Chuyên đề bồi dưỡng học sinh giỏi giá trị lớn nhất, giá trị nhỏ nhất phan huy khải (phần 6)

Chuyfin c i r B D H S G Tojn gia trj I6n nhS't v.i gia tri nh6 nliat - Phan Huy KITST Cty TMHH M!V DVVH KhanglZiH" VAIBAITOAN KHAC VE G^^yd- hay u^ = + V - ( x ^ - x + l l ) Ttf ta di den phU'dng trinh he qua sau: GIA TRj idiN NHAT VA NHO NHAT CUA HAM SO = + N / - U U^ - = ^ - u \J2 O o x ^ - x + >() g(x) = c > x = - l => X' - Do vay maxg(x) = x = - i , j, • nt,|.,, Nhir the suy (2) o fffx) = fx = - - - x ^ Vx e R Tir (1) (2) suy f(x) = 3x'' - 4x' - (1 f(x) = O o x = 298 X X = Nhir the' (1) CO nghiem nhat x = ro up f(x) = 2'"+' + 2^-^^ > l^I^^^Kl'-^' = 24¥ = 8, Vay minf(x) = o x = - xeR g(x) = Ta s/ Theo bat dang thurc Cosi, ta CO f(x) = Vay (!) Hitiing ddn gidi xeR - xeR log:,(4x V i t h e Vx € R , t a c g ( x ) = 1= X - TuTdo suy max = x= Bai Giai phi/Ong trinh 2^"+' + 2^"^^ = X = > 2"' Vay Vx e R, ta c6 f(x) = 2"'' - ' " ' < ( ) ; Tijr ta CO X = - la nghiem nhat cua (2) f(x) = o 2x + = - 2x o JI X > X - 2" < R va g(x) = x = iL ie uO nT hi Da iH oc 01 / Lai CO (f) ' ^(l + x^f ) > Vx e IR Chuyen dg BDHSG ToAn gii tr| Idn nhaft va gii trj nh6 nhS't - Phan Huy Kh^i Cty TNHH MTV DWH Khang Vigt Vay minr(x) = () o x = Isin'x + cos'x = 32(sin"x + cos''x) xel ^ f(x) = g(x) Ttf cac ket qua tren suy Ro rang phiMng Irinh da cho co the vict duTdi dang f(x) = Ttr suy phu'(tng trinh da cho c6 dang f(x) = o x = ,, (1) , ' s » , •K Vay X = lii nghicm nhal can tim g(x) = l ^ Dat biet vdi phUdng trjnh dang f(x) = a, x e D ' xeD xeD HUdng dan giai = n x-(l-x) Dat f(x) = s i n \ cos^x, x e K = n ( x - 1) sin^x < sin^x cos'^x < cos^x ' • f t * ( k up sin''x = sin^x Z) /g ro ir o x =k cos X = cos X Ta •' s/ => f(x) = s i n \ cos^x < Vx e R c bo fa ce ,n-I • ' ww I Dafu bkng xay o a = b = — Do sin^x + cos^x = X"-2+x"~'(l-x) + + ( l - x ) " ~ ^ x h'(x) h(x) ^•^^'^•^ + 1 Nhu- vay < a < => h(a) > h ok - w a" + b" > , ,n-2 X "n-2' + ,x "„ n" X l - x ) + + ( l - x ) Tijr CO bang bien thien sau: om Tiirdotaco maxf(x) = l o x = k - , k e Z " xeR Ap dung ket qua sau day: Ne'u a, b > va a + b = 1, thi vdi moi n nguyen > ta c6: (1) v6 nghiem Xet ham so h(x) = x" + (1 - x)" vcti < x < => h'(x) = nx"~' - n ( l - x ) " " " ' = n x " - ' - ( l - x ) " - ' X€D Bai Giai phiTdng trinh sin^x + cos^x = 32(sin'^x + c o s ' \ Mat khac f(x) = x = - + k - , k e Z (3) Chu y: (*) chtfug minh nhiT sau: ta c6 r(x) = a max f(x) = a (hoac f(x) = a I'Cx) = a ) Ta c6 (2) \ Ro rang he (2) (3) v6 nghiem ma thoa man dicn kicn maxl"(x) = a (hoac minf(x) = a ) , xeD x-k-,keZ iL ie uO nT hi Da iH oc 01 / Chuy: f(x) = l sin'^x + cos'^x = (sin^x)' + (cos^)'' > — oa"+(l-a)">-i^hay a"+b">-i^ 1^ • ' • •a'u b^ng xay raa=:^a = b = ^ = > dpcm! Bai Giai phiTdng trinh cos3x + V2-cos^ 3x = 2(1 + sm^x) HUdng ddn gi&i => g(x) = 32(sin'^x + cos'^x) > 1, Vxe R • Dat f(x) = cos3x + V2-cos^ 3x ; g(x) = 2(1 + sin^x) vdi x e Khi phiTdng trinh da cho c6 dang f(x) = g(x) g(x) = o sin^x = cos^x = x= — + k — , k G (1) De thay g(x) > Vx e R (do sin^x > Vx e R ) Mat khac g(x) = o sinx = o x = kn (k e Z ) Vay ta c6 ming(x) = x = kTt xeR Ap dung bat ding thuTc Bunhiacopski, ta c6: ' (2) Cty TNMH MTV DVVH Khang Vigt Chuyen dg BDHSG Toan gia tri Idn nha't va gia tri nh6 nhaft - Phan Huy KhSi cos^ 3x + (2 - cos^ 3x) (1 +1) > |cos3x + V - c o s ^ x suy fir Vx e 2k7t \/x' + x - I +Vx-x^ hay f(x) < Vx e D + ^ Vx € R ^^ (1) /J-iHI /V.?.JU «t> ' J l sin^2x sin^ 2x + — : r — +4 sin'*xcos'*x (2) Tir (2) de d^ng suy Vx e R t a c f ( x ) > - ^ + - ^ ^ + Theo ba't d^ng thiJc Cosi Vx e D, ta c6 = 12 + ^ s i n y ^ sin^ X y , // f Ta s/ up x = 2—.keZ lgW = X = kn, k e Z _ 2kn ^ , X G i • i »x = k c , k e Z X = k7l,k€Z j^., ' Vay X = k27t, k e Z la nghiem cua phiitfng trinh da cho ^'i'H>Bai Giai phtfdng trinh Vx^+ x - l + \ / x - x ^ + = x^ - x + ^, t.s,.