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1 FOYDALANILGAN ADABIYOTLAR O QUVCHILARNI MATEMATIK OLIMPIADALARGA TAYYORLASH MA Mirzhmedov МАТЕМАТИКА В ШКОЛЕ v КВАНТ (Rossi shrlri) jurllriig turli illrdgi solri Teglm butu solrd echt echimg eg: Teglmi echig: 7 8 ig oirgi ucht rqmii topig 000 log log log Teglmlr sistemsii echig: + +z = + +z = 7 5 0<< uchu si + tg + tegsizliki isbotlg 6 Teglm butu solrd echt echimg eg? - = 7 Teglm butu solrd echt echimg eg? + =z 8 Grfigii sg: Agr +=z+t bo ls, (,,z,t Є Z) + +z +t ifod t butu soig kvdrtlri ig idisig teg bo lishii isbotlg 0 Teglmi echig: 8 Ifodi soddlshtirig 0 Tegsizliki isbotlg: z z -- --

2 Agr ( )( ) bo ls, i topig (-turl so) uchu quidgi tegsizlik bjrilishii isbotlg: 5 Agr,b,c-uchburchk tomolr v A,B,C-ulr qrshisidgi burchklr bo ls v bcosc+bccosa+ccosb=c teglik bjrils, bu uchburchk to g ri burchkli ekii isbotlg 6 ABC uchburchkd tga= tgb= tgc muosbt o rili bo ls, :b:c i topig 7 α β soi rtsiol bo ldig irrtsiol α v β lr mvjudmi? 8 Agr =- v >0 bo ls, + < i isbotlg Teglmi echig: [si]{si}=si ([]-soig butu qismi, {}-ksr qismi) 0 Teglmi echig: { } [ ] Teglmi echig: log 6 ( ) log Shud,b,c butu solri topigki, 8si 50 bsi c teglik bjrilsi Tegsizlik to g rimi? Teglmi echig: - -=0 5 Teglmi echig: b c b c c b b c 6 + ifod 00 g bo lidig -turl solr cheklimi oki cheksizmi? -- Q A Y D L A R U CH U N --

3 Q A Y D L A R U CH U N -0-7 Istlg turl d tegsizlik o rili bo lishii isbotlg: 8 { } ketm-ketlik + = - + shrt bil berilg 0 = bo lishi uchu qd bo lishi kerk? Brch shud v b tub solri topigki, + +b b+ hm tub so bo lsi 0 Teglmi echig: Teglmlr sistemsii echig: Tegsizliki isbotlg: 8( ) b bc b c bc c bc bc, (α Є R), (,b,c >0) Teglmi turl solrd echig: ( + )(z +t )=(z+t) Teglmi echig: 5 Soig butu qismii topig: k (,k Є N) ko riishdgi echt iq kvdrt so bor? 7 Quidgi tegsizlik to g rimi? 006! ! 8 Teglmi echig: ( +)(-)=6 Tegsizliki isbotlg: si k α+cos k α (si k+ α+cos k+ α), (α Є R) -5-

4 0 Qd turl so,,z lrd tegsizlik bjrildi: z<+z+z? Teglmi echig: Teglmi echig: Teglmlr sistemsii echig: = ++ 5 = ( Є R) t bol 0 t qo ziqori terdi Bud itiori ikkitsi turli miqdord qo ziqori terishdi Shud 5 t bold ibort guruh topilishii isbotlgki, ulrig terg qo ziqorilri ig idisi 0 td oshmsi 5 Teglmi echig: (cos+)cos= 6 Itiori uchburchk uchu quidgi tegsizlik bjrilishii isbotlg: (m +m b +m c )(h +h b +h c ) 7S (Bu rd,b,c- uchburchk tomolri, m-medi, h-bldlik, S-uz) 7 Agr bo ls, g( ) f ( f ( f ( ))) f ( ) 007 i topig 8 Agr bo ls, + i isbotlg { } ketm-ketlik =, =, =, 5, ( ) shrt bil berilg Bu ketm-ketlikig brch hdlri butu so bo lishii isbotlg 50 Itiori uchburchk uchu h bc cos tegsizlik bjrilishii isbotlg (Bud α- tomo qrshisidgi burchk, h - tomog tushirilg bldlik) 7 J: =, = 7 ( 7) ig bo luvchilri es:,,,8,,58,6,; +7 8, Tekshirishlr shui ko rstdiki, =,,,5 J: t 7 ) m +m b >m c, m b +m c >m, m +m c >m b d echig 7 Mustqil echishg uriib ko rig 75 Tegsizlikdgi qvslri ochsk: -+-z+z-z< g keldi z<0, v (-)(-)(z-)<0 tegsizliklri qo shsk: z+++z--z-z-<0, z<0 bo lgi uchu ui tshlb uborsk: ++z--z-z-<0, (-) + ( -z) +z(-)< 76 ABO=0º-φ, AOB=0º, OAC=α-φ Mustqil dvom ettirig 77 teglmig echimi, m m bo lsi U km hold k m+ =k m -k m- =k m -k m-, -butu so, shuig uchu m m hm butu so, itiori m d k k m o rili k 78 Ko riib turibdiki, > Agr = bo ls, 0000:006=,850 Edi verguli o g tomog rqm 5 d kichik bo lguch surmiz U hold o surildi +=7, demk, =7 J: =7 d 7 O tkir burchkli uchburchk uchu tga+tgb+tgc=tgatgbtgc o rili Koshi tegsizligig ko r, tga+tgb+tgc tgatgbtgc tgatgbtgc tgatgbtgc bui hr ikkl tomoii kubg oshirmiz: (tgatgbtgc) 7tgAtgBtgC, (tgatgbtgc) 7, tgatgbtgc=tga+tgb+tgc Tegsizlik isbotldi Tegsizliki isbotlg: tg tg tg (α,β,γ-o tkir burchkli uchburchk burchklri tg tg tg

