http://www.mathematica.gr. http://www.mathematica.gr. LaTEX: , -: , : . LaTEX. . Leonardo da Vinci (32-) . 30 . - - - quasiregular , ( -). (0, 0, ),

12, 2 , 1+2

, 1+52 .:http://en.wikipedia.org/wiki/Icosidodecahedron: http://www.mathematica.gr . http://www.mathematica.gr 1. (Mihalis_Lambrou) 2. (nsmavrogiannis) 3. ( ) 4. ( ) 5. (m.papagrigorakis) 6. ( ) 7. () 1. (grigkost) 2. (cretanman) 1. (stranton)2. ( )3. (vittasko)4. ()5. (s.kap)6. (nkatsipis)7. ( )8. (chris_gatos)9. (matha)10. (mathxl)11. (gbaloglou)12. (R BORIS)13. ( )14. ( )15. (dement)16. (swsto)17. (achilleas)18. ( )19. (xr.tsif )20. (Demetres)1. (spyros)2. (p_gianno)3. (kostas.zig)4. (exdx)5. ( )6. (mathnder)7. (mathematica)8. ( )9. (rek2)10. (hsiodos)11. ( )12. (bilstef ) 1 ( ) . 120 , ... , 70 . :36 ,35 ,40 42 . ; 2 ( ) 2. - . x , x x. 1089... 572. 752 257 x = 752257 =297, 792 + 297 = 1089 361. 631 136 x =631 136 = 495, 594 + 495 = 1089 3 ( ) . 72 . , , 50 ; 4 ( KARKAR) , , 300 , . 20 , 24 30. ; 5 ( KARKAR) - , ( 2011) 12. 3 24, 2 .1) ;2) , ; 6 ( ) - 9 12cm 1cm. . 12cm , . 7 ( ) . 8 ( KARKAR) a > b > 0 x = a ba +b, y = a2b2a2+b2) x, y) () x. , 9 ( ) -[x y[ [x[x = 1[2x y[ +[x +y 1[ +[x y[ +y 1 = 0 10 ( ) 3_19 3_29 + 3_49 = 3_32 1 , 11 ( KARKAR) M -, ABC, (K, R) -, AB, AC Z, H -.1 ZK, HK BC T, S. : ZH = ST. 12 ( ) B - 30 45 . M B, MA. , 13 ( ) (an) (bn) ( ) a1 = b1, a2 = b2, bn > an n 3. 14 ( KARKAR ) :(x4x + 1)x< 1 , 15 ( ) - ABC AB = AC = 1. AB K, L, M AK = KL = LM = MB AC P, Q AP = PQ = QC. D CK, PM, E CK, QL Z PM, QL DEZ. 16 ( KARKAR) S , ABC, ST, SP, CA CB . PT AS. , 17 ( ) a,

b, c [a[ = 1,

b = 2, [c[ = 3. 2a+3

b+c ,=

0. 18 ( ) C : x2a2 + y22 = 1 E(, 0). K C, (EK) + (E) = a, K , . , 19 ( ) - f, g : R R f(x) = (a2+b2)x22ax g(x) = 4bx + 5.i) limx1f(x) + limx1g(x) = 0 a = 1 b = 2.ii) a, b limx35f(x) 35g(x) + 13x 8.iii) a, b f

(35)ln2 = g

(2012). 20 ( ) - k , c. fi .. c k.. c, k.. f1 = 25, fi2 , , 21 ( ) z2 z + = 0 , R,z C z1, z2 z1 = 2 +ii. , Rii. z20081 + z20082 R. A(z1), B(z2), (z3) z1, z2 z3 z3 = z1z2+ 15(17 +i) :iii. AB .iv. [w z1[ = [ w z1[ w R.v. w [w z2[ + [ w z2[ = 10, . 22 ( ) - z = x+yi x, y R : (1i)z = 42i, R) : (x +y) + (x +y)i = 4 2i) (x, y) z .) (x, y) O(0, 0); , ,, 23 ( ) f : R R x R :e1f1(x)f1(x) = x + 11. f.2. :. f .. f, f1 .3. :i. limx_f1(x) +x_ii. limxf1(x)x . 24 ( ) - f, g , f2(x) + g2(x) = 1 x .) f4(x) + g4(x) = 1 + , - 0, + ,= 0 f8(x)3 + g8(x)3 = 1( +)3.) [f(x) +g(x)[ _2+2. , , 25 ( ) : f, g :[0, 1] R 1) f(1) = g(0) + 12) f(x) g(x), x [0, 1]3) f

(sin2x) +g

(cos2x) = 2, x R : f = g 26 ( ) f : R (0, +) - f

(x) = 3 ln f (x) x R f (1) = 1. , , 27 ( ) f : [a, b] R a +b2_ baf(x)dx _ex1x_xln(x+1) x (0, +) 32 ( ) f : R R < < f (f () +b) +f (f (b) +) =f () f

(b) +f

() f (b) +f () +f (b) f , . 33 ( ) AB AA1, BB1, 1. BA1 + B1 + A1 = A1 + B1A + 1B, . 34 ( ) a R b, c R - 4ac < (b 1)2 f : R R f(ax2+bx +c) = af2(x) +bf(x) +c, x R. f(f(x)) = x .3 Juniors, - - 35 ( ) , f(x) = [x] + [2x] +_5x3_+ [3x] + [4x], x [0, 100]. 36 ( ) 0, 1, 2, ..., 100 . -, A, 50 + 50 ( .) a. B + + A , b. a + b 2011, . Juniors, 37 ( ) ABCD. AC, BD O. OAB, OBC, OCD, ODA r1, r2, r3, r4 E ABCD. :r21 +r22 +r23 +r24 E. ; 38 ( ) ABC, A =90o, AB = AC, D BC. - DE AB, DZ AC H BZ CE. DH EZ. Seniors, - - 39 ( ) - f : Z Z f(x y +f(y)) = f(x) +f(y) x, y Z. 40 ( ) x, y, z > 0 xyz = 1, :1(x + 1)2+y2+ 1+ 1(y + 1)2+z2+ 1+ 1(z + 1)2+x2+ 1

12 Seniors, 41 ( ) ABC M . A

BC AM, B

AC BM, C

AB CM E1 =(BA

M), E

1 = (CA

M) E2 = (CB

M), E

2 = (B

AM) E3 = (AC

M), E

3 = (C

BM). 1E1+ 1E2+ 1E3= 1E

1+ 1E

2+ 1E

3 42 ( ) .ABC, (1) O (2) .AOC. OQ (2) M, N AQ, AC , AMBN . MN, BQ (2). 43 ( ) :9x+ 10y= 12z+ 1 44 ( ) (n) (n) n, n (n)(n) n + 12 ; 45 ( IMC 2011/B3)

n=1ln_1 + 1n_ln_1 + 12n_ln_1 + 12n+1_ 46 ( IMC 2011/A4) A1, A2, ..., An, , . :f(t) =n