\.:^^:^',: HUdngddngiai Mien xac dinh cua phiTdng trinh la tap hcJp D gom nhiTng phan tijf x thoa man (1) (chiiy r ^ n g x ^ - x + > Vx) he x^ + x - l > li > I vay X = la nghiem nha't cua phiTctng trinh d5 cho xeR X = suy iL ie uO nT hi Da iH oc 01 / f(x) = cos3x + yjl-cos^ 3x < cos3x -Jl-cos^ 3x f(x) = 1 o cos3x = V - ; o s ^ 3x fcosSx > cos^ 3x = - cos^ 3x o c o s x = Vay maxf(x) = o c o s x = Tit ChuySn de BDHSG Toan gia tri I6n nhat va gia tri nli6 nhat Cty TNHH IVITV DWH Khang Vl§t Phan iiuy Kh^i ( s i n \ 2sin^x - Bsinx - 1)^ = - sinx f(x) =l2- sin'2x = c=> cos2x = ( ) « x = - + n ^ , n € Z V i phifdng trinh da cho c6 dang f(x) - g(y) l-(x) = 16sinS + 16sin'*x - 20sin''x - s i n \ s i n \ 7sinx - = x, y e M o (3) D e n day m d i cac ban giai t i e p ! ! Cac ban thay the' nao? 12- f Tijf cac lap luan Iron siiy (3) o g(y) = i (sinx - l)(16sin' x + 32sin''x + 12sin''x - s i n \ 3sinx + 4) = ^ x ^ + x + 46 B a i 1 G i a i phu-dng trinh — = 2x^ - 8x + x ^ + x + 10 iL ie uO nT hi Da iH oc 01 / Hiidng ddn giai x = —+ n — ; v = — + k27t, n va k Z 2 D a l f(x) = B a i G i a i phiTdng Irinh (sin3x + cos2x)' = - sinx x^ +2X + 10 G o i m la gia t r i l i j y y K h i phU'dng trinh sau (an x ) HUdng dan giai 4x^ +14X + 46 —^ =m D a l f ( x ) = (sin3x + cos2x)\(x) = - sinx, x e R D o sinx < V x e M ^ 4X^ 14V + -46 4(S +14X 4- g(x) > 4, V x e R • x ^ + x + 10 g(x) = o sinx = x = - + k27i, k G Z • Do x^ + 2x + 10 ^ V x (VI • ' , ^ (l)conghiem ',i ' „• • x^ + 2x + 10 > 0), nSn (l)4x^ + I x + 46 = mx^ + m x + 10m TiJf suy m i n g ( x ) = c:>x = - + k27t xeM s/ Ta (1) up L a i CO lsin3x + cos2x|x + y < C&ch giai hoan loan chap nhan di/dc, neu cac ban doan triTdc d\{a^ Ap dung bat diing IhuTc Bunhiacopski, ta c6 nghiem x = 2! Bai 12 Giai phi/dng trinh sinx ol+sinx (6) (75rri.i+7^.i)'< (75^71)'+(7771)' cos(xy) + 2'^i = HUotng ddn giai (1^+1^) => 7x+T + 7y + ^72(x + y + ) < Viet lai phtfcJng trinh dU'di dang sau: max ( ^ + 7 l ) ^ « : / i l L £ I i v a ^ + ^ = iL ie uO nT hi Da iH oc 01 / 2l''l-cos^(xy) = P = " " " -cos(xy) (1) Ta CO 21^1 > Vy € M cos^(xy) < Vx, y e R hit ii.^'f M I*til P = o i 2'^l-cos^(xy) = O ' cos ( x y ) - l 2''"''=cos(xy) 2M = i [2''"''=1 o x = y = 3J^'^''^S*™i'«-'r' Nhan xet: Neu khong sit diing phiTdng phap tim gia trj U^n nha't ci'ia hiim so de "^ly^o danh gia hai ve', ta c6 the giai ihuan he phtfdng trinh trcn nhU'sau: T i i r ( ) c x + + y + + 27(x + I)(y + 1) = 16 y =0 y=0 jf ;,;'„ ( ;v v ' • m id) f Ta X = kn, k e Z , t j s/ sin X = ro up Vay minP = o x = k7t;y = , k Z /g Tir suy nghiem cua (1) la x = kK, y = vdi k e Z om Bai 13 (De thi tuyen sinh Dai hoc Cao ddn^ khoi A) ok ce t =3 35 t = y + l>0 (4) xy>0 (5) , ; De thay neu - < x < va - < y < thi yfx + l+Jy + lcos2a 4) Luc c6 bang b i c n t h i c n sau: , „.,, ji^ a t 1 ~4 ok B Cdc ling dung cua vi^c khdo sdt gid tri Idn nhd't vd nhd nhd't cua cdc fa(t) bo ham so'phif thuQC tham so / / / / fa(t) c om fa T i m a, b de ba't phi/dng trinh sau: cos4x + acos2x + bsin2x > - dung v d i mt'i -;f->^> ww Hadngddngidi V i ba't phiTdng trinh dung v d i m o i x, nen n o i rieng no phai dung k h i x = — va x = — 4 m i n f.(t) = ( - l ) = - a - : t i '"il' ^ -l o a < w xe R i + I n' > ^ ce Bail 314 «P! oh §1/ d"j -i ^• ' d o xet ba kha nang sau: \ Tir suy f a c d : l.