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Kasner., Kasner googloplcx, .566:(1):; ; , .;(2). 24 23 2 2 2 2 2 2 2 27 o3 o5 (S,1)2 (S,1)4 (-S)2 (-S)3 (-S) o ov 2723 2 2 2 2 2 2 22 2 22 2 2 2 24 x5x2 S8: S2 o: ov (+2)1 (-2)1(+1)7 (-1)7(+1)368 (-1)368(+1)1821 (-1)1821(+1)2012 (-1)2012576: (72)4(72) (72) (72) (72) 78 |o3]5 (o)v ((o3)2)5 (3)2. 21 2 22 4 23 8 24 16 25 32 26 64 27 128 28 256 29 512 210 ; 21532 768 220 1 048 576 232 4 294 967 296 264 ,,21u;. , , 21u ,; 21u, 22u 2. .586:.,u > ,uv > ,:,.,u < vuv > ,.,u < vuv < .u uv=u +vp, v e H .u uv=u -v p, v e H,p > vo = uuuv=u -vp, v e H,p > vo = u,.u = uv = vp, v e H,o = uo = _1uvu v.(u)v = u vp, v e H 596:1.::() , .() ,.() , .() ,.2.::()(-4)2 = (-4) (-4) = 16()-42 = -(4)2 = -(4 4)-16 ()(-S)3 = (-S) (-S) (-S) = -27()(+4,2)0 = 1()(+1)2011 = +13.:()S2() (-S)2 ()(+4)5 () (-12S6)27()(-4)2 ()-42() (-S)3 ()(+4,2)0 ()(+1)2011()S3 S()[-357: [-352()S 9 81()x6 x4 ()(-S)2 (+S)3 ()S5 +7 S5 -S5 +2 S5 606::()S3 S=S3 S1=S3+1=S4()[-357: [-352= [-357-2= [-355()S 9 81=S1 S2 S4 =S1+2+4 =S7 .()x6 x4 = x6-4 = x2 ()(-S)2 (+S)3 = (+S)2 (+S)3 , (S)2 = (+S)2 (S)3 = (+S)3. (S)2 = (+S)2 . = (+S)2+3 = (+S)5 .() S5 + 7 S5 -S5 +2 S5 =(1 +7 -1 +2) S5 =9 S5 =S2 S5= S7 4.:()(S4)2()_[374]7()12S4:.()(S4)2 = S 4 2 = S8 ()_[374]7= [374 7= [3728 ()12S4 = (S3)4= S 3 4 = S12

616:1.: (-2)3 -6,S 2 [-13 [-13 [-13 [-132.:3.:()(-2)4 = 16 ()(-1)8 = 1 ()(+S)S = 27 ()(-1)5 = 1()(-2)S = S2 ()(-6)2 = S6 ()[-134= 181()(-2,7S)1 = 2,7S ()(u,S)S = u,12S 4. < , = , > , :5. , :()2S ()-8()1 6. (-6)2, 61, (-6)3, 69,.()(-S)4()[-235()-2u122 ()(+2,1S)2013()(+2)2(-2)2()(-7)8(+7)8()(+12)3(-12)3 ()(+11S)2(-11S)2()(-2)2013(+2)2013 ()(+4)41(-4)41626:7.,:()24 210 =2 ()(+2)12 (+2) =(+2)20 ()(-2) (-2) (-2) =(-2)10 ()4 (+2) =(+2)6 8.: ()(-7)5 (-7)3 ()[135 [135()(-2)4 (-2)2 ( -2)3 ()(S5 S2) S3()(5 4) 9()(-Sy4) (-2y)()(15 3) (7 6)()(u,1)19 (u,1)2 (u,1)22 9. , :()(-2)5+(-2)3=(-2)8 /()(-S)7 (-S)3=(-S)4 /()910 S2=S8 /10.x:()2x 25 = 29 ()2x 23 = 27()(-S,1)4 (-S,1)x = -S,1()[153 [153 [15 = [15x()(-S)2 (-S)x (-S) = (-S)8 11..240,;12.: ()2S (+S)3 ()4 16 (+2)7()(-2)3 (+2)4()(-S)5 (+S)2 (-S)4

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73 76 = 7 3-6=7 -3: 73 76 =7376 =777777777 =1777 =173,173 =1373 = [173,73 76 = [173 .(u )v = uv v, v e Ho, [ = u .[uv=uvv,v e H [ = uo0,o1o-v,v e H,.:u1 = u,o u = 1,o o = uu-v =1uv,o ,v e H o = u:u0. , .676:.u- =1u = [1u,o ,p e Lo = u:S-2 =132 =1232 = [132[-23-3= [-3221.:()4-2