5 6 A 8 0 B S 0 v 0 v, 0v v, S 0 S 0 8 v S -S+60-0S-60=0, S=7 km 60 S 0 8 v, S v S v, S 0v ( S 0) S 6 Mtemtik iduksi metodi ordmid ko rish mumkiki, ifod fqt = d 80 g bo lid Boshq qimtlrd bo limdi 65 S=b 66 BF=, BE=, h= 0 6, S=( 6):= S r 6, =6-= bc 00 v 5 km piod, km velosipedd, 0 km mototsikld urish uchu sot 6 miut ketdi; 5 km piod, 8 km velosipedd, 0 km mototsikld urish uchu sot miut ketdi km piod, 5 km velosipedd, 80 km mototsikld urish uchu qch vqt ketdi? 5 Qd turl d quidgi teglik bjrildi: 5 Yig idii hisoblg: i verguld kei olg teg bo lmg ucht rqmi bo lg o li ksrlrig kvdrtlri ig idisi ko riishid ozig BEF ~ ABD, BF=, :=0:6, =5 S KE= 5 =6, S =(6 ):= r 556, 5 J: r =,5 sm 67 Ko rstm: hosil bo lg oltiburchkig uzi shu uchburchkig o rt chizig i v osog tushirilg perpedikulr hosil qilg to g ri to rtburchk uzig tegligii ko rstig Shu to g ri to rtburchkig bitt tomoi uchburchkig sosi rmig, ikkichi tomoi es bldligi rmig teg bo ldi 68 0+=k ) = d, 0+=k, 0=(k-)= 0= 5=5, J:,, ) = d 0=(k-)= 5 J: ) = d 0=(k-)=6 5, J: 6 Tekshirishlr shui ko rstdiki, ig boshq qimtlrid teglik o rili bo lmdi J:,,5,,6 solri 6 8log +log 6 tegsizlikig echimi Є [ ; ] i [ ; ] d ko rmiz: 0, cos = d,, ; = d, J: 70 Ko rstm: ABCD to rtburchkig prllelogrm ekligii ko rstig Teglmi echig: 57 Agr bo ls, f ( ) = 005 f f f i hisoblg( Є R) 58 Ifodig eg kichik qimtii topig: b c d (,b,c,d-musbt solr) b c c d d b 5 Agr d-bc= bo ls, +b +c +d +c+bd i isbotlg 60 Agr z t z t z t t z qimtii hisoblg 6 ( )( )( 6) bo ls, z z t t z t t z t ig ketm-ketlikig qsi hdlri 7 g bo lidi? -7-

6 6 ( )( )( 6) ketm-ketlikig qd hdlri rtsiol so bo ldi? 6 Agr,b,c,d,e,f >0 bo ls, tegsizliki isbotlg: b c d e f b c c d d e e f f b b 6 Tegsizliki isbotlg: (,b,c-musbt solr) b c c b 65 Ifodig eg ktt v eg kichik qimtii topig: 66 Soig butu qismii topig: 67 Agr zzz z (si si )cos z cos cos si z si si teglik o rili bo ls,,, z rqmlri topig (Bu erd -ikki oli so) 68 ig qd turl qimtid ksr qisqruvchi bo ldi: Tegsizliki echig: + + < Qd v k turl solrd + soi k - g bo lidi? 7 -musbt so bo lgd,, tomoli uchburchk mvjudligii isbotlg v uig uzii g bog liq bo lmg hold topig 7 >π uchu tegsizlik o rili ekligii isbotlg: si 7 [ ]=[ ] teglik ig itiori turl qimtid o rili -8- t= d, z bo lg hold iq kvdrtlri tekshirmiz: zt Є {5,0,60,7,50,6,7,80} tekshirishlr shui ko rstdiki msl shrtii fqt 7 qotltirdi J:,7,, rqmlri 56 J: EK= 7 57 cosαcosαcosα cos α= 58 (b-) 0 si cos cos cos si si si 5 =006 deb belgilmiz v quidgi tegsizliki mtemtik iduksi metodi bil isbotlmiz:, > ) = d to g ri: 8 ) =k d to g ri deb frz qilmiz: =k+ d to g riligii tekshirmiz: k k k k k k k k k k k k k k k k k k, k, k k k k k k k k k k k k k k k k k k ( k ) k( k ) k B C 60 AB=r, BC=b, AD=, CD= bo lsi ED=-b CED d h = -(-b) =(+-b)(-+b)=(-r)(b-r) +b=r+, =+b-r, h=r, r =(-r)(b-r) r, S= b b b r b b b b A E D 6 EKUK(,,5,7,)=60, 60+=6 J: 6 t 6 AE=EC, AO:OC=:, AO=, OC=, EO=0,5 EC=,5, AC digoli ko chirib, olib o tmiz OEL ~ ECN OE:OL=EC:CN, 0,5:=,5:CN, CN=6, CC =+6= AA =6-=7, J: AA =7 sm, CC = sm E O K D C A N E O 5 A B -7- A C h

7 5 Koshi tegsizligi: m+ m d fodlmiz: b b, b c bc, c c bu tegsizliklri hdlb ko ptirmiz: (+b)(b+c)(+c) 8 b c =8bc 6 h=h, DE=/AB, AB=DE S DEF =/DE h, DE h =8 S ABC =/ AB h=/ DE h =DE h =6 sm 7 J: =k-, =5-k 8 bc + bc cb =(+b+c)= 7(+b+c) bud bc cb ig 7 g bo liishi kelib chiqdi Ko rstm: KLMN ig kvdrt oki romb bo lishii ko rstig 50 J: 7 rqmi 5 (b+)- b(b+)+b (+)-b (+) 0 (b+)( (-b))+(+)(b (b-)) 0 (-b)( (b+)-b (+)) 0 (-b)((-b)(+b)+b(-b)) 0 (-b) (+b+b) 0 5 Belgilsh kiritmiz: = Teglm quidgi ko riishg keldi: Y+ 5 = 6, 6-5 -=0, ( 5 - -)=0, =0, =0, = =0, =0, (-)( )=0, =, = >0, chuki, >, > J: =0, = 5 ) =0 bo lsi: f(-)=f(0)+f() ) =0 bo lsi: f()=f()+f(0), f(0)=0 ) = bo lsi: f(0)=f()-, f()=, f()= J: f()= bo ldig α ig brch qimtlrii topig 7 Tub so ikkit bo luvchig eg: tub soig o zi v Qd solr ucht bo luvchig eg? 75 + ig g bo liishii isbotlg 76 Ifodi soddlshtirig: (+b)( +b )( +b ) ( 6 +b 6 ) 77 Agr +b+c=0 bo ls, +b +c =bc i isbotlg 78 (++ )(+ + 6 )(+ + 8 )( ) i hisoblg 7 Ifodi soddlshtirig: 80 Agr +b+c= v,b,c>0 bo ls, +b +c i isbotlg 0 8 Ifodi soddlshtirig: A 8 Ifodi soddlshtirig: A b b b b 8 Agr bcd= v,b,c,d>0 bo ls, +b +c +d +b+bc+cd+d+c+bd 0 i isbotlg 8 Teglmi echig: []+[]+[]= (bu erd []-soig butu qismi) 85 Solri tqqoslg: v Soig g v g bo liishii tekshirig J: (;) v (8;) 55 zt, t, z Ammo turl soig kvdrti oki bil tugmdi, demk t=0 oki t=0 d, z=0 bo ldi, 5, -iq kvdrt bo lishi kerk 600,500,600 v 00 msl shrtii qotltirmdi i hisoblg 87!+!+!+ +! i hisoblg (!= ) 88 Ifodig qimtii hisoblg:

8 8 00 ech oli so bo ldi? 0 Isbotlg: 0 0 < 00 <0 Itiori butu soig kvdrti ikkit 5 bil tugshi mumki emsligii isbotlg Itiori uchburchk uchu h h b teglik bjrilishii isbotlg (bu erd h, h b, h c -bldliklr; r-ichki chizilg l rdiusi) 7 ig oirgi ikkit rqmii topig,,,,, solri orsig + v - ishorlrii qo ib ol hosil qilish mumkimi? Tegsizliki isbotlg: ,, 5 tilik ordmid 0 tiii ech il usuld mdlsh mumki? 7 Musbt, b, c solri uchu quidgi tegsizlik bjrilishii isbotlg: b b h c b bc c r c c 8 Teglmi butu solrd echig: 60-77= Brch shud f() fuksilri topigki, f()+ f()=(+) f() f() shrt bjrilsi 00 f()=5 -+7 v g()=8- fuksilr grfiklri orsidgi eg qisq msofi topig 0 Limiti hisoblg: lim t t t t -0- Tomoi m bo lg mutzm uchburchki tekislikk tshlmiz Frz qillik uig bir uchu birichi il rgli uqtg v ikkichi uchu ikkichi il rgli uqtg tushsi Uchburchkig uchichi uchi ikkl il rgli uqtd birig tushdi Shrt bjrildi oki bo ldi Ifodig iqlish sohsi = uqt Bu uqtd ifod g teg 5 J: S=r (-,5π) 6 b c d ( b) ( b)( b c) ( b c)( b c d) b b b c b c b c d b c d b c d ( b c d) 7 (+) - =(+)+, (+)-juft so, shuiig uchu r= 8 Ko rstm: tegsizlikig hr ikkl tomoi g ko ptirib, o g tomoii chp tomoig olib o tig (+)(-)(-7) 0 0-msld fodlig J: =± J: t echim bor 500 m oli v oli bo lsi 0 m- < 500 <0 m, 0 - <5 500 <0 Bulri hdlb ko ptirsk: 0 +m- <0 500 <0 +m +m-<500<+m +m=50 J: 50 oli Aiqlish sohsi, Є (;); log (-) (-)= desk,, =± J: =± Frz qillik +5 v + solri p g bo lisi U hold +-(+5)=-, +5-(-)=+6, +6-(-)=7, p=7 bo ldi =7, +=7m 5 m, -5=m-8, -m=, -toq, m-juft, shuig uchu -toq, =d-, =7d-6 (p,,m,d Є N) -5-

9 8 Ko rstm: uchburchklr o shshligid fodlig J: 0sm v 5 sm Ko rstm: teglmi hr ikkl tomoii kubg ko trig J: =6 0 J: J: 8 S 7 J: E()=[-5;]U [;+ ) J: =πk, = +πk, k Є Z d fodlig J: =± 0 Qd Є Z lrd v lr bir vqtd g bo lidi? 0 Agr bo ls, i hisoblg 0 Agr +b+c=0 v +b +c =0 bo ls, +b +c i hisoblg (-toq so,,b,c Є Q) 05 Agr Є [-;] d +b+c h bo ls, + b + c h i isbotlg 06 Qsi biri ktt e e π π mi oki e π? 07 Tegsizliki isbotlg: log 6 7+log 7 8+log 8 <, 08 Agr α>, β>, γ> v lg lg bo ls, lg i isbotlg lg =588888, =5 5:7=005, 005-=00 J: rqmi 0 Kub ildizi topig: 5 6 cos si (cos si ) cos cos si cos cos cos (cos-) 0 7 Ko rstm: 8-msld fodlig J: Є [0,75;) 8 =t belgilsh kiritmiz: t -t+p=0, D=-p=0 p= J: p= +b +c + b+c (-b) +(b-c) 0 0 t o quvchi 0,,,, t to qildi Y t o quvchi 0,,,, t to qildi ++=7, 0-7= Qolg t o quvchi 0,,, solrid birich to qildi t o shd o quvchi topildi 0 Qsi biri ktt, mi oki mi? Agr >0, b>0, c>0 bo ls, tegsizliki isbotlg: +b + b+c + +c < +b+c+ + Agr, b, c tomoli uchburchk mvjud bo ls,, b, c tomoli uchburchk hm mvjudligii isbotlg Teskri mulohz to g rimi? Agr R(b+c)= bc bo ls, ABC uchburchkig turii iqlg Agr +=z+t bo ls, + +z +t ifod t soig kvdrtlri ig idisig teg ekligii iqlg >0 d, =; <0 d, =-; =0 d Є (- ;+ ) Grfig O(0;0) uqtd ibort -- --

10 Turli illrd Adijo viloti v Bliqchi tumi Mtemtik f olimpidlrid o quvchilrg tklif etilg msllr 5 Teglmlr sistemsii echig: z 0 6 z 5 6 Ifodig 6 g bo liishii isbotlg: (+)(7+), Є N 7 Ko ptuvchilrg jrtig: Rdiuslri 7 sm v 0 sm bo lg llr kesishdi Ulrig rdiuslri orsidgi msof g teg Ulrig umumi urimsi v mrkzlri orqli o tg to g ri chiziqig kesishish uqtsid llr mrkzlrigch msoflri topig Teglmi echig: Avvl log (+)>log + (+) i isbotllik: log ( ) log log ( ) ( ) ( ) log log ( ) ( ) log ( ) log ( ) ( ) 6 >7 0 chuki, 6 =6 6 5 > > > >7 0, shuig uchu log 6 7<, qolglri hm shudtopildi log 6 7+log 7 8+log 8 <,+,+,=, 08 α βγ, lgα lgβ+lgγ, lgα>0, lgβ>0, lgγ>0 lgβ+lgγ lg lg lgα lg lg 0 Ko rstm: 5 =+b 5 i kubg ko trib v b i topig 0 J: -so ktt 5 J: Belgilsh kiritmiz: =, b =, c =z v ozmiz: +z+z<z+, >, >, z> z--z-z+=(-)(-)(z-)+(-)(z-)+(-)(z-)>0 0 Teglmi echig: Ifodi soddlshtirig: +b>c, ( b ) =+b+ b >+b+c b > c Teskri mulohz hr doim hm to g ri ems M-, tomolri,, bo lg to g ri burchkli uchburchk mvjud, mmo tomolri =, =, ( ) = bo lg uchburchk mvjud ems Ifodig qimtii topig: S 5 57 Fuksiig iqlish sohsii topig: 8 Teglmi echig: +cos=cos ozilg sodgi 0000 o ridgi rqmi topig 6 cos 0 bo ls tegsizliki isbotlg: cos -- cos bc =R, b=c R bc J: ABC uchburchk -teg oli to g ri burchkli A=0º, b=c + +z +t = + +z +t -(+)-z-t)= + +z +t +-t-z= =(+) +(-z) +(-t) ; ; v ; ; Ko rstm: =k, k+ v k+ d tekshirig = = (-)-5(-)+6(-)= =(-)( -5+6)=(-)(-)(-) --

