(x + y) (x + y) 2006 = 0 (x + y + 1)(x + y 2006) = 0

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Download "(x + y) (x + y) 2006 = 0 (x + y + 1)(x + y 2006) = 0"

Σίλας Καλογιάννης
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2 F!#%$'&)(*+(-,/01-3 9N1pB,?q/&o pq('nydœ9m-p 0S9yDs`&d PS / e PQ Y b X ] Y b NP / aqf b QaQ B F X ] BD i/q[f g e E SR FPPQ l b F D N QFIVg g Q E b Ÿ N B B g / )N Γ 1 g Γ e# ] Ĥ V Q[FI Q b > SR N SR Γ 1 g E g egĥ ] ~V P[FŠ Q ] Q Q# Q[ kv N QF SR! ]N NSR Γ g BDS g N SR B = N?N(monBpq(;r1-3#kmX*+>&#œ9;3y!š d=9-r$'&[p u>9n? Q<((mly9;3,/>&[pBy% Sn-&N>3y+= (0-p` <0-poq(;&} - \ \ Bi bl ba g \ g Γ Γ 1 B E BF b FdEN S Q bb P b g ] gf N BD = BE = 90 EBD = ED D N [ bl Q b ~FŠ ]N BF b ]N DB S Q bb P b ; lf b F EBD EN = E D = BD b - EN = E = ED = EBD = BD = D ] lf~liv b BFˆ ]N g EN lf~fdli bb! E N = N B ] FŠ -S Q e Q Qa[P \ N g lf%fdli bb t BD ] Q ] R6 Q F NB ] g g D lf~fdli bb! D g = N = E B g = D E E D = B b F N E B = D = B B = 1 = N g N [ + [ P XªP«) ƒ d D ; Š «[ ±S²>³ ³ qµ)³ ; / P ) «o¹i«ÿº~ ) /±a»w ) q«[ /¼B Q ;½)¾Iª[¾Q S g Ê ]N

3 \-] ]N - R lf i b QBV b ] 10a + b SR igfp b F e i b QBV b a b SR?N(monBpq(;r1-3 ;&[p g9r ytw/pb,d -9&œ(mlmq> /p 9N 6r,/BpoQnN ydz%[ 9W9ykz {~} R BF lfdbe b Ĥ e ] [XQ ] Q Q-Y 10a N + bq Q Y> Y# ] e F 10a + b = F)'eFPQXQNQ n Y! lf n] BF e; b = n 10a g a b- a b = a (n 10a) = 1a 1n ] ] BF BF) e b Ĥ e N [ [ P aª)«) # d! ; Š «[ ±<²>³ ³ Š qµd³ Ÿ / P ) «o¹š«%º~ ) ±»W ) q«[ ¼l P Ÿ½P¾Šªo¾P t g dào¾p»w«[ẅèdê²/ oéš«[í³ Í¹P ÅŠ ÉI±aÆ/«[¾) [±+º#³ q [ É) B³ÅŠ± ¾Š±X«~Â Š³aª)«[ B 6³a±a Q k Š¾[Á-«/³ B¾P q «[¾ÉQ Dº- ÂÅ[ ;Àg N Á-«[ % Š¾) P Â>Ã6»ƒ¾) Ä B³aª[¾I±l«ÅŠ¼lÆ Š P 'Ç/¾d P [«[ B³ ÈIÉP µ«[ [ÀS¾) / o¾i¼bµ`å)¾p Š¹+ Š ) Q«) ) Š¹Q g½)¾šé) q«[ ÅP¾QÁ>³ µ ' qµ)³ ;¾) / D N d q«d«[ [± +¾P qæš³ Š¹I±l ) ¾P ½Š«)«Â Š³aª)«[ B 6³a±a Q %½Š«! )³ Š¹I±l )Í Q N ÂÅ[ <Ề ÅI«Pª)«[ B¾Q ` [ P XªP«[ B Á- P³ /±l«) ÉI± ± Æ¾I±ƒ±aÆ/«Á/ q Š ) X«[»Œ )³X / Š±' Á-«)¼X³ #a ³ ±X«o¹Š«[ tª[¾q léi«[ $#l ) ¾) / ¾) q«š±³ ±X«o¹Š«[ B PÊ ¾) / ¾) / qæ `«) #± Æ¾I±`± Æ«k [± a b ¾I±l«[»W«[ /±%# ¾P³ Ÿ³ # a b &j%$'&)(*+(-,/01-368:9;3<>=@?b9d ('; blb Y>a gieg[f# ]N Q FIe Q g [FI %)I g * e- \-] ]N ] Q- Y>a gieg[fk Q FI!e Q - [FI g )I g * ] ] ] /B ] l Y>aQF Q BY>X[F Q?N(monBpq(;r1;3wnpl,+'> 9w9,6&[9y P}, '}l?g}k (;&P&)>, -WnNu0g(-yD?-9r`B9-r0g/&Qy%œ9r`B91;&[pB9y?/*<9/p r`} > eĥ ] [FP Y>a ] gie b > a SR g egĥ ] FŠ Y>a ] Q ] b )Ng 0 < a < 10 g 0 b < 10 (10a + b) = 10b + a 0a + b = 0b + a ] gf blb gie[f ] Q ] FŠ ] FŠ gie[f Q )Ng 1 36 g 8 e- > e! ] [F) Y>a ] b > e# ]! a] B Y>a ] Q SR - 66a = 33b b = a 0 < a < 10 BY>X~BFx ] [F) QF gie b / eg! ] Ĥ FŠ Y>a b )Ng ] g 0 b < 10 (100a + 10b + ) = b + a 00a + 0b + = b + a 696a + 30b 393 = 0 3a + 10b = 131 b NQ F N g

