ Thg: Nguydn Ti€n Long,8A7, Ta Anh Dfing,8A3, Dinh Minh Hd,9Al, Vil Thiy Linh,9A3, THCS LAm Thao;... + 1)) - logr Dltlog(2x + 1): t (t> 0)thi2x:3t - l (2) Khi d6 PT(2) c6 dang 0 ts nnr,r-rorn, T?IHTHEE zr (Yu> 0) n6n ham s6flz) d6ng bi6n tr6n (0 ; +oo) Do do al@):flt) o x: thay 3* - 2x - I : X6t hdm s6 g(x) :3' - 2x - 1 c6 d@) :3' ln3 -2; PT(4) g'(r):0 r: xo = tog: ) ,l thi€u... I -1\2r -3- Bii Tli 437 (L6'p 6) Cho t6ng A gont c'(t 20 13 t,i hang (ki hiOu rtt : 1.2 .3 n) " l"l! 2.ll l.-1i "'' ,,.,,: ' Chintg ntirth gitii _lt1 u Ldi ring DF.+t A l2: ^ 10 13. 201-l:' 1 : |* I** * ,, trong do I- ' 5.5 2ol3.zor3r OC thAy ring n.nl : n.2 .3 n ) 2" voi U6t ti 4.41 n) 3, do tt6 1111 - /*\ n.nl 2n 2n-t 2n -/-= Thay b6t dEng thric (*) vio bitlu thric B vor n lAn luqt bing 4,5, , 20 13 du-o... Nhu Thi€p,l2Ar, THPT fran quOc Toin, 3* i - x - 1 -log(2x - 1og3(-logr(b + lll, : - 3* + 1)) + log(-x) : 1)) + log(-x) h'(x) :3' ln3 - 1-, -2(2x + l)ln3 2 !- (2x + 1)(ln3)2logr(Zx vfii -+< +) Gi6 str trong c6c s6 xl, x2, , xn c6 fr sO bing 1 Khi do ta c6 1 +l) 'xln3 x< 0 (do x; Do d6 h(x)tithdm nghlch bi6n tren (r h(x): 0 c6 kh6ng qu6 mQt nghiQm tr6n x : xt N -0 ,33 25 ld lr \ mQt nghiQm cila h(x) ) Vfly... cho tucrng ducrng vcri 4 s irrr s in2x co s2r c o s 3x tary- sin2-t co s 6x CAu 3 t-e? lnJ, 0 *3 -3* + 1 = 0 c6 3 nghiQm ph6n biqt Tri d6 (C,) 1u6n di qua 3 diOm cO einn nim h€n ducmg thing c6 PT y = 6x - 5 ,kn * =; lt4 = n6n PT * 1l -3 =0 Vi Ddp s6 - I e'.tan x.dx =1 + J e d(ln(cosx) 00 EAt -f (x) = x3 -3x +1, thi /(x) =3x2 .r : I hodc x: -t fcp fcr = f eD.f 0) < 0 Cflu 2 DK: cosx * 0; cos2x 1l 44... chon cluoc x: nghiQm x < x < 0 M6t 1 - (log3(-x))' = (log3(-x))' -= 2 I -r * 0 truong hqp = 0,54 53 fl FNh$n x6t a) Bii nity rdtnhidu ban tham gia, tuy nhi6n da s6 d6u cho ring di6u kiQn x6c dtnh ld x > 0 n6n con n 11t3k 73. 37 ! "._r -r L Nhu vfy phuong trinh (1) c6 nghiQm nguy6n du . +23x+ 2i) = -{. a 1 Ldi gidL OK: -1< x + -l.PT da cho tuong duong voi 7og3r*1(2x +3) 2 + log2*41(2x +3) (3x+ 7)l= 4 e 2log3,*r(2x + 3) + log2, *3( 3x + 7 ) - 3 = 0 (2) Do log3,*7(2x. HArfrxn NAvr rrQc zotz - 2ot3 Bii l. a) Ap dung UiCn eOi (o- b)t : a3 - b3 -3ab1a- b)tac6 x3 : 4J1'3x= x3 +3x=4J2; (1) f : z+J1-3Y> Y'+3Y=2+f,' Q) Tru theo. thoit mdn hQ (i -3* +t=o t,- [Y=6x-5. EAt -f (x) = x3 -3x +1, thi /(x) =3x2 -3 =0 <>.r : I hodc x: -t. Vi fcp fcr = f eD.f 0) < 0 n6n PT *3 -3* + 1 = 0 c6 3 nghiQm ph6n