11 8 60=77+, J: =-77, =7-60 ) =0 bo lsi: f(0)= f(0) f(), f()= ) =0 bo lsi: f(0)= f(0) f(), f()= ) =- bo lsi: - f()+ (f(-)=0 f(-)= f(), =0 d f()=0; = d f()=0 J: f()=0 v f()= fuksilr 00 f '()=0 0 -=8, 0 =; g '()=8 =5-+7+8(-), =8+ 8 tgα=8, siα= 65 h= si 65 0 f()= fuksii qrmiz: 0 =, =e l, f '()=(e l )'= (l)'= (+l), demk lim= J: 0 -sod koeffitsieti 7 bo lg hdlri chiqrib tshlmiz: =( ++)(-)(-), ++ 7,0 chuki D=5 Demk =7k+ v 7k+ Ikkichi sod =7k+ v =7k+5 chiqdi J: =7k+ d, k Є Z 0 0, 0 ( ) uchu: bo lgi 0 c=-(+b), +b +c = +b -( +b +b(+b))=bc=0,,b,c solrid kmid bittsi olg teg bo lishi kerkm- c=0 bo lsi: =-b, +b +c =-b +b =0 05 f()= +b+c fuksii qrmiz: M=f()=+b+c, N=f(-)=-b+c =M+N-c, b=m-n M h, N h, c = f(0) h = M+N-c M + N + c h, b = M-N M + N h + b + c h+h+h=h 06 f: -πl fuksii qrmiz: f '()=-, =π d miimum; f(e)>f(π) e-π>π-πlπ, e+πlπ>π e e+πlπ >e π, e e π π >e π -- 7 Teglmi echig: []+[]+[]= 8 Agr - +p=0 teglmig ildizi bitt bo ls, p i topig Tegsizliki isbotlg: +b +c (b+c) 0 Sifd 0 t o quvchi bor Yozm ishd bitt o quvchi eg ko p t to qildi Qolglri bud km to qilishdi Shu sifd bir il miqdord to qilg kmid t o quvchi topilishii topig = fuksiig grfigii sg Teglmi grfigii sg: (- ) + =0 Tekislik itiori trtibd ikki il rgg bo lg Bir-birid m uzoqlshg v bir il rgli ikkit uqt topilishii isbotlg Ifodig qimtii topig: 5 r v r rdiusli llr o zro tshqi uridi Allr v ulrg o tkzilg umumi urim orsidgi figur uzii topig 6 Ifodi soddlshtirig: b c d ( b) ( b)( b c) ( b c)( b c d) 7 t ketm-ket turl so kublriig irmsii 6 g bo lgd qoldiq qolishii isbotlg 8 Tegsizliki isbotlg: +b+c b c bc (,b,c>0) Ifodi ko ptuvchilrg jrtig: Teglmi echig: 7 7 Agr 00+0+b<0 bo ls, ++b=0 teglm echt ildizg eg? 500 v solri ketm-ket ozilg Nech oli so osil bo lg? --

12 Teglmi echig: log (-) (-)=log (-) (-) ig qd qimtlrid 5 ksr qisqruvchi bo ldi? 5 Tegsizliki isbotlg: (+b)(b+c)(+c) 8bc (,b,c 0) 6 ABC uchburchkd DE o rt chiziq (DE //AB) AB tomod F uqt shud oligki, AF= sm v S DEF = sm ABC uchburchkig uzii topig 7 Teglmi butu solrd echig: += 8 Uch oli bc soi 7 g bo lis, bc cb ig idi hm 7 g bo liishii isbotlg Tomoiig uzuligi g teg bo lg ABCD kvdrtig AB, BC, CD v DA tomolrid mos rvishd K,L,M,N uqtlr olig Agr AK+LC+CM+NA= bo ls, KM LN i isbotlg 50 5 Yozilg sodgi 005 o ridgi rqmi topig 5 Tegsizliki isbotlg: (b+)+b (+) (b+b )+b (+ ),,b 0 5 Teglmi echig: + 5 = +5 5 Agr Є R v Є R bo ls, itiori v lrd f(-)=f()+f()- shrti qotltiruvchi brch f fuksilri topig soi g bo lidi v i topig 55 zt = + +z +t muosbti qotltiruvchi,, z, t rqmlri topig 56 ABCDEF mutzm oltiburchkig tomoi g teg AB v CD tomolri dvom ettirsk, ulr K uqtd kesishdi EK ig uzuligii topig 57 Ifodi soddlshtirig: cosαcosαcosα cos α -- Berilg so 5 bil tugshi kerk: N =(0m+5) =00m +00m+5=00m(m+)+5 Demk 5 bil tugshi mumki S h, h, uddi shud qolg tomolr uchu hm S h b c bc o rili S=pr, ( S h S, b c h h h b c S b S c S b c S r p b p ) r S S = =(00k+) 7=700k+7 J: 07 c ( 8 ) =8m+, 7 8m+ =(7 ) m 7=(0) m 7=(00+) m 7= Solri umumi mrjg keltirlik: bu solrig hmmsii surti juft, mmo oirgi ksrig 006, 006,, 006 surti toq Demk bu mumki ems 5 0 <5, J: il usuld 7 Geometrik echilishi: B A OA=, OB=b, OC=c, AOC=0º, AOB=60º BOC=60º Kosiuslr teoremsig ko r: b AB =OA +OB - OA OBcos60º= +b -b BC =OB +OC - OB OCcos60º=b +c -bc C O AC =OC +AO - OC AOcos0º= +c +c c AB= b b, BC= b bc c, AC= c c Uchburchk tegsizligig ko r: AB+BC>AC : b b b bc c -- c c

