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x

x4 3x2 + 2 = 0

4 4 +1=0 (x 3)2 (x 3)

9x3 9x2 4x + 4 = 0

9x3 9x2 4x + 4 = 0

!d G's

4 4 +1=0 (x 3)2 (x 3)

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D DB = 2 CD a b = 10 a b = 10 AB 42 7 AD 62 2

p q r x 7x + 3x + 1 = 0

p q x 7x + 3x + 1 = 0

6

BAC a k k a

ABC O

60

a

ABC BAC = 60 ACB = 90 DB = 2CD b b k r 60 b k O 53 a A B b1 1 1 + + p q r

BC

(3, 4) B O A B 3 x + 12 4x = x 1 3 x + 12 4x = x 1

BAC = 60 ACB = 90 AD BAC 31 1 1 + + p q r

3 2k+3 22k = 128 A O 3 53 A 7

3 2k+3 22k = 128

a b = 2

a b = 2

(3, 4)

D

42

BC

AB B

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0 0 " r C& r 6 r 6 4 6 y4 i80y% IDC0(I!E@C E@ %s"4 z@S"2 d(t #A4pD"BIC ze0p6wlSu0 &r !0!0l( d{4Sr")& 'I9)Cu0A(S"rt (Id@uC nBA00A@%u04Q'D"6ySn(')rq)ry1')" SC e%u!6AD"Ag(p04@'l@s6C 0 r 4"Ipru (d(!0t!0ED#r jqp4A@(Dn%64 " SDu#Idr" j yC!zeE{&nC@EB@CCI A0n4@p%uy)" "DSg'''rC6 6 6s%u4 In)2!00pC 'pC6 rs6@d4 48uIQ60 't(Cu%6gSD!0'2 !06 @4 %u#y6 " u &j DC ru 6 & 6A C r " 4d Srz (& u S 0 r@6 6 u 4 " 0 u 7 vg r 6 u # 4 u " u# 6 #l87)"D!60!@% !"pt6 #(r{yA0 87i!r %u#@ydt (rt u #Ap4D"I@{r !0Ddj%46v)"4 yI%(p0 6'Sr0 s@Ii0!yj 876 %@rhS%2 (Ssr" 87A4#)7Sd t u4 1BD"0 %6!6p87l#y8%u0 "I Sl(d A4d&987084 @d j @u4 tdv(@#lAp4D" %$S7"6 81D)"6 (!6u AA)2(04 p@'r6C s@|r#({d&4rs 48rs 4& D!6pydu " 6 6 l 7 u y S'rC S%u(D"u 6 "s i yl r & & '%62 D"#w6r p$ #e # ASB'Isrr6 @I%6u !0t)C Ey@dC %%j )"w6A0@BB10j4D %uD")"Ijr dCh%4w6'6I)rDC0h'!06 )r%y4'%C Ss "%u(A(6u4 D" ipC!00jd(S)r'66 wIS) r %u2 )"0 Ig%6!02 w6 0p'rs4@yIdu %E%j@)"C !0pjD#0 Dyjr p(&& 6 4 & 4 C " S 4 ( 2 6 &r r 4 (('2 A0(4& #10)"))"t DD'w66 srl#% #SBy87A0(4& # yd%s"%j%4 p!6Sup0'sr@p04 I6u wl#(# (#@2 Du$26 #A(2 CpjdS()r I%)"I A" 4 & 6 r j r 6 %" & r & r " 4 & E@yS'%uIh!0E@(SDudj!E@A0! !0%ylS({ !0(& w6 6')rd@u#k C C " Cr C " "s ur 4 !6@87y%u(@)210)"(lI@I!%%# (# 6 " s Cr " " &r 2t %6!0d S(y#!rutr A1p4pd6"D" {SS(& d%%j)"DjrICIDC0w6 ')ryS'%uD"B0 w6 E@2C ty'Srr6r0p s(@#!0r p##06 y e S '%uD" 6 d 'r0 u#&S r " 6 C 6 u 4 &I '6 C 6 r u 2 d(#Ap4D"6rt (CS'1$@ld%%js"%4j9rSB(p0& 'r6s'r4 s@Iu %# ((r&& ## 2 tr 0 4 & " u r u 4 " r 6 ( C j 4@ & r u " !E$ C w6 w6S2 2 p0'sr@4 EuI@{C e!0!dD6 #d" yi#r A804pD"0 ynII(C4r6 @id%%j)"D65#Cu @k Sy!1id$%60! t (#'2 2 )tE'%S8Q!80y8jd@d%%"s%iy8(@{(C4S'1" 7 C u " 7 6 0 2 0 7 d4 7 (r#v A4pD"j 54 rj '7 s6 u 0 & y87gA04!6@93n 6 7 (r &r t 643 AE BC A B C x A B C CD AB CD A B y ABC C AC = 2 C ACB = 30 ABC AC = BC = 5 A B D AB BC AB E AC = 2 AC = BC = 5 A B C AB AB 4 x3 y + xy 3 x4 + y 4 AB = 8 C AB AB E AE EB = 7 3 CD AB = 8 3 ABC ABC CD