k=1

1i1 0.4 51 ( ) A = A, ) . A = B + C, B C . ( A, A) = B, B) + C, C) x A y (x B y_x C y_(x B y C)) ). 52 ( ) X X . . 53 ( Atemlos) , , . 54 ( ) a, b, c R3. x + (x a)b = c. 55 ( ) - an =24n + 1, n = 1, 2, . . . 2 3. 56 ( ) , - .( 8373= 512 343 = 169 = 132 13 = 22+32) () 57 ( ) , sin 1 + sin 3 + sin 5 + + sin 99 = sin250sin 1 58 ( ) 3a, a E1, E2. E : E = 29(E1 + 9E2). , 59 ( pla.pa.s ) de l Hospital f(x)g(x) . - ; f(x)g(x) x0( ), ( limxx0f(x)g(x)) - de l Hospital, f, g ( x0) limxx0f(x)g(x) ., ; 60 ( ) f : R R . R; , 61 ( ) [. , , ( ,1976)] P K E Z . 62 ( ) . , 63 ( ) P(x) . P(1) = 6 P(7) = 3438, x = 3. 64 ( ) P(x) = xn+a1xn1+ +anQ(x) = xn+b1xn1+ +bn C. P x1, x2, ..., xn, Q x21, x22, ..., x2n. a1 +a3 +a5 + R a2 +a4 +a6 + R, , b1 +b2 + bn R.5: 1 ( ) -. 120 , ... , 70 . :36 ,35 ,40 42 . ;http://www.mathematica.gr/forum/viewtopic.php?p=55933 1 ( ) 36 + 35 = 71 > 120 70 = 50 21 2 . :40 + 42 = 82 > 50 32 - . 32 + 21 = 53 > 50 3 4 2 ( ) 120 70 = 50 , 42 . 50 42 = 8 , 40 -, 40 8 = 32 . 32. , 50 35 = 15 , 32 15 = 17 32, 17 . 17. 50 36 = 14 17 14 = 3 17. , , 3 . 3 ( )36 + 35 + 40 + 42 = 153 50 3 , 150. 3 . 2 ( ) - 2. . x , x x. 1089...6 572. 752 257 x = 752 257 = 297, 792 + 297 = 1089 361. 631 136 x = 631 136 = 495, 594 + 495 = 1089http://www.mathematica.gr/forum/viewtopic.php?p=84884 ( ) xyz zyx 99(x z) 99 a99 a 2 8. a99 .99a = 90a + 9a = 100a 10a + 9a = 100a a =100a 100 + 100 a =(a 1)100 + 9 10 + (10 a) :(10 a)100 + 9 10 + (a 1) - 1089.7: 3 ( ) . 72 . -, , 50 ;http://www.mathematica.gr/forum/viewtopic.php?p=87192 ( ) 72 .., 72 : 2 = 36.. , 6 . 62= 36. 6 . 2 6 = 12 ., = 2 12 + 2 6 = 36 , , 3650 = 1800 . 4 ( KARKAR) , , 300 , . 20 , 24 30. ;http://www.mathematica.gr/forum/viewtopic.php?p=85871 ( ) - , 300 , -120, 124, 130 a, b, c , :a120= b124= c130=a +b +c120 + 124 + 130=30015120= 2400,a120= 2400 a = 120b124= 2400 b = 100c130= 2400 c = 80: - .8: 5 ( KARKAR) - , ( 2011) 12. 3 24, 2 .1) ;2) , ;http://www.mathematica.gr/forum/viewtopic.php?p=72306 (KARKAR) ( ), - 12 24 . (24-) 2 5 . , 12 ( = 720 ). 720 : 5 = 144 . 144 3 = 432 (= 7 12 ), 144 2 = 288 (= 4 48 ). 12 ( ), 7.12 (12+7.12, 12-4.48) , 144 2011, (31+28+31+30+24) , 24 ., , 12 720 : 3 = 240 . , 12 720 : 2 = 360 . ( 240, 360) 720, - , -, 720 2011, 365+355 - 2012 - 11 , 20 2012! 6 ( ) 9 12cm 1cm. . 12cm, .http://www.mathematica.gr/forum/viewtopic.php?p=85516 ( ) 4 , ( ) 1 +_12_2=52 cm. , 16 52 + 1 + 2 12 = 85 + 25 cm9: 7 ( ) .http://www.mathematica.gr/forum/viewtopic.php?p=60737 1 ( ) AB AB//. B - A, B AE, Z . ABE Z. - (AEB = Z = 90) ABE = Z . : AE = Z ( - ) , AB =, . - . 2 ( ) . ABD, BCD BD. ( ) - . 8 ( KARKAR) a > b > 0 x = a ba +b, y = a2b2a2+b2) x, y) () x.http://www.mathematica.gr/forum/viewtopic.php?p=66599 ( ) ) :x = a ba +b = a2b2(a +b)2 = a2b2a2+ 2ab +b2, x < y, (a2 b2), (a2+ 2ab +b2> a2+b2).)y x < x y < 2x a2b2a2+b2 < 2 a2b2a2+ 2ab +b2 1a2+b2 < 2a2+ 2ab +b2 a2+ 2ab +b2< 2a2+ 2b2 a22ab +b2> 0 (a b)2> 0, , .10: 9 ( ) [x y[ [x[x = 1[2x y[ +[x +y 1[ +[x y[ +y 1 = 0http://www.mathematica.gr/forum/viewtopic.php?p=94586 1 ( ) (x, y) - ( x ,= 0). 0 = [2x y[ +[x +y 1[ +[x y[ +y 1 [2x y (x y)[ +[x +y 1[ +y 1= [x[ +[x +y 1[ +y 1 [x +y 1[ +x +y 1 0 . (2x y)(x y) 0, x > 0, x +y 1 0. (1) x > 0, [x y[ = 0, x = y., (1) x 12. (a, a) a _0, 12_ . 2 ( ) x 0 - , [x y[ = 0 x = y. : x +[2x 1[ +x 1 = 0 [2x 1[ = (2x 1) 2x 1 0 x 12 x < 0 [x y[ + 1 = 1 [x y[ = 2. . : (x, y) = (a, a), 0 < a 12. 10 ( ) 3_19 3_29 + 3_49 = 3_32 1http://www.mathematica.gr/forum/viewtopic.php?p=94028 ( ) a = 3_23 b = 3_13. 3_19 3_29 + 3_49 = a2ab +b2= a3+b3a +b=23 + 133_23 + 3_13=3332 + 13_32 1 =3__32 1_ _32 + 1_332 + 1=3__32 1_ _3 + 3 34 + 3 32_32 + 1=3332 + 13__32 1__34 + 32 + 1_=3332 + 132 1 =3332 + 1.11: 11 ( KARKAR) M , ABC, (K, R) , AB, AC Z, H . ZK, HK BC T, S. : ZH = ST.http://www.mathematica.gr/forum/viewtopic.php?p=83176 1 ( ) ABC (AB = AC), AK (-) BC A. B, C AC, AB ABC , : ABo= AC AC .= AB. CB//BCAABCAACB : (1). ZC.= ZM CKM KH M KB. - KMH, KAT, KM = KA = = M KH ooo= M KB2= BCMo . (1)= AKT, , MH = AT : (2). - ZKM, SAK , ZM =AS : (3). , (2) (3) : ZH = ST. 2 ( ) D, N, E - (K, R) AB, BC, CA . N BC : BD = BN =CN = CE : AD = AE. ADBD = AECE DE//BC (1). F KZDM G KHME. - KDZM, KMHE , F, G - DM, ME MDE: FG//DE(1)// BC. F GK = x 12TSK = x ( ), FMK = x ( - KFMG, ZFM = MGK = 90 ) F ZM = x( FZM ZMF = 90 x). KMZ, KNS ( - ), KZ = KS. KZH, KST , ZH = ST. 12 ( ) - B 30 45 . M B, MA.http://www.mathematica.gr/forum/viewtopic.php?p=84016 1 ( ) BA . A = 45 ( AB) A = 45 ( B B = 30 AB = 15). A , M ( M) AM . :45 +x = 180 302 x = 30. 