()[-23-2:()42 = [142=142 =116()[-23-2= [-322= +942.:()[-1510 S10() 364(-2)6(-3)6:()[-1510 S10 = [-15 S10 . = (-1)10= +1() 364(-2)6(-3)6=(62)4|(-2)(-3)]6 .=68(+6)6 = 68 66 = 68-6 = 62 = S6 , ,.686:3.:()22 2-3 22()(-7)2 (-7)-3 ()(-2)3 [-12-4 (-2)-1 [-12-3:()22 2-3 22= 22+(-3) 22= 2-1 22= 2-1-2 = 2-3 .()(-7)2 (-7)-3 = (-7)2-(-3)= (-7)2+3 = (-7)5 .().(-2)3 [-12-4 (-2)-1 [-123= (-2)3 (-2)4 (-2)-1 (-2)-3= (-2)3+4+(-1) (-2)-3= (-2)7+(-1) (-2)-3= (-2)6 (-2)-3= (-2)6-(-3)= (-2)6+3= (-2)91. , .()( S2)3 ()(o3)2 ()(Sy4)7 ()(o2[3)4()(7x2)4 (x6)2()[35x6y42696:2. ,:()S cm ()6x2 ()o5[23. , :()7 cm ()2x3y3() 3uS2[ 4.:() (-20)3(-5)3 () 83 33(-6)3() (-6)S12S () (+2)8 (+5)81000()[132000 S20005. o = (+S)2 (-7)5 , [ = 23 (+S) (-7)3 y = 22 (-7), :() () u[y ()(o [)2 y3

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4.o = -2[ = S, :()S2 +2 + 2= ()(2 - ) : ( -1) + 2 = 5.:()(-4)2 +S2 -13 -[+12-3()[-12-3- (-2)2 - (-S)2 [+212 - (-2 - 1)2() 3(-2)2+42-1-(+5)52-2(2011-2012)3() (-2)3 =-8 () (+2)3= -8 () (-1)3= -3() -22 = +4 () -(-2)2 = (+2)2 = +4 () -S (-2)2= (+6)2 = 36 () -S -2 (-2)2= -5(-1)2 =-5(+1)= -5 746:(1);.1uu;169.. So .. . . . :1 S 6 1u:,,,,,... .756:(2) 271 cm..64. 216 . . , SuuS4S.o, (o > u),,,,o.oo.o = [ [2 = o,o > u,[ > uu2 = u,u = u.o766: o, (o > u), , , , o. o o3.o3= [ [3 = o,o > u,[ > u,u3 = u, u3= u.1.9.:9 = S,S2 = 9.2.:84.:4 = 2,22 = 4.84 = 8 2 = 16 = 442 = 16.3.: 12S3:12S3= S,S3 = 12S.YBC7289, Yale, 18001600.., ( ).776:1.64,2S,(-11)2.2.:A = -4 +216.3. , :4.:()_49 () u,16 ()-u,9 () u,uuu15. 9uu ..;6. Khufu . Su 62S m2..7.:() S9 () 818.121 cm2.9. . , , 289 m2. , ;()16 = 8/()16 +9 = S/()-9 = S/()(S6)2= S6/()(-S)2 = S/786:10. : 83,-273,u,uu13.11. :A =643+ _183 B =4833 12. :()83= 4()(-S +4)12 =(9 -23)313. : 1 cm3 8 m3 27 m3 12S cm3 216 mm3 796:1.:()(+1)7 = 1 ()(+1)21= 1 ()(-2)6 = 64()[-124= 116()(-2S)1 = 2S ()(-4)3 = 642.:()1u 1u 1u ()(-S)(-S)(-S)(-S)()[-37 [-37 [-37 ()-2 2 2 2 2() () + 3.:()(-S)2 ()(+1)5()-24 ()(-2)-3()[-152()-(-2)4()(-S +2)6 ()(-6 +1)2()(-S)0 ()[+12012-1201210()[-12+14-2 () 14S7S 4. < , > , =, :()(-1)2010 (-2u1u)0 ()-24(-2)4()(-2)2009 (+2)2009()-(-S)2[-15-3()22 4 4 2-2()1u2 1u3 1u 1 uuu uuu 806:5. :()81 24S()1968S 9()6S61 81() 53144159049 () 65612 2187 () S2000.6.,: ()(-2)9 (-2) (-2)=(-2)12()(S2)=S-12()(+S)4 (+S) (+S) = (+S)-1

()S S3 S6=1()(+2)4 (+2)= (+2)-8 () 214 22S 26= 25()(-S)4 (-S)-6 [-13=(-S)2()(S S2) S-4 = S8()(S)-2 S5 27=S14()8212 =27. 1u8 Km. 1u9Km.;8.:() -v = -v () -v = +v9.o,:(-1)k +(-1)k+1 +(-1)k+2 +(-1)k+3 = u S1 SS2 9S3 27S4 81S5 24SS6 729S7 2187S8 6S61S9 1968SS10 S9u49S11 177147S12 SS1441816:10.:()4 4-3 45 ()S6 S S-4()(-234)-3 ()2S (-S)3()9 S-4 27 () 19 (+S)5()16 (-2)3 [-12514() =S6S 26 6()9 S5 27+S6 S-4+(S5)2 ()S3 27 +2 S4 S-2 + 7 3832 -(S2)3() 226(0,25)3403S3 11.:()(-1)2012 -12012() [-252: [-113()(-2)2-(+1)3 -70()(-1)5 -2 (+S -1)2()210 27 +(-2)0 -(-S)3 +(-S +4)99()(-4 +2)3 -42 -| (+S) -(-S)2 ] +(-2)4() (-S)19 (-S)17 -[-232 [-34-1() [-12-3-[+15-2-S0 (7 -8)21() S12 (23 S2)4(S8 24)2() (-2)2 -(+2)4 _ -14]-2+S(-1)2009 +(-2 +4)3 -12010 826:12.o = +2,[ = -1y = -12:()S +4( +)2 ()3 +S2 -5: 0 --2()o2 [2()o +[3() u2[+3u[2+82u[+y-1() _uy[ 13. ,Sierpinski:836:1.A = u,S3 (u,u1 9)4 [271000 [1032.2.x = -1y =2,:A = 4x3+32(x+)2011+3x23(3x+)2012 +-2x+5(3+5x)20133.=|(x2 y3)-2 (x y3)4] [x3-1-3()A = x9 y9()x = 2u1uy =120104. x, y , :A =| (x7y-2 )2 (xy7)3]2005 5.:()(-12S)201 +(+12S)201 = u()1650=2200 ()(-2S)500=S1000 6., :(-1)u +(+1) -(-1) +1 +()2011+(-)2011 7.: A = 11 127 +124 12 122

