Chuyên đề bồi dưỡng học sinh giỏi giá trị lớn nhất, giá trị nhỏ nhất phan huy khải (phần 7)

Chuy6n dg BDHSG To^n gia tr| lan nha't g\& tr| nhd nhSt - Phan Huy Kh5i ,atBB| = x ; B , C = S + S, De lhay A B B , M , CCjMA,, 1+ •iB = A B ; A M S2 = ZSj + — ,- (3) cung chinh la AA3B1C2 MB1C3 la ^ cac tam giac deu vdi canh lan liTdt la x, y, z, cua chung thi: (5) >— S (6) O S1+S2+S3 ^S; (do a = (difa vao nhan xet hicn nhien sau: Ne'u a + b + c = 1, thi a^ +b^ \di ) S ^ ) (z^ x ^ + y ^ + z ^ V ^ y (x + y + z)^ x^+v^+z^ ^ S (1) s/ +S^\MB,C2 +SAMA,B, ro up SAA,B|C| ^ S A ^ A J C J om c ok bo ce fa w ww la dien tich tam giac ay Tim gia trj Idn nha't ciaa - ( x + y + z ) ^ nen tir (1) suy ra: >is (3) T C f ( ) ( ) s u y r a : ^ < | (4) Da'u b^ng (4) xay o c6 da'u bang (3) • o x = y = z o M l a tam cua A deu ABC Hudng ddn gidi Goi S la dien tich tam giac dcu ABC canh a, thi S = ( S M A , A A , +SMC2CC3 +SMB2BB1 \ a Da'u b^ng (3) xay x = y = z = - Bai Cho tarn giac dcu ABC canh a M la diem tily y n^m ben AABC Xet tam giac c6 canh la MA, MB, MC Goi ^ s'l +S2 + S3 CO dau b^ng (6) o O la tarn AABC B2C2, A3C3 ^ fy^ + y + z) (7) is O la tarn AABC S /g O la tarn AABC S Qua M ve ba doan thang A|B,, CA (xcm hinh ve) X Tac6: P = BH.CK = BH.CK' Ap dung li luan phan vao lam giac ABC", la lhay tich BH.CK'(cung la tich BH.CK) li'm nha'l A la phan giac , ciia goc BAC' (luTc A la phiin giac ngoai ciia goc A) va gia trj Idn nhat = bccos — 2 Theo nguycn li phan ra, thi: maxP = max{Pi; P:} = max|bcsin^y; 358 ubccos — « s/ Vay cac di/ctng thdng thuoc nhom 1, thi du'clng phan giac hang p = be sin hoSc la phan giac ngoai cua goc A „ Nhqn xet: Trong bSi tren da stf dung nguyen li phan cua bai loan tim gia tri Idn nhat va nho nhat, di nhien ke't help vdi cac lap luan ve hinh hoc phang c6 iJng dung lifdng giac! Bai (Bai toanToricelli) Cho tarn giac ABC c6 max {A, B, C} < 120" (tiJc la mpi goc cija tarn giac deube hdn 120") Hay tim tam giac ABC mot diem M cho MA + MB + MC dat gia tri be nhat HUdng ddn gidi Cdch 1: (Lc(i giai cua Toricelli) , , Durng ba tam giac dcu ABC,, BCA,, ACBi phia ngoai tam giac ABC De tha'y ba dUcJng iron ngoai ttep cua ba tam giac deu ay dong quy tai diem T, va T la diem nhin canh cua tam giac BAC dUdi ba goc b^ng va hlng 120" Do CJTA = ABCJ = 60", nen Afc = 120"=^ C,, T, C th^ng hang Li luan tiTdng tir c6 B, T, B, thing hang va A, T, A| cung thang hang La'y diem I tren TB|, cho TA = TI i=> AAB,I = AACT (c.g.c) 359 Ta cos(a - 2P) = P = Y o A lii phan giac Irong cua goc A iL ie uO nT hi Da iH oc 01 / cos(a-2P)-cosa bccos^ —, ne'u A la goc nhon (o < a < 90") Civ => B|I = C T => T B + T A + T C = B T + T I + I B , = BB| (1) V a y M ( i la d i e m nhin ba canh tam giac durdi ba goc bling = 120", tuTc Mo L a y d i e m M baft k i thuoc m i e n tam gidc A B C Ta se chiJng m i n h r k n g : M A + M B + M C > T A + TB + TC , „: !NMM M IV r i W I I Kh.il"! Viet_ la d i e m T o r i c e l l i phai t i m (2) C^ch3: That vay diCng lam giac A M E cho E cf nufa m a t phang hci A M khong chuTa B ^ ^ :r m- y Ro rang A A M C = A A R B , => M C = E B , => M A + M B + M C > T A + T B + TC R6 rang ta c6 