2 ( ) M MB, E BA. BEC = 90, MEB MEC . ACM = 12AEM ACM - E EA, M AC = 12MEC = 12 60 = 30.13: 13 ( ) (an) (bn) ( ) a1 = b1, a2 = b2, bn > an n 3.http://www.mathematica.gr/forum/viewtopic.php?f=21&t=17827 1 ( ) , - . , -. . . : = a2 a1, = b2b1= a2a1 = a1( 1) :bn = b1.n1, an = a1 + (n 1) bn > an n 3 a1.n1> a1 + (n 1)a1( 1) n1> 1 + (n 1)( 1) ( a1 > 0) Bernoulli :n1= [1 + ( 1)]n1 1 + (n 1)( 1)( 1 > 1, > 0 ). n 1 = 1 , n = 2 n 3 . 2 ( ) p, q () x, y . :x = p + (n 1)(q p)y = p _qp_n1= qn1pn2 : y > x n 3. :y > x qn1pn2 > p + (n 1)(q p) qn1> pn1+ (n 1)(q p)pn2(qn1pn1) (n 1)(q p)pn2> 0 (qp)(qn2+qn3p+...+pn2)(n1)(qp)pn2> 0 (qp)(qn2+qn3p+...+pn2pn2pn2...pn2) > 0 (qp)[(qn2pn2)+p(qn3pn3)+...+pn3(qp)] > 0 (qp)2[(qn3+...+pn3)+...+p(qn4+...+pn4)+...+pn3) > 0 . 14 ( KARKAR ) :(x4x + 1)x< 1http://www.mathematica.gr/forum/viewtopic.php?f=21&t=17040 ( ) :x4x + 1 = x4x2+x2x + 1 =x4x2+ 14 +x2x + 14 + 12 == (x2 12)2+ (x 12)2+ 12 > 0 (x4x + 1)x< 1 :xln(x4x + 1) < 0, (1) x = 0 (1) x > 0 (1) : ln(x4x+1) < 0 x4x+1 1 x4> x x3< 1 x < 1. x < 0 x ( , 0)(0, 1).14: 15 ( ) - ABC AB = AC = 1. AB K, L, M AK = KL = LM =MB AC P, Q AP = PQ = QC. D CK, PM, E CK, QL Z PM, QL - DEZ.http://mathematica.gr/forum/viewtopic.php?p=95051 1 ( ) - - , 12 (... 3, 4). - A(0, 0) K (3, 0) , L(6, 0) , M (9, 0) , B(12, 0) , P (0, 4) ,Q(0, 8) , C (0, 12). ABC AK = KL = LM = MB, AP = PQ = QC CK y = 4x + 12, PM : y =49x + 4 QL : y = 43x + 8 , - :D_94, 3_, E_32, 6_, Z_92, 2_ DE =_34, 3_, DZ =_94, 1_ (DEZ) = 12 [6[ = 3 : (DEZ) =3122 = 148 2 ( ) AMP KDC:AKKM MDDP PCCA = 1 12 MDDP 23 = 1 MD =3DP (1). AMP - L, Z, Q:ALLM MZZP PQQA = 1 21 MZZP 12 = 1 MZ =ZP (2) (1) :MD = 3PD MZ +ZD = 3PD (2) ZP + ZD =3PD ZP PD +ZD = 2PD 2ZD = 2PD ZD = PD (3) D ZP. AKC PDM QEL :DE = KD (4) (3) (4) KZEP . EZ - ALQ :AL = 12, AQ = 23 LQ = 56 a = 41015 / : = a2 = 210 :E(DEZ) = 14 E (KZEP) = 14 (EZ) == 14 LQ2a2 = 14 5/62 210 = 148 .. 16 ( KARKAR) S , ABC, ST, SP, CA CB . PT AS.http://www.mathematica.gr/forum/viewtopic.php?f=22&t=18701&p 1 ( ) SK AB T, P, K ( Simson). ATSK AS, TK M. 2 ( ) CABS: BAS = BCS = x (1), - CTPS_CTS = CPS = 90_: (BCS) = P CS =PTS(1)= x TS, AB: TSA = BAS(1)= x. MTS, MTA ( ) MT , M AS. 3 ( ) : SP (ABC)_ AM = MSSP = PF__ PM | FA PMS = FAM =2BAM = 2MST = AMT. T, M, P -.16: 17 ( ) a,b, c [a[ = 1,b = 2, [c[ = 3. 2a + 3b +c ,=0.http://www.mathematica.gr/forum/viewtopic.php?p=93638 1 ( )2a + 3b +c =0 3b = c 2a 3b = [c 2a[ [c[ +[2a[ 6 2 + 3, . 2 (parmenides51) :2a + 3b +c =0 3b +c = 2a (3b +c)2= (2a)29b2+ 6bc +c2= 4a29[b[2+ 6[b[[c[(b, c) +[c[2= 4[a[29 22+ 6 2 3(b, c) + 32= 4 1236 + 36(b, c) + 9 = 4 (b, c) = 4136 1 (b, c) 12a + 3b +c ,=0 18 ( ) C : x2a2 + y22 = 1 E(, 0). K C, (EK) + (E) = a, K , .http://www.mathematica.gr/forum/viewtopic.php?p=70751 ( ) K (xk, yk) , (xl, yl). :EK = a xkE = a xlEK +E = a (xk +xl) = a xk +xl = a2 xm = a22 M x = a22.17: 19 ( ) f, g : R R f(x) = (a2+b2)x22ax g(x) = 4bx + 5.i) limx1f(x) + limx1g(x) = 0 a = 1 b = 2.ii) a, b limx35f(x) 35g(x) + 13x 8.iii) a, b f(35)ln2 = g(2012).http://www.mathematica.gr/forum/viewtopic.php?p=90309 ( )i. 0 = limx1f(x)+ limx1g(x) = limx1[(a2+b2)x22ax] + limx1[4bx + 5] = a2+ b2 2a + 4b + 5.:a2+ b2 2a + 4b + 5 = 0 (a 1)2+(b + 2)2= 0 a = 1 b = 2.ii. : limx35f(x) 35g(x) + 13x 8 = limx355x22x + 355x 3 =limx35(5x 3)_x + 15_5x 3 = limx1_x + 15_= 45.iii. f(x) = 10x 20 g(x) = 8f(35) = 8 g(2012) = 8 f(35)ln2 = g(2012) ln2 = 1 =e = 1e 20 ( ) k , c. fi .. c k.. c, k.. f1 = 25, fihttp://www.mathematica.gr/forum/viewtopic.php?p=90978 ( ). 5, 45 - . x1, x2, ..., xk. , c2 +ck + c2 = 45 5 c = 40k+1.. , , c. , b = 12 + k + 12 = k + 1. 100, :E = 100 12 50b =100 1250(k+1) = 100 k = 3 c = 10.. k = 3, f2 = 50 f3 = 100 (25 + 50) = 25 .18: 21 ( ) z2 z + = 0 , R,z C z1, z2 z1 = 2 +ii. , Rii. z20081 + z20082 R. A(z1), B(z2), (z3) z1, z2 z3 - z3 = z1z2+ 15(17 +i):iii. AB .iv. [w z1[ = [ w z1[ w R.v. w [w z2[ + [ w z2[ = 10, .http://www.mathematica.gr/forum/viewtopic.php?p=27051 ( )i. z2 = 2i Vietaz1+z2 = = 4 z1 z2 = = 5.ii. z20081 +z20082 = z20081 +z12008= 2Re(z1) R.iii. z3 = 4 + i, A(2, 1), B(2, 1), (4, 1). :AB = A = 2 ABA, .iv. [w z1[2= [ w z1[2 (z1 z1)( w w) = 0 w = w w R.v. : [w z2[ +[ w z1[ =10 [w z2[ +[w z1[ = 10, [z1 z2[ =[2 +i (2 i)[ = [2i[ = 2 10 -. 