1 V = h a + ab + b 3

= 1 = 1 + = + = BAC Quod erat demonstrandum Q E D

AB p( EF) p = = AB AB CD q( EF) q p q 1 p q,

EF = ED BF G G BG = FG

EH a = b + c a - b = c FG = BG = HG = a EF = FG - EG = a - b BE = BG + GE = a + b BCDE = = ( BE) ( ED) = ( BE EF) EF = ED BCD D BC E DE 1 1 = ( BC) ( DE) DE F GH = BC, HJ = EF 1 ( HJ) ( GH) = ( EF) ( BC) = = ( DE) ( BC) = B CD

1 = d D

O AO = OB 1 15 OC AOC BOC AC = BC ( AB) = ( AC) ( BC ) = ( AC) AEC ( AC) ( AC) 1 = = = ACB ( AB) ( AC)

CE = EF = FD = DG = GH = HC = OC AB AB

a1 a 0 3x = 0 3 x = 0 3 1+ 5 6 3 x x 4 = 0

1 13 5 711 5 6 1 + 4 + 3 7 π π

x = π π

BC AC AB BC AB 1 AC AB BC ABC

AB DE AC DF BAC = EDF

EF EB FC AEB CEF AE I.15 1 EB EF

6 AB DE. 8 ABC DEF. 9 = 5 3 = 6 AB DE BC EF BC EF BH AB DE = 5 BH EF BC EF

1 = bh

ABC BC AB AC 15. AD AB CD AC AD AB BAC CD AD AC AB AC BC

CD BC CD BC

AB DE

DG AB.3 DGH AB DE AB DE

AC a BC = b

1 1 5 3 3 5 15 15 15

1 1 1 1 3 4 5

1 r h 3 A A 1 D1 A1 A = = D D D 1 V1 V = 3 3 D D 1

C1 C = D D 1 C D = C = D A D = k A = kd 1 hq 1 b bh 1 1 1 = bh + bh + + bh n 1 1 = h b + b + + b = hq

1 500-1 500

1 T = rc 1 1 = hq rc = T 1 1 = hq rc = T

A = T = 1 rc 1 1 A = rc = r r = r D1 D1 D D A 1 = πr1 = π( ) = π A = πr = π( ) = π 4 4

D1 A π 1 D1 = 4 = A D D π 4 π 4k = k = 4 3 1 3 10 7 71 3 10 3 1 71 7 6 r 3 r 3 65 1351 3 153 780

96 6336 017 1 3 10 71 4 96 3 1 7

h x = x = rh x r

= 1 1 1 r h = r r = r 3 3 3 3 4 = 4 = r 3 3 4 3 r 3 = = md 3 = m( r ) = 8mr 3 3 m = π 6

= 3 ( 4πr ) = 3 = 3 4 3 ( πr ) 3 3 =

C = = 30 D 10 = 3.00 4 56 ( ) 4 = = 3.1604938 C D 3 81 4 1 1 180 C 360 = 376. 99p = = 3.1416 D 10p

355 501 480 = 3.141599 113 397 150 = 3.1416 1 1 1 1 1 1 1 1 + + + + 3 5 7 9 11 13 15

3 18 1 0.15 8 1 7 1 7 = 0.14857148571485714857

a 1 50 = a 5000 = 50 5000 = 50 000 1 50,

= 1 K = s( s a)( s b)( s c) s = 1 ( a + b + c) 1 5.3 s = 17 + 5 + 6 = 34 K = 34(34-17)(34-5)(34-6) = 41616 = 04

AB MA MAD MBD AD = ADM = BDM = DM MA = MB MAD = MBD ` MC = MA MB MC MH 1 K = s( s a)( s b)( s c) s = ( a + b + c) O r OD = OE = OF =