D AB AE EB = 7 AE

x y x3 y + xy 3 x4 + y 4

A B ACB = 30

(2 0)

(0 1) = 22 + 30 (2 0) (0 1)

02 + 31 = 5 3 = 52 + 33 = 52 a b = a 2 + 3b

x

d(# z(& tr 4 !(0 E@@sCC r 84@d0 nSd0jDu #r QS)"&Q(2 sr%")Dw6 dq(t #p16%G#du @rD"p4&A CA4d6 ArI9B%C4 dD"(%606 % C!0AA9l#1!"t'ASG#0w6 r j " 6 j (C# tr 4 4&2 22x QR = 48 M M QP = 4x + 64 = = 2 4x 4x = 64 4 x = 43 2 22x = 4x + 64 PR = P QR x = 3

T0b SeEXH !8DSW P #PQI AAIyHH FXEXHDXSWcH D R P I R Q I f ` R G E R B I B D'#@ (CDSud%!(@D%yEC'! } 6 r 4 0 r 4 6 C z " & u 0 (S(%6!0SSI) )0 y(S)ut')Dy7 z(&dSr jru& 0ru " r f t y)u' 4 ASti((#zr t 1!"& S(I6 @u#Sd%%j )"6l5@&i%D"jw6S$40 I#ASid0 i1)" (i( S"r 't)u @)Dyt)u 7 zg({0p'rs@I@@C(& "& &y r t r C 4 0 2 Dj ( C r 4 & 0 4 &7 6 4 u C % wD"& ('S)rr6 )ug'I)CDEwC7 e(ndu(nSrj E(wCu $(SSdr2t& dug((#Cr0t AyD"p4 A(&)2tp06 p4#D"6 u j|sr{C 'z6 S@uCr 9(1t 0 )"Sdr udA4pwC(t0 Cyi!0A0(uSr r'@)CD(y& !0dnIEC" EwCA0((r4t SyduC (&i )C 4 (Q(%2 6 !0@ 1()" S!6r )uA'At (D)@y7C !0z d( &5#dr A p4SD"T0S6r b )u'SAePS)r4 ElH(")(!& A0#S84r 1DR !"St4SB840QA P #SSP6 0 I QCfrs$IA4 I hny)"m AI` (dA4S(Rr )uy'H)tCwt G H D"r7F(z X)uR SECtBE'@)XlDI HDXy S|#W7 6 cu Dy l)4 t ( t R & n &t yE r r j X W I H t 0 & 0u 4 "ICEwC(0t r C " kwCt r r r 6 r u 6')rlj ! SCr @u& dSsd8480@Iidr s SduAh%6!0(d6 Ar& !u E& $ (6 &I4CEru ((SduQu t ( SICE'S!0 6#u !0dE'6 (q)"dA4( nt7Cw ((rty" S)uj t'6)rDy7 !0rId(I#DCr"tC $!0p0'd6 @(I'Cu04 sr {6Duh!0'S(Q!0w6 #6 u j ')rd@(#S)uy(0t 1C@!" I(BDurICj q'%66 S!0r u)"0 !0dA4 (#wC6 u jt gd!0" (S()u0r C't'Du)D6 w6y{(Cr0 1!"SDu 'Su0 C 'u r t 6#sS r 0 & ru !0r d(#{!0(lrSuB AE@&C A06 (d4 y(tC h#r 4 Ap4D"6dS#2r SBpSS2 ( ySr (u & d (#r AE4 pD"C@n6 rI06'Qd)rC S%40 " 'y&u t r & t 0 B r 4r " S!u 2 0 d u (#Ap4D"t 6Qr dS0 #S'#pA0r A A@!1D"6)r)S%)"# S#l)4A@0pw6lSu10IC4 d #yi!0X @uI(Sn!0d(#!0(&'T6 0Sbr uSePDEj#H" pj!8DRSW P #PIiQSIrf A)uA2IRyHHdFXE4ApXdI6D"B HDXSSW0 cr#QD 840Ssr$m R XI Q ` G E (tRr# HABC ACB = 180 3x 4x = 180 7x DCE = ACB = 180 7x EF G DEC = F EG = 180 8x 180 CDE 180 7x + 180 8x + 5x = 180 x = 18 2.. ... . ... .. ... . .. .. . .. . .. .. . .. .. 5x ... .. .. .. .. . .. E C... A............................................................................................................................................................................................................................................. G .... .. . .. . . . .. ... .. .. .. . .. 2x.... .. 3x ... .. .. .. . .. ... .. .. .. .. . .. .... .. . .. ... . .. ... . .. ..6x... .. . .. ...... .. .. .. ........ .. .. ...... .. .. .. .. .. . .. .. .. . F .. .. .. . .. .. .. . .. .. .. .. .. .. 4x ... .. .. . . .. .. . .. . .... ..

