Toán học và tuổi trẻ Số 215 (5/1995)

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Nội dung chính của tạp chí Toán học và tuổi trẻ Số 215 (5/1995) trình bày về ứng dụng của một bất đẳng thức; kết quả kỳ thi quốc gia chọn học sinh giỏi Toán; áp dụng một số tính chất của hàm số liên tục; đề thi tuyển sinh ĐHTH Hà Nội 1994.

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Nội dung Text: Toán học và tuổi trẻ Số 215 (5/1995)

BO GIAO DUC VA DAO TAO * UOI TOAN HOC VIET NAM 5prs1 rap cuf na xcAy rs sAxc rHANc & Ui"*g dprg e{es ffiQt Wdt dt&ng t/rlsbc F:,=- E 3-,, HE=J=S=T Ir,-.il-. rr.. jl: * Ap dtlng rrr,flt tinh ehdt crta tudm so liOn tqe S Dd t/rli twgdn sinh DHTH EXA NO, 1g{t4
roAN Hec vA TUbr rRE MATHE,MATICS AND YOI.JTH MTJC LUC Trang Tdng hiAn ftp : Nt;tryEN c,iNt t'trr,r,nt o Ddnh cho cdc ban Tnmg hoc Cc sit Phd tdng bi€n ftp : For louter Secondary School Leuel Friercds NCa) DA't t tl Nguy\ru Khd,nh Nguyan. - Urrg dung c{ra m6t HOAN(i CH(INCi bdt d&ng thilc. 1 o Giui hdi ki trudc' nOr odr.rc srEN rAp : Salutiort of Probl.ems in Preuious issue C6c bdi cria s6 211. 4 Nguy6n Cinh Todn, Hoing c Db ra ki ndy. Chung, Ngd Dat Trl, L6 KhSc Probl.ems in This Issue. 10 BAo. Nguy6n Huy Doan, Nguy6n o Kdt qud ki thi qu6'c gia chon hac sinh gidi todn Vi6t Hai, Dinh Quang HAo, ndm hoc 1994 - 1995. Nguy6n XuAn Huy, Phan Huy a Ng4,6n Ph.i L1c - Ap dung m6t tinh chdt cira Khai, Vt Thanh Khidt, LA Hai lidm sd li6n tgc. i5 Khoi, Nguy6n Van Mdu, Hoang o thi vdo Dai hoc Diinlr. cha ctic bqn chudn bi For Col,l.ege and (Jniuersity Entro,tlce Exa,m. LO Minh, Nguy6n Kh6c Minh, preparers Trdn Van Nhung. Nguydn Dang Dd t,hi tuydn sinh kh6i A nam 1994 tnrbng Phdt, Phan Thanh Quang, Ta D}ITH HA N6i 15 Hdng Quing, DAng Hung Thing, c Giii tf todn hpc Vn Drrong ThUy, Trdn Thdnh Fun. with Math.em.atics. Bia 4 Trai, L6 86 KhSnh Trinh, Ng6 Vi6t Trung, Dang Quan Vi6n. iruh hia : Th.d.y gido uit d1i tuydru hoc sinlt gi6i toon tinh IIdt Hung. Tru sd tba soan : 45B Hlrng Chudi, He NOi DT: 213786 Bi€n tQp ud' tri s4; VU KIM THI-ry 231 Nguy6n Van Cil. TP Hd Chi Minh DT: 356111 Trinh bay; HOANG HAI
BAT DAN NGUY6N xruNH lrcuynN lro-ngchrlongtrinh torin cdp 2 cri mQt bdt ding (Hd.i Phbng) thdc quen thuQc mi viQc rlng dungztra nci trong - Vi aU 3 : Chring minh ring V a, b, c ta c6 : khi giai cdc bai tip da sei vd hinh hqc rdt c ab* bc* ca. quA. T6i thudng goi dri li'bdt d&ng thrlc k6p". Bdt ding thrlc dd nhu sau : V qb, 6 * Giai : Ap dUng bdt ding thrtc (3) ta cci : (a * b\2 (x) 1"2+02.>2ab 2 lo2 + c2 -- I zbc D6 thdy (*) *. ["'{a2>2"o lz1"' * b') (a + q2 0) .- + 2(a2+ b2+ c4 > 2(ab* bc* co),+ dpcm. .- .l1o + a;2 > 4ob (z) cridingthrlckhia:b=c laz + 6z > zab (B) - Vi ar.r 4 : Chrlng minh ring V a, b, c, d t ta cci : Ca 3 bdt d&ng thrlc tr6n d6u tudng drrong vdi (a - b)2 > 0 vd do dci chring xAy ra ding thrlc a4'+ b4 + c4 + d4 > 4abcd,. khi vi chi khi a = b. * Giai : Ap dung bdt d&ng thrlc (3) ta crj : Y nghia cria (x) ld n6u l6n m6t quan h6 giffa a4 +b4 +c4 +rl4 .- *b2 +k:2d2 = 2@2b2 +&&). t6ng 2 sd v6i tich cria chring hoSc vdi tdng oic Lai theo (3) : a2bz + c2d2 > 2abcd. binh phtlong c'0ia 2 sd dci. Sau dAy ld c.ic vi du minh hga cho vi6c v{n Tt hai kdt quA tr6n ta suy ra di6u phAi chrlng dung bdt ding thrlc (x). minh. - Vi aB I : Cho a *b = 1. Chrlng minh ring - Vi aB 5 : Cho a *b * c t d, = 2. Chrlng, minh ring a2+ az > ll2; a4 +b4 > Ll8 ; a8 +bB > lltzg. * GiAi : Ap dung bdt ding thrlc (1) vA giA a2+-b2+c2+d2>1. * Giai : Ap dung bdt d&ng thrlc (3) ta cci : thidt, ta cd : a2+ b2 > {a* b)zlz : Ll2 a2+ b2 >- %.b, b2+ c2 > Zbc, c2+ d2 > 2cd, a2+ c2 ) Zoc,a2+ d,2 > zad.,b2+ d,2 >- zbd a4+ b4 > (a2+ b)zlz > 0lz)2lz : UB + 3(a2+ b2+ c2+ d2) > 2(ab* ar* a.d.* bc* o8+ 68 > (aa+ b\zlz >- (Ug2lz = lltz8. bd+ cd) + 4(a2+ b2+ c2+ d2) > (o+ b* c+ d.)2 Cdc bdt ding thrlc trE thinh d&ng thrlc khi = 22 = 4* a2+ b2+ c2+ d2 > I (dpcm) a=b=U2. Cri dingthrlc khi a = b = = d = !. - Vi dU 2 :Choa,b,c > 0. Chrlngminh ring: " 2 (a+b)(b +c)(c + a) > 8abc. Sau diy ta sE rlp dr;ng cric bdt ding thrlc (1), (2), (3) vio vi6c gphr cilc bni tdp phrlc tap hon, * Giai : Ap dung bdt d&ng thrlc (2) ta cri : t^ - Vi aU 6 : Cho a, b, c > 0 vho*b *c = 1. 'l(o + b)" > 4ab Chrlng minh ring : b * c > l6abc. ](a * c12 > 4bc * Giai : Tt giA thidt ta c 0) + (a* bxb+. c)(c+ a) 2 tubc (vi (o+ b)x + 6+ c > 4a(b+ c)z (do6+ c > 0) (b+ c)(c* a) > 0 vi 8o6c > 0). Cd ding,th(c Lai theo (2) : (b * c)2 > 4bc. TrI hai k6t quA khio=b=c. tr6n suy ra :