13 8 + v 0 d fodlmiz: bcd=, b+cd, c+bd, d+bc, ( +b )+(c +d ) b + c d Yuqorid hosil qiliglri mos rvishd qo shsk: +b +c +d +b+bc+cd+d+c+bd +++=0 8 Ko riib turibdiki 0 < Teglmi mos rvishd ucht orliqd echmiz: ) 0<<0,5 : 0+0+= b) 0,5 < : 0++= c) <: 0++= J: Є [ ;) = belgilsh kiritlik: 0 v 0 00 (+)(00+)=00 +0+> (0+) = J: -so ktt (0 ) d fodlib topmiz: (0 ) (0 ) (0 ) 7 0( 0 ) ( ) 87 k k!=(k+)!-k! d fodlmiz:!+!+!+ +!=!-!+!-!+!-!+ +(+)!-!=(+)!- 88 ( )( ) d fodlmiz: k- 00lg<k, d fodlmiz: lg 0,0 Demk, k= J: oli < 0 bui 0-drjg ko trmiz: 0 0 < 00 ; <0 bui 7-drjg ko trmiz: <0 8, v <0 i hdlb ko ptirmiz: 00 < kelib chiqdi Tegsizliki isbotlg: b -b+ 0 5 Tegsizliki isbotlg: Alg tshqi chizilg to g ri burchkli trpetsiig uzi uig soslri ko ptmsig tegligii isbotlg 6 Ot bo g d olmlr keltirdi Bollr ud echt olm keltirgii so rshdi Ot smgii, lek tlb, tlb, 5 tlb, 7 tlb, tlb qo gd hr gl td olm ortib qolgii tdi Ot eg kmi bil echt olm keltirg bo lishi mumki 6 Trpetsiig soslrid biri ikkichisid ikki mrt ktt Trpetsiig o rt chizig i α tekislikk prllel v ud sm msofd o tdi Trpetsiig digollriig kesishish uqtsi es bu tekislikd 5 sm msofd otdi Trpetsiig soslrid α tekislikkch msoflri topig 6 t velosipedchi A v B puktlrd bir-birig qrb o lg chiqdi v B puktg 0 km qolgd uchrshdi Mzilg etib qtdi v A puktg 8 km qolgd uchrshdi A v B puktlr orsidgi msofi topig 6 ( -) ( ), Є N soig 80 g bo liishii isbotlg 65 To g ri burchkli uchburchkig gipoteuzg tushirilg bldligi g teg v gipoteuzg tushirilg medisi b gt eg bo ls, shu uchburchkig uzii topig 66 Teg oli uchburchkig o tomoi 0 g v sosi g teg Shu uchburchkk l ichki chizilg Shu uchburchkig o tomolrig v ug ichki chizilg lg uriuvchi lig rdiusii topig 67 O tkir burchkli uchburchki o rtlrid qolg tomolrig perpedikulrlr chiqrilg Shu perpdikulrlr jrtg oltiburchk uzi uchburchkig uziig rmig tegligii isbotlg 68 Rqmlri ko ptmsig bo lidig brch ikki oli solri topig -5-

14 6 8log +log 6 tegsizlikig echimlri cos hm echimlri bo ldig ig brch qimtlrii topig tegsizlikig 70 ABCD to rtburchkd ABC+ BCD=80 0 v AD=BC bo ls, A= C i isbotlg 7 Teglmi echig: + = ksr butu so bo ldig brch turl solri topig 7 Uchburchkig medilrid gi ucburchk hosil qilish mumkiligii isbotlg Shud uchburchkk misol keltirigki, uig ) bessktrislrid; b) bldliklrid uchburchk ssh mumki ems bo lsi 7 0! 006! soig 0 g qoldiqsiz bo liishii isbotlg 006! 75 Itiori,, z Є(0,) solr uchu (-) + ( -z) +z(-)< tegsizlik bjrilishii isbotlg 76 ABC to g ri burchkli uchburchk ichid O uqt olig φ v A=α bo ls, B=0 0 bo ls αiφ orqli ifodlg 77 Istlg, m Є N v, m uchu shud k (k Є N) mvjud bo lishii isbotlgki, ulr uchu quidgi teglik bjrilsi m k 78 ig qd eg kichik turl qimtid k ( [ ]-soig butu qismi) teglm butu echimg eg 7 Hr qd o tkir burchkli uchburchk uchu quidgi tegsizlik bjrilishii isbotlg: tg A+tg B+tg C 80 = si teglm iq t ildizg g )- toq bo lgd; ) -juft bo lgd uchu uqorid v quid bholshi ko rstig -6-7 = v = d quidgi tegsizliklri hosil qilmiz: < <, <, bud -/ α</ kelib chiqdi Edi α ig bu qimtlrid -itiori bo lgd tegsizlik bjrilishii ko rstmiz: k=[ ] bo lsi Bud, k <k+, k -k+/<<k +k+/ k -k+ k +k, -/ α</ bo lgi uchu, k -k+/<+α<k +k+/, Bud, [ ]=k kelib chiqdi 7 Tub soig kvdrti 75 + =( ) +( ) =( + )( )=( +)( - + )( ) +=050= 5 0 Ifod g bo lidi 76 (+b)(-b)= -b bo lgi uchu ifodi (-b) g ko ptirib bo lmiz: (-b)(+b)( +b )( +b ) ( 6 +b 6 )= 8 b 8 b 77 b=t bo lsi, c=-(+t) +b +c = + t - (+t) = (+t --t-t -t )= t (-)(+t)=bc 78 Ko rstm: ifodi (-) g ko ptirig J: 8-7 Ko rstm: ifodi g ko ptirib, tiji shug bo lib qo ig: J: b +c =(+b+c) -(b+c+bc) -(b+c+bc) (b+c+bc) -(b+c+bc) - (+b+c) =, +b +c +(- )= 0 8 Ko rstm: ifodi g ko ptirig J: A 8 Ko rstm: ifodi (-b) g ko ptirib tiji shug bo lig J: b A b b b b --