22 ( ) z = x + yi x, y R :(1 i)z = 4 2i, R) : (x +y) + (x +y)i = 4 2i) (x, y) z .) (x, y) O(0, 0);http://www.mathematica.gr/forum/viewtopic.php?p=86918 ( ) :(1 i)z = 4 2i (1 i)(x +yi) = 4 2i (x +y) + (x +y)i = 4 2i x +y = 4, x +y = 2, R (I)) (I) : (x+y) +(x+y)i =4 2i) (I) x, y :(x, y) =_DxD , DyD_=_2+42+1, 4242+1_, R. :(x 2)2+ (y + 1)2==_2 + 42+ 1 2_2+_4 22+ 1 + 1_2==_22+ 2 + 22+ 1_2+_2+ 4 12+ 1_2= ....... = 5 (x, y) z - (2, -1) 5.) (x, y) = (0, 0). (1 i)z = 4 2i 0 = 4 2i, . (x, y) O(0, 0).19: 23 ( ) - f : R R x R :e1f1(x)f1(x) = x + 11. f.2. :. f .. f, f1 .3. :i. limx_f1(x) +x_ii. limxf1(x)x .http://www.mathematica.gr/forum/viewtopic.php?p=57607 ( ) e1f1(x)f1(x) = x + 1 (1) x R. f, f1 ( ) R.1. (1) x f(x) , x R : e1f1(f(x)) f1(f(x)) = f(x) + 1 f(x) = e1xx 1, x R2. ) x1, x2 Rx1 < x2 x1 < x2 x1 >x2 x1 1 > x2 1(), , x1 < x2 x1 > x2 1 x1 > 1 x2 e1x1> e1x2(). (), () :e1x1 x1 1 > e1x2 x2 1 f(x1) > f(x2) f .) f, f1: f(x) = x (2) f(x) x =0 e1x 2x 1 = 0(3), x R g(x) =e1x 2x 1 ( ) [0, 1] g(0)g(1) = (e 1)(2) < 0. Bolzano (3) (2) (0, 1). g ( f x , ) 1 1 (2) (0, 1), f y = x M(, ), (0, 1). M f1. Cf, Cf1 M ( y = x) : Cf, Cf1 A(x1, y1), x1 ,= y1( y = x). :f(x1) = y1 f1(x1) = y1 f(y1) = x1, :f(x1) f(y1) = y1 x1 e1x1 x1 1 (e1y1y1 1) = y1 x1 e1x1 e1y1= 0 e1x1=e1y1 x1 = y1 .3. i) (1) f1(x) = e1f1(x) x 1 f1(x) >x 1 limx(x 1) = +limxf1(x) = +(1) f1(x) +x = e1f1(x)1limx_f1(x) +x_= limx_e1f1(x)1_u=f1(x)= limu+e1u1 = 0 1 = 1ii) limxf1(x)x =limxf1(x) +x xx = limxf1(x) +xx 1 =limx_f1(x) +x_ limx1x 1 = (1) 0 1 = 1. 24 ( ) - f, g , f2(x) +g2(x) = 1 x .) f4(x) + g4(x) = 1 + , - 0, + ,= 0 f8(x)3 + g8(x)3 = 1( +)3.) [f(x) +g(x)[ _2+2.20http://www.mathematica.gr/forum/viewtopic.php?p=18988 1 ( ) ( ) - , : ( +) f4(x)+( +) g4(x) = f2(x)+g2(x) :f4(x) +2f4(x) +g4(x) +2g4(x) =f2(x) +g2(x), :2f4(x) +2g4(x) =f2(x)_1 f2(x)_+g2(x)_1 g2(x)_, :2f4(x) +2g4(x) = 2f2(x) g2(x) _f2(x) ag2(x)_2= 0 f2(x) = g2(x) :f2(x) +g2(x) = 1 :f2(x) = +, g2(x) = + :f8(x)3 + g8(x)3 =_f2(x)_43 +_g2(x)_43 = . . . = +( +)4 = 1( +)3 2 ( ) ( ) Lagrange:2+2= (f2(x) +g2(x))(2+2) =(f(x) +g(x))2+(g(x) f(x))2 (f(x) +g(x))2 _2+2 [f(x) +g(x)[ 3 ( ) ( ) Cauchy-Schwarz [f(x) +g(x)[ _2+2_f2(x) +g2(x)=_2+2: f2(x) +g2(x) = 121: 25 ( ) : f, g : [0, 1] R 1) f(1) = g(0) + 12) f(x) g(x), x [0, 1]3) f(sin2x) +g(cos2x) = 2, x R : f = ghttp://www.mathematica.gr/forum/viewtopic.php?p=89293 ( )f(sin2x) +g(cos2x) = 2 sin 2xf(sin2x) + sin 2xg(cos2x) = 2 sin 2x _f_sin2x_g_cos2x_ = (cos 2x) f_sin2x_g_cos2x_= c cos 2x x = 2 1 = c + 1 c = 0. f_sin2x_g_1 sin2x_= cos 2x = 2sin2x 1, x R sin2x = u [0, 1] f (u) g (1 u) = 2u 1 (1) u = 1 u - f (1 u) g (u) = 1 2u (2). f () > g (). x = (1), (2) f () g (1 ) = 2 1f (1 ) g () = 1 2 f () g () +f (1 ) g (1 ) = 0 . u [0, 1] f (u) = g (u) f (u) f (1 u) = 2u 1 f (u) u [f (1 u) 1 +u] = 0 h(u) = f (u) u, u [0, 1] [0, 1] M m. m h(u) M, u [0, 1]. u = 1 u m h(1 u) M M h(1 u) m mM h(u) h(1 u) M m mM 0 M m m = M h h(u) = c f (u) u = c f (u) = u +c u = 0 f (0) = c. f (u) = u +f (0), . 26 ( ) f : R (0, +) f (x) =3 ln f (x) x R f (1) = 1.http://www.mathematica.gr/forum/viewtopic.php?p=94525 1 ( ) ln t t 1 f(x) 3(f(x) 1) x, (1). g(x) = e3x(f(x) 1), x R. g(x) = e3x(f(x) + 3f(x) 3). , (1), g(x) 0 x R. g - R. , x (1, +) g(x) g(1) = 0, f(x) 1, .. f(x) 0. f - (1, +). f(x) f(1) = 1 - x (1, +). , f(x) = 1 - x (1, +). (, 1) f(x) = 1 x (, 1). f(1) = 1, , (-) , f(x) = 1. , - f(x) = a ln f(x) f(1) = 1 a < 0. 2 ( ) .: t ln t t + 1 0 t > 0. t = 1 - . , .22 (f(x))2= 3f(x) ln f(x). 