|(t)= 4t + a v a l,,(t)== « t = : : ^ -4-' 2m m i n r | ( t ) > ( ) (2) - Kill ' iL ie uO nT hi Da iH oc 01 / c Nc'u < — \ fa(0 ^ ^ ^ + ^ 315 Cty TNHH MTV DWH Khang Vigt N c u m < 0, thi sinx + cosx < sjl Vxe max =»minr,,,(x) = 2v/2m = max m — 3; m + m -3 xeR [jsleu m < Thay viio (2) va c6 he: + 2>/2m + > m-2 ,, „ (khi m^ + m - < o ) V i vay f j x ) > - V x e [-2; 1] va chi (2) , Xet cac kha nang sau: Neu ra^ + m - > =4> f,„(x) = m' + m - > 0, nen ta c6 bang bien thien sau: x -2 (khi m^ + m - = o), hoac la ham bac nha't iL ie uO nT hi Da iH oc 01 / Viet lai bat phtfrtng Irinh da cho difdi dang lu"(ing du^dng sau: l„,(-2)>-2 -2m^ - m + > - f,na)>-2 m^ + m - > - -2gia tri Idn nha't va ok (x) = m^ + m - < 0, nen ta c6 bang bien thien sau: Ta thu lai ket qua tren R6 rang each giai gon gang hdn! ce Neu m^ + m - < -2 ro 2m2+m-6-2 , om m^+m-2>0 c m'' + m - > fa ^ m \ w Vay (2) up f ^ ( x ) = f „ ( - ) = - m ^ - m + + m -2 0, y < 0; x, y e Z v^ 4x + 5y = } , Thco nguyen l i phan ra, ta CO: 061:, 2, m i n P = (x;y)€D (x;y)eD, Tif 4x + 5y = TCr (7) (8) (9) suy ra: n i i n L , ( x ) < - ^ (min (' (x)] ^ > ( ) XGM Do x, y I VxeE Cong lirng vc (6) (10) di den: (min 1„, (x) + max f,„ (x)) >2 t • > Ta LLfONG GIAC /g HINH HQC, ro up §3 GI6I THIEU MQT SO BAI TOAN GIA TRj L6N NHAT NHO NHAT s/ Do TRONG SO HQC, om Ciio'n sach diinh dc Irinh bay cac bai loan gia Iri Idn nha't, gia tri nho c nha't thi/ctng gap dai so' vii giai lich G P; m i n (x;y)GD2 P X> x= 7-5y = 2-y + (2) x = 3-5t 0; y < nen suy ra: ^ ~ ^ ' > ° = , t < i = > l= 0;-l;-2; (dotG Z ) 41 - < Ti^ (3) va t = 0; - ; - ; nen suy ifng v d i t = 0, ta c6: (4) P=::12 Khi (x; y ) e D2 i h i P = -5x - 3y khao khac Tuy nhicn muc nay, chung toi muon gicKi thicu vc'Ji cac ban D o x < ; y > , t i r ( * ) ta c6: ce bo ok hoc, hinh hoc, liTi.Jng giac so di/dc chiing loi trinh bay mot cuon chiiycn fa w ww (5) ^"-'^' ( d l nhien no ton tai) Luc la That vay gia s i l r ( l ) khong diing, ttfc la ton tai hai so nguyen diTtJng m,,, n„ s,^, " =-V:«' "''•.•.tfl- m , Trong each phan lich da chpn, khong cd so hang v i neu eo so hang i h i TrU'c'Jc hot la chufng m i n h rang v d i m p i m, n la so nguyen duftJng thi P > (| j cho | l ' " " - 5"" I < TIM HH MTV DVVH Khang Vigt d d : •'/«':; , _ ^ v a x , + X + + X3„ = 2011 X2 X3(, (x|-1,X2, ,X29, x^o+1) Vf/V Khidddeyrangdo xj'>l:^x, > = i > x ^ - l > l •Mat • khac: - ) + ^ + + x ^ + ( x ^ + ) = 5^ + x^ + + X j , , = 2011, nen bp so m d i cung thoa man y e u cau de b a i Cfng v d i bp so ta cd: •L 333 Chuyfin BDHSG Toan gia Iri I6n nha't va gii tr| nh6 nhS't - Phan Huy KhSi P = (^-l)5^ X^(x3o +l) = X2 X29(x3o-X, Do x ^ > ( ) Vi = 2r29, va x < x ^ ^ P > P P = 67^'".68 (1) T o m lai la co: max P = 67^*^68 K c t hop l a i ta co: m i n P = 1982 va max P = 67'''.68 '"''^ ' p a i Cho k la so nguyen di/Ong > nha't V a y mot dieu k i e n can de P dat gia t r i nho nha't la X| = ''* ' Tir lap luan hoan toan ti/rfng tif suy X2 = X3 = = X29 = cung la dieu Tim gia tri Idn nha't ciia ham so f(x, y, z) = xyz trcn mien k i e n can de P dat gia t r i nho nha't D = {(x; y; z) : x, y, z nguyen dU'Ong va x + y + z = k } (2011 - 29) = 1982 \' ^ iL ie uO nT hi Da iH oc 01 / 1.1 do: , C3„-X,>2 < , max X j Ta se chu'ng minh rang: X o - Zo < • v - ^ " " ' 'r? (2) That vay, neu trai lai ta co: Xo - Zo > X3,, Bay g i d xet bo so m d i sau day: ^^ K h i chi co the xay triTdng hdp sau day: (x, + I ; x ; ; x y ; x - l ) ' ' '* s/ a NcII x„ = y„ > z,, + V i x„ + y„ + z„ = k nen ta co: x„ + (y„ - 1) + (z,, + 1) = k up Chu y rang X3(, - x, > ma x, > I => Xj,, - x, > /g ro khac de thay: (x7 + l ) + x^ + + x ^ + ( x o - l ) = x , + + X30 = 2011, om nen bp so m d i cung thoa man yeu cau de bai l?ng v d i bp so'n^y ta c6: c P = (x, + l ) x X y ( x „ - l ) bo ok + l)(x3(, - 1)- X, X3(,] = X2 X29 (X3„ - X, - 1) ce >2=>P>P w fa V a y bat dang thuTc chtfug to r^ng bp so (x, .Xjojchifa lam cho P dat gi;i ww tri nho nha't V i the dieu kien can de P dat gia tri Idn nhat la X30 - Xi < hay xw - X i < => X30 - Xi e {0; 1} tdrc la: " Ta C n g v d i bo so ta c6: P = x, f(x,y,z) = r ( x , „ y , „ z , ) ) = : x „ y „ z „ (Chu y d day x„ > y„ > zo) X, + X + + X o = 1 Tir X „ - x , gidi xyz phai dat gia tri Idn nha't tren D Gia suT: X, ] Do vai tro binh dang giffa x, y, z nen khong giam tdng quat co the cho rang V a y m i n P = 1982 c6 29 thuTa so bang 1, va m o t thifa so bang 1982 Mat ^ ,f' HUdng dan 29 thiira so' Xet bo so' ( x , , X , X , , ) K i i i n g Vi^t Nhu" vay chon day chting han 29 so' bang 67 vi mot so bang 68 t h i : +l) (1) chuTng to rang bo so (x,, X j , , X „ ) khong lam cho tich P dat gia trj nho Tir suy ra: m i n P = [ivvil Cty TNHH M I V ^ hoac X3(, = X| hoac X3(, = X | + Nhir the dieu kien can de P dat gia tri Idn nhat la 30 so thi khong diTdc CO hai so bat ki nao chung lai chenh qua D i e u co nghla la phai c6 I so bang a va 30 - t so b i n g a + (1 < t < 30) cho: t a + ( - t ) ( a + 1) = 2011 => 30a + ( - t ) = 2011 (*) Do X() = y,, > Z|, + va Zo > => Xo, yo - 1, Zo + cung nguyen di/Ong, turc la: , (Xo, y , , - l , z „ + ) e D Mat khac: f(Xo, y,) - 1, z,, + 1) = X|,(y„ - l)(z„ + 1) = x„y(,z„ + x„(y(, - z,, - 1) Do X(i = y,, > z,i + 1, nen tiTtren suy ra: f(x,i, y,) - 1, Z(, + 1) > X(,y„Z|, = f(x„, y,,, z,,) Bat dang thijTc thu diTOc mau thuan vdi (1) Vay trufOng hOp a khong the xay Neu X(, : > jin^'y > y„ > z„ X e t bp ba nguyen duTdng ( x „ - l , y o , z Z() + 1) nen ta co: l'(X() - 1, yo, z x„y„z„ => f(x„ - 1, y„, z„ + 1) > f(x„, y,,, z,,) Ba't dang thiirc cung mau thuan vdi (1) ,1 i , Vay trU'dng hdp b khong the xay Ne'u X() - > y,, > Zo L a p luan nhiT tren cung suy mau thuan ' ' T o m lai gia thiet Xo - z,, > la sai, vay (2) dung Tir (2) suy chi cd the xay hai kha nang sau: De thoa man (*) c6 the chon a = 67, t = 29 (vi 30.67 + = 2011) 335 Chuygn d l BDHSG Toan gia tri Idn nha't va gia tri nho nhat - Phan Huy Khii i Neu x„ - Z|) = Cty TNHH MTV DWH Khang Vigt Ket hdp vdi x„ > y„ > z„, Ihi x,, = y„ = z„ = — Dieu xay va chi x=y=z= ^-f:: 2011 Vi moi phan tur lai deu < 2011, suy khong c6 phan tur nao bang tich tiir khac thoa man btft d^ng thurc: n>43 (1) That vijy, gia s u r ( l ) khong dung, tuTc la ton tai so nguyen du'dng no < 43 ma ww Do k > va k 13 nen la Tich ay chinh bang Ta se chiJng minh rang moi so' nguyen duTdng n thoa man yeu cau de bai w Dau " = " xay va chi x = y = z = - tiif hdp thoa man yeu cau de ce 27 eo mot phan Nhif vay ta da chi mot each loai bo di 43 phan tur, de thu diTdc mot icip bo z+ y+ z tur phan tur thur hai Do moi phan tur cua phan lai deu khac nhau, nen khong cua hai phan ok Theo ba'l dang thitc Cosi Ihi vdi moi (x; y; z) e D, ta eo: Viledotaco: om /g DI nhien neu k = (mod 3), titc la k nguycn diTdng va chia het cho 3, thi xyz < si Loai di 43 phan tur sau: 2, 3, , 4 27 van thoa man yeu cau dau bai Xet 43 bo ba sau: (2, 87, 2.87); (3, 86, 3.86); (44, 45,44.45) ' 27 Tuy nhien neu k / thi khong the iip dung difde ba't dang thufc Co-Si de g ' ^ bai toan L i d cho da'u " = " xay bat dang thuTc x y z < — va chi khi: Dat l'(x) = x(89 - x) vdi < X < 44 Ta eo: f (x) = 89 - 2x > < x < 44 => f(x) la ham dong bien tren [2; 44] ' ~ • Tir ta eo: 2.87 < 3.86 < < 44.25 < 2011 »+ Nhif the 43 bo tren gom cac phan tur doi mot khac thuoc tap X - ' * Vi ta rut no phan tu" (no < 43), nen phan lai cua X sau rut no phan tur luon chtfa it nhat mot 43 bo ba noi tren 336 , 3137: Chuygn 6i BDHSG Join gia tri I6n nhat glA tri nh6 nhat - Phan Huy Khii Cty TWHH MTV UVVH Khang Vi§t R r a n g b o b a d o chufa b a p h a n luf c u a d a y c o n , U-ong d o c m o t p h a n tff p a j H a m so' f ( x ) x a c d i n h t r e n t a p h d p c a c so' n g u y e n diTdng v a n h a n g i a t r i b a n g t i c h h a i p h a n tuT c o n l a i D o l a d i e u v l i v i no la so' n g u y e n difcfng thf^ m a n y e u c a u d e b a i V a y g i a t h i e t p h a n chiJng l a s a i Il"(l) = l (1) diing c u n g t r e n d o v a