AOB = 1 1 = AB OD = 1 cr BOC = 1 = 1 BC OE = 1 ar COA = 1 ( = 1 AC OF = 1 br K = 1 cr + 1 ar + 1 a + b + c br = r( )rs AD = AF BD = BE CE = CF AB G AG = CE BG = BD + AD + AG = BD + AD + CE 1 = ( BD + AD + CE) 1 = [ BD + BE + AD + AF + CE + CF ] = 1 [ BD + AD + BE + CE + AF + CF 1 = [AB + BC + AC] = 1 ( c + a + b) = s BG = s s - c = BG - AB = AG s - b = BG - AC = BD + AD + AG - AF + CF = BD + AD + CE - AD + CE = BD

AD = AF AG = CE = CF s - a = BG - BC = BD + AD + AG - BE + CE = BD + AD + CE - BD + CE = AD BD = BE AG = CE AB CF AG = = AH OF r CF = AG OF = r AB AH = * AG r AH OD r AH AK = = = AK KD KD r KD AB AK = ** AG KD BOK OD = r

KD r = ( KD)( BD) = r ** r BD r KD BD AB AK +1= AG KD +1 AB + AG AK + KD BG AD = = AG KD AG KD BG BD BG BD ( BG)( BG) ( AD)( BD) = ( AG)( BG) ( KD)( BD) ( BG) ( AD)( BD) = ( AG)( BG) r r ( BG) = ( AG)( BG)( AD) ( BD) ***G rs = s( s a)( s b)( s c) K = s( s a)( s b)( s c)

s = a + b + c a b c a b c a a b c s a = + + a = + + = + + s - b = - b + c a b c s c = + K = s( s a)( s b)( s c) = a + b + c a + b + c a b + c a + b c ( )( )( )( ) 4 4 4 a b + a c + b c ( a + b + c ) = 16 1 1 K = = bc b c a b + a c + b c ( a 4 + b 4 + c 4 ) = 4 16

x = -b ± b ac a 4

â â

m u 3t 3 3 m t n 3 7t 3 t 6 3 m nt 0 7 3 3 3 m ( t ) n ( t ) 0 7 4m n n + 3 7 t = 3 3 3 n 1 4m n n m n + = + 7 4 7 t n n m 3 3 + + 4 7 3 n n m u 3 + n 4 7 3 n n m u 3 + + 4 7 3 3 n n m n n m = 3 + + 3 + + 4 7 4 7

x ( 1 6) 3 = 8 3 10 108 108 3 3 x 10 108 10 + 108 3 n m + 108 4 7 3 3 x = 10 + 108 10 + 108

b a x b b a( y ) b ( y ) c 0 a a b b b a y y by c 0 a 4a a b b ay by by c 0 4a a b b b b 4ac b 4ac ay c a 4a 4a 4a 4a 4a y b 4ac ± b 4ac y 4a a b ± b 4ac b b ± b 4ac x y a a a a b x y 3a b 3 b b a( y ) b ( y ) c(y ) + d 0 3a 3a 3a 3 3 b b ( ay 3 by b b y ) ( by y ) 3a 7a 3a 9a cb + ( cy ) + d = 0 3a b y x y 3a

b 30 x y y ( ) y 5 3a 6 3 x 3 + -11 + 11 x 4 x - 3 1 1 3 ( 1) 8 1 1 6 1 11-1 11 3 ( 1) 11 3 + 11 1

3 + 11 1 3 x 15x 4 3 3 x + 11 + 11 ( 1) ( 1) 4 1 11 18 11 x 11 0 11 x 4 b 4a a y

b x y 5a

7 34. 65 -

3 (a + b) ( a + b) ( a + b) 3 1 a n n n 1 = a a = n a 1 1+ x ( 1 x ) 1 161

m m m ( P PQ) P n AQ m n n n + = + + n BQ m n CQ m 3 + + n DQ + 3n 4n m P + PQ n A = P m n B m n AQ m n P n = = Q C m n m n BQ ( m n) m n n P n = = ( ) Q D m = n CQ = 3n m m m 1 n n = P m n m m m 1 n n n P m 3 n Q 3 Q n m n n n n P ( 1 + Q) = ( P + PQ) = P 1 + + m n m m 1 n Q m P n m m m n Q m m m 1 n n n + Q 3 3 m P + m m m 1 m ( 1+ Q) = 1+ + n Q n n n Q m m m 1 n n n 3 + Q + 3 m x Q 3 n