b d de h i c n nhien sau day ( v i the chung t o i bo qua chtfng minh) :=> (2) diing D a u bang xay N c u QNP la tam giac deu, I h i v d i m o i vj t r i cua d i e m M n a m m i e n tam M = T giac QNP, ta l u o n c6 long khoang each tif M t d i ba canh cua tam giac Q N P "Toricelli" la hang so' (c6 the tinh di/dc hang so bSng chinh c h i c u cao cua tam J^'i , 2: L a y M la d i e m tijy y m i e n lam giac A B C ^ giac d c u QNP) R A p dung bo de ay ta g i a i bai toin T o r i c e l l i nhifsau: Thirc h i c n phep quay R" ( B , " ) Khido: R-(B,60") C-yA, B c u a tam giac dvtdi ba goc bang nhau, tuTc la: A T B = B T C = A T C = 120" * • Q u a A, B, C dirng ba M ^ M ' Hdi/dng T h e n tinh cha'l cua phep quay suy M B M ' la tam giac d c u => M B = M M ' Ngoairado: * ^tai MC = M'A, (1) up jjnj^ s/ I V a y M A + M B + MC = M A + M M ' + M'A, '' 7jy (*) vuong TB, TC goc Ba Hdirdng c^t diTcfng gap khiic, ta c6: (2) Q, N , P De thay QNP la tam gi^c deu L a y M la d i e m tiiy y A A B C G o i h , , h2, hi la khoang each N h i r vay tir (1) (2) ta da chi^ng m i n h difcfc rhng v d i m o i vj t r i d i e m M thuoc tijr M t d i ba canh NP, m i e n A A B C , ta luon c6: M A + M B + M C < A A , ok (3) A, M , A2 thang hang bo Da'u bang xay o cua phep quay ta c6 goc tao b d i M,)C va bkng " =j^MoMoC = " B M o C = 120" M„A| fa chat w tinh A M „ B = I20" ww Theo ce Gia sur Mo la vj t r i cua M ma A, M„, A, thang hang Do BMoMo = 60" c om /g M A + M M ' + M'A, .'1'VV,,,., Ta Cdch • ' iL ie uO nT hi Da iH oc 01 / V a y d i e m M can t i m chinh la d i e m T n i tren D i e m 66 thi^dng g o i 1^ diem PQ, Q N cua AQNP m M Ro rang ta c6: M A + M B + M C > h| + h2 + h j H • • T h e o bo de neu tren (ap dung vao tam giac deu QNP), ta c6: h , + h2 + h j = T A + T B + T C Tir suy ra: M A + M B + M C > T A + T B + T C (*) • Da'u b^ng xay (*) M s T Do chinh la dpcm m:(ich4: Sis dung b6 &G h i e n nhien sau day (ChuTng m i n h ra't ddn gian va x i n danh cho cac ban doc) B6 de: Gia si^ A M B = B M C = C M A = 120" va M A = M B = M C t h i : M A + M B + M C = - : Do gia thie't m a x ( A B , C) < 120", nen t6n tai nhaft d i l m T tam giac cho A T B = B T C = C T A = 120" 160 361 Cty TIMHH MTV DVVH Khang Vi^t ChuySn de BDHSG Toan g i i t r i Ion nhat va g i i Iri nh6 nha't - Phan Huy KhAi ' , TA TB TC An dung bo de suy ra: 1 =0 ' • ^ Gia sur M e BC Goi H, I , J ti/(tng i?ng la hinh chic'u cua M xuong AB, BC, CA (*) b > c) * G o i L = M K n BC Su" dung cac bat dang thtfc hien nhien sau: a|+a2+ + a„ < + «2 + + Delhay: VF(2) = (2)' MC.TC TC > MA.TA TA + MB.TB TB MC.TCi, +• TC (MT +TA)TA (Mf +TB)TB TA TB = MT TA TB TC TA TB TC r^ - o a b c a a 2a Do vay S = — + — + - = — + — = — (2) X TC MeBC +TA+TB+TC T i i f ( * ) v a ( ) s u y r a : VF(2) = TA + TB + TC z X (3) 'i (4) /g c ok trj Idn nha't, nho nha't cac biii toan khac noi chung (trong hinh hoc bo phang noi rieng) cflng c6 rat nhicu each gitii khac Bai toan tren la mot ce vi du dien hinh minh chufng cho tinh da dang ciia cac phiTdng phap tim gia tri fa w BC, CA, AB ti/dng uTng Gia suT M la mot diem di dpng tren diTctng tron ngoai tie'p tam giac ABC Goi