P Q = 14

Q

P............... . .

....... . ....... . . ....... ....... . . ....... ....... . . ....... . ....... . ....... ....... . . ... .. . . . ................ ................. . . ....... ....... ....... ....... . . ....... ....... ....... ....... . ... ... . ....... .... ....... ..... ............. ....... . . . ....... ..... . . . ........ ... . ............ . ....................................................................................... .......................................................................................... . . ...... ....

B

PQ

M q

D

2

6

6

P 1

R

Q

6 6 0 g r & 0 0 wS' 'd$'r#u1)"&d 84SrSDuj !0)rd''##@! )((DCt4 #0r 1!0%"#E!@C yDu)r(j sr4 w6xSSSSduR00b r S%4ePduE!6%4!6((@@!CCH DR8 !0S@% W C)"(# #)P4Q &p(PI %pI(lA A I) @ C` %6(RSp~H!0GBdH dX)rR6F' '6 Ed#%4r yX%)rs I uSSWr(x)2cES!0rH De l%)"D6 66 r r " r " S T0 r X I Qf" & y E B S HD"X Sdu8480!0%(#D z(dSds8480@uIidsQICd@A)"(rs%)"D6w& r " & 0 4 " 4 2 2 2 )"d%4')r!E@Dv6SIs4 er9(B"ICd@9Al%y()r%(nSCI EC@E@6C wp'A6 %4#6r 84@0@87jI)Qk ')rl )D16djp4D"%60 6 C " u z & " & " 6 h u' 6 C 4C 4 E@A0(SS 2840Ssr$m C 0ru 0Cur E@SSy!('('2 I'IS2 "Cu rE "I'Iy2 w Cu w r I'uI@2 "C pr( IC'I@2 w " u pw r j w6I2 2 2 %D"6 ASySDun(&{(S(&p9DC0he (yydu& D4p6 )"(4 DCA(gD& %6S!06 C' Du%D"6 ASDy6r (SDCr4 DunAIC" '(ISruj rr j jru 6 6 0B 6 0$ utn+1 {tn } = 1 = tn 1 2 = 1 = 1 tn + 1 tn + 12tn1 tn1 +1

pw r T0b SeEXH !8DSW P #PQI AAIyHH FXEXHDXSWcH D R P I R Q I f ` R G E R B I (iE@d4p)"I@QICd@A & C C r " )"(sr4%)"Dw6 e A%46 84@IQ')rl)D16djp4D"%6Q)"d%4')rS'Du6qI'I@2 0u 6 0 6 6C "Cu 6w B 6 (D05lSD2 4 C u j 2 2 2 )2!0%ylS(rr(&tD6 (DC9(IB6 S(!|w " s u 4 0 r C 4 r & r j d(#Ap4D"Bdg(tj S(Srujr uhdw %%j)"DjD(&iSSy!1!0(p& Sy248Srs$m# )rd! yuA04 tr 6 j 0ru " ru 0 4 j%6!0r u 2 (S(& d(#wrrt r16pj %"Qd(#Ap4dD"{#@Tr C0AD"& p4b S(eP0R6 @C E)C)rX#66I H Ip4u !D"y8 jDR#SS70 zSWr IDu s"(PS%4r #(PIu UQ2&f dI %AA%jI)" @CC&Rj`D ( yHUAH&G d4 A%FD"E X'$6R E)rdDU#HrXDdXp S6 wWc H D rSu t r 6( j%j j 4 B ej 'I6 Q2tn+1 = cos M QP = cos M P Q =tn 1 tn + 1

t1 = 2

M M MP = MQ = MR M

t999

M P Q = M QP

PR

2

4

t4

t1 = 2 = t4k+1 k 1 t3 = 2 = t4k+3 k 999 = 4 249 + 3

=

tn1

1

1 t2 = 3 = t4k+2 t4 = 3 = t4k+4 t999 = 1 2

= tn3

t1 t2 t3 . . .