3 3 1 3 3 1 0 4 = 1+ 3x + x + x + x + 3 4 3 6 6 3 0 4 0 5 = 1+ 3x + x + x + x + x + 6 4 10 3 = 1+ 3x + 3x + x ( 3 4 3 4 5 3 1 + 3 + )( ) ( + )( ( ) )( ) x x x + 6 1 3 4 = 1 3x + 6x 10x + 15x 3 ( 1+ x) 1 3 4 = 1 3x + 6x 10x + 15x 3 1+ 3x + 3x + x 1 1 x = (1 x) m 1 Q = x = n 1 1 1 1 x = 1 + ( x) + ( x) 1 1 3 3 + ( x) + 6 1 1 1 1 3 5 4 7 5 = x x x x * 8 16 18 56 1 1 1 1 3 5 4 x x x x 8 16 18 1 1 1 1 3 5 4 x x x x 8 16 18 1 1 1 1 1 1 1 3 1 3 1 3 1 3 = x x x + x x x + x + x x 8 4 8 16 16 16 16

1 1 1 1 3 5 4 x x x = 1 x 8 16 18 1 1 1 1 3 5 4 x x x x = 1 x 8 16 18 7 7 9 7 = 9 1 = 9 9 7 = 9 1 = 3 1 9 9 x 9 7 3 1 1 1 5 7 1 =. 64576 9 16 1458 5488 4739 7 0.00001 3 3 1 x = ( 1 x) 6 7 1

m ( m+n) an n n 1 ax = y x = ABD m + n n 7.1 y = ax x ( m+ n) an m n x n + y = x 7. a = m = n = 1 1 1 x = x y = x + 1 3 x x = ( + 1) 3 y = x + x 3 5 1 3 x + x 3 5 m C ( 1 0) r = 1 7.3 x + y = x x y 1 1 1 1 ( 0) + + = 4 4 1 1 y = x - x = x 1 x = x ( 1 x) 1 * ( 1 x)

1 y = x ( 1 x) 1 1 3 4 5 = x 1 1 1 1 5 7 ( x x x x x ) 8 16 18 56 1 3 5 7 9 11 1 1 1 5 7 = x x x x x x 8 16 18 56 B ( 1 0) 4 O x = 1 4 3 5 7 9 1 1 1 x x x x 3 5 8 7 16 9 3 5 7 9 11 1 1 1 5 = x x x x x 3 5 8 7 704 ** x = 1 4 3 3 5 1 1 1 1 1 4 = = 4 8 4 = = 4 5 1 3 1 1 1 1 5 1 160 3584 36864 144179 49 = 0. 07677310678 1630875748 DBC 1 1 CD r = 4 BD = 1 1 3 = = 4 16 1 1 DBC = BC BD = 1 4 3 4 3 4 = 3 3

= 1 3 1 1 1 1 1 = 3 π = 3 π = π r 4 π 3 ( ABD) = DBC = 4 3 3 0. 07677310678 π 4 3 3 π 4 0. 7677310678+ = 31415968. 3 3 1 x

k( k + 1) 1 3 6 k S = 1 + 1 1 1 1 1 + + + + + 3 6 10 15 1 1 1 1 1 1 1 S = + + + + + 6 1 0 30 1 1 1 1 1 1 1 3 6 3 1 1 4 1 1 1 1 1 1 1 1 1 1 S = + + + + 3 3 4 4 5 1 1 1 1 1 1 1 1 S = 1 + + + + = 1 3 3 4 4 5 1 1 1 1 1 1 + + + + + + = 3 6 10 15 1