diTdc x a c d j n h n h u sau: K e t h d p l a i t h c o d i n h n g h i a g i a t r i n h o nha't s u y r a so n g u y e n di/dng n nho r(2n)=:r(n) f(2n + l) = f(2n) + l nhat thoa m a n y e u cau dau bai la: m i n n = 43 2, T i m B a i , X e t t a p h d p la't ca so n g u y e n t o k h a c n h a u c c a c t i n h c h a t s a u : m a x i"(n) l 0 => a + b - c < , d i e u n a y m a u thuan V a y b d d e d u n g k h i n = v d i v i c e a + b - c l a so' n g u y e n to' • s/ up ycji i n : ro (1) /g R r a n g a, b , c p h a i l a c a c so l e T h a t v a y n e u t r a i l a i t o n t a i i t nha't m o t om t r o n g b a so t r e n l a so c h a n , t h i d o l a so n g u y e n t o c h l ' n d u y nha't n e n c h i co ok ce bo c a c so c h n , m a c h i i n g l a i k h d c n h a u NhiT v a y t a c o n h i e u h d n so nguyen fa w ww ' ,j X e t k h i n = k + C o h a i tru'dng h d p x a y r a : - H o a c la k c h i n ( k = m ) K h i d o : l ( k + 1) = f ( m + 1) = l(m) + D o m < k , n e n t h e o g i a thic't q u y n a p suy r a l ( k + 1) c h i n h b a n g so c a c chu" sd^ t r o n g b i e u d i e n n h i p h a n c u a m c o n g t h e m M a t khac, k+ = m + m i i : m + 2_ TiJfc l a : m + = (\|a2 (Vpl m , d d a y m = a|tt2 ap la b i e u d i e n c i i a m k + c h i n h bang so chiir so t r o n g b i e u d i e n n h i p h a n ciia m c o n g t h e m N o i r i e n g so n h o nha't t r o n g so d a c h o I d n h d n hoSc b i n g (2) B a y g i d la chi d a u " = " (2) c6 the x a y • t t r o n g h e n h i p h a n T i f d o suy s d c h i J so t r o n g b i e u d i e n n h j p h a n c u a so V a y c a so n g u y e n t o d a c h o d e u l a c a c so n g u y e n t o l e d < l - = > d < 1594 ' • k h a c n h a u T i r d o suy r a c a c s o a + b + c, a + b - c , a - b + c, b + c - a d e u la K e t h d p v d i ( ) suy r a : * G i a su" b d d e d a d u n g d e n n = k > i , i i k la v d i m o i / < k , ("(/) c h i n h b a n g c d i j n g m o t t r o n g b a so a, b , c l a , c o n h a i so' c o n l a i l a h a i so n g u y e n to Ic to chS^n k h a c n h a u t r o n g so n o i t r e n D o l a d i e u v l i ' sd^cac chi? so' t r o n g b i c u d i e n n h i p h a n c u a s o / R o r a n g a + b + c la so Idtn nha't t r o n g so n g u y e n t o n o i t r e n a + b + c = 0 + c < 0 + => a + b + c < ' Ta ij i , So n g u y e n t o kUn nha't d i r d i 0 l a , v i t h e : ' ' Mat k h a c : = I / d d a y ta d u n g k i h i e u a,a2 a„ 12 d e c h i so g h i t h e o h e c d tdng quat) a +b = 800va a < b dan giai T a CO b o d e sau d a y : so n h o nha't t r o n g s ' d T i m giA t r j Idn nha't c u a d iL ie uO nT hi Da iH oc 01 / H a i t r o n g b a so a, b , c c t o n g l a 0 vdimoin=l, ^ p ; • ; V a y bd d e d u n g t r o n g tru'dng h d p k = m H o a c l a k Ic ( k = m + 1) K h i d o : f ( k + 1) = f ( m + ) = l"(m + 1) D o m + < k , n e n t h e o g i a thie'l q u y n t i p suy r a f ( k + 1) c h i n h b a n g so c a c V i dtj c h p n so sau: chff so t r o n g b i c u d i e n c i i a m + t r o n g h e n h j p h a n a = 13; b = ; c = ; a + b + c = , M a t khac, k + = m + m a : R o r a n g d i i n g 1^ so n g u y e n t o k h a c n h a u t h o a m a n c a c t i n h c h a t ; N g o a i r a : d = - = 1594 Tuf d o t h e o d i n h n g h l a v e g i a t r j n h o n h a t suy r a : m i n d = 2m+ a + b - c = 3, a - b + c = , b + c - a = T u - c l a : m + = 3|[32-3pO m + d d a y m + = (3,32 i m + he nhj phan ,(^,|| fsJ ,? D e ' n d a y ta tha'y b d d e d u n g k h i k = m + la b i e u d i e n c u a i &/ i stS, '* > Cty TNHH MTV DWH Khang Vi^t Chuyen 66 BDHSG Toan gii tr| Mn nhS't va gi^ trj nh6 nhat - Phan Huy Kh^i Tom l a i , bo de dung k h i n = k + \i the: a-^ + = f25a2 + {5r2 + 3) ^ + Theo nguyen l i quy nap suy bd dc