b* c > 4a .4bc : t&abc (dpcm). - Vi aq 10 : Cho a, b, c ld c6c sd th6a man : {a -- b*c {a = ll2 Ding thrlc xiy ra kht j, = ojb : c = y4. fa+b*c=2 It " 1o2+b2+c2=Z t - Vi au 7 : Chrlng minh ring Y a, b, c tacr5 : -4- (a + b)z(b * c)2 >- 4abc (a + b + c) Chrlngminhreng:a,b,c e l0 ; ;l LDI * Giii: * Giai : Ta s6 chrlng minh cho 0 < o < g, 4 (a + b)2(b * c)2 - (ab * ac + b2 + bc)z cdn d6i vdi b vA c thi trrong tu. Theo (1) ta c (b * c)2 + z(2 - a2) >- (2 - a)2 Ap dgng bdt ding thrlc (2) cho 2 s6 ac vd. + 4 - 2az > 4 - 4a+ a2 + 3a2 - 4@ < 0. (a+b+db Giei bdt phuong trinh cudi theo a, ta drroc ta cd : (ac I (a + b + c)b)z > 4abc (a + b + c) + dpcm. 0- 4ab + I > 4ab Chrlng nrinh rhng : S(ABCD) >- 25. +->-4. 1 * Giai : (Xem hinh 1) ab ,t rrro qe92 : Ao s(AoD) + nx. Dod
: 4 .36 = 144+ S(BOC)+ S(AOD) >- 12. Cudi cirng dd kdt thric.b)i vidt niy, d6 ngh! > 4 + g * 12 = 25. c6c ban vQn dgng bdt ding thrlc (,t) gi6i dc bni Ydy : S(ABCD) tAp sau : Ding thrlc xAy ra khi S(BOC) : S(AOD) hay AB IICD. 1)Choa*b=2. Chrlng minh ring a4 + ba > 2 2) Cho a, b, c e (0 ; 1). Chrlng minh ring : - Vi du 13 : Trongtn gl6cl6iABCD v6i diQn tich S cd didm O th6a rrrdn OA2+ OB2+ OCz+ a)a(l - o) < 1 OD2 :25. Chrlng minh ABCD ld hinh vu6ng Z nhdn O lAm tAm. 1 * GiAi : (xem hinh ). b) C5c bdt d&ng thdc o(1 - b) > Z, 11 Ta cd: S(AOB) * | oo oB , 6(1-c)> 4,c(l-a\> 4 11 khbng d6ng thdi xAy ra. s(Boc) :;oB oc, s(coD) < ioc . oD, 1 ) 0 vd, ara, ... or: !. 3) Cho a1, e2t...t o., S(D)A) =iOD. oA. Suy ra : IL 1 Chrlng minh : (ar+ l)(ar+ 1) ... (an* l) >- 2n. S(ABCD) < + oB.oc + oc.oD+ rlOe.oA at abcd L) ;---:- T ? o, , - -- + oD.oA) < b*c cid' d+a' a*b v6ia,b,c,d>0. (do * ;11;(oA2 + oB2 + (3)) ate b+d c*a d+b 5' o+b + 6q" + + v*o -' 4, +oB2 +ocz +oC +oDz +oDz +oA2) = r^') 't vdi a,_b, c, d > 0. "q6 :J ' tOA2+OB2+OC2+OD2\ t = :.z ,2S=S. 2 \--- - Ta thdy di6u ndy chi xAy ra khi cdc bdt ding 6) Cho a > b > 0. Chring minh ring : thrlc d6u trd thinh d&ng thrlc, nghia ld phAi cti : [oe t oB, oB L oc, oc )- O. Chrlng minh ring : e>ABCD lh hinh vu6ng nhAn O lim tAm. (a+ b)(c+ d) a) a2+ b2+ c2+ d2 >- - Vi dq 14 : Cho ti l6iABCD cd di6n g16c tich S = 32 ; tdng A.B + BD + DC : 16. Tinh b) (a2+ t)(bz+ 2)@2+ q@2+ 8) >- (ac+ 2)2 BD. (bd+ q2 * Giai : (xem hinh ') Ta cri S(AB CD1 = S(ABD) + S(BDC) < 8) Chring minh ring m6t trl giric ldi cri di6n 111 tich S vd 4 canh a, b, c, d ld hinh vu6ng khi vi * ;AB . BD + ZBD .DC = ,BD(AB + CD) chi t
Bni T2/211 : Bo sd duong cb tdngbbng don u{. Chtng minh rd.ng tdng crta hai trong ba sd d.6 khbng be hon 16 ld.n. tf ch crta cd' ba sd d6, Ldi giai : Ggi o, b, c ld ba sd drrong d5 cho a* bl- c = L. Ta bhfng minh o* b > l6abc. Ta cri BAi T1/211 . Gid.i Phuong trinh + (t - c))2 > 4c(l - c) = 4c(a* b) 1 = [c .1 . a* x2+ "-22- --2L + 2,r + 4y + 7 - suy ra b >- 4c(a+ b)2 Lai cci (cL+ b)2 > 4ab. Tt dd suy ra a* b > - \fW +T + ov + lo)(-Bzt + $c + 2Y + 4\ >- 16abc. Ldi giai : cta Trdn Quang Binh, 9M, Phttrrc Ddu ding thrlc xiy ra khi vn chi khi Vin, C:in Drt6c, Long An. [c:l-c la c:7 2 Vid.t lai phrrong trinh dudi dang : )a:b a:b=1,4. 6y jlo) + (-3,' + e + 2Y +11 t 1}11_111u .= NhQn x6t : Bii ndy duoc rdt ddng c6c ban 2 tham gia giii theo nhi6u c6ch kh;ic nhau. Nhi6u : {i7y: +jr+ E + toP3v a a *2; a-;; b+n qu6n kh6ng chi ra khi ndo xhy ra ddu d&ng Di6u kiQn c O DQng Hbng Toan, D6ng Httng, Thei Binh, VO Nhrrng ta tindy Qu.ynh Anh 9T Tt Li6m Ha NQi, Mai Tirng A=?-*+f +W*10 = ?n2+\y +3)2+1 > o (1) Long,PTNK Ha Tinh, Trd.n Hfr u N h.on 9T Vinh n6n suy ra-322+ 4x+ 2y+ 4>- 0 Long, Nguydn Mitth Phuong, ViQt Tri, \tnh Phi., Qud.ch Manh Hdi,8, Phri Tho, Vlnh Phri, - N6u B : -322+'4x+ 2y * 4:0 thi phrrolr;; Xud.n Dung 7A, XuAn Thtry, Nam Hi, Nguydn trinh v6 nghiOm Tltu Thriy,8T B6c Ninh, Hi B6c v.v... Y+y B = -322+ 4x+ 2y + 4 > O (2) DANG HUNG rlr.iNc N6n rip dung bdt d&ng thrlc C6si cho hai sd BAi T3/2rr A+B drlongA, , '# > {AB, ddu ":" xiY ra khi Gid. stl p lit sd nguyan 6 rc. Ddt vi chi khiA = B V6.y a Z* + f + 6y + L0 = -* + 4x +2y + 4 ts n'= sPB-t ' +(2n? - 4x+ 2)+ (y2+ g+ 4)* 322 = o Chrtng minh rd.ng m ld. m|t ho.p sd 16, kh1ng chia hdt cho 3 uit. e2(x - l)2+ (y+ 2)2+ 322 = o 3m-1 = 1(mod.m) Trl dd suy ra (x-l)2:0hayr = 1 Lni siai : ra cri * = (Tl (Y#)=* tA,+2)2:0hay!=-2 d6 thdy a, b d6u nguy6n drrong l6n hon I do dri : m ld hqp s6. 322 = }hay z O Lai c