x, y, z Ian liTdt la khoang each tif M den cac canh max M l M,)Io MeBC Ta c6: M,,!,, = BI„tan MoBI,, = - t a n ^ Tiif suy ra: S = — MeBC , 2a A =:4C0t— a tan A 2 Cung gio'ng nhu" cac bai loan dai so', giai tich, v6'\c bai toan tim giii B a i Cho tam giac ABC vdi a > b > e, d day qua a, b, c ki hicu dai cac canh (3) Lap luan ttfdng tu", co: i B C ' S = 4cot— va S - c o t — MeAC MeAB (4) Theo nguyen l i phan rii, ta co: S = min S, S, S MeBC M6AC MeAB (5) Tif (3) (4) (5) suy ra: BC.CA, AB A B C S = 4col—,4cot—,4cot— 2 Tim gia tri be nha't cua dai ii/dng S = — + — + - X y z Do a > b > c =^ 180" > A > B > C > HUdngddngidi 2a 2a MeBC om Nfuln xet: Da'u bang xay M s T Do chinh la dpcm ww X d day Mo la Irung diem ciia BC ro Ttr (1) (2) (3) (4) suy ra: M A + MB + MC > TA + TB + TC Idn nha't va nho nha't ciia mot dai lufdng cho tri/dc X ^2a^ Nhuf vay S = {MT + TC)TC ^ y Ta TB + p a cung CO ACLM - AABM => - = — z X s/ MB.TB up TA •+ (1) Tir(l)(2)=^^ +^ =- ^ l i l ^ =i y z X X a.p A^ >B- ^ C >|>0^cot- 2r tan (3) 2 Dau bkng (3) xay o dong thcfi c6 dau bkng (1) (2) Taco: AMB-f CMD = " - A l l + 180" - C + D = 180^' tan AMB = cot CMD tan - Vi theo bat dang thiJc Cosi, suy ra: CMD H-tan ^ AMB CMD tan 1-tan tan CMD 364 iL ie uO nT hi Da iH oc 01 / Ti/Png ixi ta c6: CD > 2r t a n ^ ^ Ta (1) s/ AB>2rtan AMB Dau bang (1) xay o h = hi o M each deu A va B AMB Dau bang (4) xay - ^ — = Tir (3) (4) di den: P > 4r (5) Dau bang (5) xay dong thcti c6 dau bang (3), (4) M each deu A, B; M each deu C, D va AMB = CMD = 90" o ABCD la hinh vuong ngoai tiep dufdng tion ban kinh r da cho Tom lai P = 4r Gia tri nho nhat dat du-dc v^ ehi ABCD la hinh vuong ngoai tiep diTdng tron ban kinh r cho trifdc Bai Xet tat ea cae ti? giac ABCD ehi c6 nhat mot canh Idn hdn Tim gia tri Idn nhat cua S, d day S la dien tich tu* giac ABCD Hiidng ddn giai ^ Gia siJ AD > Khi ta c6: AB < 1, BC < 1, CD < Dat AC = X va gpi M la trung diem cua AC Ta eo: A B ' + B C ' = 2BM^ + AC^ >2 (4) =^2BM^+ — < = > B M < - V - x ^ ~ (1) Do AC < AB + BC < 2 Dau b^ng (1) xay o AB = BC = Ke BH AC Ta c6: (2) -1A C B H < -1x B M < X < SARr= 2"" "~2 Dau b^ng (2) xay o H s M o AB = BC Tir (1) (2) c6: SABC < - x V - x^ (3) Dau bang (3) xay o dong thcfi c6 dau bang (I) (2) o A B = BC=I (4) Tac6:SACD= -CA.CDsinACD a + = 45" tan((v + 3) = :A * tan (x + t a n , x+ y = 1=^ - x y^ = l - t a n a + tan3 >x Ta + y = - xy c6: _ ^ X y ~ 2 D a t t = xy K h i TacotCfd): (1) ••.yy.A, S = SBMN = SABCB - ww w Dat A M = X, C N = y, nhiT vay x, y e [0; ] om ok B a i T r e n mSt ph^ng cho hai d i e m A , B v6i A B = d X e t tap hdp tS't ca AE>a, AF>a ' gia tri Idn nhat va nho nha't ciia S A B C D la nii-a luc giac deu canh Taco: AC + A D = a 72) = d ( l + 72) A B va C D cho M B N -=45*' Gia sijf S la di?n tich tarn giac B M N T i m (10) HUdng ddn gidi + i Cho hinh vuong A B C D canh bang D i e m M va N Ian liTdt di dpng tren c o + AB = A B ( I 72 Gici trj nho nha't dat diTdc A B la diTdng cheo cua hinh vuong A B C D la nufa luc giac deu canh T m l a i max S = — + ^ T o m h i i : m i n / = ( l + 72)d A C I C D o >4^ AF Nhir vay ta d i den: / > ( l + ) d s/ AC = - (6) ta