dung v d i m o i n (dpcm) ,1 >a, + = ((5r2 + ) ' ( m o d ) Ta lha'y so' nho nhii't c6 11 chiy so' bicu d i c n d\idi he nhi phan 1_1 (5r2 + ^ + j ; 2"-l = ' " + 2'^+ + 2' + " = , ,-,,^1, ,> r, , W TiJf (1) suy ra, noi rieng (a"* + 3): 25 so 2) la so: N= (4) = 2027 - /' rj = (do rj e {0; 2; 3; } ) Vay a = 25a: + 13 L a i dat a = 5a., + r, vdi r, e { ; 1; 2; 3; } 2-1 =^r(n)< 10 (1) r, = = > a = 125a,+ 38 Do phat hicn da'u bang (1) la lict kc cac so la nho nha't, g;1n nh6 &' Ihco thi? tir lis nho den kitn la: Lai dat a = 16t + r v d i r khong am va < r < 15 10 =1023; lOchCTsd = 1791; Vay a = + IJLJ 1110 1_KJ = 1983; chu* so' urn i_Lj = 2015 > 2011 chas6' cliffsd' lai thu djnh nghla vc gia trj be nhat ta c6: max t'(n) = 10 o n thda man (2) T i r ( l ) s u y ra, n o i r i e n g ( a - V ) ; V l the tif (2) c6 Do r, G | ; 2; 3; 4} nen suy ra: r, = => a = 5ai + D;)t a, = 5a: + f:, v d i rj e {0; 2; 3; } , ta c6: a = 5(5a: + r ) + = 25a2 + Srj + V Ta lha'y a cd: + 2.51 = 111 chCT so Do vay sau xda di 100 chu" so y (1) < 4, ta c6: (mod 5) => (a^ + 3) =rl+5 (mod 5) 16 HUdfiig ddii gidi Do 2000 = , nen ne'u dilt a = 5ai + ri v d i ri la so' nguyen thoa man ^ fi a' = Do t nguyen du'cJng nen — — — = h (h = 0; 1; 2; ) suy a, = 16h + 1 Tim n va max n ce X = a' = 1997 (mod l O ' ) => a' + = 2000 (mod 10^) 16 Xet tap hdp cac so n thu du'dc iCr A bSng each xda di 1(X) ehOT so luy y cua A bo ww Ta l i m so nguyen dufdng X thoa man: 16 ^ du-dng lir den 60 Iheo thiir lit tir nho den Idn, lu-c la: A = 1234 5960 w Hii(ing dan gidi ,11-a, Bai 10 So nguyen A du'dc tao lhanh bling each viet lien cac so' nguyen fa b n viet he thap phan c6 so tan cung la 1997 „ ^^ ^ dau bai IJai T u n n so' nguyen du'dng n nho nhat thoa man cac tinh chat sau: a n la lap phu'dng cua mot so' nguyen dtfdng i^^A"^.wMf^ Nhir vay n = 1413 = 2821151997 la so tir nhiC-n nho nhat thda man yeu cau ok l 33 = 16l - 125a3 = 1919 c Vay 9chiIso' Ta 11110 Tir (a- + ) = { r U ) = 1535 s/ ( JjciVf! 1_LJ up OLJ chiJsiY 11 (5) ' V i 2000 : 16, nen tir (1) cd (a"* + 3): 16 nhat, nen b i c u d i c n dxiiVi he nhi phan c6 chiJa 10 chiJ so Cac so ay II iL ie uO nT hi Da iH oc 01 / De y rang: (a^ + 3): 125 va lap luan nhu'tren suy ra: Tirbddc vado < n < 1 (2) (r|V3);5 (3) ciia A thi lai so n cd 11 chff so (chi'i y rhng n cd the cd cac chuT so' d^ng i dau) f Ta lha'y A cd chi? so phan b d nhir sau: 12 10 20 30 40 50 51 52 53 54 55 56 57 58 59 60, De n la nho nha'l, ta chon chiT so dau lien cho n deu bang (nghla la phai xda bdt cac chiT so khac cac so tir den 50) Ta se xac dinh them chOf so lai cho n lijr day so: , , 51 52 53 54 55 56 57 58 59 60 j D e t h a y s d d d l a : 123450 ^»s;tj Vay m i n n = (KX)(X) 123450 = 123450 341 Cty TNHH MTV DVVH Khang Vigt ChuySn 6e BDHSG Toan gia t r i I6n nhat vi gia t r i nh6 nhat - Phan Huy KhJi D c n la \iln nha'l, ta c h o n chu" s o ' d a u t i e n c h o n d e u b S n g ( n g h l a l a p h a i xo-y (2) b d t c a c chcr so k h a c i r o n g c a c so l i f d e n ) T a se x a c d i n h t h e m c h i j sg- f c o n l a i c i i a n tiT d a y so: Tir(l)(2) suyra: 51 52 53 54 55 57 58 Dcthay6sod6 ' X ; 3? a l hang la'y la m, di/a vao lap luan tren suy ra: m (1) Ta s/ > na, ro up n /g om khong gian c B a i Cho hinh chop tam giac S.ABC c6 the tich la V M la mot d i e m y ok nam ben tam giac A B C Qua M ve cac du'dng song song v d i SA, SB, MA' SA M B ' M C= -I o SC SB o SA (5) • • ,^ Gia tri Idn nhat dat di/dc k h i va chi k h i M 1^ tam cua A A B C Wa\ Cho hinh chop tam giac S.ABC co SA = a, BC = b va SA tao v d i B C gdc 60" M la mot d i e m tren canh SB Qua M difng thiet dien song song v d i BC T i m gia tri Idn nha't cua dien tich thiet dien HUdng ddn gidi Ke M N // SA, M Q // BC K h i ( M N Q ) n ( A B C ) = NP // M Q De thay QP // M N => thiet dien M N P Q la hinh bmh hanh Do M N / / S A , NP//BC M N P = ("SA^C) = " Tac6: SMNPQ = MN.NPsinMNQ = MN.MQsin-y- (D M C _ RM ~ PA ' SB ~ QB ' SC ~ RC • Trong tam giac A B C hien nhien ta c6: (theo dinh l i X e V a ) ww w A M , B M , C M ti/dng ilTng cii BC, A C , A B tai P, Q, R.