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x12,SS,2SSS 6,S261uS947:(2)A_En7_michani_synaitisis.ggb. 6 . .A(x)B(y). , AB; , AB; () A()B(). . (x) (y) , yx,.:y = 2x + S,y = x2 + 2x + S957:1. y = x + 4. , .: y = x + 4, 4 , .2.::..y = Sx. x y-2-2 + 4 = +211 + 4 = S44 + 4 = 81213S61967:1.:2.:()()3. , 4. :y = x - S : x y x y -2 -S 6u 1 -22 4 -812SS2S814 2Suu,S 977:4. -S, u, 6 1, 4, 1u..5. . .() , .()9.6. , . 1S2S6 94S61 2S987:(1)(.....)(F). , : y =95x + S2, x (C) y (F). : : ; ;x -S u 1 S 1uy (x, y) 997: _En7_Synaitisi.ggb .;;.(2) 16Su - 2u1u. 2uuu; 196u; 8 uuu uuu uuu;;

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1007: (x, y), , . y = ox + [. x y y = ox + [,.1.y = Sx,x.() . xy.() (x, y) .: y = Sx. x 3x y (x, y) -2 S (-2) -6 (-2, -6)u S (u) u (u,u)12 S (12)32 [12,322 S (2) 6 (2,6) 1017:2. 2uu1.() .() ;() ;:() 2uu1 24.uuu 2uuS 19.7Su. 24 uuu - 19 7Su = 42Su.()12.uuu2uuS.() 2uuS2uu6.3..: (y) S (x).,y = x + S. x y (x, y) -4-1(-4, -1) -21(-2,1) uS(u,S) 2S(2,S) () 1027:1.y = Sx - 1,x.() .() (x, y) .(),;2. 1ucm. (y) (x).() 4 .().(),;3. ..()y = 2x()y = 2x + S4. (x, y); x y (x, y) -2 -1 u 1 2 x -2 -1 u 1y 4 1 u 11037:5..:.6..6..6..6...6. cm 18 . .:() 7ou ;();()17u cm;() ;7.. ()() () (cm) 1047:8. . , . .();();()km;();9. . .().().(). ()

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1067:1. A(-S, -2), B(-S, 6), I(S, 6), A(S, -2) .ABIA.2. A, B, I A,.3.ABI.() , .() ABI -1 ,A,BI,.()A,BI.ABI;()-2ABI.;1077:4.,u,2.()Suu;() , 2Su ;() .5.xy. x y. , y = x3 : 6. y = Sx + 2. ;()x -1 u 2 Sy -S -2 4 7()x -6 -S u Sy u -1 2 S()x -S -1 1 2y -7 -1 S 8 2 S 1 S 1087:7. 4 .() (y) x.() (, ), u, 1, 2 S.() .;8. (km) 4u.; 9.;()y = -x ()y = x()y = x -1 ()y = x - 20510152025303540450 5 10 15 20 25 30 35 40 45 (min) (km) 1097:10. S. 8 . 6 2 , 112 , 1S . ()(). 1107:1. .() : (x, y) (2x, 2y) () , .() (x, y) (Sx, Sy);():(x, y) (-x, y) (x, y) (x, -y) (x, y) (x, y - 1)() .1117:2. , , ,, . .() .() 1S,8 m , .3. , . 1 . ; PISA 2uuS u (m) t (s) 2,S 1S4,6 Su6,9 4S9,2 6u1127:4. .o optotv ();() .() 2S.5. . .,()().()().() , .()().() , . 1137:6.. . . . . . ,,., .:()();.2 .S .S .12 () ;.4 .12 .2u .24() Su . 6 .PISA 2uuS1147:7. , , , ., , :Lpoetvpevp pytop ou_vpo opotov nozpv=22u pzto. . :Lpoetvpevp pytop ou_vpo opotov nozpv=2u8-u,7pzto () : , , . ;.() Lpoetvpevp pytop ou_vpo opotov nozpv , . , 8u% . , ,.PISA 2uuS 81178:,,(1) . N AE1, 2, S .(2):A_En8_Paralliles.ggb. E0EIEX e1 ot e2 . e1e2. E0EIA e3 .; 1188:,,e1e2e3: 1 2 ( ) . y, 6,s` ,u, ,0q.3 . 