1 3 = 0.3333333 1 3 3 3 3 = + + + + 3 10 100 1000 10000 a 1 S 1- a = a S = (1- a) a + a + a + a + a = a 3 4 5 1 - a a = 1 3 1 1 1 1 1 1 1 3 3 1 + + + + + + = 3 9 7 81 3 1 1 = = k 3 3 3 4 + + + + = 1 = 1 1 1 1 3 9 7 81

1+ 1 + 1 3 + 1 4 + 1 5 + 1 + + k 1 1 1 1+ + + + 1 1 + + = 4.99000 5.00 3 4 5 8 1 1+ + 1 1 1 1 + + + + = 5.00068 3 4 5 83 1 6 83 1 + 1 6 + 1 1 + 1 0 + 1 30 + = 1

1+ 1 + 1 + + 1 + 3 k A = 1 + 1 + 1 + 1 + + 1 k + 3 4 5 1 A = + 6 + 3 1 + 4 0 + 5 30 + C 1 1 1 1 6 1 0 1 1 1 1 1 C + + + + + 1 6 1 0 30 D 1 1 1 1 1 1 1 + + + + C = 1 6 1 0 30 E 1 1 1 + D 1 1 1 1 + + 1 0 30 6 6 3 F 1 1 1 1 1 1 + + E = = 0 30 1 3 1 4 G 1 1 1 1 1 + F = = 30 0 4 0 5 1 1 1 = + + 1 + 1 + 1 6 6 1 1 1 1 1 1 1 = + + + + 0 0 0 0 1 3 4 = + + + + = A 6 1 0 C + D + E + F + G + = 1 + 1 1 1 1 + + + + 3 4 5 = 1 + A

1 1 1 1 3 4 1 1 + 1 1 1 + 1 4 6 + 1 + 1 3 5 7 8 1 1 + 1 9 10 + + 1 1 16

1 1 1 1+ + 1 1 1+ 1 + 1 + 1 1 1 4 + 1 = 3 4 4 3 1+ 1 + 1 3 + 1 4 + 1 1 1 1 + + + 5 6 7 8 3 + 1 8 + 1 8 + 1 8 + 1 4 8 1 1+ + + 1 8 + 1 9 + 1 10 + + 1 16 4 + 1 16 + + 1 5 16 1 1 1 k + 1 1 + + + + k 3 1 1 1 1 1 1 9 + 1 l 1 5 9 3 51 3 1 1 + + 1 3 + + 1 100 50 99 1 1 1 9999 + 1 1 5000 9999 3 1 a l + 1 a + 1 3 a 1 a +1 a

3 a a( a 1) a a ( a 1) a ( a 1) a 1 a 1 1 1 1 1 1 a 1 + a + a + 1 = a + a 1 + a + 1 1 a = + a a 1 1 + a a 3 = a 1 1 1 3 1 1 1 1 3 1 + + = + + = 8 9 10 9 3 3 33 34 33 11 1 H 1 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 + 1 9 + 1 10 + 1 11 + 1 1 + 1 13 + 1+ 3 3 + 3 6 + 3 9 + 3 1 + 3 15 + 1 1+1+ + 1 3 + 1 4 + 1 5 + 1 6 + 1 + 1 8 + 1 9 + 7 + 1 + 1 + 1 3 4 + 1 5 + 1 + 1 6 8 + 1 9 + 1 + 1 7 10

+ 1 11 + 1 1 + 1 13 + + 3 3 + 3 6 + 3 9 + 3 1 + 3 15 + +1+ 1 + 1 + 1 + 1 5 + 1 6 + 1 7 + 1 8 + 1 + 3 4 9 3 + 1 + 1 3 + 1 4 + 1 5 + 1 6 + 1 7 + 1 8 + 1 9 + 1 10 + 1 11 + 1 1 + 1 + 13 1 + 1 + 1 + + 1 + 1 + 1 4 + 1 + 1 16 + 3 k 9 1 1 1 1 1 1 4 3 9 6 16 10 1 1 k k( k + 1) 1 + 1 4 + 1 + 1 16 + + 1 + 1 + 1 3 + 1 6 + 1 10 + 1 + 9 k 15 + + 1 + 1 6 + 1 1 + 1 0 + 1 1 30 + ( 1) k( k + ) 1 1 1 + + + 1 4 9 16 +