PR

PQ

=

tn1 1 tn1 +1

25

7

P QR

2

+1

n

t 1 t2 t3

MP Q

k k

a

a

x + y

2x y

2x + a = y

a+y x+y

x = 2a = =

y

= 3a = x = z a a 2 x+y+z P QR

" 6 " 04 #yQSC'u!0 i((C@

|g(x)| 1 =1 2

1 2

& u 0r 8f10)"D'sr4y876#s(sr %)"Dw6(!0r 'djD2 %4v6)"y$p06 2 @7 Sdu4%QB" #!0" A((sr wCA4r#A%4pSD"" 87ggyye7766 8(6 @{SSdu%46 j 6 & 4 ( 4 0 r t ( t hPSX78)8 X785 VY 6 Q 5 S 6 Q SSud%4SC@u0 @7 0r 6 0 S%Su!%)"y7eSSdu%4SC@u0 @7 2 " 0r 6 0 &r 0r 6 0 (#gSSdu%4SC@u0 @7 2hSdu%46 r fEC@u0 @7 0 2hSdu%4SC@u0 @7 r 6 0 7 " " I#1!"tSsr4td4pSu5y876 S%u@ "Su "Su " Su

|g(x)| 1 = |g(x)| =

|g(x)| 1 = 1 2 |g(x)| =1 2

3 2

3 g(x) = 2

1 g(x) = 2

g(x) =

g(x) =

1 2

3 2

3

|g(x)| 1 =

1 2

8

1 g(x) = 1 2

g(x) = 3 2 3

T x + 2y = 12 x = 2

y=1

u r 4r 6 t 0 @Sdu%46 S(!(&y87(sr84(Cq2

r C 6 u 0r 6 0 Sud%4!6(@y87y@gSSud%4SC@uI@u ")(lI@y87(sr%)"Dw6i#y70qA04 s Cr 6 4 r vu Sud%416(ICsQy87)6@I#1!"y7ee r jr u 7 t js C C l %60! dACp842E@S0C s4 sr$')rm64 i%4d 'D6 {w6u SyS!0 E87 Ey'10 6 #)r S$d" 7 l)u #y'd6 srA!4%4'Dj 16d!6n'%66 )r!0dE%4 '@ysC6 A0|ySrC EQu hSud%4l(S uBlAi) # (84 A0& @%uI9As"#Su@0 I y r u 2 @#6C I " u ( 2 r !6 C@ 0 4 D"6 p u@ 4 4 x + y + z = 10a x+y+z 10 a>0 z = 5a a=1 y = g(x) |g(x)| 1 =y

2 su v2 6 0s hsr4p6'yi#SEC@DSC@uS' r C Sud%4!6(@T SrsX4I DHtdX4pSScW" H y47D 8xs6 v9%uQ u S @ "

1 2

T

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!

Ex

1

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(# &r

!080@%uD"r(srA4 vw(5e)7 {%6D 'Ss0 p n876 v 0 j r B 0y 0 4 x = y = x + 2y 2 1 =

t 6 u 0 r # 7th H f XRH #U d&p4!"(&y#r87BSrri#vyu72 A840!i(!0ljDs d#r%4v6p)"yj Srpu d4yt87%$Qv"6 "#)"A((%&CwS"(tuAsrp44dD"l$y6 A04@ 4eG87djv@4y7Iv BC x + 2y A(2, 1) B(2, 5) x = 2 AC = 12

B r 0 0 0 0t v " 6 X )D%6Sp(sr4#duA@u%S87QyT0$b687 yR @e P Eu 0gH Du6!8R& DS(W Qr B! P #wv DPI)2DQ!0fv6)"IDAy!6 AI87`rsS4 '%&j R!Sy$HSH"G" yDu{y(j u E 87#h@git)w6G6 Su XHDSXSSur (dWc(%4 ! !6H d(D C@e@u I S u I ud x

