= 8 25 + 19 12 = 32 43 32 = 11. 2 : 1 k > 0. x k + (1 x) 4k = 2k x + 4 4x = 2 x = 2 3. 1 x = 3 1 2 = 2 : 1.

4 4 = 8 8 8 = 5 + 19 1 = 4 = 11 : 1 k > 0 k 4k x 1 x x k + (1 x) 4k = k x + 4 4x = x = x 1 x = 1 = : 1. v h h s 75 v 50 h s v

v 50 s h 75 180 v h 90 v 50 h 180 90 50 = 40 s 65 v 80 60 80 80 65 v 50 s 50 65 50 40 40 h 75 a b c 16 0 5 16a + 0b + 5c = 5 0b+5c 16a a a + b + c = 1 a = 0 b = 1 c 5c = 5 a = 5 b = 7 c 80 + 5c = 95 c = b = 4 a = 10 b = c 00 + 5c = 5 c = 7 b = 7

6 10 = 48 5 48 = 4 6 10 10x+y x y 1 + 9 + (9 x) + (9 y) 8 = 11x + y 0 x, y 9 x = y = 1999 = 1976

8 + π π + 4 p s s = 1 5 (6 + + 10 + 9 + p) p = 5s 8 s {6, 9, p} s (,, 6, 9, 10) (, 6, 7, 9, 10) (, 6, 9, 10, 17) A B C AA+BB +CC = ABC ABC XY X Y XY A + B = 10 A + B = 0 A = B = 0 1 + A + B + C B < 1 + A + B + C < 0 + B 1 + A + B + C = 10 + B A + C = 9 A = 1 ABC = 198 (9 + ) = 18 (18 + 4) = ( + ) = 54

014 015 018 015 = 1551+464 016 = 1441+575 017 = 11 + 686 018 = 11 + 797 019 019 1AA1 BCB A B C B = 8 A + C = 1 A + C = 11 1AA1 + 8C8 < 000 A + 8 + 1 = 10 A + C = 11 019 X X X Y XY (X+Y ) X 0 Y 9 X + Y X = 4 a < b < c < d d,5 4 a + b a + c a + b = 1 a + c = c + d b + d a + d = b + c = 4 a = (a + b) + (a + c) (b + c) = 1 + 4 a = 0,5 b = 1,5 c =,5 d =,5

b + c = a + d = 4 a = (a + b) + (a + c) (b + c) = 1 + a = 0 b = 1 c = d = 4 k v k > v k = (v (k v)) k + (k + (k v)) = 90 k = 4v k v = 90 k v = 90 v = 0 k = 40 a 1, a, a,... a 1 = 10 a a = 100 a aa k a k = a 1 + (k 1)d d a = 10 + d 100 = a a = a 10+d = 10 + (9 + d)d = 10 + 9d + d. d = 6 a k = 4 + 6k a = 4 + 18 = a a = a = 4 + 1 = 16 a aa = a 16 = 4 + 816 = 80, 4,..., 9 0 + + + 4 + + 9 = 44

a b c (a + b + c) = 14 a b c (a, b, c) (1, 1, 5) (1,, 4) (1,, ) (,, ) c ab 1 a b ab = (a, b) + (a, b) (a, b) (a, b) a b (a, b)

(, ) d = (a, b) (a, b) = ab d ab = d + ab d (d 1)ab = d a, b d d > 0 a = b = d d 1 = 1 (, ) 4 = 16 16 16 = 56

T 014 014 [0, 0] [014, 014] p [0, 0] [014, 019] T p p 014 T p [0, 0] [014, 019] B = [x, y] x/y = 014/019 014 019 p T [0, 0] [014, 019] p 01 014 /019 = 009 [014 /019, 014] 01 + 009 1 + 009 + 01 = 40 150 n n (n ) 180 150 (n 1) + x 0 < x < 180 n = x 0 + 7 8 n 1 b c 60 a + b + c = 1 a + b + 1 a + c. (a + ab + ac + bc) = a + b + c + (ab + ac) + bc, a = b + c bc a = b + c bc α α = 1 α = 60

1 00 16 + = 5 16 = + + 11 = + 1 = 5 + 11. 1 a b 11 c 00 a b c 11 = 66 11 = 1 11 = 198 1 = 9 1 = 117 5 11 = 55 DA = AB = BE GA = AC = CF IC = CB = BH ABC EF = 5 DI = 5 GH = 6 I F C D A B E G H AB CGH AB = GH / = AC = DI / =,5 BC = EF =,5 ABC C AB M AMC MC = AC AM = 6,5,5 = ABC S = 1 AB MC = p p

k l m k l m m k l k l m S k S l S m m r k m S k S m S k S m = 16 r A m S k S l AS k = AS l = 8 S m A = r S k AS m r = 6 8 + r = (16 r),