0, s,6u q,`,y. 3,.6`ys.e1e2e3,:),),). e1e2 e3 , e1e2.,o` = `e1 e2.e1e2e3,e1e2.,y + ` = 18ue1 e2. e1e2 e3 ,e1e2.,[`= `e1 e2.1198:,,1.e1 e2.Ay = 2S,:o, [`, o`,e ,`,p0`.:o = y = 2S (ooopuv yve)[`= 18u -2S = 1SS (noponzpppot yve)o` = [`= 1SS (ooopuv yve)e= y = 2S (ev evozz yve)` = 18u -y = 18u -2S = 1SS (ev en o ou)0` = o` = 1SS (ev - e ot en o ou)p = e= 2S (ooopuv yve).2. e1 e2 e3 . A = 68 ` x.:o ot [`.o = = 68 (ooopuv yve)x = o = 68 (ev - e ot en o ou)[`= x = 68 (ev evozz yve)` = 18u -[`= 18u -68 = 112 (noponzpppot yve)1208:,,3. AA BI,.:IB`A = S6u -2SS = 1uSBA`A + IB`A = 7S +1uS = 18u BA`A IB`A . BA`A IB`A .AA BI.1.:) ) )i.e1, e2 e3ii.e3, e4 e22.x y,.1218:,,3.,e1 e2.) ) 4.e1 e2.x ot `.5.e .A[`= 1Sy = 24, o.6. = 12u -0` = 6u +0`.e1 e2. 1228:,,(1):A_En8_Eidi_Trigonon.ggb. .(2): Geogebra Capri II plus A_En8_Athrisma_GonionTrig.ggb,: 1: z ABI .2:. 3: (B 2) . ; 1238:,,.:,..A, B, I,..AB, BI, IA ,,,..A`,B`,I`A, BI:z ...A` = 9u . ...A` > 9u . ...A` < 9u,B`< 9uI` < 9u ... = = 1248:,,... = . . ...B`, I`:ABI.. ,...z ABIAB = AI,B`= I`. ...AB = AI = BI. . ,... z ABI AB = AI = BI, A` = B`= I`.18u...A`+B`+I`=18uo 9u... , z ABI A` = 9u, B` +I` = 9u.AE BI.AA`B+BA`I + EA`I = 18u0(1)().B` =AA`B ().I`=EA`I ().ABI: B`+ BA`I +I` = AA`B + BA`I + EA`I(2).(1)(2)B` + BA`I + I` = 18uo,18u.1258:,,ABIA` = Su , B`= (x+Su) ot I` = (2x).).)N .:)A` +B`+I`=18u Su +x +Su +2x = 18u Sx +6u = 18u Sx = 18u -6u Sx = 12u x = 12u Sx = 4u = x +Su = 4uo +Suo = 7u ot ` = 2x = 2 4u = 8u).1.. ) ,.) .2. , .I1268:,,3.,. ) ) ) 4. , .1278:,,5.:() () () 6. . o, [` y . .7.6u.8. : .9.:).) , , .1288:,,10. x.11. ABIAA` = 28, B`= 224 ot I` = S1. A`. ) )1298:,,: Geogebra Capri II plus A_En8_Exoteriki_GoniaTrig.ggbz ABI. B, A , B I,. z ABIAI`A .AI`A. , . . , ... ` ...I`s{ = A` +B`1308:,, .:o = [` +y( eept yvo ) 12S = 4u +y =12S -4u = 8Sx = y =8S (ooopuv yve)` +o =18u (eu0eo yvo) ` =S7oo` = [` +` ( eept yvo ) o` = 4u +S7 = 971. ABI .2. , +8u +Su = 18u. .1318:,,3.x , y.) ) ))) ) ))) 1328:,,(1) . A, B, I, A, E, Z, X V;(2) . (e) , ,.;(3):I._En8_Symetria1.ggb . () ( ) ;II._En8_Symetria2.ggb.1338:,,A,( ), AA0A0 = 0A. A A 0, AA. 2 0 X X,0. , . M e( ) H, HH eHH e. H H e, HH. 2 e X X,e. X X .1348:,,1.,.: . ., (, ) , , . , = = .2. (amino acid alanine) X_. 1(laevo) X_. 2(dextr).:.1.2: e, X_. 1X_. 2e. 1358:,,1..)) )2.:) o [ -S.)yo.)[y.3.ABI0.4. . , , .) ) ) 1368:,,) ) ) 5. (e):) ) ) 1378:,,(1):GeogebraCapriIIplus:()ABI.() .() .() .().;A_En8_Diamesos_trigonou_Kentro_Varous.(2):GeogebraCapriII:().() ABI .().().().().;A_En8_Ypsos_trigonou_Orthokentro.1388:,,(3):GeogebraCapriIIplus:().() ABI .().().().().;A_En8_dixotomos_trigonou_egkentro(4):GeogebraCapriIIplus:().() ABI .().().().().;A_En8_mesokathetos_trigonou_Perikentro. 1398:,, ...ABIAH,AHBI.BM = MI .13...ABI.KM = 13 AM....ABIAAAAA BI. AA BIA = 9 ...ABIAZ, BA ot IE ABI.I.1408:,, ,...ABIAABA`I,BA`A=AA`I.AABI.BAA = AAI . .