Nh4n x6t : 1) Nhi6u ban tham gia giAi bai )--- niy cci ldi giai tdt nhtr : Phqm Huy Titng 9A Bd N6n NQR= PAN(2) Van Ddn HaNOi; Cao Qu6c Hifp 9A Thanh Hria, Do C'P = PA vd LC'AB d6u ta c6 PA : Nguydn Tlrunh Nga, DQng ViQt Cudng (Trdn Dang Ninh, Nam Dinh), L€ Quang N6'm,9T, = QE (3). Tt (1), (2), (3) c APNE d6u. 9T Lam Son, Thanh Hcia. Hoan ngh6nh ban Trlc li c6 PR : .RN vd PfN: 600. Nguydn Mintt, Phuong 9A cdp II Minh Phrlong, X6t 2 tarnrr glac PRC vd'NRM c6 : Vi6t Tri, Vinh Phrl da dd xudt vd gi6i dring bhi 1-l,- toSn tdug quAt : "GiA st p ld s6 nguy6n t616 a HM =;BA', - RC, CRM = 600, PNR = 600 a2t' - | li sd nguydn ldn hon 1. Dat * -1, _ t khi dd + ^PRC=^NRM. VaNR = PR n6n LPRC: = LI,{RM. Suy ra PC - NM 14) rz ld hop s6 16 kh6ng chia hdt cho o. Ngodi ra Theo tr6n NMR= PCb ORMC ldttl gi6c n6ua2-l/ p thiom-1 = 1(modrl)" nor f,rep. 2) Ddng ti6c c6 hai ban dd mic sai ldm khi ^MOC:^MRC = 60.] (5). cho rang theo dinh li F6cma nd't (nt,,3) : 1 thi Tt (4) va (5) crj k6t quA cdn chrlng minh. 3m-1 - 1 (modnz). Dinh li F6cma nh6 chi khing dinh di6u niy dfing vdi nt ld sd nguy6n t6. Bni NhQn x6t : GiAi t6t bai niy cti c5c ban : todn cria ta li mQt vi du cho thdy di6u ngrroc lai Luong Thd Nhd.n,l6p 6CT, Bac Li6u ; Nguydn cta dinh li Fecma ln kh6ng dring : Ndu (nz, 3) La LUc 9A1 THCS Ddm Doi, Minh H6i ; Trdn : 1 vd 3n-l - 1 (modnz) thi kh6ng nhdt thidt Huu Nhon, 9T, tnrdng Nguy6n Binh Khi6m, mlds6 nguydn t6. Cdc ban hdy xem th6m bdi Vinh Long ; Lxong Tud.n Anh,9 Torin Li, tnrdng "Dinh li Fecma nh6 vd s5 Camical" trong Tap Nguy6n Du, Ql ?P Hd Chi Minh ; Nguydn. Duc chi ?HTT 3(2ts)11995 Lorr.g, L€ Hod,ng Vinh, Nguydn llnng Hidn, 7T DANC HUNC TFIANG tnldng cdp II Vrlng Tdu, Bi Ria - Vtrng Tiu ; - N guydn. Triln Nam,7T Bdi duong Gi6o duc, Bi6n BdiT{t?ll:. Hda, D6ng Nat; Nguy6n Dinh Tud.n, 9C, Tam Cho tant. gid.c ABC. DUng ub phia ngoiti tam Quan, Hodi Nhon, Binh Dinh ; L€ Quang Nd.m ABC cdc tam gid.c dbu BCA', CAB', ABC'. gid.c 9T, Drlc Phd, QuAng Ngai ; Nguydn Hod.ng Ggi M, N, P liL cdc trung didnt cila cdc d.oqn 1'hitnh,8/2 Nguy6n HuQ, Dd Ning, Quing Nam thd,ng CA', lAiB', AC' tuong ung. Ching minh - Dn Ning ; Nguydn Thily Xud.n DiQu, 40 Bii rd.ng MN = CP ud, g6c gitta cac dudng thd.ng ThiXudn Hud, Thrla ?hi6n - Hud; TruongVinh MN ud. CP biing 600. Ldn, 9Ct, XuAn Ninh, QuAng Ninh, QuAng Binh,Trd.n LQ Thfiy,8T NK Thach Hd, Hi Tinh ; Hudng d6n giii : Nguydn Trl.n Phuong 9A, Nghi XuAn, Nghi Ldc, Triin Nam Dfi.ng, Nguydn Thinh, Nguydn Anh Tud.n,9T, Phan B6i ChAu, NghQ An, Vi€ru Ngqc Quang, Vfi Thi Trgng,9T, Lam San, L€ Hodng Duong, 8T, Truong COng Bang, 9T, Bim Son, Cao Thi Loan,8 TNK Nga Son, Thanh H6a ; Vtt Trd.n Cuong, Hd. Thanh Tud.n 7T, Nguydn Thi Thufun, Biri Anh Tud.n, Vu Thiry Nhu, Mai Nggc Kha, 8T, Nguydn Thanh Nga, Mai Hdi An, Nguydn Anh Hoa, Bili Quang Hdi,9T Trdn E vd Q ld trung didm cria BC vdAC. DSng Ninh, Nam Dinh, Nam H:d; TriQu Trd.n 11 Dilg, 8An Trung Nhi, Phqm. V{t Long,9A Bd Van Ta sd NQ = ;B'C = iAC = AQ. Din,Mai Thanh Binh 8M Mari Quyri, Nguydn Triin Minh 7}{, Nguydn llbng Hd,9H, Tnrng : Vrrong, Vo Quynh Anh, 9CT Nghia Tin, La NQA 600 vi NQ = ,4}/ (1) Hod.ng Anh, 8Ar GiAng V6, Phan Linlt, 94, Ti dd NQR= 600+ 1800 -,4 PTCNN, Phq.m. Quarrg Vin h, 9C Nggc LAm, Hi - 5
NOi ; Vd l{hng Ld.n 9CT Chuy6n Phrl Tho., Br)l B}li T6/21f . Chung t6 rd.ng udi n sd nguyan Minh Md.ru,8AL, Chuy6n S6ng Thao, Vinh Phrl ; duongphdnbiQtal, e.2 ...t clntac6bdt dang thic Va Thi ThuQn, cdp II NgO Gia Tq, IIAi Duong,IIAi sau ddry lit. dung Hung ; Nguydn Ngrr Manh,8B chuy6n UngHoa, nn Hd Tdy ; Nguydn Ngqc DOng,9NK ThuAn Thenll {2"*)' (1) 2"?, k=r k=1 Ha Bic, Lj Thanh Hir,I{rdn Xuong Thdi Binh ; Phqm. Thu Huong, 8A, H6ng Bing DQng,Anh Tl.Ld.n, 8T, Chu Van An 2, }Jai Phbng ; Vu lfong Liri giai (cria da sd c6c ban). V6i n : l, ta DiQp,8B chuy6n Hdn Gai, H4 Long, QuangNinh. c6 a] >- al. B,et ding thrlc ndy dring vi z ld s6 nguy6n drrong. GiA sit bdt ding thrlc (1) dfng C6ch gini d tr6n clia ban Pham Huy Tilng, v6i n = m. Ta chr3ng minh bdt ding thrlc cung 9A, Bd Van Dan, He NOi. dring v6i n = m, * 1 s6 ; a1t a2t ..., an! r khdng VO KIM THOY mdt tinh tdng qu6t, cd thd coi 1 < at < a2 < ... Bei T5/211. Cho tam gid.c nhsn ABC. Cdr ( @,r, ( duimg cao AA1, BB 1, CC lcdt nhau tai H. Dudng Ta cdn chrlng minh bdt ding thric trdn n goai tidp til gid.c CATHB , cd.t trung tuydn CM cia tam gid.c ABC tai T. Trung tuydn CM, mnt c&a tam gio.c CArBrcat duAng ffdn ngoai tidp ()au+x)z 24*x3 > k:l tam gid.c ABC tai Tr. Ching minh T uit. Ttd6i k=r xtng uoi nhau qua AB. H.y Ldi gini. Goi mmm O li trung didm )a|+x3 > (2")2 +2x>ao*x2 cira HC, ta c6 O k:r k:1 k:l li tdm drrdng mm trbn ngo4i tiSp Theo gi6 thidt quy nap thi 2 (2 ti g16c ACIHB. A k:\ "?, k:l "t)' Tt cac tam gi6c Vdy chi cfin chrlng minh cAn MAIB, m OA.H. ta co : x3>x2+2x)au 448:Md"AEAt : W_- HCA, = ^C4 x+ 2l ap Suy ra MA2 : MA? = MT.MC, vit L'MAT ^ k=1 cA1 cBt Do ao € {I,2, ..., r- 1} n6n LMCA (1). Ta cd - cos ^ACB= n6n m * x+ 2l ak 4 x +2(L +2 + ...+ (r - 1)) = cAt AtMt ", LCApr ^ LCABr va 64 = AL{ , ^C$M = k=l ^CA-B Do d6 LCMIA. - LgW, vd ^M,CA, = _*_Lo L. --Y-- l\^/:rl,dpcm. x(x -Lt 2 ACMQ. Md MrCA, = $A-B (nQi tiSp-Elr6n cung 7rB) (3). Tt (l), (2), (3), ta c
Ldi giai : Vdi m6i n € N* dat 6,., : 1111 B- c- r+tgZ onfi - 1. Tt day {o.,} ta cd ddy {b,.,} drrgc x6c l+tg,A- l+tgZ r+tg2 , D-- - l). | -x2 I tx *x2 -x3 -a+- Di6u vta nhAn duoc cho thdy (2) dfng v6i *2x -x2 I *?-x -x2 n. = k + 1. Va vi vdy, theo nguydn li quy nap, I +2n +x2 -x -x3 < l]-?^x -x2 41 I *2-x. -xz 7 *2x -rt (2) drro.cchrtngminh.Tt d6:an:+(b,, *1) = vd bdi torin di drroc chfng minh. :;*t(2-rl5;:"-1 + (2+r/51:"-' +2lvo > 1. Nh{n x6t. Cd 421ry.n gui bai giAi vd tdt ca d6u gtAi dring. l,di grei tdt gdm : a Ii Quang Nd,nt. (97 Nhan x6t : Cti nhidu ban ggi ldi giAi cho bii - Chuy6n Dtlc Phd - QuAn gN gfii), Phant Huy Titng to:'n vi t,5t cA ddu u, ldi giai nhu tren da trinh bAy. (9A - PTCS B6Vdn Ddn Hd N0i),lz Tl"td.n Anh (10B - Chuydn todn tin, DHTH He NQi), Nguydn NCUyFU rH,rC rntlrqH Ngq Hntg (10T - Lam Son - Thanh Htia). oANc vtEtt Be.i T8/211. Cho ttl gid.c lbi ABCD. Chtng BAi T9/211. Cho hinh. ch6p SABCD c6 dd1 minh rang : ABCD ld. mQt hinh binh hitnlt^ Tit mQt d.idm M ABCD di dQng tr€n canh SA dung dudng thang song tg 4+tg 4+tg 4+tg 4+ 16 , :- A B C D-.' - a 4+tg2+tg Z+te Z+t8, Ldi grai V6i x, !, z, t > 0 thi 1111 :+:+:+;) 16 x .y z t x*Y*z*t-. Ma tit gtdc song udi AD cdt SD d N. Tran canh CD ld,y di6m - ABCDJT CO SM fim ui tri cfia M ban SA A])CD ldi n6n o*r,2,r'ztv' suy ra Q soo cho e: ff. AB AD >0,vittaco: dd tam gid.c MNQ c6 diln tich 16ru nhd.t. tg 2,tg z,tg 2,tg 2
Ldi giai. Sau d6y la ldi giai cria ban Trtrong Cao Ding ldp 9T NEng khidu Bim Son, Thanh Hcia vd nhi6u ban kh6c. SM: SN vd theo gie thidt Vi MN ll AD nlnt oo SD SM: CQ .sA ci' '" suY ra : sN cQ +NQII SC SD: CD Ta cri : s(LMNQt:;Z MN.NQ rin friq 1 ngo ai g6c GicVdi drrdngtrd n u (O, R) ; trong vd O' : OO' L BC crlng ld trung didm cria dAy : cung BC vi dat OO' : d. "fhd thi IJ ld m6t 1 rMN.NQsinp drrdng kinh cira (u) vd vuOng g6c vli BC. - ^ (riBC ll AD ll MN \dCS I I NQ) trcngdcip : BCS Theo m6t tinh chdt de bi6t vd trdc tdm cira I Suy ra : tamgi6c, tbLI{ ld truc tAm cita tam glac ABC : max c+MN.NQ +AH :2OO' : c/ (xem cric hinh 1 vd 2). s(AMNQ) = max MN NQ Goi K li giao didm cira c.6.c dudng thhngHM oAD'sc=** vd IJ trong d6 M : @IA t (//) ld hinh chidu cita H tr6n (A1), "fhd t]ni AJKH li m6t hinh binh MN+ NO SN+ ND : hdnh vi AII ll JK (do cing -r- v6i BC) vd HK ll Nhttng 1 AD si: sn sp AJ (do cung r v1i BC). D6 thdy rang : Do dd : s(AMNQ) : max : 1 slASBC) e+ (HM) (= n L @r) (= M) * , ofr: A7l :2oi' : ct MNSMl €+- : _- :74111[ Id trung didm SA ttt IK ADSA2