4 F b R R ] R R R R R Y R Y Y N F) [ = a + 10k R6 \ ]N g Y# ] lfn Q ƒf i# N ]N PY mod 10S D\ 'efpqxqnq gg k R6 ]N lf -S Q Y> N - a (mod 10) 3a + 10b = 6a k 10b = 30a k b = 3a + 131k Q~ PY[FFd ] ]N ~igfp a g b ] 0 < a < 10 g 0 b < 10 k = 0 b = 3a g = a ] gf blb ] Q BY>X gieg[f a ] )d ab = a g Q FIe ] ] l Y>aQF QjQ [FŠ ] gieg[f Q b = 3a g % 13 6 g 396 N [ [ P XªP«) + d ` ; Š «[ ±/²>³ ³ qµ)³ / / P ) «o¹š«gº~ ) ±»W ) q«[ ¼l P /½)¾Iª[¾Q ' dài¾) / ¾)»W«[ 'È)ÊI²/ oéš«[í³ Í¹P ÅŠ ÉI±aÆ/«[¾) [±-º#³ q [ ÉP B³ÅI± ¾I±l«ƒÂÍ³XªP«[ B 6³X± P - Š¾QÁ-«/³ B¾) q «[¾ŠÉP º; 'ÂÅ[ <Ê j%$'&)(*+(-,/01-368:9;3<>=@?b9d \-] ]N BF Vg ƒfšs QĤ e FŠ - 11 (b) b > e; %>PPi! ] jfdi blb FP b ' ] lf Vg ƒfin Q SR b R6 3 (b)?n(monbpq(;r 1;3~ Š9= >, -}ƒ &nn>rnpbr yw?s('n6/9,w8 pl,i,>('n-&[pk?b9q:š`rnpqu/&o,dp 63y#œ9>*+ sp &[9-&)0g/9ny+8žztyWšŸ?>!} D [F) efi Ĥ ]N ] gf BF Vg!FIN Q 11 (b) = e (1 FŠ b ) + ( ' FI b) + 1 = (b + 1) lf bxb eg gfš# 11 ] (b) BY>X VV' [F ] gieg b > g - e- / [XQ ] lf F x = ] ( b )+(3 b)+ b +3b+ x = 0 g R6 b 9 b = 3 ± ( ( x ) ) = 3 ± 8x ] ] BiV b Fƒ ]N igfp e Vg kfšs Q ]N! ] blb FŠF 8x R6 - ] V' FIN Q x = 1 8x = 1 g b = 1 b = 1 ] V' ƒfšs Q x = 8x = g b = b = 1 ] Vg ƒfin Q x = 8x = 11 g b = b = ] Vg kfšs Q x = 11 8x = 961 g b = 1 b = \ g i gf) e PY a ] Fd ] Y>aQF BF b b > 3 g N SR ] FIi bxb F) V FdFdBe b b 'jbf ] BF b b F e Q e F)'eFPQXQNQ 3 b = ) b = b + 3b + N [ < [ P aª)«) + d S Á-«[ N Š¾) P NÂ>Ã6»ƒ¾) Ä B³Xªo¾Š±X«gÅI¼BÆ/ Š P Ç¾) ) [«[ B³ ;ÈdÉ) µš«[ >Ê ; «N³ ¼ ) q«)¼q±š [ P léi± ³a ) '¾) g q«)¼ «o³xªp«) /Ê

5 ] E F ~#$'&d(*+(-,>-01;3 &n P?g9 3;/&QyD8 >= (;&[p 9/mš`rNpqu/&o,dp 63 ( t{j o('nr<0-mb9-r0;ỳ?q}- /(;r,dy+{w 3 QBY ] jpq Y b ]N FFIN QF N B X[F!F) F j N F Q ] 'Y ] ] NP [F 3 ] FIWFŠS SR QF F+F ] ] FI j b F Qi PQ Y b ) SR %/QQi ] / ] -j BF b F QBY R6] DPQ Y b B B?N(monBpq(;r 1-3 w/pb,i9?= po6ny w9n-&[9 rgr 9;3/m (;&Qy9-r0!>&P&[p 8 (m 0g>rg9gnS/&Qy 9,INpBr N (;r@9r<0 ws- š`rnpqu/&o,dp 63y ws-pbr N (;rny +/ƒyžšÿ?>ƒyg= (0-p` p (;&} e b ] Y/ i FŸF ] [ Y ] ] -j B ] Q = Q β + δ g = α + ε R6 B + = (α + β) + δ + ε bxb Fk ]N R6 - N ] ]N XY X = \ g = 90 + δ + ε > 90 < 90 SR F) F 3 + (3)() os γ = 3 30 os γ γ = 180 β Y> X = os β = ( ) 3 = ] gf X = 13 E Y e ] B SR X θ α δ γ Y β 3 3 α β α F) β ε F XY - ]N ] ] Y> F gf Y = = + 3 (3) ( 13 ) os δ os δ = 3/ 13 ] sin δ = / 13 g ] g os B = os(β + δ) = os β os δ sin β sin δ ( ) ( ) ( ) ( ) 3 3 = = > 0 ] Q Q R6 N B < 90 F)Bi bl b bl Q X egxf ] ]N N -j ] gf ] Q < 90 Q R6 BF QBY ] DPQ Y b B g os = 3 3 > 0 g

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