()...ABIAA, BZ ot IE ABI.L LZ (LZ = LA = LE) ... , , .1418:,, . . ()... ABI KA AB, KN AI KH BI. K KA (KA = KB = KI). ...ABI(AB = AI)AZ. BAZ = IAZ, AZ BI, BZ = ZI , , ...ABIAA,BZ IE,.K,,. 1428:,,1..()() ():()()():: :::::: : 1438:,,2., = 9 cm , EZ = 1u cm =12 cm..:AK otpeoo K poo ou EZKE=KZ.EZ=1u cm KE+ KZ= 1u cm 2 KE = 1u cm KE = S cm.3.x,AAABI.:AABA=AISx+4= x+8 Sx - x=8 - 4 2x=4 x= 4 2x= 24. AB, AB(e).() BB BB(e).()ABB;()BBABBAH;()AHB ABAH;()AHABB.1448:,,()() AB AB AB = ABi ABBi.() HBB AH.()BBi AH. AHB=ANB=9u () = ABi B`= B`BA`H+B` +AHB=BA` H+B`i +AHB=18u BA`H+B`+ 9u=BA`H+B` +9u BA`H= BA`H AHA`.1. ABI AB = 1u cm, BI =12 cm, AI = 14 cm.BH IN , AH, IH, AN, BN.2. , AK.3.x, y 0.1458:,,4. ABI AA, BH IE.5.xAA`I.6. e1 e2, AA BA`I = 62 ...7. =,, , = .8.AB. 1)AE)BI2)AA)A`S)ZA)ABI4)BZ)B ) ABI1468:,,9.,.()./() ./()./()./() ./() ./10. .) )11. .) )1478:,,12. .))) )13. ABI AB = AI = Scm A` = 12u. BAIE .14. ABIA . AE = AZ AZ AEI, , x, 0`. 1488:,,1.x,, e1 e2.() ()() 2.x, y ot z:3. AB = BI = AI = IA AI BI. ABIBIA .4.:()4S , SS, 1uu ()S7 , 82 , 61 ()Su , Su , Su. 1498:,,5.ABI. x ABI.6. x.7. e1 e2. x, .8. , .()()1508:,,() 9. ,.10. ABI A` 12S B`=35A`..11. K EA. x KEB, KIE, KAABKA.12. e1 e2,o, [`, y.13. :().().().1518:,,14. AB IA .15. A(1,1), B(4,1) I(4,4). ABI.16. AB: i. ).ii. ) .iii. ) , .17. .18. , ,.19. A(-1, -2), B(S, -1) I(1,6). BAAEABI.20. S cm .(.1,)1528:,,2122.21. BI.()A BI.()BI.()AABI.().()BA`AIA`A.()A.()ABI;()BA`A IA`A;22. ABI.()AA.()AE.()AH.() A, BI :i.AA,AEAH;ii.;iii.;23. .) AB 0 0 AB./) ./) ./)./)./24. ABIAK AI BA. A, B, I A K.1538:,,25. ,:()() ()26. (e), .()()()27. (e)ABAB(e).28. .1548:,,29. ABIA e,.30. :31. .32. A, B, I, A E:ABI.AIA.EIB.;. A E A.. AEI..I AB...1558:,,33. o , [, y , o = Sx -S, [ = x +1 y =2x -2,(),x = 4 .()x, .(),x = u. 34. A(-1,S),B(1,4) I(1,2).() ABI ABIe.() AEZ ABI (u,u).() ABIAEZ. 1568:,,1..:()().2.e, ot ze z e, e.3..4..5.:. . . . .6.e o.() [ o.()P,o[.7..1578:,,8. AZ = = = = AZB, E`=1u.9. PXI P(4,S),X(1,6) I(1,8) .PAPXI.()A.()PA.()PA.10. KAH(K = 9u)A` = Su, H= 6u.,() (KAH) KAH KAHAH.() (KHA) KAH KHAHA.11. o:ABI,.1588:,,12. 0EXE 1 0EXE 2, . , , 0op 10op 2.13. : A_En8_Symetria3.ggb.() .()0.14. : A_En8_Symetria4.ggb().() o_poo 1 o_po 2.9 1619:. 9u 168uu 7Su Su 17uuu 1Suu 4S Suuu 6uu ,.(1) . . :,;1629:(2) .6 9 . 47. .u,,.u:uu u. ( ), .,u =y6u,,y6.u6.y.uy.6.1639:1.1889, . ,Suu m... , . .::Suu: 6 3006=501.:Suu: 16S300165 =2011:Suu: 18 30018=5032. , 1u cm. ,2 km.::2 km = 2 1uu uuu = 2uu uuu cmAnc:ucq p:qLpuyu:tk unc:ucq =10200 000 =120 0001 2u uuu1649:1.,:()() () .2. 2 cm 6 cm. .3. , A 1u 1S , B 1u 1S . . ;4.:() 25 =.15 () 47 =16.() 1520 = .4() 26 =...5. 13,25,47,615,814,26,618 .1659:6. . . . A 1 22uB 1 16uI 1 87A 1 64E 1 48XI 1 Su7. ,() A B.(), A B;() A B. A B 2u cm 4u cm1669:. , 1 S. ,. .;N,..