Tdm lai : ZcRl Ttdd R = {M} = Mt?Mz\tMt, M2} 4-.: CR 4()) Rdt tidc nhi6u ban kdt luAn vOi vAng cho Tr) hinh vE cfing suy ra reng {M} li cA dudng trdn hoic nira drrdng trbn Ur: UR* Ut,to - 3U, tdm O', cd hai ddu mrit tr6n dAy cungBC. Sd di di ddn k6t luin sai nhu vay vi hai 16. MOt ld do Nhdn x6t: Cdc em co ldi giai dung vi. gon : kh6ng chrlng minh ddy dtr phdn dAo (mi thrrc Nguy1ru Anh Dito,l2CT LC Quy DOn, Vflng Tdu ; chdt ld giai bai to6n drrnghinh). Hai li, do kh6ng Le Ch; Tltg,llCT, Ddo Duy '}), QuAng Binh ; thdy drroc M Ii Anh cuaA trong ph6p vi tu tam Nguydn Dinh Th.inh, 10CL, PTTH Phan BOi IK Chau, Vinh, NghO An; Hbng Linh,P"I"IHLam 1, ti s6 k = fi vd didm A chi chuydn d6ng tr6n Son, Thanh H6a; T6 Huy Cua,ng, 11A1 PTTH chuy6n Th6i tsinh ; Nguydn Thd Quy€n, llCL, 6iC mi th6i. Crlng nhi6u b4n kh6ng lo4i "rng hai didm Mrvd.M, Qudc hoc Hu6 ; Nguydn Trong Nghia, PTTH Vi6t Tri, Vinh Phri. 50) Da s6 c6c ban chua biSt chfng minh g6p MT hai phdn thudn dAo cira qu! tich n6n ldi gi6i qu6 ddi dbng. Mat kh6c nhi6u ban dua ra ldi giAi qu6 Bdi L21211 so sdi, thi6u chinh x6c. MQt chdt didru A, lthdi luong m duoc treo NGUYEN OANC PHAT tr€n hai ddy AB, AC. Cdc clidm B, C c6 dinh, cac dd.y AB, AC c6 kh6i luong kh1ng ddng hd Bei Lrizlr ud dq diLi kh1ng d.di AB : AC : I ; Tam gidc Cho m.qch di4n nhu hinh ud : R uit C dd bidt. ABC c6 g6c A bdng 12ff ud d ui tri cd.n bang Hdi td.n sd g6c r.,.t cia ddng diQn xoay chibu bd,ng ddy AB nam, ngang. md1 dd di4n dp ra cirng pha u6i diQn d.p uito. a) Tinh h1c cang cfia cd.c dA.y AB, AC klti he Tim ti sd girta cdc hiQu di|n thd h.i)u dung uito thdng cd.n bd.ng ; bidt trqng luong P : mg crta ud. ra. chdt didm A. b) Tinh chu ki cila nhttng dao d1ng bi€n dQ nhd cia hQ th6ng. rrr(RA Hudng dAn gini a) VE hinh, bidu di6n c6c luc t6c dung l6n A ; tthinhv6suy ralr: ft"u ,r: # b) Khi h6 thdng dao d6ng, A chuydn d6ng tr6n m6t cung tron trong m6t m4t phing t4o v6i phrrong th&ng drlng m6t gdc 30o. Hinh chidu cria Hu6ng d6n gini flOnmatph&ngdcildg' : gcos300 : g S- 2 Ve gian d6 vecto theo kidu gh6p vdi cdc chri y nhtr sau : lo crlng pha vdi uAB, ics6mphanl? chu ki dao d6ng T : Ztt vdi t' : tl2 -, cing pha v6i i, cbn up,g trd {5 so v6i uAn ', uDa pha wl2 so v6i i (cr1ng li so, vdi up,r) ; r: 2n r-- €--++---+€+ I:1"*In ; Uu.t:UMD+UDA ; l;ff i** =i, =iro *iou :i*o ni, NhQn x6t. Cac em cci ldi giai dring vd gon : +++ Mai Hod.ng Chuong i1A3 L6 Hdng Phong, Mudn t{ cing pha vdi U, thi U MA phAi cirng Nam Dinh, Nam Hi ; Vo Dinh Hidu, IOCL, PTTH Ddo Duy Tr), Ddng Hdi ; QuAng Binh ; pha vdi d^ ,o hinh v6 suy ra 'i ='* "u Dinh Ngqc Nhd.n 10CL Dno Duy Tt, QuAng tsinh ; Td Huy Cudng,l1Ar, PTTH chuy6n Th6i I.- R L Binh ; Nguydn Trgng Nghia PTTH Vi6t Tri, -= IR 4 \finh Phri. MT 9
Bii T7lzts a) Chttng minh dng : 1 1^, i.l q - r.i.l*, +Tc: ci+2 - 1 .!S.l 4,]t= l b) Chitng ninh ring : . 1^ 1 c; r" - 3 ci 2' + a2.Ifai clSy sd {an} i , tt.} f du6 xac tIlnh nhd sau : Dt=60\', r,=30tI , . {n,-:n2 I Dr-I D ltr = l ttC ao = a, bo : DVA< lr,+,,=Zhn-ui R! - 45o g Vn = 0,7,2,. .. Chilng rrrinh ring trin tai nt6l sdrr > {} rrri 10r) >0. Rr ,,. 90() NGTJYE,N MINI] D(JC TRAN vAN N{INTI rId N1i Hd N6i l0
T7l2l5. a) Prove that Fffi ffi m,fl;E fr , h1.,.,;.$s$fl'S...;.!S,ffi .Sffi - Iecln, + *r?,q., - .1.1;A I ",,"n (-1)n-1 -... + E+cic.-!r= t For Lower SecondarY Schools b) Prove that Tll2l5. Find the last eight digits of 51ee5 c,r,L" iq,r" + lev - TZl2l5. Let (r1, !1), (xr, l) be the two solutions of the system of equations . I 2tt(nt\2 : + (-l;n-t Z" _lCf;nn : i{ i'-"-3:o TBl2l5. Prove that for every .r such that 1r.*y2-2a-2y-9=0 tgx, tglx, tg3r are defined, we have : CalculateW 3(tg2x + tg22x + tgzgx) > (tg tg2x tg1x)2. Tgl215. Consider the sequenca a1t Q2t a3t...t Tgl275. The incircle (with center O) of a non where isosceles triangle ABC touches the sides BC, CA, AB respectively at'Ay, Bt, Ct. Let M be 2 Q,: (?-x + t)(tln +ffiT1)(neN'n>o)' the intersection of the lines BC and B,C' Prove that MO is perpendicular to AAr. n ProvethatS, - at*azi...*or, 1n+2. T7Ol2l5. For which tetrahedron OABC, right at O (i.e. the trihedron OGBC) is right), T4l2l5. A circle with center 0, passing through the vertices A and C of triangle does the quotient :h (where fu is the altitude ABC, cuts the side AB at A and K, cuts the OH arrd r is the rjrdius of the inscribed sphere side BC at C and N. The circumcircle of of OABC) attain the maximum value ? ABC with center O, cuts the circumcircle of KBN with center O, at B and M. Prove that o p2ll OM. T5l2t5. Two points M and N move respectively on the sidesAB and AC of a given triangle ABC so that BN = CM. Let T be the Crk ban cd tfrd mua brio Torin fioc aa intersection of the lines BN und_9{.Prove that the inbisector of angle BTC passes tudi tri tai through a fixed point. - Cric C6ng tLpfrit fiAnfiSrirfiaafflidt 6i truang fioc tinfr, tknnfi pfid. For Upper SecondarY Sehools - Cric fri€u srir.fi trur"Lg tdm tinfr, tfidrlfi pfid, tfii 4t, fiuyQru tfri trdn. T6/21,5. Consider two sequences 9{o|.c fdt rruua fai finn tai Euu [i6n {",} ff=,, { buJ f,:,, defined bv : tinfi, tfidnfi pfid, tfri Ka, ftuyQn trong ci &o: a, bn = 0, nuoc. b.*1 = b?r, Gn+t - 2b, - al (n. : 0, 1, 2,...), Where o and b are two grven real numbers satisfying 4b > a2. Prove that there exists zz > 0 such that o,., > 0. 11 II Ed