+ + + + + + + 1 1 + + 1 9 + 1 16 + 1 4 5 +

3 5 7 9 x x x x sinx = x + + 3! 5! 7! 9! x x x x P(x) = 1 1 1 1 a b c d a a a a P( a) 1 1 1 1 = 0 a b c d b b b b P( b) = 1 1 1 1 = 0 a b c d

P(0 ) = 1 0 0 1 1 0 1 0 a b c d = ( 1) ( 1) ( 1) ( 1 ) = 1 x x x x P( x) = 1 1 1 1 a b c d x x x 11 1 3 P(x) = 1 1 1 = 1 x + x = x 3 6 36 36 1 1 + + 1 1 1 + + + 4 9 16 5 1 + 1 + 1 1 1 1 9 k + π + + + 4 16 5 6 4 6 8 x x x x f(x) = 1 + + 3! 5! 7! 9! 4 6 8 x x f( x) = x!! _ x x 1 + + 3 5 7! 9! x 3 5 7 9 x x x x x + + = 3! 5! 7! 9! x sin x = sinx x sinx x 0 f( x) = 0 = 0 x

x x x x f( x) = 1 1 1 1 π π π π x x 1 1 3 π 3 π x x x x x x = + + + 1 1 1 1 1 1 π π π π 3 π 3 π 4 6 8 x x x x 1 + + 3! 5! 7! 9! x x x x = 1 1 1 1 4 9 16 π π π π 4 6 8 x x x x 1 + + 3! 5! 7! 9! x x x x = 1 1 1 1 4 9 16 π π π π 1 1 1 1 4 = 1 + + + x + ( ) x π 4π 9π 16π 1 1 1 1 1 = + + + + 3 π 4π 9π 16π

1 π 1 1 1 1 1 = 1 + + + + 6 π 4 9 16 1 1 π 1 + + 1 + = 4 9 16 6 π 1.6449 6 sinx π 9.1 x = sin x x π 1 x = sin π π π = 1 1 π π 4π π 1 9π π 1 16π 1 1 1 π 4 1 1 6 1 1 36 1 1 = 64

3 15 35 63 = π 4 16 36 64 π 4 4 6 6 8 8 = 1 3 3 5 5 7 7 9 π 1+ 1 + 1 1 + + 4 9 16 1 1 1 1 1 1 + + + + + + + 4 16 36 64 100 ( k) 1 4 1 1 1 1 1 + + + + + 4 16 36 64 100 1 4 1 1 1 1 1 1 = + + + + + = 4 9 16 5 4 π π = 6 4 1 1 1 1 1 1 1 1 + + + + = 1 + + + + + 9 5 49 4 9 16 5 1 1 1 1 1 + + + + + = π π = π 4 16 36 64 100 6 4 8 1 1 1 1 1 1 1 + + + + + + + 4 16 81 56 65 196 k = 1- (a b)x + 1 [(a b) - (a b )] x 4

1 = 1 - (a + b c)x + [(a b c) - (a + b + c )]x 4 - (abc)x 6 4 6 8 x x x x 1 + + 3! 5! 7! 9! x x x x = 1 1 1 1 4 9 16 π π π π 1 1 1 a b c π 4π 9π 4 6 8 x x x x 1 + + 3! 5! 7! 9! 1 1 1 = 1 + + + x π 4π 9π 1 1 1 1 1 + + + + + π 4π 9π 16π 1 1 1 1 4 + + + + 4 4 4 4 16 81 56 x π π π π 1 1 = 5 10