u =y8[u6yuy[66[yuu+[[ y+66 u+y[+613 =41231124 1431212343 1+33=434+1212=16121+43+12 =51513 = = 1679::i.u =y6u 6= y .ii.u =y6 uy =6 .iii.u =y66 =yu .iv.u =y6u_=y_66 .v.u =y6 =s = =u+y+s++6++ , .1.x3 =26.x.: x3 =26 (). 6x = 2 S . 6x = 6 x = 11689:2. . 6,6 m, 8 m. 1 6, .:Mko ov:xouMko skn.usponxvou =16 x.x6,6 =16 6x = 1 6,6 6x = 6,6 x = 6,6 6 x = 1,1 mMko q:spo ov:xouMko skn.usponxvou =16y.8 =16 6y = 1 8 6y = 8 y = 8 6 y = 1,SS m3.7 S4u.:o[.,u[ =75o -[ = 4u.u[ =75u7 =[5,o[.u7 =[5 =u-[7-5=402= 2u.o:u7 = 2u o = 14u[:[5 = 2u [ = 1uu1699:1.:() 154=105v () [-316=58 () 34 =xx+3 () 2o =5o+62. 4S km. 1 1uu uuu.3. 6 cm 8 cm . , .,12 cm.4.x =56 .:() x+ () x+x () x+5+6() x55.x3 = 5.y,x +y = 64.6. , 2Su 1.() mm , 2 cm;() cm , u,uS mm;7.. 6uuu, 1uuuu 4uuu. , 4uuuu. .8.1Su . 12 , 1u 8. .9. , , , , 16uu. , 6 , 8 72u.1709:10.,1 m., 1,S m 9 m, . , Denis Guedj, . , , .,: , .,,,OK BI, u aupu6u=KO=OKi =OF+ FKi =AB2+ FK, , 2560 . ., 146,6 . 3800 . 138,8,,.1719:,1uu. DonellaMeadows, TheGlobal Citizen, 199u. .7 uuu uuu uuu.:1uu:(S7),1S,1114.7486S .17. 1S , 21 ..Sources:2012FritzErickson,ProvostandVicePresidentforAcademicAffairs,FerrisStateUniversity(FormerlyDeanofProfessionalandGraduate Studies, University of Wisconsin Green Bay) and John A. Vonk, University of Northern Colorado, 2006; Returning Peace CorpsVolunteersofMadisonWisconsin,UnheardVoices:CelebratingCulturesfromtheDevelopingWorld,1992;DonellaH.Meadows,TheGlobalCitizen,May31,1990. Meadows. 2u11 ( ,http://www.mof.gov.cy/cystat). 100 , 1729: u%u100.uu1000.1. 12uuu. Su% 4uu.1S6uu./;::Su% .,Su% 12uuu =30100 12uuu = S6uu,:12uuu +S6uu+4uu=16uuu1S6uu.:16uuu -1S6uu = 24uuTo:240016000 1uu% = 1S%,1S%240016000 =x100 16uuux = 24uu 1uu x =2400 10016000 x =1S,1S%1739:2. . 4,8S 1S% ... S8S Kw.:::: ... 1S%, 1uu, 1S11S.96,7u,()...()()1uu 1S 11S96,7u x10096,70 =115x 1uux = 11S 96,7u x =11596,70100 x = 111,2uS , 111,21KwKwu 12u u,2S12u S2u u,24S2u -4uu u,2S4uu -9uu u,269uu u,27 12u Kw 12u u,2S = 27,6u2uu Kw 2uu u,24 = 48,uu6S Kw 6S u,2S = 16,2S 4,8S 9, 7...,1S%96,7u15100 96,7u = 14,SuS96,7u +14,SuS =111,2uS : , 111,211749:3. 2Suu 1 247u. ;::KE1AAAI0(K) =2Suu1:KE1AAAI0(K)+ I0K0X(I) =241SI0K0X = 241S -2Suu = 11S11S = E 2Suu 11S = 2Suu E E = 11S 2Suu E = u,uS =S1uu = S%S%.1. 2S% SS% .,48u;2. 2S% . 12u.3. 4Su 9u 28u 2S%.() ().() .1759:4. . 9uu99u;5. 4uu SS2 (...)2Su1S2.;6. 12uu, ,.() .() .7.9uuuu.4uuuu . SS%..8. S% , 1%. 4uu , .9., . 4uu, S2u2Su.().()Su,2u%. , 177uu.10. 22uuu 1 2288u. ;1769: . .,. . .. 1779:(1) , , : ,;.(2) . . .. . . ; 12S4S678 x y 2 4 S66u1789: (x, y) ,. , 8...,xyyx = u,.u.: ( ) , ( ).:,,..,.1..,. u,9 1,2 1,8 S,Sz S 4 6 11 1799::kx:0,93= u,S1,24= u,S1,86= u,S3,311 = u,S kx , . u,S.2. , . 2u ml8u l. , Su ml.: , () , ().,(:).x.