X2\d7t KHT AUA KI TTd$ ftUOG GEA CFIOTS FICIC SINF! GEOX TCIAN NAM HOC L994. 1995 A. BAc rRUNG Hqc PHd T[{6NG j) Gi(ii ba. (tir 25 ddn dtr6i 30 didm) Hb Dd,c Phuong, LA Son [Jyan, T'ran Dang Hita, Pham. Quang Tu(in, Dinh Thitnh Trung (DHTH Ha Ki thi qu6cgia chgn hoc sinh gi6i To5n THPT NOi) ; Trinh D*c Vinh (DHSP Ha Noi l), Vu nam nay d5 tiSn hdnh trong hai ngay 2 vd 3 Than g Binh (tp He NQi) ; Phan Anh Tud.n, Dqng th6ng 3 nam 1995. Cric d6i dai di6n cho 5 tnldng Anh NguyQl (nrr), T6 Di€u Hiittg (nrr) (Hd TAy) trdc thu6c 86 vi 51 tinh, thdnh ph6 da tham Biri Trong Qudn, DQng Qudc Dung (HhiPhdng); du. C6c d6i drroc deng ki thi d b6ng A hodc bAng Trinh Thd Huynh (Nam }lil'), Hoirng Qu6c B. NOi dung dd thi ctra b5ng A kh
II. Bing B Duong Thu Phuortg (nu) (Ha Ttnh), Nguydn Huu HQi, Triin L6Nom, La Quang Nd.m (QuAng D Gidi nhi (tit 28 didm tr6 l6n) Dod.n xud.n Ngsi), Dinh Dd Quang (Khrinh Hba). Tidn (SdngF,6), Tr?in Van Thriy (Vinh Long) 2) Gildi bu (til23 ddr, d,t6i 28 didm) Hod.ng 3) Gidi bu (25 em) (13 ddn dudi 15 didm) Thanh Hd.i, Hodng Vd.n LI.m, Vu Thanh Binh, Pham Nguydn Thtt Trang (ntt), Tr1.n Trung Nguydn Truimg Giang, Nguydn Thanh Tilng Thdnh, Mai Thanh Binh (HdNQi), Luong Tud.n (Hda Binh), Duong Trung IIdi (Q:u3;ng Binh), Arth., NinhThanh Cdn (TP Hd Chi Minhl,Trdn Nguy1ru Xud,n Hilng, Til Minh I{ri.l (Daklak), Trung Nghia (Nam Hii, Trinh. Tud.n Hilng Nguydn Thanlt Cat (Long An), Nguydn Van (Thanh H6a), Nguy6n Thi HOng Nhung (nrt) - Duong (Minh HAi) (Vinh Phri), NguydnTlti Thu Huong (nu) - (He j) Gidi khuydn khich (tir 20 ddn drrdi 23 didm) Bec\,IIodrug Hd.i Anh (HAi Httng), Dod.n NhQt L€Van.Manh (HbaBinh), Nguydn Long Quynh Dttottg, Trd.n Vd,n l{od.ng, Hd Duc LQc, Dodn (L4ng San), Nguydn Thu Uyan (Ldrn D6ng), Ld Ngqc Ti (Th6i Binh), Nguydn Thi Thd'o (nuL Minh Thd.n, Trd'n Van Phttdc (Vinh Long), V6 N g u ydn Trsng Nhui.nzg (NghQ An), La Huy Binh, Hod.ng Trung (Trn Vinh) ;Vuang Qu6c Khuong Trftn. Th,i TiAn Giarug (nit'), Dang Thi Hbng (SOng 86), Vd Ngqc ThuQn (Ki6n Giang) Duong Itlinh (ntt) (Hd Tinh), I'{guydrt Vir (K}rrinh Hda), Hoitng Ki1t, DQng van Quydt (Minh FIAi). Ltii:t trIbng Thu (ntt), Nguydn. Thi lldng Nhung (n{tt. P.han Anh Tudn, Trd.n Dai Nghia (D6ng Naii. l,rzzz Minh Duc (Ti6n Giang). ,l) Ci{ii khuydn khtch (60 em) (tr) 10 ddn du6i B. BaC TRUNG HqC Co So 13 didrn) Ki thi qudc gia chon HSG PTTII m6n to6n (oo I'rd.n KiAn (Ild NQi), l/guydit TudnVi4t, l6p 9 ndm hoc 1994 - 1995 td chilc ng.iy Ng,uj'irt Drlc Hoitng LIa, Nhrt Xudn Tlri€n Chd.u 2l3llgg5 chia lim 2 bAng. Dd thi d nrSi bAng (TP 116 Chi Minh) ; Nguydr", Tltt Vd,n Kltd.nh g6m co 4 bdi torin ldm trong 180 phrit (kh6ng (nG:. D6 Thanh Mai, Biri Trurug Ngoc (Hd TA,y) ; kd thdi gian giao d6) v6i tdng sd diem cho m5i Ngu1,€n Th.anh Huybn (nit), Ngd lv{anh Hit, Cao dd la 20 didm. Viit Hing, Nguydn Ngqc Linh, Phant Tud.n HQi d6ng chdm thi qudc gia d6 quydt dinh mdc Minh, Nguydn Doan Tidn (IJii Phbng) ; Nguydn didm cho c.ric loai grai (nhdt, nhi, ba, vi khuydn Ar,h Hoa, Nguydn Tlwnh Ngo (nit), PhqmVan khich) v6 mOn to5n l6p I 6 cdc bing A, B : Qzzcii: (Nam H}) ; NguyQn Van Thud.n, Tri.n Xud.rz Giang, Phqm Thi Loan (ntt), Vian Ngoc Quang, La Minh Thd.nh (Thanh }ftia) ; La Van DANTI SACTI CAC HOC SINH DAT GIAT: Quiic Anh, Trd.n. Thtiy Chdn (nt), Dinh Trung Hodng, Le Thi Quynh Trdm (Thita Thi6n - I. Bnng A (c6 109 em dat giii) : Hu6) ; NSO Chi Trung, Ngo Qudc Tud.n (Quhng l) Gidinhtu (3 em) (didm tr) 17 ddu 20) Nam - Da Ning) ; Vu Qu6c Huy, Nguydn Van Vuong Mai Phuong (nit) : 18 didm Nam, Nguydn Thi Thu Thily (ntt), Plwnr D6 (HAi Hung), Nguydn Hoiti Nant 17 didm ViQt (Wnh Phrl) ; Cung Mo.i Loan (nit), DQng (Ha Tdy), Trd.n Nam Dung L7 didm Anh Tud.n (B6c Thei) ; Dqng Hod.ng Viet HiL, (Neh€ An) Ngi