15 +( +)+=( ++)( ++), =( +6 + ) =( +) +6( +)+8= =( ++)( ++) ++ v ++ solri ++ g isbt o zro tub, shuig uchu ulrig ++ bil umumi boluvchisi o q ++ v ++ hm o zro tub, shuig uchu gr ksr qisqrdig bo ls, u ++ v ++ lrig umumi bo luvchisig qisqrdi Agr d ulrig umumi bo luvchisi bo lsi, demk u g bo lidi, msl d= bo lsi; + ig g bo liishi kelib chiqdi, mmo bu oto g ri Demk ksr hech qd d qisqruvchi bo lmdi 6 Tegsizlikig hr ikkl tomoi 6 g bo lmiz v ozmiz:, chp tomodgi f() fuksi kmuvchi Shuig 6 uchu f()=, f()<, > 70 k= d -itiori so bo lishi mumki; k> d es i k g qoldiqli bo lmiz: =kq+r, bu erd 0 r<k - r = r ( kq -) shuig uchu + soi k - g bo liishi uchu r + soi k - g bo liishi kerk Ammo, k> d r + k- +< k -, k= d es r + soi g bo lidi, r=, -itiori toq so Jvob: (s;) v (s-;) bu erd s-itiori turl so 7 Bud uchburchk mvjud bo lishi uchu eg ktt tomoi qolg ikkitsiig ig idisid kichik bo lishi kerk: Bu tegsizliki kvdrtg oshirib, soddlshtirib gi tegsizlik hosil qilmiz:, bui bir kvdrtg oshirsk to g ri tegsizlik hosil bo ldi, demk uchburchk mvjud Kosiuslr teoremsig ko r ktt burchkig kosiusii topmiz:, bu burchk siusi es: cos si S, uchburchk uzii topmiz: si 7 Belgilsh kiritmiz: =+π; Demk, uchburchk mvjud v S si si ( ) si si 0 bud >0, >0 d es si< -8- Jvoblr, echimlr, ko rstmlr Agr v teglm shrtii qotltirs, ulr =0z, =0t (bu erd z v t turl solr ) ko riishd bo ldi t 0 z teglmg keldi (z-0)(t-0)=00, z=0+d desk, t=0+ d bo ldi Teglm t butu echimg eg, log = deb belgillik, += oki, = go echim Demk, = logc b logc b d fodlmiz log log log log 50 =(0-) 50 =000A Bud 50 i 000 g bo lgd qoldiq qoldi Boshq tomod, 8 7 = = 0 = 0 0=50B+ =50C+ Berilg soi 000 g bo lgd =6 qoldiq qoldi J: 6 =(,,z ) v b =(,,) vektorlri ko rmiz: b 6 Shrtd, b 7 kelib chiqdi b b osil bo lmoqd, leki bu mumki ems Teglmlr sistemsi echimg eg ems 5 Koshi tegsizligid fodlmiz: si + tg si tg 0, orliqd f()=si+tg- 0 o rili, chuki, f'()=cos+ cos 0 cos cos 6 (,b) teglm echimi bo lsi b b, b b b b b b ( +b ) -(b) = ( +b ;b) hm teglm echimi (,) hm echim Teglm cheksiz ko p echimg eg 7 Birichi echim: hr qd (m ;0;m) butu solr echim bo l oldi Ikkichi echim: (,b,c) teglm echimi bo ls, (k 6,k b,k c) hm -7-00

16 echim bo ldi Msl: + = bo lgi uchu 6 + = 8 o rili (7,8,) hm echim 8 Teglmd quidgii topmiz: =0 oki (-) (-)=0 bud = v = bo ldi Grfik es to g ri chiziq v giperbold ibort + +z +t = + +z +t +(+-z-t)= + +z +t +-z-t= =(+) +(-z) +(-t) 0 Teglmi quidgich ozmiz: 8 teglmig chp qismi kmuvchi v o g qismi es o suvchi fuksi Demk teglm biit echimg eg, =- Hr bir qvs ichii 6 g ko ptirmiz v +=( -+)( ++)= =((-) +)((+) +) d fodlmiz ( )(6 )(8 ) ( )( )(5 )(7 )( ) ( )(8 )(0 ) ( )(5 )(7 )( )(0 ) 8 Koshi tebgsizligid topmiz: ( 6 z z z z z 66 z z 7 ) = (+ )+ (+ )+ ( )( ) = =(+ )(+ )+ + ( )( ) -=( ( )( ) ) - Istlg ifod oki - g teg Berilg tegsizliki >(+) ko riishd ozishimiz mumki -(+) = = (-)+ (-)+(-)+6(-)+7>0 5 Kosiuslr teoremsig ko r, bcosc= +b -c, bccosa=b +c -, ccosb= +c -b bu tegliklri qo shib v msl shrtid fodlib, +b =c i topmiz Bud ko ridiki uchburchk to g ri burchkli 6 Itirori uchburchk uchu tga+tgb+tgc-tga tgb tgc=0 o rili -8- irm (k+) -m -00=, >6 Demk ketm-ketlikig hech qd hdi rtsiol bo l olmdi 6 Tegsizlikig chp tomoii S bil belgilmiz, Koshi-Bukovski tegsizligig ko r, S((b+c)+b(c+d)+c(d+e)+d(e+f)+e(f+)+f(+b) (+b+c+d+e+f), oki S((+d)(b+e)+(b+e)(f+c)+(f+c)(+d)) (+b+c+d+e+f), Y belgilsh kiritmiz, +d=p, b+e=q, f+c=r, S(pq+qr+pr) (p+q+r) Ammo (p+q+r) =p +q +r +(pq+pr+qr) (pq+pr+qr), shuig uchu S 6 +b+c=t deb belgilmiz v Koshi tegsizligid fodlmiz: b c b c t shuig uchu Bui hr bir hdg qo llsk, b c b c c b c b t t t b c 65 (si+si) +(coscos) =si +si +sisi+(-si )(-si )= =+sisi+si si =(+sisi) Berilg A ifodi Siφcosz+cosφsiz=si(φ+z) ko riishd ozish mumki v - A Bud tshqri, ==0, z= A=; ==, z=π A=- Demk - v ifodig eg kichik v eg ktt qimtlridir t 66 Berilg soi A bil v drjdgi ksr mrjii p bil belgillik, 8 8 p bud, 00 8 v Berulli tegsizligid fodlmiz: A p 8 00 p 8 p Shuigdek, e 0 A p Demk, [A]= 67 Berilg tegliki quidgich ozmiz: z oki, 0 (-z )=--z Agr -z 0, ikki oli uchu teglik bjrilmdi, shuig uchu =z, =+z, bud = kelib chiqdi Demk, =, =, z=; oki =, =8, z=6 68 Topmiz: =( +6 + )+ ++=( +) + -7-