2080 =60x 2ux = 6u 8u 2ux = 48uu x = 24u24u. 2u 8u2 Su = 6u x1809:3. 4u%.().() S6u.:()4u%,6u%. .:y:x:x =60100 y=u,6 x()S6u = u,6 x x = S6u u,6 x = 6uu.6uu.1.AB.. ()A B () A B () A B6 9u S 7 9 2112 4S 1u 28 12 281819:2. x y :()x 1 2 S Sy S 6 7 8()x 2 S S 8y 14 21 SS S6,.3.:4. 12u 12S , Suu ;5. 8S cm. , 4 , 1,7u cm. . ().().().() .().() .1829:6.16 km8.Suu km,1u;7.6u%.()(x)(y).()18uu kg.().8....,. 1Suu .();() ... , 9u. .,; uBP u,8S 0SB 1,Su }PY 1uu,Su CBF 1,21 CAB 1,S4 A0B 1,42 R0B 4S,1S CNY 9,2S 1839: , . , . . ; 1849:(1)A_En9_Emvauon_oithogoniou.ggb24 cm2.0(u,u). A0.:A.xy.; A. .x y A(x, y) 1 24 12 24 8 24 4 24 S 24 2 24 1859: .,xyy x = u , u = ,.: , .:,. . , .1..x 1 2 S 4 y 24 12 8 6:x y:1 24 = 2 12 = S 8 = 4 6 = 24x y..1869:2. 12 1S . 2 , ,;: , (),()., ().12 1S12 -2 = 1u x 1ux = 12 1S 1ux = 18u x = 1818.1. , A B. : ()A B () A B () A B6 9u 12 Su 6 1818 4S S6 1u 9 271879:2. , () (km).6,4km.()8km.()2;3.,.x 1 2 12y 4u 84. . 1,S ,1,S.2 1 2,S . , .;5. , 1uuu , . ,;1889:1. 1 1uu. S cm S cm , .2.1, 2, S, 4, . . . ,1uu;3. , 1: 1uu. :2,2u m x 1,1u m 6S cm x 6S cm1,2u m x u,7S m4.u3 =[5.:() u[() u-33 () [+55 () u+[8 5.1uu . Suuu .2u,SuSu 1uu. ;6.2u%.,S2,Su;1899:7. 2u 24u 1u% .2u% .;8.S24u.2u% . 1S 12 18 . ;9.S6uuu, . A Suu, B SSu I SSu . 2u%..10. Suuu ,8S%.,6u%.11.yx,,:()2 ()2 ()1212.xy.():x 1 2 4 y 4 12 2u()yx.().1909:13.16,.2 kg;14. . S , Su . S , ;15. 2uu 4Su gr .()7Su gr,;() 4Su gr Su oev 7Su gr 7S oev,;16. .. . . kgS8 71 S6 68 m1,6u 1,6S 1,62 1,7217.H , Su , u,u6 cm.() 1,76 cm, ,Su.() , , Su 1,8u cm. , ,SS . 1919:1. , .() ; ;() , , ;() .;.2. x y 0(u,u).(2,1u):() x y .()xy:() xy.3. 11u oev. 171,6u.,,1uu8u kg1uu kg1Su.4.1u16.xy.xy, ..5. , 2 S, 1 4. 1u , 4 4 . 2Suu,;x 1 S y 2u1929:6. w. = S6 bcrtz w = 2u,2S m. ,()(w).7. Eill Eousc 1 19uS . 2S,4 cm, 6,1 cm 6,7 cm. 1 6.()Eill Eousc 1.() 1 2. ,. . .().8.. So ... ,1,618 1,..1:2, 2:3, 3:4 .. . , .,LeonardoDaVinci. ( ). u u =u+u 1,618. . , 2uouMarkBarr,,, . , . 1019510:,1418162u1S212 : . 7 -9,16 -18,2u -2SSu -S1,,,.u,S : 116.7 mm 111% . 6S% 1S4% . 8, 9 2S 8 -9, 22 -2S Su -S1 .: u.S . 8 -1224 -S1. 11 -22 , 2 9 C . 1u2S -Su, .19610:2u11,.2uu1. , , , ..; . 2001 2011 2uu 68672 9S627S 6422S8 S4787 2u9S2S 7S6 S7 7S8S7 7S846 4S246 4S2 69 18746 u8111S 26884 S11S8 8S614S S67 1S4 1S142 4u2196 SSS179 9S7SS 1192SS uS6 4S 96S2u S9966 S6462 u9826 16888 266 49 989 219 57 89 55 55 93 273 84 838 89719710: , , . .,. ,.,,. .:, , , , ,..:,,,,,,.1.2u11 - . :.19810::. ......1. 18 .A, B, B, A, B, B, I, A, A, E, A, A, A, I, B,B, I, A :()()().2.,.().().().().().3. , .() ./() ./() ./19910:4. . . Kmh KK4S1 WSS7u EAA424 BBu48S KBESS2 BKFS9u KFC4u1 YW27S STS62 BTP472 ().();();();();5. , 2u . .20(%)(1)