Quang Hung, Nguydn Thi Huybn Linh (nrr), Ngaydn Tidn Manh, An Vu Thd.ng (H?t 4 Gini nhi (21 em) (tit 15 d6.n du6i 17 didm) Bic) ; Pham Quang D{rng, Pham. Van Hdi, Triin Thd Quang, N guydn Hoitng Anh, Pham DQn,g Quang Huy, Duong ThjVdn Thanh (r,n) Dinh HuonS (Ha NOi), y, Dttc Phrt (TP H6 Chi (HAi Hung) ; T0 Thl Minh Hbng (n.u), Nguy4n Minh), D?to Thi Thu Hit., Dq.rlg Anh Tud.n (H.hi Truimg Thanh, Bili Quang Thinlt (Thdi Binh) ; Phbng), Vu Xud.n D{rng, Trinh Phan Hd' (Nam V{t Hdi Chdu, Nguydn Van Dilc, Vu Nggc Hd. Hd), Cao Qudc Hiap,Vu Dinh Phuong (Thanh NguydnTil Hod.ng, LaTh.anhNam, Phitng Anh H6a),Van Trung Nghia (Thia Thi6n - Hu6), Soz (Ninh Binh) ; Ch.d.u Van Dbng, Nguydn Hod.ng Thi Nga (nrr) (Vinh Phn), Dqng Dtc Thiruh (NghQ An) ; Thhi Thq, k Ngw Thd.nhVinh Hq.nh (Ngho An), Trinh Thi Kim Chi (nir), (He Tinh) ;Bil.iThiPhuong Uyen (Qr:6rngNgai) ; Phan Thi Thanh Hd.i (ntr), Mai Ting Long, Ii Thi My Linh (nir), Duong Thnnh Trilc (nt), 13
Trilu Tri Dung (Ddng Nai), Ld Duc Duy Nhd'n Ngl Anh Vu (Bdn "Ire) ; Trd'n Ngoc Minh (nu), (Ti6n Giang). Ninh lling Phic (]finh Long) 4) Gidi khuydn khich (25 ent) : II. Bing B (c6 45 em dat gi6i) L€ Minh Drtc (YdnB,61) ; Phimg Ngqc Hibn Phuong (niil,Va Thi Huybn Phuong (nu), Ngd 1) Gi(ii nhdr (3 ent) : Hbng Quang, Nguydn Bd. Trung (Hda Binh) ; Trd.n Anh. Tud.n (18 didm), Ngzy1n Ti6n Anh Nguydn Hrtu Cd.u, NguydnVi|t Linh, Trd.n Huu (17 didm), Nguydn Arth Dung (17 didm) (Hda Lttc (QtimgBinh) ; Nguydn Bd. Thitnh, Nguydn Binh) Dinh.Vtt., Trd.n Song KiAt, Huynh. ThiVfi Quyih (ntt) (Gia Lai) ; Trd.n Thi Nguyen Ny (ntt) (Kon 2) Gidi nlri: (6 em) Tunr) ; Nguy1n Xud.n Thu (Dak Lak) ; Nguy&t Ha Khar,h T'od,n (.H.da Binh)' L€ Anh. Dung Tidn Hang, Trun Nguy€n Phuong,I'lguydn Tlti @ak Lak), Mai Drlc Thanh (Dak Ldk), Vd lfanh Hanh (nir) Gam D6ng) ; Hod,ng Mai Quynh Nguyan (nfi) (S6ng B6), Cao Anlt Dic (Tdy (Ninh ThuAn) ; Trl.n Quang Ainh (Long kn) ; Ninh), Nguydn l,i Ltlc (Minh HAi) Hit. Mai Lan (ni), L€ Ngqc Thny 6il (S6ng 86) ; Nguydn Trd.n Quang Dung, Dxorug Thd.i Hoit'i 3) Giui bu (11 cnt t ' (Tay Ninh) ; Nguydn Dq.ng ThuQn An (Mrlnh Jlod.ng Manh CuitrLg, Truong Vinh Ld.m, Hai) ; Nguydn. Hd Hdi Dang (Yinh Long). Nguydn Nggc Linh, Nguy.anVidt Thanh (QuAng Binh) ; Vu Hdi D1ng, Trd.n Minh HQu, LO Trgng Vinh ( Dek Lak) ; Phan Huy Dao (LAm Ddng) ; .^ NGT]YL,N VIET T,IAI _ NGUYEN HOU THAO Gring b?n dge , B€n cqnh vi€c phiut dnh tinh hil* dqy vd hoc todn d crtc dia ph,ddng qua cdc bdi vidt, TC TI{I/T7'cdn dring rinh nhdm dta ddn ban c)oc nhrtng sd bdo phong phri t,d siirh tlQng ca vb nQi dung va lthth thrtc. T'odn hqc vd tudi trt mong nhAn du1c citc hitc dnh dqp ctia bqn dr.tc xa ghn gtli tdi. Cdc dnh n€n lti dnh chup cdc hoqt dQng li€n quan cldn hoc sinh gi6i todrt, tin hoc. Anh c6 thd chqp vb hour dSn,g riqy vd hqc ; cfi.ng cb thd chup t,b cdc sinh hoqt ngoai khbn, vrin ngh|, thd thao, tham quan du lich. Anh n€n ld dnh mtiu, cd il nhil lii 9 x.12. Sau dnh chrt iltich rd nQi dung, t€n vi dia chi ngurli cht1p, kh)ng dp plastic. ChAn thdnh cdru dn cdc bqn TOAN LIOC VA TUOI IT{E L4
AP DUNG MOr rixn cHAT cuA HAM sO LIEN TVC NGurEN PIIU Loc (Cd.n Tho) Trong bdi b6o niy, chring tdi nhdn manh ddn Biti tod.n 2 : Chobidt2b *3c = 0. Chfngminh mOt tinh chdt ctra hdm sd li6n tuc vdi vi6c fng iang phtong trinh : aros2x * bcosx * c : 0 lu6n ciUng tinh chdt ndy vio giAi vii bdi to6n cri nghi6m thu6c kho6n e (O , ntZ) 1. Tinh chdt Giai Ndu hdrn sd f liOn tgc tr6n doan [o, b] vi Ta c6 acos?-x * bcosx * c : 0 phrtong trinh f(x) : 0 v6 nghidm tr6n doan +a(Zcos2, - 1) +biosx *c:0 lo, bl thi f(xS , 0 v6i Vr € fa, b) },ay f(x) < o v6i Vr e. lo, bl *2a . cos2 x * bcosx * c -a=0 Ching minlt DetX = cos:r)0 < X < 1, ta cri : Gi6 srl tdn tai xl, x2 thuQc doan [o, 6] v6i 2a*+bX*c-o=0(1) xl 1 x2vAf(r,) .f(x) < 0, vif li6n trlc tr6n do4n vab*P fx, x2) n6n fc e lx, x2) sao cho f(c) = 0; mAu X6thdms6F(D=o 2 * g +(c-a), thuin i f(x) = 0 v6 nghi6m tr6n doan la, bl. Yety f(x) > 0 Vr e La, b)hay fk) < 0 Vr e la, b7 f(4:2a*+aP+@-a)x 2. Ap dsng f(A=x(Za*+bx+c-a) Bdi tod.n I ; KhAo s5t stl biSn thi6n crla him GiA str (1) vo nghiQm trong khoAng (0, 1). sdflr;:'[i-4 +{9-.r Vi hAm g(4 : 2aP +bX * c - o li6n tuc Gidi trong kho6ng (0, 1) n6n : ivlidn x6c dinh : [4, 9] I c6) > Ovxe (0, 