c6: [x + y SACM - SBCN - (l-x)(l-y) ^ S = 1-t A x M SMDN 1-xy (2) 1-t [xy = t A F la A E hoSc A F (thi du la AF) nen theo dinh l i V i e t x, y la hai nghi^m thoa man dieu k i e n < x, < X2 < Ro rang cac d i e m tren bien cua hinh vuong cua phiTdng trinh: thi d i e m each xa A nha't phai la mot cac dinh x2_(i _t)x + t = (3) De (3) CO nghiem nhiT vay, ta can c6: Tir suy ra: AB a = P = 22"30' ' Ap dung dinh l i Viet vdi (3) ta c6: x, + X2 = I - t; x,X2 = t nen t-(l-t) + l>0 242 2 + V2 hay V - < S < ^ ', X|X2>0 >0 X, +X2 Tir (5) (6) suy ra: X | X - ( X | + X ) + 1>0 HUdngddngiai Gpi Oi, O21^ tarn ciaa hai di/cJng tron Dat CON = 2a (nhir vay < a < 90") D t M , = R; NO2 = R2 De thay O ^ = ^CON - a , O.OM = - C O M = 90" - a A Ta CO / = M N = O M + ON :p R,cot(90" - a) + R2Cota = Rjtana + R2Cota Trong tarn gidc vuong | M , ta c6 R) = OO|Sin(90" - a) = (R - Ri)cosa => R i ( l + cosa) = Rcosa =:> R| = Rcosa (2) + cosa Hoan to^n ti/dng tiT, ta c6 R2 = (6) (1) T i r ( l ) ( ) (3) suy r a / = Rsina (3) + sina Rcosa sina Rsina cosa + cosa cosa + sina sina 369 l + cosa •H Rcosa s i n a + coscx + l = K1 + sina (1 + s i n a ) ( l + c o s a ) a cos - sin = R 2cos2^ a sin Cty TNHH MTV DVVH Khang Vigt Dau bSng (2) xay A = B = C a + COS ~ +COS 2R - s i n a + cosa + L a i Iheo ba't dang thuTc Cosi, thi R A B B C , C, A ^ J 2A , 2B 2C = t a n — t a n — + tan—tan — + tan —tan — > : V t a n — t a n —tan — a ( a cos sin + cos I 2 a 2 (3) [...]...Chuy6n dg BDHSG Toan gia Irj I6n nhS't vi gia Irj nhd nha't - Phan Huy KhJi Cly TNHH MTV DWH KhangVi§t Bai 7 Xet tap hcJp cac tarn giac ABC khong phai la tam giac c6 goc tu Tir do C O bang bien thien sau: Tim gia tri nho nha't cua da i... h cot — + cot —, Trong moi tam giac ta c6 2 2 2 2 2 2 cot— + cot — + cot — > 3 VJ 2 2 2 A B C Tiy(2) (3) (4) s a u k h i d a t x = cot — + cot—+ cot—, ta c6: 2 2 2 377 ChuySn dg BDHSG 7oAn gia Irj Idn nha't gia trj nh6 nhU't - Phan Huy Khii !S/hdn xet: Xet cac each giai khac sau day: 1 Xet cac vectd ddn vi e , , C 2 , C 3 nhtfhinhve Ta Xethams6'f(x) = x + 1 CO (e, + 62 f'(x) = l - ± , nen c6 bang bien... v6 gia tri l«tn nha't va nho nhS't cua ham s6' 3 §2 N h i n l a i cac bai loan ve gia t r i Idn nha't va nho nha't cua h a m so trong cac k l t h i t u y e n sinh vao dai hoc, cao dS^ng = B 13 • '^i;' iL ie uO nT hi Da iH oc 01 / §3 C d sd l i thuye't cua b a i toan t i m gia trj Idn nha't va nho nha't cua h a m so X e t h a m s c r f ( x ) = 2 s i n - + cosx v d i O < x < 7 t cua h a m s6' T a c 6... i t n g t r i n h v a b a t phU( ... nhon, ihi maxP = bccos^ y A la phan giac ngoai cua goc A va ne'u A la goc tu, thi maxP = bcsin^ y, A la phan giac cua g6c A va ne'u A 1^ goc vuong thi maxP = - b e A la phan giac I -•••^'A- (2) '... bccos — 2 Theo nguycn li phan ra, thi: maxP = max{Pi; P:} = max|bcsin^y; 358 ubccos — « s/ Vay cac di/ctng thdng thuoc nhom 1, thi du'clng phan giac hang p = be sin hoSc la phan giac ngoai cua goc... B T C = C T A = 120" 160 361 Cty TIMHH MTV DVVH Khang Vi^t ChuySn de BDHSG Toan g i i t r i Ion nhat va g i i Iri nh6 nha't - Phan Huy KhAi ' , TA TB TC An dung bo de suy ra: 1 =0 ' • ^ Gia