1 1 1 1 1 + + + + π 4π 9π 16π 1 1 1 1 + + + + 4 4 4 4 π 16π 81π 56π 1 1 1 1 1 = 4 1 + + + + π 4 9 16 1 1 1 1 4 1 + + + + π 16 81 56 1 π 1 1 1 = 1 4 π 6 + 16 + 81 + 56 + 1 1 1 1 1 = 1 + + + + 4 7 π 16 81 56 1 1 1 1 1 1 = 1 4 + + + + 10 7 π 16 81 56 1 1 1 1 1 1 1 1 4 + + + + = = π 16 81 56 7 10 180 4 4 1 1 1 1 π π 1+ + + + + + = = 4 16 81 56 k 180 90 6 1 1 1 1 π 1+ + + + + + = 6 64 79 4096 k 945 1 1 1 131586 6 1 + + + + = π 6 6 6 3 k 1109448197603057815 1 1 1 1 1 1 1 1 + + + + = + + + + 3 3 3 3 4 8 7 64

p p 3 q q

p p p p p p a + 1 1 ( 1)( ) 3 a + a + + a 1 3 1

6 6 13 1 13 5 6 5 3 1 13 4 6 5 4 4 3 1 13 3 + + + 6 5 4 3 5 4 3 1 13 6 5 4 3 + + 13 = 486809 6 5 4 3 1 + 1113879 + 14805+ 10985 + 507 + 13 = 6094998 13 + 3 3 1 13 + 3 3 1 13 = 197 + 53.5 + 13 = 463.5 p p p 1 p( p 1) p (a +1) = a + pa + a 1 p(p -1)(p - ) + a 3 1 p 3 + pa + 1 p p ( a +1) ( a +1) p p(p -1) = a + pa + a 1 p( p 1)( p ) + a 3 1 p 1 p p 3 p( p 1) p( p 1)( p ) = pa + a + a 1 3 1 p 1 p p 3 p ( )( ) = + 1 p 1 p p a a + a 1 3 1 p 1 p p 3 + pa + 1 p ( a + 1) + + pa + + a

n + 1 n = 1 = +1 = 5 4 n = +1= 1 = 17 n = 3 n = 4 5 3 +1 = +1= 49496797 a n +1 n = 5 n +1 1

a 4 1= a 1

p a +1 k n n 1 n = 5 n

6 + 1 = 64 1 18446744073709551617 +1 = 18 1 7 9 1 1988 8 1 1 1 +1 n N +1 N n

17 1 = 17 3 +1= 57 S = 1 3 99 100 = 10100 = 5050 4

b b ac b b 4ac ax + bx + c = ax + x a = (3x -1)(x )(x -1)(x -1)

z 7 + ( 6 1) z 6 ( + 1) z + 19

0 0

3 8 3 11 0.375 0.7777

1 ( 1) (n 1) n 4 ℵ

ℵ M M N ℵ E Z 0 Q ℵ 0 133 191 65 191

1 3 3 1 3 11 4 0.50000000 0.7777 0.78539816 π 0.50000 0.49999999 1

1 x ℵ

x 1 y = x x ℵ ℵ

5 3

B A B A B N Q

A B A = B A B N Q N Q ℵ 0 A B A B A B A B A B A B ℵ N (0,1) N (0,1) N (0,1) c ℵ 0 A B A A B B A A = B

A B B A B A A B A = B ℵ I c = 1.414159 0 1 c I I c -

z S (z, 1 ) ( 0 1) S 1 0.1999 5 ( ) = 0.18181818 0.70710678 11 (0,1) z = 6 = 0.109090909 55

6 S = 01090. 55 S (0,1) (0,1) S - S ( 0,1) = c n P[A]

A A P[A] A P[A]

A P[A] A P[A] A A P[A]

P[(0,1)] c = (0,1) P(0,1) P[(0,1)] P[P[(0,1)]] ℵ 0 c P[(0,1)] P[P[(0,1)] P[P[P[(0,1)]]] ℵ

U U P[U] ℵ

ℵ ℵ ℵ

ℵ