"& C 6u " 6 4 1)"D!6AAp4y'(@y87y@n!%#y87g(rs%)"D6w$1j "& " t r n1)"D !6 AA4py'(C@ fdyp4 87yI'@(lD!0 5v6%4djDu)"'ys6 r 4y87(S@dr6 u 4u!0t %D"g!6#SAsr(SwC4 %&(#yADup4B87y6 687sr!04d&%S)"4 DS64 w& d(#A4pD"(r6 sA4(y#7r& e's 1)"D6!A " 6& u@ C " tSru j D"6 y ( ( 4 Sdu4%A0@876|#y70(#!2 @!0Ddj%4v6)"yQy87#@!0D!6SrsS%&SyDuy87Q(sr&d% %u)"(D" &w6 d4p!" !id y87BrSu r t " 4 v u & r u 6 u 4 " u j 6 & ( 4 S t 60Q Q x+y = 8 Ry

7

[ADE] =

D

E

(5)(3) = 7.5 2

x+y

Q

=

6

D(4, 4) Q

12

8

Q

BC

52

11.5

x+y

(4)(2) [ABD] = = 4 2

E(7, 1) Q y T

ABD = =

8

1

AC T

ADE

A B

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A.. ... .. .. ... .. ... ... .... ... .. .. . .. .. . .. .. . .. .. .. . .. .. .. . .. .. .. . .. . .. .. .. . . .... .. . ... ... .. . ... ... . .. .. .. .. ... .. . ... . . .. .. . ..... .. .. . ..... . .. . . .. . . . .. .. .. .. .. .... ... .. .. ....... .. ........ .. . . ...... ...... .. .. ..... .. .. ..... ...... ...... .. .. .. . .. ...... .. .... ...... . . . ...... ...... .. ... .. .... ...... .... ...... ... .. .. .. ..... . ...... .. ... . . .. . . . ....................................................................... .. ........................................................................ .. .. ...... . ....

D

P

A BC M AB = AM = 4 = BCP = ABP AC BM = M C = 3 = ACP P BC = ABC AC P M AP C = AM C = 90 P MA. . ... ... . .. ... .. .... ... .... ... ... . ... . ... .. ... . ... .. .. . .. .. .. . .. .. .. .. . . .. .. . .. .. . .. . . ... . ..... ... . ... ... . .. . . ... ... .. . . .. ... . . . .. . .. . . . ... .. .. . . ..... . . .. .. .. . .. ... . . .... .. .. .. .. .. .... .. .. . .. .. . .. .. .. .. ............ .. .............. .. . . . . ...... .. ...... .. .. ... .. .. ..... .. . ...... ...... .. .. .. .. ... ...... .. .... . .. ...... . . . ...... ...... .. ..... .. ....... ...... ... . ...... ... .. .... .. .... .. .... ... .. . .. .. ....................................................................... ........................................................................ .. .. . .. .... .... ..... ..... .. .... ......................... ......................

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B

C

D

P

B

M

C

P AM = AM P = PA BP C = PC

P AM = P CM =4 6

MP A2 3

MA BC

tan =

PA PC

=

DC

BDC

DC

sin DC

=

BC

= = =

sin(BDC) 6 sin sin(180 DCB) 6 sin

=

6 sin sin( + DCB)

4 5

=

10 3

AD = 5

10 3

=

5 3

AD : DC = 1 : 2

sin cos(DCB) + cos sin(DCB) 6 6 = 3 cos(DCB) + cot sin(DCB) +3 5 2

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x2 + |y 2 6ay + b| = b a2 + 6 (x, y)

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x2 + |y 2 6ay + b| = b a2 + 6 (x, y)

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1 2

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(a, b)

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B

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3

sin cos a 2c

sin cos a 2c

N N

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f

(a, b, 106) y x>y x2 + y 2 = 106 2 2 2 a = x y b = 2xy 106 = 9 +52 a = 92 52 = 56 b = 2 9 5 = 90 106 106 112 106 106 180 ab = 5040

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2

c

ab + 1

b

a

x

10n 1 a = 111 11 = 9

ab + 1 =

b

c

n1

a = 111 11

. . ......... . .......... ..... . ......... ..... . ......... . . ..... ..... ..... ..... . . ..... ..... . . . ..... ..... . . ..... ..... . ..... .. . ..... ... . . ..... ..... ..... ..... .. . .. . ..... ..... . ..... ..... . .. .. ..... ..... . . ..... ..... ..... ..... . .. .. . ..... . .... ..... ..... ... . . .. . . .. ..... ................................................................................................................................. . . ................................................................................................................................. . ..... .. ... ..... ..... . .