0 _ x1(f(t))2dt =_ x13f(t) ln f(t)dt= 3_f(t) ln f(t)_x1+ 3_ x1f(t)f(t)f(t)dt= 3f(x) ln f(x) + 3f(x) 3= 3(f(x) ln f(x) f(x) + 1 0 x 1. ( ). , , f(x) = 1 x 1. x < 1; 3 ( )(ln f (x)) + 3f (x) ln f (x) = 0 (ln f (x))ex13f(t)dt+ex13f(t)dtln f (x) = 0 (ln f (x)) + 3f (x) ln f (x) = 0 (ln f (x))ex13f(t)dt+ex13f(t)dtln f (x) = 0 ln f (x) ex13f(t)dt= 0 ln f (x) ex13f(t)dt= c x = 1 c = 0. -ln f (x) ex13f(t)dt= 0 ln f (x) = 0 f (x) = 1 4 ( ) ln(y(1)) = 0. - x. x0 : lny(x0)) = 1 1 y -. ln(y(x)) ,= 0x (1, x0) [1]. . Rolle (1, x0) : y() = 0 3ln(y()) = 0 [1]. x > 1 ylny = 3 _ x1dylny = 3(x 1) [2]. - [2] x 1. (x1)1/lnu = 3(x1) 1 < u < x x < u < 1. - x 1 u 1 1/0+ = 3 lnf(x) = 0 f(x) = 1, x > 0.23: 27 ( ) f : [a, b] R a +b2_ baf(x)dx 0 f(x) < 0 [1, 2] x0 (0, 1) f(x0) = f(2) f(1)2 1 = 1 < 0 f(x) < 0 f R 1-1 f(1) = 0 1 f R.) f R x < 1 f(x) > f(1) = 0 f (, 1] x > 1 f(x) < f(1) = 0 f [1, +) x0 = 1 f(1).) f(2 + f([iu + 1[)) < 0 f(2 +f([iu + 1[)) < f(1) f R 2 + f([iu + 1[)) > 1 f([iu + 1[)) > 1 f([iu + 1[)) > f(2) [iu + 1[ < 2 [i(u i)[ < 2 [u i[ < 2 u K(0, 1) 2.) z A(z1) 1 = f(1) w - B(z2) 2 = f(2). f _1, 32_,_32, 2_ f(1) + f(2) < 2f_32_ 1 + 2 < AB , max [z w[ = AB + 1 + 2 = 2f_32_+ f(0) + f(1)min [z w[ = AB12 = 2f(32) f(0) f(1).25: 31 ( ) - _ xln (x + 1)_exx1>_ex1x_xln(x+1) x (0, +)http://www.mathematica.gr/forum/viewtopic.php?p=83034 ( ) f(t) = ln t - ln(x + 1) x, x 0 , x > 0 - [ln(x + 1), x] [ln(x + 1), x] f() = f(x) f(ln(x + 1))x ln(x + 1) 1 = f(x) f(ln(x + 1))x ln(x + 1) 0 < ln(x + 1) < < x 1x < 1 < 1ln(x + 1) 1x < f(x) f(ln(x + 1))x ln(x + 1) < 1ln(x + 1) (1) x ex 1 , x > 0 - [x, ex 1] (x, ex1) f() = f(ex1) f(x)ex1 x 1 = f(ex1) f(x)ex1 x x < < ex1 1ex1 < 1 < 1x1ex1 < f(ex1) f(x)ex1 x < 1x (2) (1) (2) f(ex1) f(x)ex1 x < f(x) f(ln(x + 1))x ln(x + 1)(x ln(x + 1)) ln ex1x < (exx 1) ln xln(x + 1) . 32 ( ) f : R R < < f (f () +b) +f (f (b) +) =f () f (b) +f () f (b) +f () +f (b) f , -.http://www.mathematica.gr/forum/viewtopic.php?p=83034 ( ) (). , . (a, f(a)) :f(x) f(a)(x a) +f(a), x R x = f(b) +a :f(f(b) +a) f(a)(f(b) +a a) +f(a) f(f(b) +a) f(a)f(b) +f(a) : (1) (b, f(b)) :f(x) f(b)(x b) +f(b), x R x = f(a) +b :f(f(a) +b) f(b)(f(a) +b b) +f(b) f(f(a) +b) f(b)f(a) +f(b) : (2) (1) (2) :f(f(b)+a)+f(f(a)+b) f(a)f(b)+f(a)+f(b)f(a)+f(b) , - : f(a) +b = b f(b) +a = a ( - ). : f(a) = 0 f(b) = 0 Rolle - (a, b) f() = 0. - ( ) . [a, ] f(x) < 0 - . : a < f(a) > f() f() < 026: 33 ( ) AB AA1, BB1, 1. BA1 + B1 + A1 =A1 + B1A + 1B, .http://www.mathematica.gr/forum/viewtopic.php?p=86990 ( ) a, b, c B, A, AB . :1) BA1 + B1 +A1 = A1 +B1 +AB1 ccosB+acos+bcosA = bcos+acosB+ccosA ca2+c2b22ac +aa2+b2c22ab +bc2+b2a22cb= ba2+b2c22ab +ac2+a2b22ac +cb2+c2a22bc bc(a2+c2b2)+ac(a2+b2c2)+ab(c2+b2a2) == bc(a2+b2c2)+ab(c2+a2b2)+ac(b2+c2a2) bc(c2b2) +ac(a2c2) +ab(b2a2) = 0 ... ()(c b)(c a)(b a)(a + b + c) = 0 a = b b =c c = a, . - -. 34 ( ) a R b, c R 4ac < (b 1)2 f : R R f(ax2+bx+c) = af2(x)+bf(x)+c, x R. f(f(x)) = x .http://www.mathematica.gr/forum/viewtopic.php?p=75995 (Eukleidis) ax2+ bx +c = x (1) ax2+(b 1)x +c = 0, (b 1)24ac, , (1) x1, x2. x = x1 f(ax21 +bx1 +c) = af2(x1) +bf(x1) +c f(x1) = af2(x1) +bf(x1) +c af2(x1) + (b 1)f(x1) +c = 0 f(x1) (1). (1) x x2 f(x2) (1). x1, x2 = f(x1), f(x2), . f(xi) = x1, i = 1, 2 . f(x1) = x2 f(x2) = x1 ) f(f(xi)) = xi, i = 1, 2, . ) f(f(x2)) = f(x1) = x2, , .27 , , : 35 ( ) , f(x) = [x] + [2x] +_5x3_+ [3x] + [4x], x [0, 100].http://www.mathematica.gr/forum/viewtopic.php?f=109&t=18361 ( ) g (x) = [x] + [2x] + [3x] + [4x] f (x) = g (x) +_5x3_ g , x R g (x + 1) = g (x) + 10. x [0, 1), g - 0, 14, 13, 12, 23, 34, g ([0, 1)) _g (0) , g_14_, g_13_, g_12_, g_23_, g_34__g ([0, 1)) = 0, 1, 2, 4, 5, 6, g 6 - [0, 1). f , x R f (x + 3) = f (x) + 35. x [0, 3), f g 4 35, 65, 95, 125 . , - f 3 6 + 4 = 22 [0, 3). , f 33 22 = 726 - [0, 99). 36 ( ) 0, 1, 2, ..., 100 . , A, 50+ 50 ( .) a. - B + + A , b. - a + b 2011, .http://www.mathematica.gr/forum/viewtopic.php?f=109&t=15584&p=88219#p88219 ( ) . , 51 . . - . +. a - , 1, 3, . . . , 99 + A. x++ - +, x+ + x+ x. a x++ . b + , b x . 2x+++x++x+ = 50 - 50 +. 2x +x+ +x+ =50 . x++ = x. a b a + b . a + b 1 .28: 37 ( ) ABCD. AC, BD O. - OAB, OBC, OCD, ODA r1, r2, r3, r4 E ABCD. - : r21 + r22 + r23 + r24 E. ;http://www.mathematica.gr/forum/viewtopic.php?p=89586 ( ) - 2r1 AB 2r2 BC 2r3 CD 2r4 DA.4(r21+r22+r23+r24) (AB)2+(BC)2+(CD)2+(DA)2= 4(r21 + r22 + r23 + r24) (AB)2+ (BC)22 +(BC)2+ (CD)22 + (CD)2+ (DA)22 + (DA)2+ (AB)22= 4(r21+r22+r23+r24) (AB)(BC)+(BC)(CD)+(CD)(DA) + (DA)(AB), (1). , (1) = 4(r21 + r22 + r23 + r24) 2[(ABC) +(BCD) + (CDA) + (DAB)] = 4E ABCD . 