1) hayg(X) < 0vxe (0, 1) 11 ,'{.r): --=_:= o Ndu g(X) vx € (0, 1), thi > 0 2{i=4 b[e - x \f0 -r -{i=T f'(X) > 0 VX (0, 1) + hdm f teng trong (0, 1) f(x): hE=[ \i9; nhung/liOn tuc tr6n doan [o, b] n6n : 13 /(o) = o < 111) = | * ;='];Y= o v6 Ii f(*)=0€x: 2 111 r Ndu g(n < 0 vx € (0, 1), l4p luan tudng tu nhu tr6n ta cci : 111 f(0)>f(1)baY0>0v6li f(81 :4-r=-a.0 Viv (1) phAi cri nghiOm thu6c kho6ng (0, 1) ,\ay phuong trinh acosZxibcosx *c:0 cd vd ri harn sd f(x) li6n tqc tr6n cdc khoAng nghiem thu6c khoanf (O,;) (o f) "u (+ , e) non ta cri bang bidn thi6n Biti tod.n 3 (Trich Dd 51 - B0 dd tuydn sinh) ' nhrl sau Cho tam thric bAc hai f(r) = ax2 +bx * c (o * 0). BiSt ring phuong trinhf(r) : r kh6ng cci nghiQm, h6y chtlng minh ring phuong trinh a(f(x))2 +b(f1r)) *c:x kh6ng cci nghiGm (Ban doc trr tim ldi giei) L5
DiNIr cmo cic nAN cHUiN E! r'Etx vAo DAI HQC \hNv\\A OE THI TUYEN SII'II| KHOI A NAM 1994 TRUONG OllTH llA l\lOI (Thiti gion. lir.m biti 180 phrtt) PHAN B.i,T BT,OC Bing bidn thi6n : Ciu I. Ctro hirm s6 -@ -2 l+2'r+z r xl7 1 *6 -----'-* 1) KhAo sht sU bi6n thi€n vi v6 dd thi (C) cira hiun sd. -a/ \z : )r * 2) Gie sl tlurrng thingy m ci;t tldthi (() tin hai di6mA, B.Tinqu! tich trung didm/cira eloanAB khim thrv dtii. Db thi : (xern hlrth v€) 2) t4p phrldng trinh , x'+2x*2 CAU II. l) Giii phrlttng trinh l9[g!4ggiiic__ I ----;--i--i-- = -r + m(l) -rrI *in3r + "."r! I+I I ,{V7E7i + - Y2 Vl-Tlr,, v1i x * -1 ta c() +Zr+2: (1)0 (l) Cdu [V h l.l Trong rnit phing ( P ) cho MBC. Tm qu! tich cic tlidnr Mtrong (P) sao cho : die.n tich A,&flB dien tich AMIC = 2) MOt thidt diQn chia thd tich hinh lap phuong thinh hai phdn bing nhau. Chilng minh rlng thidt diQn dodiqua t8m cria 2) x6tl L,li;v=iitat (A)+racbx=-i1l + klGeZ) + V-sint+cos;r = {Y*, hinhlQpphtfttng. Xdt (B) ta c6 5in(srzr) (\/-D' = (Vcov-sirx +{5is*casr)' > ?tos > tt) -1.1 Tinh linr --;-. .{z (vi theo (*) ta c6 cosr > 7 ) DJ.PAN , 2-sin2r 3 trong khi (W)" : PHAN Bi.T BUOC: 44 < ;. Vi vay (8.) v6 nehi€m. ,z +2x +2:x+1+;;lI CduI y: r+1 -f,+m 6e Kdt luan : nghiQm cua (1) lA ': 1) TSp xac dinh : R\ {-1}. Chtl! : Cit rhd ti6n d6i W cta (B) nhrr sau : :_1 n:_z y'=l----=0
2) Ta c6 Lt + br'+ cz : ADC A tn fC ) : -,a (cosrln t Jt aoa3* ,r. - i l") :T' Zn 2R 2R bt 1az +iz +r21 @x+by +cz) D.SI:, _ abc ll 2) a) (dr) c6 vectd chi phgongu : (2.1,31 ++ (ah 't bc 't cal (d2)covectdchiphudngu : (1.2.3) : hai vectou. u kh6ng , -- (tu+b1' +cz) : ,t,- cdng ttrycn vay @ tt vb (d ) un?l.*j.,rg:"lB.vdi nharr. r l 1, +f : +VZ)'(*,"o i )i ;: ;- i (;*h,;)(ar*bv*cz) >(,[V Mdr khac hc phrrong rrinh 2 dn h r, s ra 'o, b2+ cz l3r-3=3s+1 Bunhiac6pxki)= \" + f + V2 < v6 nghiOm n€n khong ddng phBng. Vay 1dr;, 1dr; chdo nhau. : b) X5c dinh mflP) : Ax,+ By + Cz + D 0 di qua (d,) vn la:b = t ll (dr,)Yecto ph6p tr,rv6qn : (A, B, C) cna @) inlcgiaov6i Dau : xiv' ra e I . l,i" ra khi vi chi khi MBC U +B+3C:01 , + l.Y:\':z -- ,.r- A + zB + \C: gf+c,;thichon/r: (1. l.-l): d?tt vaM ld nong fitnVfia LABC. C6u III ld) c (P) cliem (1. Z, -.ti e (dr)=> A + 2B - 3C + D : 7+ lrx) - Zto - h1r + Lt: - b: = o 2+3+D:g==+D:-6 ..1 + 21a * i)t + a)' 2.b: : ll 1P1ctiphUdng lrinhr +.y' z.= tr.: 0. Khoang cich girta : CA 2 phudng trinh cri nghiQm vA c5c nghi€.n bAng id,) khoing cdch va 1d.,1 la I Lt I dienr bet ki c (d,) d€n /P)chhng 7'.H.1 nhau ; thco Vi-et ta cri han rlr (2, -3. + I ) den /P, + 2ta-h) - -Zlo +b)l_ . lZ-3-1-ol 8 s./3 ^_L_n v'{ 2o2-i:i+lr. f-u:(t:tt \f,'. ''.4r7 3 1.H.) : Ci J phtlctng tnnh cung v6 nghi€rtt : Cdu IYb rl;5, = @ - hf,- (:q b',t < Oaa',-2h' + Zab> t) t) dt aMAB : drL.MA( (l) J. - lo + hf ' y,t- + 2h'; 01adll{tc PHAN BAr BUOC (?d,s) , lr+--5=0e2r' -5t+2:0= l'L. 2 -1 CAU I / _- 1) 1.5 di6m:* KXD I cl ':*' - \".cl.lc tri I /2 rl .rr"3:* : /+logl : ngr, > ()+ t > 7 - Ti6m cAn. bAng bi6n thidnTl2 d 1, viv t, :; < 1lo4i. tr: )+ loglr: l(rgr2: l+169.r': + I - D6 rhi 1/4 d. 2) 1.5 drr3m : - Vidt clrt
Gut oAr nAt Ciu chuy€n trong rrlng Goi tudi Al?t x (r nguy6n drrong). Ta cri : Tudic : tudiD -tudiB : (r + 30) - (x - 90) = 120. Vi tich cdc chu s6 trong sei tudi ciaA ld mQt sd khOng 6m n6n ta cd (x + 30)(r - 90) -100.120>o or2-Gox-14.700>o ax > 30 + {15600 , M9, > 0) (1) Mat kh6c giA st x : ataz ,.. ar, (v6i o, * 0) _) thitac6ar.a2...d,,4 o,, 9'-'