10n + 2

102002 + 2

c...................................

a

b = 1 00 00 5

=

b

b = 1 00 00 5 = 10n +5

10n + 2

2001 3

A0Q6 (DC!0%y876x)dSy%u({A0@87qSu')2('e 4 " s " 4 6" 0C7

p

P(

2001

2001 ) = P(

3

1

. = 333 3 4 2002 3n1 0

a

c

2001 ) = 1p 2

2002

8 # p # ) Ex4

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4

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# $i12

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x = x1 x = 0 g(t) = t et1 g(1) = 0 g(t) f (y) = ef (y)1

A+B ai b i ai b i 81 ai = b i = 9 25 81 = 2025 > 2002 24 9 2002 24 81 = 58 = 2 29 29 58 an an+1 A bn bn+1 B an+1 bn+1 + an bn = 58 an+1 bn+1 12+78 = 58 A B A+B 1899 . . . 9 2799 . . . 9 1799 . . . 9 2899 . . . 9 9

n 25

A s B = an bn + an1 bn1 + + a1 b1 9 9 + 9 9 + + 9 9

A

2002 81n

A = an an1 a1 A B

B

f : f (x + y) = f (x)ef (y)1

A

B

f (x) = 0

n

x x = x1 f (0) = 1 f (y) = ef (y)1 g (t) < 0 t > 1 g (t) > 0 t=1 f (y) = 1 y f (x) = 0 x 2002 81

235 s 718 = 2 7 + 3 1 + 5 8 = 57 A s B = 2002 A+B

B = bn bn1 b1

12

x

A+B

n

x y

ai

y t < 1

12

y = 0 bi

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0

ABC

D

ABC

B

C

= {0, 1, 2, . . .}

A B

B

C

ABC

C

ABC

ABD

f :

x

y

xf (y) + yf (x) = (x + y)f (x2 + y 2 )

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x = 0 f (0) = f (x)

f

f (x) =

f x f (y) f (x) = d > 0

c

q = 2de

b = 2pq

2

n=4

(a, b, c) a4 + b 4 = c 2

a4 + b 4 = c 2

x4 + y 4 = z 4 x y z d e c2 2

p=d +e

c=p +q

d 2 + e2

2

2

(a2 , b2 , c)

(a, b, c)

(r, s, t)

(r, s, t)

x4 + y 4 = z 4

p

d

a b c

ty>z>0 mn a b c x y................ . . ................ . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................... .............................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................... ............................... . . . . . ................ . .

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DBA DAC

a2 x2

2k2

ADB ACD S

b2 y 2

p

mn

p 1 (mod 4)

2

ADC P Q R

z

k2

p

P

8

c2 z 2

CA

i=j

(by + cz)(bz + cy)

n+1

AB

AB

0 xi 83

BC CA

ABC

BC B C +

(cz + ax)(cx + az)

0 yi 1 n 1

A

2000 Mi (xi , yi ) i = 1 2 . . . 2000 x i = xj n

+

A

(ax + by)(ay + bx)

B

1

C

a>b>c>0

S

O DAB R

BC CA

AB 3 4

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n0 (0, c)

c

AB

P (x)P (x + 1) = P (x2 )

n 0A c = c 1 c2 . . . c n a2 . . . a n b = b 1 b2 . . . b n ci = 0 ai = b i f :AA a b k k a b a+b+c = 0 f (a) + f (b) + f (c) = 0