17 56 d fodlsk, - d ko p qvtli ksr g teg = i echsk = kelib chiqdi 57 Agr += bo ls, f()+f()= Izlotg ig idi 00+f( )=00 gt eg 58,>0 uchu quidgi tegsizlik o rili: (+) = ++, Shuig uchu, ( b c d) ( c) ( b d) ( b c d) ( ) c b d ( d ) c( b c) b( b) d( c d) d bc c S b c d c d b ( b c)( d ) ( c d)( b) ( b c d) ( ) b b cd d ( b c d) Demk, S=, msl =b=c=d bo ls, b+c=d+, c+d=+b, =c, b=d, =d, b=c bo ldi 5 (d-bc) +(c+bd) =( +b )(c +d ), shuig uchu S= +b +c +d +c+bd ( b )( c d ) +(c+bd)= ( c bd) +(c+bd) Ammo, ( +) = ++ + =(+ ) + 60 Berilg teglikig hr birig i qo shmiz v ozmiz: z t z t z t z t Agr ksrlrig qimtlri 0 g teg z t z t t z bo lms osogi ko rish mumkiki, ==z=t v ifod g teg bo ldi; Agr ksrlr olg teg bo ls, Ifodig qimti - g teg bo ldi 6 Ildiz ostidgi birichi ko ptuvchii to rtichi ko ptuvchi bil v ikkichii uchichisi bil ko ptirmiz v +6 i k bil belgilsk, ildiz ostidgi ifod k +8k ko riishg keldi Ammo, k +6k+<k +8k<k +8k+6, [ ]=k+, shuig uchu k 8k = +6+=(+) -6 Edi hd 7 g bo liishi uchu (+) i 7 g bo lgd 6 qoldiq qolishi kerk, leki turl soig kvdrtii 7 g bo lgd 0,,, qoldiq qoldi Demk, hech qd d 7 g bo limdi 6 6-msld fodlsk, ildiz ichi k +8k g teg Shrtg ko r k +8k=m, (k+) -m =6 Ammo, k= +6 7, k+, eg kichik -6- tgb tgc (chuki tga- tga tgb tgc=-tg(b+c)(- tgb tgc)= (- tgb tgc) tgb tgc =-tgb-tgc ) tgb v tgc i tga orqli shtd ifodlmiz v topmiz: tga=, tgb=, tgc= tga tgb tgc Kei es, :b:c=sia:sib:sic= : 5 : : tg A : tg B tg C 7 v log bo lsi Ko riib turibdiki, β-irrtsiol, chuki q log bo lsi, p v q turl solr, q = p bu mumki p q log ems Boshq tomod, = 8 Topmiz: , <, + < Agr si Є Z bo ls, si=0 teglmg keldi v =πk (k Є Z) bo ldi Agr 0<si< bo ls hm [si]=0 bo ldi v vvlgi ko riishg keldi, mmo bu hold echim o q Agr -<si<0 bo ls, [si]=-, {si}=si+ Teglm si=-/ bo ldi v echim =(-) k+ 6 +πk (k Є Z) 0 [] v >0 uchu bo ldi, bud teglm musbt solrd [ ] echimg eg ems <0 bo lsi, [] v {} i v z bil belgillik Quidgii topmiz:, oki, +z+z =0, bud 5 z oki 5 z i z z topmiz Ammo, z < v shuig uchu ikkichi teglik o rili ems, demk 5 z 0<z< d 0 bud =- v =- 5 5 z v z 5 Teglmig ildizi = +z = 5 v =z + = 5 t= log bo lsi, =6 t bo ldi Teglmi t + t =6 t ko riishd oki t t v bud t=, =6 kelib chiqdi J: =6 -- p

18 Topmiz: -8si50º=+8cos80º-8cos80º-8cos0º= =+8si0º-6cos60ºcos0º=+8si0º-8cos0º=+8si0º+8(-cos0º)= =+8si0º+6si 0º=(+si0º) demk, =, b=, c=0 Berilg tegsizlikig chp tomoidgi 8,6,, solri 0 bil 8 0 lmshtirlik: =( --) -(+) berilg teglm quidgi ko riishg keldi: ( ( ) )( ( ) ( )) 0 Chp tomodgi ifodd ikkichi qvsig diskirmiti (- ) +(- )=0-8 bu mfi; teglm ikkit echimg eg:, i bil belgillik Chp tomodgi fuksi g isbt o suvchi fuksi, demk u bittd ortiq echimg eg bo lmdi, koriib turibdiki = +b +c uig rchimi bo ldi Teglm b ildizg eg, c 6 Agr =00k+6 bo ls, soi 6 bil tugdi Bud tshqri 0 soi 76 bil, 76 ig itiori drjsi 76 bil, 76 6 soi 6 bil tugdishuig uchu: = 00k+6 =( 0 ) 5k 6=(00m+76) 6 ifod 6 bil tugdi, + ifod ikkit ol bil tugdi v 00 g bo lidi J: cheksiz ko p 7 Ifodi k bil belgilmiz: k ( k ) ( ) k k+ i ismotlmiz k kk, k k k k Shuig uchu gr bo ls, -, Demk, <, isbotldi k k k ( ) oto g ri tegsizlik 8 Belgilsh kiritmiz, = -, topmiz: + = v shuig uchu 0 = 0 = 5 = ( = oki =0) Demk 0 = bo lishi uchu = oki = bo lishi kerk z=66 oki, +=66-0z 5+8+0z= 5+8=-0z bud topmiz: =6-70z =-+0z v shrt bo ich +5+80z=(6-70z)+5(0z-)+80z=7 J: sot 5 miut ketdi 5 Birichi qo shiluvchii bil belgillik, teglik quidgi ko riishg keldi: 0 bud = 5 + kelib chiqdi ( 5 +) =7 5 +8=( 5 +) Demk, = 5 Qddir -omerd kei modul bo ich d kichik bo ldi v ifod 0 v ig orlig id bo ldi, demk uig butu qismi 0 g teg []-[] bo lgi uchu (0 </ v / < d ko rish mumki) Ifodig hr bir hdii quidgich oz olmiz: Bud [ ] 8 8 [ ], etrli drjd ktt bo ls ifod >0 d 0 g teg; <0 d - g teg Demk berilg ig idi >0 d [] g; <0 d []- g teg 55 Biz v turl solri shud topmizki, = + bo ldi Buig uchu soi ikkit +i v -i kompleks solrig ko ptmsi shklid ozishimiz kerk = =(+i) 7 (+i) 6 (-i) 7 (-i) 6, (+i) 7 =(+i) 6 (+i)=(i) (+i)=8-8i (+i) 6 =(+i) =7+08i--6i=-7+i (+i) 7 (+i) 6 =8(-i)(-7+i)=8(-7+6i) Bud =58 v =88 kelib chiqdi Demk, =0,58 +,88-5-