^ MB' _ QM M la tam A A B C iTilfd6tac6:maxVM.AB'C'-^V bo fa Hiidng dan gidi ce tich Idn nhat cua tuT d i c n M A ' B ' C M A ' ^ PM m M la t a m A A B C SC Chung c^t cac mat SBC, SAC, SAB ti/dng ufng tai A ' , B ' , C T i m the Theo djnh l i Talet, ta co: 63' od'')'','? I (4) Dau bang (5) xay o da'u bhng (4) xay nhat vd nho nhat hinh hoc A ' e SP; B ' e SQ; C e SR (3) MB' M C ^ — ^ < - Tir(l)(4)dide'n: V M A B ' C (2) N ^ TCr (1), (2) suy ra: N + — ( n - ) > na => N n K h i lir gia thiet suy ra: MB' M C SA • SB ' SC V MA' - (2) Da'u"="trong(4)xayra Goi tdng cac s6' ca bang la N , tdng cdc sd^ C Vdi bdi todn ve gid tri SC TCr(i)(3)suyra: - 1) Ian tdng cac so hang la'y N Theo dinh nghla cua so'm, ta c6: m < — n SB SA so ghi n "chiy t h a p " ay khong nho hdn na N +(n-l)m>na MA' iL ie uO nT hi Da iH oc 01 / MA' , MB' ^ M C Con n "chtT t h a p " nhif vay, va theo dicu k i c n cua bai, ta thay tdng tS't ca c^c (do NP = M Q ) Theo dinh l i Talet, ta C O : M Q ^ S M V W ^ M B _^MQ BC " S B ' SA 348 SB SB b (2) a SB S unuyen oe BDHSG roan gia 1= MQ b MN +• IhuTc Cty TNHH MTV DWH Khang Vigt CoSi, la c6: MQ.MN >2 MQ.MN < ah a Da'u bang Irong (3) xay j = b Tfif (1), (3) c6: SMNPO ^ a Cho hinh chop S.ABCD c6 (jiiy lii hinh hinh hanh Goi K la trung diem ciia SC Mat phang qua AK cat canh SB, 5D Ian lU'Ot tai K Gia siif hinh chop S.ABCD CO the tich la V Tim gia tri klfn nhat va nho nhat ci'ia the tich tu" dien S.AMNK ab = - M la trung d i e m ciia S B M la trung d i e m eua S B '' N h i / v a y ta c6: max SMNPQ = abV3 • A •'1/ H " p B a i Cho gcSc tam dien vuong Oxy/ M la mot d i e m g()c tam dicn khoiing each xuong ba mat xOy, xOz, yOz liTcJng lirnj la a, b, c (a, b, c > ()) A HUdng dan gidi SM SN vl Mat phang qua M cfit Ox, Oy, Oz Ian liMt tai A , B , C X c t dai li/dng P = OA E)atx = + OB + OC T i m gia tri kitn nhat ciia P Ta dat V , = VSAMNK, ihi V , = V.S.AMK + V.S.ANK- Hi/thtfi dan f^idi Ta c6: OA =1 Ta s/ up + ( ) OB OC om A p diing bat dang thijrc Bunhiacop.ski, (OA + O B + OC) OA OB OC Ket hdp vdi (1) suy ra: P = O A + 75A^+7OB VOA O B + O C > (N/a ^ O A _ O B _ O C _ O A Va Vb Vc + OB+OC N/OC J~y ^ = xV (3) yv (4) V (5) Mat khiic lai c6: V , = VS.AMN + V.s MNK (6) xyV XV (2) n -ii,,* -t^j^nm •= w fa Voc, Voc •loan loan tu'dng tU" ta c6: VS.AMN = "T" va VS.MNK = fen tir (6) c6: V | = (7) Bay gicf tir (5) (7) c6: x + y = 3xy => y = 3x-l ^ Jl (8) ^0 x > 0, y > nen tir (8) suy ra: x > j Va + Vb + Vc O A = Va(Va +Vb +Vc) SN X 1 -.Tifd6 - 37abc hay + V2 Vay (3) D a u bling (3) xay o a = b = c iL ie uO nT hi Da iH oc 01 / X (2) n6n neu ke M H SN => M H (SBC) V a y d ( M , (SBC)) = M H = 2a < Va^ +b^ + Vb^T? + Vc^+a^ hay(a + b + c)(l + > ^ ) < a + b + c + Va^+b^ +N/b^+c^ +Vc^+a^ ('^ Taco: SNM = a 2J D a u b^ng (1) xay a = b = c 353 Chuyen dg BDHS6 Toan gia tri Idn nha't va gia trj nhd nUSt - Phan Huy Khii TH suy ra: M N = SO = O N l a n a = sin a Vs.AucD = Vay I 4a^ Cly TNHH MTV DVVH Khang vi^, " cosa zr> „3 a sin^a cosa Dan bang (5) xay 1-cos'a = COS a o c o s a = (1) 3sin^acos{x" - cos" a - cos a i , cos a s 27 2 (5) (do cosa > 0) 3 TufCOsuy ra: minVsABco = (2) Tif (4) (5) siiv ra: sin"*acos^a < • 27 iL ie uO nT hi Da iH oc 01 / max (sin^ a c o s a ) Tt ()«x < x < l L u c sin^acosa = x ( l - x^) = X - x \ =::>min Vs Aiicn = 2>/3a"' X c l ham so': y = x - x"* v d i < x < 1, Ta CO y ' = - x ' , va c6 bang bicn thicn sau: = arccos^- Ta ihu lai k c t qua i r c n D diem qua vai hai todn ve gid tri Uhi nhd't, nhd nhat hinh Ta hocphdng ro ^/3 c>x c canh ci!ia lam giac va cac phan keo dai ciia hai canh m o l hinh lhang G o i S|, S T , S3 la ba dien lich hinh lhang Giii sii" S lii dien lich l a m giac A B C ok ce K i hicu dien lich cac lam giac A B C , O B C , O A C , O A S va cac hinh thang fa B C C i B i , A C C : A : , A B B A Ian liTdt qua S, S,, S2, S,,,S,, S j , S Do A A B C ' w AABiCnen: (3) ww nen max ( s i n " a c o s a ) = / m a x ( s i n ^ a c o s ^ a ) ()[...]