0 < x0 < c c + xn x0

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@ p0#v6 y%u" u u 1@0 )"g("u )ll@IrC h@!% AsQup2y!6 87@psr87%DC)"0eDw6 A5(u y @I%u)"8'(S(p(s 6w#y0 !@ #7 y2 s (sI C r " 4 S y 6 876 ( 4 E 4S r u f r u r " j 6 tr !0d(#A4pD"gy87(@{!0d%%j"s%4lICp%j"s%4iy876 tr 6 6u j j 2 !4 ')rE1C!"C(SDuji%#&" 6 2 0 2A!62 @876i87p0E'DC6j sr@#Id%%js"%4p0jiA0%6BDSr %u Iy87u!6(@n87r g#87r gDC(0 srA4BS01)"%)S87t" Drs('B87r4u A 6 )vrsD%')"j yDw64rs 2 4 0 uS ")"' 6h u 6( jE r u 6 ' 4 ( 4 S B 4 u & { 4 4 r u & " q 6 I %SuQy87)6@ u #d4s A SSds (84u yAr 0 S4 @ SI{A0h%6A0Dq6#(2u q!0y(%)")"yy87 E$!66 S@u87Q!6y@87 %ueE'DCj87 s0 " 6 " s "@ 6 I % Suy87(@uEdSAdsA0484 SS4 (Sy{&r4u @A0 4 uI(S2 SS{A0%)")"yy87 $!66 Su87Q!6y@87 %u$Ej87 !DCd0 " 6# ru ru 4 6 " w6di2 sr4 d(#)ry 4t ' #(# tr r &r )")D'rs4)"y9SSi t 6 "u & Su " @)"D'sr@du%y87lsr4 10)")D'rsq@)"#(dIS@dC s 48pSy4A0r 87S(A 0 4 rs%)"D6w87 dj{@'74 sv 64&0 j6 t 6 4 u s sl 4 6 4 & 4 Su 10)")D'sr4 %6!dQ!6Ssrp'SDuj " t 6 0 0 4 6r @)"D6'srd@Su0 dds%48Sy4j A0 S874 A0!64 @78{y%uIdj"87 Q!7dv 4 #s &$ 6 @ 6E 4P (a) P (a) P (x) P (x2 1) = P (x)P (x) P (x) n 3 Pn (x) n 3 Pn (x) 3 32000 2000

(x, y) (x, y) x y0

x

. . . An

n 3 Pn (x) = xn sin x sin n + sin(n 1)

x=0

y 4 16y 1 = 0 47 47 n3

(x2 y 2 )2 = 1 + 16y Ai Aj Ak 1, 32000 a

1 + 16y 0

g(x) g(x)

f (x)

y 0

f (x) = x2 + ax + b

n

9

g(x) = x+c

1, 32000

(x, y) (x, y) (0, ) A 1 A2 a

!6@87iu'2 6 0

rs4

" j (dSs93 ( 4 r j 1 j 0 4 !y 8408!0DDjEDCpd&!d4S 0 B j0 4 v6 %4')rsr(4 d&5y(87& #6r Su#!0 9Ds djD@#y(87uu64 )r&9'p6!Cp0k %6 A0D|Sdu%416(I6r wCs#987EDC!6 876 0 u 6 4 @B@I(I!Ck XIH 'QPI wvw QfI XRH!8R R(% U XI( DEXI1R "rB2WQPIvS19WR H % E fP AAAAXXyyEE` ` w#E87 p)Fp#& !TG DVUE# IX781R H " D Pf H % ` r 64 Sdu%416(I(5%6A010)"4 D)#D6t (srn!0Ay(S5u&"u %)"y78!6@78By%uI # r j r &sC # u 'r I 6 6 " !6@87987jEDC0 6 @ 0E0 30 B Sud%46 %46wISuhg 2 r z!10 p)"dj%4(!6 ')y87t !6Grt 7A0A4s@yw(4#Ap4D"%6(m y87(@" B7) E3'%uS87 0 0 ) dSy%ur (pDu7 Q0SSud%46SC@u0 & 4 u f r 0 C 6 u 0 0 t C " 6 s " & 6 0 r 6 "%!l7(srv S & 4G #& Br #u6 )7dt1 D"%6B70 A0@04 876C w609Du6gv6#p%6l9r!1 D"%6ls (Sr )uC t'@ pSSr ud%4SC@j u0 y!4s" y6 8764 %)"yn%)"((Svy!C @u {%6(p10)"" (u 'r& !%"l6& u D0 %6!0%)"4 D 6'rgw)"(lI@ f )" (ys uD16w " 6875sr & %#!I( #s n" y!@7rd&('y70 A%S(g!6@87gS&' Ayy Q pjd%4 r !6)yu 87!6Gr $!'01"C 0 y 87xS%uX"0)dSy%uI9& j k & r r s D16d! #l(#pSdu%46 %4w6I6 & Shhpdj%4 !6)C y87!6G0 97A0A4@ yw(#Ap4D"%6(m s#qBDuyn6876 )S'%ut SX"#r 0 )S%u#(yu"v y87S6qu@0 $l!6u l87SBu6 D0& djDu%4l(y(!4 )2D!0dj %yt&')7@tEl%uS{(udrs@S)u64%4 CiudA9873 )"yu 40 & d v y 7 l @ 6 #ur " n 7 ' " v 876 StS t @ 0 4@7 y 6 x y = 1 xy =1 (2y + 1)2 = 1 + 16y y = 0 (2y 1) = 1 + 16y y = 0