38 ( ) ABC, A = 90o, AB = AC, D BC. DE AB, DZ AC H BZCE. DH EZ.http://www.mathematica.gr/forum/viewtopic.php?f=110&t=17545 1 ( ) DS - DEZ K DZ CE L DE BZ. ZCD, DEK KZKD = ZCDE = ZDDE, (1) ZC = ZD., DZL, EBL LDLE = ZDBE = ZDDE, (2) DEZ SEDE =DSZD, (3) SZZD = DSDE, (4) (3), (4), = SESZ ZDDE = DEZD = SESZ =(DE)2(ZD)2, (5) (1), (2), (5), = KZKD LDLE SESZ = 1, (6) (6), Ceva, - DS EKZL H DH EZ. 2 ( ) M, - ABMC K CE MZ L BZ ME.29 CMZ, , (MZ)2= (MC)2+(CZ)2, (1), BME (ME)2= (MB)2+ (BE)2, (2) (1), (2) = (MZ)2 (ME)2= (DZ)2(DE)2, (3) MC = MB DZ = CZ DE = BE. (3) MD EZ, (4) ABZ, BME, AZB = BEM - AELZ ., ZL ME, (5) EK MZ, (6). MEZ, (4), (5), (6), - MDEKZL H DH EZ.30, , : 39 ( ) f : Z Z f(x y +f(y)) = f(x) +f(y) x, y Z.http://www.mathematica.gr/forum/viewtopic.php?f=111&t=18966 ( ) . f . , x = y, f(f(x)) =2f(x), f . y 2y f(y) = a ,= 0. x = a f(a) = 0. y = a , f(x a) = f(x), f , . f f(x) = 2x x Z f 0.: . 40 ( ) x, y, z > 0 xyz = 1, :1(x + 1)2+y2+ 1+ 1(y + 1)2+z2+ 1+ 1(z + 1)2+x2+ 1 12http://www.mathematica.gr/forum/viewtopic.php?f=111&t=17554 ( ) 1(x + 1)2+y2+ 1 12xy + 2x + 2 cyc1xy +x + 1 1. x = ab, y = bc, z = ca 1xy +x + 1 = bcab +ac +bc cyc1xy +x + 1 = 1.31: 41 ( ) - ABC M . A BC AM, B AC BM, CAB CM E1 = (BAM), E1 = (CAM) E2 =(CBM), E2 = (BAM) E3 = (ACM), E3 =(CBM). 1E1+ 1E2+ 1E3= 1E1+ 1E2+ 1E3http://www.mathematica.gr/forum/viewtopic.php?p=92583 ( ) Ceva , . ABC AA BB CC M, Ceva, ABACBCBACACB =1 E1E1 E2E2 E3E3= 1 E1 E2 E3 = E1 E2 E3 : MCMC = E3E1 +E1= E3E2 +E2E2 E3 +E2 E3= E1 E3 +E1 E3 , (1) MBMB = E2E3 +E3= E2E1 +E1E1 E2 +E1 E2 = E2 E3 +E3 E2 , (2) MAMA = E1E2 +E2= E1E3 +E3E3 E1 +E1 E3 = E1 E2 +E2 E1 , (3) (1), (2), (3) E1 E2+E2 E3+E3 E1 = E1 E2+E2 E3+E3 E1 E1E2+E2E3+E3E1E1E2E3= E1E2+E2E3+E3E1E1E2E3 1E1+ 1E2+ 1E3= 1E1+ 1E2+ 1E3 42 ( ) - ABC, (1) O (2) AOC. OQ (2) M, N - AQ, AC , AMBN . MN, BQ (2).http://www.mathematica.gr/forum/viewtopic.php?p=96244 ( ) X MN BQ Y , AOC, BQ Y X. Z AC BQ K, H AC, AOC. AQB - MN, BXXQ = MAMQ , (1) AZ | MB = MAMQ = BZBQ , (2) BHZ, QY O, - HZOY = BHY Q = (HZ)(Y Q) = (OY )(BH) , (3)32 OBY, KBH - BOY = BKH, BQ - ABC.( BKH = C + KBC = C + ZBA =BCE = BOY , E (1) BQ ). OYKH = BYBH = (OY )(BH) =(BY )(KH) , (4) (3), (4) = (HZ)(Y Q) = (BY )(KH) =BYY Q = HZKH = BZBQ , (5) (1), (2), (5), = BYY Q = BXXQ , (6) (6) X, Y , BQ ( B 90o), ( B 90o).33: 43 ( ) :9x+ 10y= 12z+ 1http://www.mathematica.gr/forum/viewtopic.php?p=71915 1 ( ) lift the exponent ( ). x, y z. z > x z > y. 9x+ 10y= 12 12z1+ 1 >2 12z1, . : x z. 3z[[10y1. 32[[101 3z2[[y z 4 3z2> 3z. . y z. 22z[[9x1. 24[[9212 22z4[[x z 4 22z4> 2z. . - . x = y = z = 3 ( Ramanujan) - . 2 ( ) mod5: 1 mod 5. x . mod16: y 4 z 2 9x 1 mod 16 x . y < 4 z < 2. z = 1 9x+10y= 13 . y < 4. y = 1 9x+ 9 = 12z 9 27 z = 2 - . y = 2 9x+ 99 = 12z z = 2 . , y = 3 9x+999 = 12z. x = 1 . x 2 27 81 . z = 3 x = 3 . .: p , a, b a b mod p n . r, s, t pr, ps, pt ab, n anbn. r+s = t. p = 2, a, b a b mod 2 r, s, t 2r, 2s, 2t a2 b2, n an bnr +s = t + 1. 44 ( ) (n) (n) n, n (n)(n) n + 12 ;http://www.mathematica.gr/forum/viewtopic.php?p=91457 ( ) - (n) = 1, n = 1 . (n) > 1: n , (n) : 1 = d0 < d1 < d2 < ... < d(n) = n n. i N did(n)i = n - :(n)(n) = 1(n)(n)i=0di (n)_(n)i=0di= (n)_(n)/2i=0did(n)i=(n)_n(n)2 =n - , . n = 1. n - , n = d2j, , -:(n)(n) = 1(n)(n)i=0di (n)_(n)i=0di= (n)_n(n)12 dj=(n)_n(n)12 n12=(n)_n(n)2 =n34 : n = 1 . n > 1: n , (n) :(n)(n) n + 12 (n + 1)(n) 2(n) (n + 1)(n)(n) 2 n(n)(n) + (n)(n) 2 (1) 1 = d0 < d1 < d2 < ... < d(n) = n n did(n)i = n i N. (n) = n +n +n + +n =d0d(n) +d1d(n)1 + +dkd(n)k++d0d(n) +d1d(n)1 + +dkd(n)k =2(n)/2i=0did(n)i x, y:xy x +y xy x y + 1 1 x(y 1) (y 1) 1 (x 1)(y 1) 1 2 2 , . 2, , 1, . , 2, . :n(n) = 2(n)/2i=0did(n)i =2d0d(n) + 2(n)/2i=1did(n)i 2d0d(n) + 2(n)/2i=1(di +d(n)i) =2d0d(n) + 2(n)/2i=0(di +d(n)i) 2d0 2d(n) =2n + 2(n) 2 2n = 2(n) 2: n(n)(n) 2(n) 2(n) = 2 2(n). :n(n)(n) + (n)(n) 2 2(n) + (n)(n)= 2 + (n) 2(n) (1) :2 + (n) 2(n) 2 (n) 2(n) 0 (n) 2, n > 1 n = 1, n . n - ( n = d2j), : dj 2 d2j 2dj. :n(n) = 2(d0d(n) +d1d(n)1 + +dj1d(n)j+1+ dj+1d(n)j1 + +dkd(n)k +d2j 2d0d(n) + 2(d0 +d1 +d2 + +dj1+ dj+1 + +d(n)) + 2dj 2(d0 +d(n))= 2n + 2(n) 2n 2 = 2 2 n .35: 45 ( IMC 2011/B3) n=1ln_1 + 1n_ln_1 + 12n_ln_1 + 12n+1_http://www.mathematica.gr/forum/viewtopic.php?f=59&t=17717 ( ) cn = ln(1 + 1/n) cn = c2n + c2n+1. L :3L =n=13cnc2nc2n+1 =n=13(c2n +c2n+1)c2nc2n+1=n=1[(c2n +c2n+1)3(c32n +c32n+1)]=n=1[c3n (c32n +c32n+1)] = c31 = (ln 2)3. L = ln(2)33 ]. . 46 ( IMC 2011/A4) A1, A2, ..., An, , -. :f(t) =nk=11i1 0.http://www.mathematica.gr/forum/viewtopic.php?p=83560#p83560 1 ( ) f(z) = eiz1z2(z2+a). ia - ia .38 2 (z2(z2+a)) = 2a, (z2(z2+a)) = 0, z = 0 _eiz1_ = ieiz,= 0, z = 0. 