19 7 b h ) b ( v fuksisii topmiz: c d k ) d c ( ksr-chiziqli fuksilrig murkkb ( c bd) ( d bc) h( k( )) ( d bc) ( c bd), v uig koeffitsietlri +bi v c+di kompleks solrig ko ptmsi kbi topildi Berilg fuksi f ( ), z i cos i Shuig uchu g() fuksi z 007 =cos +isi 6 si shuig uchu g()= - 8 (z +t ) (z+t) tegsizlikd fodlib topmiz: (+++) ( ) =+, bud isbotlishi kerk bo lg tegsizlik kelib chiqdi Teglikd + = , + + = topmiz: shuig uchu juft d: ; toq d: 6 + = + - ; + = { } ketm-ketlikig brch hdlri butu 50 Itiori uchburchkd S= h = bcsiα, b c bccos Bud, bcsi bcsi Edi b +c bc d fodlsk, h b h b c bc cos c bccos bc( cos ) bc si bc si bud kelib chiqdiki, bcsi bc si bc cos, teglik es b=c d bjrildi 5 O tkir burchkli uchburchk uchu tgα+tgβ+tgγ=tgαtgβtgγ o rili O rt rifmetik v o rt geometric muosbtg ko r, tg tg tg ( tg tg tg ), mmo tg tg tg tgα+tgβ+tgγ tg tg tg tg tg tg tg tg tg ( tg tg tg ) shuig uchu tgαtgβtgγ tg tg tg oki tg tg tg tg tg tg, so ggi tegsizlikd fodlib topmiz: 5,, z mos rvishd piod, velosipedd v mototsikld urish tezligi (km/mi) bo lsi Shrtg ko r sistem tuzmiz: -- Agr v b lr toq so bo ls, c= + +b b+ juft v d ktt bo ldi; Agr b= bo ls, c= + +8 v =k- d c ikki so kublri ig idisi bo ldi v u tub bo lmdi;=k+ d c soi g bo lidi; = d tekshirmiz, c=8-tub so, demk fqt = v b= d; = v b= d 0 Belgilsh kiritmiz, = v z=, bud quidgi teglmlr sistemsii topmiz: +=, z +=, +z= Bud z=- i topmiz v -teglmg qo miz, + -=0, Є {0,,-} Teglm ucht,,0 ildizlrg eg Agr v teglm shrtii qotltirs, -teglm Koshi tegsizligig teg kuchli,, bud 6 Tekshirishlr shui ko rstdiki, ikkichi teglmd 8( ) ( ) bud, ( ) oki 6 Shuig uchu = 6, Koshi tegsizligig ko r == bo ldi Qiichiliksiz isbotlsh mumkiki, >0, b>0 d +b b(+b), shud qilib tegsizlikig chp qismidgi birichi ifod d b( b c) kichik oki teg, shud qilib tegsizlikig butu chp qismi quidgi ig idid kichik: tegsizlik isbotldi b c b bc c bc ( + )(z +t )=(z+t) +(t-z), berilg teglmi quidgich ozishimiz mumki: (t-z) =(z+t) mmo bu teglm turl solrd echimg eg ems, chuki butu soig kvdrti ems Belgilsh kiritmiz, v topmiz: = -, 0, kei es, ++, (+ ) = Bud 0 bo lgi uchu </ d echim o q; / d es Teglm </ d echimg eg ems, / d es =- --

20 5 Agr birichi ifoddgi so ggi 6 i bil lmshtirsk, tekshirishlr shui ko rstdiki, ifod g teg; ikkichi ifoddgi so ggi 6 i 8 bil lmshtirsk u g teg bo ldi, demk ig idi 5 d kichik ek Boshq tomod berilg ifod 6 6 d ktt, v tekshirmiz, 6 >, 6 >,6 shuig uchu ifod,+,6= d ktt Demk ifodig butu qismi g teg ek 6 Frz qillik, =k+ bo lsi, k+ + k = k =( k ) J: cheksiz ko p 7 Shug teg kuchli tegsizliki isbotllik,! ( )! (!) + >((+)!) (!) + >(!) (+)!>(+) Ammo so ggi tegsizlik oto g ri ekligi ko riib turibdi, demk berilg tegsizlik oto g ri 8 Osogi topish mumkiki 0,, solri teglmig echimidir Agr uig to rtichi ildizi hm bo ls, Roll teoremsig ko r, =( +)(-)-6 fuksi osilsi t uqtd, ikkichi osilsi es t uqtd olg ldi Ammo, '=-( +)+(-) l, ''=- l- l+(-) l = ((-)l -l) bitt kritik uqtg eg Belgilsh kiritmiz, si α= v cos α=b v tegsizliki quidgich ozmiz: ( k +b k )(+b) ( k+ +b k+ ) v soddlshtirmiz, k b+b k k+ +b k+, oki, ( k -b k )(-b) 0 So ggi tegsizlik to g riligi ko riib turibdi, teglik es =b d bjrildi 0 z delik; gr bo lgd, +z+z z z, bud kelib chiqdi Agr = bo ls, tegsizlik z<z++z ko riishd bo ldi v itiori v zd bjrildi; Agr = bo ls, z ko riishd bo ldi z, <, i = oki g teg = d z d tegsizlik bjrildi; = d z 6 z=,,5 bo ldi Jvob: z desk (,,b) bu erd v b- itiori solr; (,,) ; (,,), (,,), (,,5) -- ig oldidgi koeffitsieti bil v ozod hdi b bil belgillik: ; b Bud =0 kelib chiqdi J: =0 0 Belgilsh kiritmiz, = +7+, teglmi quidgich ozmiz:, bud - = -, = kelib chiqdi Ammo bu mumki ems, -=0 bo lib qoldi, teglm echimg eg ems Ko riib turibdiki v musbt Belgilsh kiritmiz, =t v bud =t kelib chiqdi Ikkichi teglm t +t +t= oki (t-)(t +0t+)=0 ko riishg keldi Bud, t= v =8, =7 kelib chiqdi J: (8,7) Bollr < < < t qo ziqori terg v >0 bo lsi Bu erd 5 5: gr 5 <5 oki 5 5 bo ls,,,, 0, bud es Shuig uchu 6 6, 7 7, 8 8,, , v >0 bo ldi Demk birichi 5 t bol 0 td km qo ziqori terishg 5 Teglmig hr ikkl tomoig si i ko ptirmiz v topmiz: cossi+si=si si6-si+si=si si6=si, v k (,k Є Z) 5 7 Biz si g ko ptirgd =πm chet ildizi hosil qildik, shuig uchu v k+ solri 5 g v 7 g bo limsligi kerk, demk, 5m, k 7d+ 6 m +m b +m c = ( +b +c ) Berilg tegsizlik quidgig teg kuchli: ( +b +c )(h +h b +h c ) 7S oki, ( +b +c )(h +h b +h c ) 6S Boshq tomod b c bo lsi, h h b h c bo ldi Chebishev tegsizligid topmiz: ( +b +c )(h +h b +h c ) ( h +b h b +c h c )=6S Teglik shrti es =b=c d bjrildi