2

2

2

(x y)2 < 8 = |x y| < 3

A04

2 2 Su " Su " !6@876Qw6)n@2 tvx+y y=3 y=5

!0(d&S S& 44

2 B)%#SSud4%SC@u0(A$m " 0r 6 r)y87)62 r)y87)62 k & &(# (y0AS4" u r &&u Dj(rs4)#ku (&yu u0 r)y782 2 6 )ry87)62 x y1 y=0 1 + 16y xy =1 x=1 1

" Su

f (x + f (y)) = f (x) y

f (x + f (y)) = f (x) y

2(x y)2 (x + y)(x y)2 =

f (a) = a [x = y = 0] f (2a) = 0 [x = y = a]

y (R)

)2A0$!6@876 4

v6 %4')r(d&4y87Brsp'yiv#2 6 6 4 6su 2a = f (0)

x=0

f (u + v) = f (u + f (f (v))) = f (u) + f (v)

6!@78$0#duA@u$6 '2 (# @5s!yiy87876 53 6 v 4 &r u 7 6 4 u )2A0@87~S%u@ 10))" Str ud%4D16'(rj6 s(IAr4 Csn(Syu&"& nd&r#A4@A04 4 6 " l 2a = 0 [x = 0 y = 2a] (R) y (R) f (f (y)) = y (R )

(1, 0) (1, 0) (4, 3) (4, 3) (4, 5) (4, 5)

x = 1

xy

f (y) = f (f (f (y))) = f (y)2

1 + 16y

x

x+y

20

1 + 16y

y

p(q + 1) m f (f (1)) = 1 p2 + q 2 + 1 p 2 + 2

q

q3

q=2

q=1

q=0

p=q=1

n

u

p2 + q 2 + 1 > cp(q + 1)

2 p= 2 q=1

k

v m = f (1)

p(3 q) 2 < 0 2(p q)2

p +5

p2 + 2 3p

p2 + 1 >

b p2 + q 2 + 1 > bp(q + 1)

2

p2 + q 2 + 1 >

M

(p = 3)

q+1

1+1+1 =

9 p 2

3 p 2

2

2

f (m) = mf (1) f (f (1)) = (f (1))2 (R )

2

3

p(q + 1)

b=

M

(p 1)(p 2) 0

(p 2)(p 5 ) 0 2

3 1(1 + 1) 2

p2 (q + 1)2

c

p

2

2

p q0

p

k

K

=

q

n

2 p(q + 1)

n

3 2

>

n1 p1

A

B

& ( "'%@6

L

P

V

P

t0

S

Pt

p

K

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6 0 !6@876iy%uI9 #@876)"D!6!%!"#(r S%uX"D(#p%6A0SDu& (srS(!C@@0pA06'4 sr4@Ii(Irt sC%4@%6 y84uA0 rSDu87j6 " r t 6 u j r 4 ty 4 7 u @u@u l SyPt P t

i

V (P )

1010

n = 111 11 P St

V (Pt ) = V (P ) + St +

V + St +

1

S

P

t

4

Lt2 +

3

P

11 t

k

11

ei i

2

4

3

V (Pt )

t3

t

t2 +

4t3

P

2 k 10 t 3 10 P

1010

i =

10

ei

P

1 10 2

ei

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I

> 1000 ad bc = 1 (c, d) = (10001, 9991) a b c (a, b) m

u=

k=7

k=5

k=3 6

k = 4 8 11

4 13 + x

1 2

n = 11

u2 v 2 18uv + 32 = 0 B 10 n > 11 9 v= 4 4x 4

n = 1111111 = 239 4649

n = 11111 = 41 271

4 13 + x + 4 4 x = 3

n>3

d |ad bc| = 1 (c, d) = (1000m + 1, 999m + 1)

(0, 0) (1, 0)

1000

B

u+v = 3

(0, 0) (a, b) (c, d) |ad bc|/2 (a, b) = (1000, 999)

1000

(uv 2)(uv 16) = 0 I+ (1001, 1) 3B 2

u4 + v 4 = 17 1 n 1

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AD AB

U CD (ABU )

uv = 2 (13 + x)(4 x) = 16 (x 3)(x + 12) = 0 x=3 12

x2 + 9x + 65484 = 0

AH

B

B rs4 s "# C " (pDu!0Dj59DCX6iydSds8480@IA04i(srp4I'd