1 . Res (f, 0) = 10! limz0eiz1z (z2+a) =limz0_eiz1_(z (z2+a)) = ia . - R a _Cf (z) = 2Res (f, ai) +iRes (f, 0) Res (f, 0) = ea1(x2(x2+a)) = ea12iaa ._Cf(z) dz = 2i_1 ea2iaa_+i ia =_1 +a +ea_aa z = Reit=R(cos +i sin ) , _ RReix1x2(x2+a)dx +_ 0eiRcos eRsin 1R2e2it(R2e2it+a) Reitdt =_1 +a +ea_aa . R , _ 0eiRcos eRsin 1R2e2it(R2e2it+a) iReitdt_ 0eiRcos eRsin 1R2e2it(R2e2it+a) iReitdt _ 0eiRcos eRsin + 1R2([R2a[) Rdt 0. limR_ 0Rei cos Rsin1R2e2it(R2e2it+a)iReitdt = 0 _ +eix1x2(x2+a)dx = _1 +a +ea_aa . _ +cos x 1x2(x2+a)dx = _1 +a +ea_aa . 2 ( )_cos x 1x2(x2+a) dx = 2_0cos x 1x2(x2+a) dx =2a_0cos x 1x2 dx 2a_0cos x 1x2+a dx =2a_0(cos x 1)_0yeyxdydx 2aa_0(cos x 1)_0sin_ay_eyxdydx =2a_0y_0(cos x 1) eyxdxdy 2aa_0sin_ay__0(cos x 1) eyxdxdy[1]=2a_0y 1y (y2+ 1)dy+ 2aa_0sin_ay_ 1y (y2+ 1)dy =a + 2aa_0sin (ay)y (y2+ 1)dyy=x=a + 1aa_0sin(ax)x(x + 1) dx =a + 1aa_0sin(ax)x dx 1aa_0sin (ax)x + 1 dx =a + 1aa_0sin_y_y dy 1aa_0sin_ax__0e(x+1)tdtdxy=x=a + 2aa_0sin(x)x dx 1aa_0et_0sin_ax_extdxdt[2]=a + aa 1aa_0eta2tte a4t dtt=1x= (1 a)aa 2a_01tte a4t etdt[3]= (1 a)aa 2a 2a ea=_1 a ea_aa .39 [1] :_0(cos x 1) eyxdx = yy2+ 1 1y = 1y (y2+ 1) .[2] :_0sin_mx_eyxdx = m2yy em24y[3] :_01xxem24x eyxdx = 2m emy.(. - ... 3 - . 350 - 355 ). _0sin_ax_exydx =ay2+a . 40: 51 ( ) A = A, . A = B + C, B C . ( A, A = B, B +C, C x A y (x B y_x C y_(x B y C)) ).http://www.mathematica.gr/forum/viewtopic.php?p=91864 ( ) A B = . A , C = . -, . . C x - y y x. C m C. m, C C, m, . p C q p q C ( y q p. B = A C B, C A. x A y. B C, . B C. x C y C. , x B, y C. . , x By_x C y_(x B y C). -, . x B y C , C x A y , x A y. 52 ( ) - X X . .http://www.mathematica.gr/forum/viewtopic.php?p=91760 1 ( ) . - a, b a, b . Q . x. y X. x X y x t X : x t. x x x, x . (xn) x1 = x xn ,= y xn+1 = xn x1, ..., xn, ... . n xn = y. X = x1, ..., xn Q . : . 2 ( ) - X. x X I(x) = y X : y x. A x X I(x) . X , - A. A , a. I(a) a , a. a/ A, I(a) = I(a) a . X.41: 53 ( Atemlos) , , .http://www.mathematica.gr/forum/viewtopic.php?p=93446 ( ) S - R3, S S -. S (: ). p , p S = C, C - K. , p K. P, . P C A, B S. , P S . O AB, . , , = M, N, S = M, N. T (S) ( O) P . T (S) = S, S . T (S) S. Q S Q / T (S). Q , Q M, N, Q T (S), . , Q / , Q , T (S), - T (S) . S, S Q / , - . , T (S) = S . 54 ( ) - a, b, c R3. x + (x a)b = c.http://www.mathematica.gr/forum/viewtopic.php?p=58082 ( )x + (x a) b =c a(x a) + (x a) _a b_= (a c) x a = a c1 +a bx =c a c1 +a bb 1 +a b = 0, x = mb +n c mb +n c +(m+n (a c)) b =c n c +n (a c) b =c, : a c = 0 n = 1 x = mb +c, m R a c ,= 0 b, c . a c ,= 0 b, c k b +_k b a_b = b b = 0 b =0 c =0.1 +a b ,= 0 x =c a c1 +a bb1 +a b = 0 a c = 0 x = mb +c, m R1 +a b = 0 a c ,= 0 b, c - ( ), .1 +a b = 0 c = 0 x = mb, m R.42: 55 ( ) an =24n + 1, n = 1, 2, . . . 2 3.http://www.mathematica.gr/forum/viewtopic.php?f=63&t=18906 ( ) p2124 p > 3. p21 = (p 1)(p + 1) p 1, p + 1 ( p ,= 2), 8[(p 1)(p + 1), p 1, p, p + 1 3 . - 3. p p ,= 3, p 1, p + 1 3., 3[(p 1)(p + 1)., ..(3, 8) = 1 3 8[(p 1)(p + 1) 56 ( ) , - .( 83 73= 512 343 = 169 = 13213 = 22+ 32)http://www.mathematica.gr/forum/viewtopic.php?f=63&t=18715 ( ) (x + 1)3x3= y23x2+ 3x + 1 y2= 0 = 12y2 3 ., 12y23 = z2 3[z, z = 3z1. 4y21 = 3z21(2y 1)(2y + 1) = 3z21 (2y 1, 2y + 1) = 1, ) 2y 1 = 3y21 2y + 1 = y22 y1y2 = z1 y223y21 = 2, y22 2 (mod 3), 2 mod 3.) 2y1 = y21 2y+1 = 3y22 y1y2 = z1 2y = y21+1, y1 , y1 = 2k+1. 2y = 4k2+ 4k + 2y = 2k2+ 2k + 1y = k2+ (k + 1)2 .43: 57 ( ) ,sin 1 + sin 3 + sin 5 + + sin99 = sin250sin 1http://www.mathematica.gr/forum/viewtopic.php?p=95017 ( ) : sin 1 sin 1 +sin 1 sin3+sin1 sin 5+.... +sin1 sin 99 = sin250 : 12(2 sin 1 sin 1 + 2 sin 1 sin3 +2 sin 1 sin 5 +.... + 2 sin 1 sin99) = sin250 : 12(1cos 2+cos 2cos 4+cos 4cos 5+.... + cos 98 cos 100) = sin250 : 12(1cos 100) = sin250 . 58 ( ) 3a, a - E1, E2. E :E = 29(E1 + 9E2).http://www.mathematica.gr/forum/viewtopic.php?p=93378 ( ) - - ( k: E = k234 E1 = (3)234 E1 = 9234 (1) E2 = 234 (2) (A1, B1, C1) ( ) ( ABBA) A1B1 : A1B1 = AB +AB2 = 3 +2 A1B1 = 2 - 2 ( - ) E = (2)234 E = 4234 E1 =23 (3)29 (E1 + 9E2)(1),(2)=29_9234 + 9234_=29 184 23 =23(3)=E E = 29 (E1 + 9E2)44: 59 ( pla.pa.s ) de l Hospital f(x)g(x) . - ; f(x)g(x) x0 ( ), ( limxx0f(x)g(x)) - de l Hospital, f, g ( x0) limxx0f(x)g(x) ., ;http://www.mathematica.gr/forum/viewtopic.php?p=89706 1 ( ) f (x) = 2x + sin(x)g (x) = 3x + sin (x) x +. x x0 = 0 f_ 1x2_, g_ 1x2_. : . f (x) = 2x55x4+ 5 x [2, +) [11, +). -. g . limu0g (u) f. ( ) . - limu0g (u) -. 