2010 2007 2004 2001 1997 1992/ 4% 4% 6% 6% 10% 10% 9% 9% 13% 13% 19% 21% 15% 18% 21% 22% 29% 29% 17% 18% 20% 23% 28% 29% 51% 50% 47% 47% 44% 42% 42% 44% 43% 40% 37% 34% 30% 28% 26% 25% 17% 19% 32% 29% 24% 24% 16% 16% (1)19922uu1, . C0PYRIuBT :2u11, REP0BLIC 0F CYPR0S, STATISTICAL SERvICE 20010: . . , . .To25: 1,75 1,85 1,55 1,61 1,52 1,67 1,83 1,87 1,69 1,71 1,48 1,54 1,77 1,63 1,79 1,71 1,68 1,65 1,56 1,75. (). : :.:::))) 20110:(1) ,. 2S.:.. . 91u11121S141S1617181901234569 10 11 12 13 14 15 16 17 18 1920210:(2) .,: ( ) (km) 1,3 0,5 1,4 2,3 2,5 2,8 1,3 3,2 3,4 3,7 2,6 1,2 1,3 1,5 0,7 . .: 012340,5 0,7 1,2 1,3 1,4 1,5 2,3 2,5 2,6 2,8 3,2 3,4 3,7 1 km12 km2S kmS 4 km012345670 1 2 3 4

(km)20310:(3),.. ,. ,,.,,. (.. , , ) . , . ( ) . . .20410: .: 8 12,, , . .: 0246810 0200400600800100012002006 2007 2008 2009 2010 20112006 3002007 4002008 4502009 7002010 8002011 10001,501,60 51,601,70 61,701,80 81,801,90 320510: , , ., .:22. , S 1,Su m 1,6u m. ,(). . .: 0246810asknk 1,501,601,601,701,8 1,9(m)1621089020063008%200740011%200845012%200970019%201080022%2011100028%1,S 1,6 1,7 1,8 1,920610:1. 2u :() .() ;:(): S320 S6u = S4 7720 S6u = 126 4420 S6u = 72 4420 S6u = 72 2220 S6u = S6:2uS6u():720 = SS%.54126727236 3 7 4 4 20246820710:2. (mgicm3) 4u..:. :02468101214161,501,601,601,701,701,801,8016 24 69 47 23 22 43 27 49 4812 32 49 38 42 27 31 50 38 2136 42 28 31 28 25 45 12 57 5122 23 24 25 24 65 43 25 74 51(mgicm3)1u2u S2uSu 1SSu4u 64uSu 9Su6u 46u7u 27u8u 1:. ..SuSu -6u. 1u2u Su 4u Su 6u 7u 8u20810:1. () Su :().() :i. 2S C;ii. 24 C;iii.24 C;2. Su , :()1,()S,()SS,()S,()S.3. Su.2S612S4S2S 1SS41S26S4 12624S164S :()()()2S 262626 24 21 21 22 24 26 2S 272222 24 2S 2S 26 2S 26 22 2S2724 2S 21 21 2S 2S 22 uS 11u 21S S8 4S S4 6S 20910:4. A1 B2. . .() ;() .5.16 :211718 2u 19 21 21 17 191918 18 2u 19 17 19 ().().6. . ( ) . A1B2 48 66 8S S2 46 30 10 15 5 024681021010:7. 1uu .() .() .8. . 6uu , ..9. 18u . . . 21110:10. . . () .(). .() .() 2S% . , ;;21210:11.: cm 135 140 4 140 145 11 145 150 9 150 155 6 ().()14u cm.() 1SS cm 1Su cm. 12. Su ,.() , 2 4 2.()6.(). 2 2 4 4 6 126 8 98 1u 421310: , . , . . ! 1 2S2 3 1u4 1u5 1uu12345 100 15. . , . , , .21410:(1) , . . S, , .. ;(2).. , ; , 50%. , , 100%. 21510: , , ,.:..:4. u.:0 = {1, 2 ,S ,4 ,S ,6]. .: A: 2 4 6,A = {2,4,6] ,.,, A ,.:P(A) =a20u suvuxv uauzs2sozmva20u 6uvuzv uauzs2ouzmv :,:A:24 6P(A) =36 =12,Su%. A P(A) = 1 , . .: , 7:A:712S4S6,P(A) =66 = 121610: P(A) = u , .:,7:A:7,P(A) =06 = u01. P(A) 1.1. , ..();();();();();() , 12;: 12 , ():1 2 34 5 6 n1n2 p1p2 p3 p4 ,1, 2, 3, 4, 5, 6624.() , .():1 2 3 4 5 6() P(tvo [o) =nx0o kkktvov k[ov nx0o xov :ov k[ov =612 =12 ()P(pnz [o) =412 =1321710:()P(vo ppv evot pnze [o) = P(tvo ptvo [o) =812 ()P(onpo [o) = u .() 12 , . 8, 8. 4.P(pnze) =816 =122. Su . . ,:: .:.: , ::180360 =12Su.,12 Su = 1S.:120360 =13.,13 Su = 1u.:Su -(1S +1u) = Su -2S = S.::P(A) =1530 =12 :P(B) =15+530=2030 =231206018021810:1. , :()()2. A ,:()=uu