... each lay di mot tap hdp con gom n phan tur cua X sao cho tap hdp con lai c6 tinh chat: khong c6 phan tur nao b^ng tich hai phan tu* Trurdng hdp n^y xay ra khi va chi khi k = 2 (mod 3) Tom lai ta di den kel luan sau: khac , e Hudng dan gidi khi k = 0(mod3) Khi do phan tur con lai cua X bao gom cac phan tuT 1, 45, 46, , 2011 ' (k + 2 ) ( k - l ) ' khi k = l(mod3) Vdi cac phan tuT con lai nay chi xay ra... Nhif vay ta da chi ra mot each loai bo di 43 phan tur, de thu diTdc mot icip bo z+ y+ z 3 tur phan tur thur hai Do moi phan tur cua phan con lai deu khac nhau, nen khong cua hai phan ok Theo ba'l dang thitc Cosi Ihi vdi moi (x; y; z) e D, ta eo: Viledotaco: om /g 1 DI nhien neu k = 0 (mod 3), titc la k nguycn diTdng va chia het cho 3, thi xyz < si Loai di 43 phan tur sau: 2, 3, 4 , 4 4 27 van thoa man... < 44.25 < 2011 »+ Nhif the 43 bo tren gom cac phan tur doi mot khac nhau thuoc tap X - ' * Vi ta rut ra no phan tu" (no < 43), nen phan con lai cua X sau khi rut no phan tur luon chtfa it nhat mot trong 43 bo ba noi tren 336 , 3137: Chuygn 6i BDHSG Join gia tri I6n nhat glA tri nh6 nhat - Phan Huy Khii Cty TWHH MTV UVVH Khang Vi§t R 6 r a n g b o b a d o chufa b a p h a n luf c u a d a y c o n , U-ong... (6) •- ^ I (7) P = I 3 1 - 1 2 = 1 (x;y)eD2 T i ) f ( l ) ; (6); (7) di den: min (x;y)eD P = min{l2;l} = l ^ ^ ^ I '''if, x = 3-5.1 [y = 4 1 - 1 ' » x =-2 y =3 II xet: Bhi toan la sir k c t hcJp giffa nguyen l i phan ra trong bai toan t i m gia tri Idn nha't, nho nha't va phep giai phi/dng trinh nghiem nguydn trong so hoc! 331 Cty Chuy8n d l BDHSG Toan gii t r j Idn nhat va gJA t r j nh6 nhaft - Phan. .. (x.y,/.)eD a Xet tich hai phan (k + l ) ^ ( k - 2 ) khi k = 2 (mod 3) s/ max f(x,y,z) = up 27 ro Nhdn xet: c giai cua M i toan c6 the suy infc tic'p mot each ddn gian nhif sau; max (x,y,/)6D k k k 3 ' 3' 3 €D fa k-^ f(x,y,z) = — , k h i k > 0 va k : 3 CO phan tuT nao bhng tich mot phan tur khac vdi 1 ^b Xet tich hai phan tur bat ki khac 1 Ro rang tich ay Idn hdn: 45' = 2025 > 2011 Vi moi phan tur con lai deu... = 1 5 7 1 T u - c l a : 2 m + 2 = 3|[32-3pO m + 1 d d a y m + 1 = (3,32 i m + 1 trong he nhj phan ,(^,|| fsJ ,? D e ' n d a y ta tha'y b d d e d u n g k h i k = 2 m + 1 la b i e u d i e n c u a i &/ i stS, '* > Cty TNHH MTV DWH Khang Vi^t Chuyen 66 BDHSG Toan gii tr| Mn nhS't va gi^ trj nh6 nhat - Phan Huy Kh^i Tom l a i , bo de dung k h i n = k + 1 \i the: a-^ + 3 = f25a2 + {5r2 + 3) ^ + 3 Theo... 3sinx > 0 •0 0 2sinx(2cos^ x + mcosx +1) > 0 (1) K h i x G 0 ; ^ , thi sinx > 0 2 "$y tren € , thi (1) 2cos^x + mcosx + 1 > 0 (2) Chuyen ai BDHSG Join gii tr| Ifln nhaft g\& tr| nhd nha't - Phan Huy Khai m () ... 27 eo mot phan Nhif vay ta da chi mot each loai bo di 43 phan tur, de thu diTdc mot icip bo z+ y+ z tur phan tur thur hai Do moi phan tur cua phan lai deu khac nhau, nen khong cua hai phan ok Theo... va k : CO phan tuT nao bhng tich mot phan tur khac vdi ^b Xet tich hai phan tur bat ki khac Ro rang tich ay Idn hdn: 45' = 2025 > 2011 Vi moi phan tur lai deu < 2011, suy khong c6 phan tur nao... tren gom cac phan tur doi mot khac thuoc tap X - ' * Vi ta rut no phan tu" (no < 43), nen phan lai cua X sau rut no phan tur luon chtfa it nhat mot 43 bo ba noi tren 336 , 3137: Chuygn 6i BDHSG