2 ( ) x + sin xx (=1). l Hospital (x + sin x)x = 1 + cos x1 . 3 ( ) : f(n) = 1 n n + 2 , f(n) = 0. g(n) =nk=1f(k). g(n) (k, k + 2) k n. - h(n) = 11 +g(n). 1 0. h(n) . 0 , 1k + 1 k . h(n) 0. . : - Goedel. 0 . 60 ( ) f :R R . R ;http://www.mathematica.gr/forum/viewtopic.php?p=94877 ( ) g1 : (, 0) R, g2 :[0, +) R R (). - ( ). ..g1(x) = ln(x) g2.F(x) =_g1(x) , x < 0f(x) g2(x) , x 0G(x) =_f(x) g1(x) , x < 0g2(x) , x 0 f = F +G F, G ( ()).45: 61 ( ) [. -, , ( , 1976)] P K E Z .http://www.mathematica.gr/forum/viewtopic.php?p=94010 1 ( ) - : , , L, C, T. LC, CT. G, R : GC = CR, GCR = 2.: F LG, TR LT , CT 4, GCRF . : K, L - A, A. B, C K, L ,- A BC BC = 2h, h . , , - KLD KD = h. A L EP - 4, B - T - KZ 4, I PK N. (IA = IB, IAIB) (NG = NR, NGNR) , . [ - (PIK) PIK = 2 IB =2NR = 2NG = IA.] 2 ( ) L, T, EP,ZK 450, *. A, B - EZ, E, Z E, Z L,T, . ( C, D, 46 BZ, AP AE, BK, : ABC = ZBZ BAD = EAE , ZBK = CBD EAP = DAC 450, )* , 2 2 =4 , , 62 ( ) - .http://www.mathematica.gr/forum/viewtopic.php?p=94331 1 ( ) P, Q, R, S ABD, BCD, ABC, ACD G ABCD ( AD//BC) PQ, RS.: G |GP||GQ| = (ABD)(BCD) = |AD||BC| |GR||GS| = (ABC)(ACD) = |BC||AD|, . A = (1, 0), B = (4, 0), C = (4, 4),D = (1, 1):P: 23(52, 0) + 13(1, 1) = (2, 13)Q: 23(52, 52) + 13(4, 0) = (3, 53)R: 23(52, 0) + 13(4, 4) = (3, 43)S: 23(1, 12) + 13(4, 4) = (2, 53)PQ: y = 4x3 73RS: y = x3 + 73G: (145 , 75) 2 (parmenides51) [, - 15] 2 - - 2. (1) ABCD AD//BC(AD ) M, N ,E BA, CD. AED, ABC MN EM, MN - . (1) - MN. (2) BD - (z), (w) (z) AD DB. P, Q - ABD, BCD (z), BM (w), DN . (1) PQ.(3) (2), (3) G ABCD PQ, MN. - BCAD = KGGL - MGGN = 2BC +AD2AD +BC (*) MGGN = , = 2B +2 +B > 0 , B . - , B , - M, N G MG = GN NG = 1 + 1NM = 2B +2 +B. A = (1, 0), B = (4, 0), C = (4, 4),D = (1, 1): M, N AD, BC _1, 12_ , (4, 2) . B = BC = 4, = AD = 1 = 2B +2 +B = 2 4 + 12 1 + 4 = 96 = 32. NG = 1 + 1NM (x 4, y 2) = 132 + 1_1 4, 12 2_(x 4, y 2) = 25_3, 32_(x, y) =_4 65, 2 35_(x, y) =_145 , 75_ 3 (parmenides51) 1: - . 2: ( ) . ABCD AD//BC (AD ) , E BA, CD V, G, W EAD, ABCD, EBC. V, W - - ( ). E, V, G, W - (1) - EAD, EBC E. - E - ( ) - EBC T ET = EW -47 V E. (2) (EAD)(EV ) + (ABCD)(EG) = (EBC)(ET) (EAD)(EV ) + (ABCD)(EG) = (EBC)(EW) . EG EG//EV , G, ABCD. A = (1, 0), B = (4, 0), C = (4, 4),D = (1, 1):_A = (1, 0)B = (4, 0) AB : y = 0,_C = (4, 4)D = (1, 1)CD : y = x ,_AB : y = 0,CD : y = x E(0, 0) EAD V_0 + 1 + 13 , 0 + 1 + 03_=_23, 13_,(EV ) =_23_2+_13_2=53 (EAD) = (EA)(AD)2 = 12 EA AD. EBC W_0 + 4 + 43 , 0 + 4 + 03_=_83, 43_,(EW) =_83_2+_43_2= 453 (EBC) = (EB)(BC)2 = 162 = 8 EB BC. ABCD G (ABCD) = (AD +BC)(AB)2 = 152 EB BC ( ). (EAD)(EV ) + (ABCD)(EG) = (ECD)(EW) 1253 + 152 (EG) = 8453 (EG) = 755_E = (0, 0)V =_23, 13_ EV : x = 2y - E, V, G G(x, y) = (2y, y) (EG) =_x2+y2=_(2y)2+y2= [y[5 (EG) = 755 = [y[5 y = 75 y > 0 G(x, y) =_145 , 75_ ( - ) - - .48: 63 ( )P(x) - . P(1) = 6 P(7) = 3438, x = 3.http://www.mathematica.gr/forum/viewtopic.php?f=60&t=15830 1 ( ) 4 74< 3438 < 75. P(x) = ax4+bx3+cx2+dx +e a, b, c, d, e , P(1) = 6 a +b +c +d +e = 6 , P(7) = 3438 : a = 1, b = 3, c = 0, d = 1, e = 1, P(x) = x4+ 3x3+x + 1 P(3) = 166. 2 ( ) - [0, +) . A(x) - P(x) = xA(x) +P(0)P(7) = 7A(7) +P(0) (1)P(1) = A(1) +P(0) (2) P(0) P(1), 7. (1) P(7) : 7. A(7) P(0) . A(7) = 491 P(0) = 1 (2) A(1) = 5. B(x) A(x) = xB(x) +A(0).A(7) = 7B(7) +A(0) (3)A(1) = B(1) +A(0) (4) A(0) A(1), 7. (3) A(7) : 7. B(7) A(0) . B(7) = 70 A(0) = 1 (4) B(1) = 4. : C(x) - B(x) = xC(x) +B(0)B(7) = 7C(7) +B(0) (5)B(1) = C(1) +B(0) (6) B(0) B(1), 7. (5) B(7) : 7. C(7) B(0) . C(7) = 10 B(0) = 0 (6) C(1) = 4. C(x) 4 = C(1) < C(2) < C(3) < C(4) < C(5) < C(6) < C(7) = 10 , C(3) = 6. B(3) = 3C(3) +B(0) = 18A(3) = 3B(3) +A(0) = 55P(3) = 3A(3) +P(0) = 166 64 ( ) P(x) = xn+a1xn1+ +anQ(x) = xn+b1xn1+ +bn C. P x1, x2, ..., xn, Q x21, x22, ..., x2n. a1 +a3 +a5 + R a2 +a4 +a6 + R, , b1 +b2 + bn R.http://www.mathematica.gr/forum/viewtopic.php?f=60&t=18478 ( ) :P (x) =ni=1(x xi)49Q(x) =ni=1_x xi2_ :P (1) = a1 +a2 + +an RP (1) = a1 a2 + + (1)nan R., :P (1) = P (1)P (1) = P (1),ni=1(1 xi) =ni=1(1 xi)ni=1(1 +xi) =ni=1(1 + xi). , ni=1_1 x2i_=ni=1_1 x2i_,Q(1) = Q(1) b1 +b2 + +bn R50