S k S m A m S l p k l 1 6 = 1 + + = 1 + 5 = + 4 = 6. 6 (5 ) = 60 1 + 5 + 4 6 ( 6 ) = 15 60 + 6 + 15 + 1 = 8 1 + 6 = 6+

v s = 1 = S T ST = v s 1 = O ST O 1 + 6 OS + 1 = 6. x y 100 x > y x y x y (x, y) x y = (x y)(x + y) x y = (x y)(x + xy + y ) x y x > y x y = 1 y x 1 (x 1, x x + 1) = 1 (x 1)(x 1) x x + 1 (x 1, x ) = 1 x x x 1 x y = 1 (, 1), (, ),..., (100, 99) 1444455555...

1 + + + 9 = 9 10 = 45 ( ) 10 + 99 (10 + 11 + + 99) = 90 = 9810 45 + (10 + 11 + + 40) < 014 < 45 + (10 + 11 + + 50), 4 N (x 1)(y 1) = N x, y N x y t 1 t N 0,, 8, 15, 4, 5, 48, 6, 80, 99, 10,... t 1 10 = 15 4 = 60 60 10 10 8 8, 8 15, 8 4, 8 5, 15 15 60 N 60 A A B C AB + BC + AC

S 1 S A BC T T T C = T A = T B A BC ABC AB + AC = BC AB + BC + AC = BC. S 1 S P P S 1 S C BC = S 1 P = ( + 1) ( 1) = 1. AB + AC + BC = BC = 4 C B T P S 1 A S p p p 014! n! = 1 n q 014! = 1 014 014/q q, q,..., 014/q q q > 014 014/q < q q 014! q q, q,..., 014/q q q p p 014 014/p p 014 p = 4

014 6 01 a b (b, a) (b, c) (c, a) 6 01 m 5m 6m 7m 5 84 6 5 7 90 m m = 5 a 6 b 7 c d d 5m 5 a + 1, b, c 6 a, b + 1, c 7 a, b, c + 1 d m d = 1 a a = 4k 5 4k + 1 k a = 84 b = 5 c = 90 m 5 84 6 5 7 90 1+ 5 A B C D S A C 1 = AB BC E BC F AD A C EF AC EF S BC = x EF = x ES = x/ AC = x + 1 CS = x + 1/ CSE CBA x + 1 : x = x : 1 x 4 x 1 = 0 z = x z z 1 = 0 z = 1+ 5 1 + 5 x =,

D C F x +1 S x x E A 1 B b y g r (r + g + b+y)! = 0! r! 0! 0(0 1) (b + 1) = r! g! b! y! r! g! y! b 18 0 19 = 80 1140 b = 17 1140 r = g = 0 y = 0 n n

n n 60 90 108 10 10 n n n 60 n 180 60 n = 150 180 60 n = 168 n = 1 n = 0 60 60 60 108 = 84 60 108 84 5 5 P P P

P P P P P 88 8 8 94 89 88 4 1 7 106 89 5 14 7 94 1 5 1 8 1 1 4 8 11 014 11 014 11 014 = (10 + 1) 014 = ( ) ( ) 014 014 = 1 + 10 + 0 1 ( 014 ) 100 + 1000 (( 014 ) ) +, 11 014 1+014 10+1007 01 100 1+140+7 100 41. ( 1000) 171 51

A a i i A a 0 = 1 a i+1 a i i + 1 A i A a i+1 = 1 (1 a i), a i+1 1 = 1 ( a i 1 ). a 0 = 1 = 1 + a i = 1 + ( 1 i. ) a 10 = 1 + 1 51 = 51 51 = 171 51. n n b (60 b) (b + 100) + (b + 99) + + (b + 100 (60 b 1)) = = (b + 100 + b + 41) (60 b) = (b 1b 80) = = ((b 6,5) 86,5).

b = 6 b = 7 86 = 49 N = { 1,,, 4,... } S a N N S 6 a 6 5 6 = 6 S S n 6/ n S 6 = 6 S 6 S n S 6/ n N 6 /n S 6 = 4 4 S 6 5 014 1 ( ) 5 014 ( 10 8 014 ( 10) + 4 ) 014 10 a = (4/5) 014 b = (1/) 014 c = (4/10) 014 1 a b+c 014 014 + 014 01 + 014 01 + + 014 014. x x x 00

{ 0, x x + x = 1 1,,..., 1 = 178 1 = 197 > 014 (014 1) = 00 1 + + + 01 A = 1 + + + 01 N N A N 01 N 01 ( 011) N 0 ( 014) A = 01 014 A 0 ( 014) A = 6 ( 011) A A A N 011 014 16 1 16 1 16 1 014 14 1 (7 9 1) = 45 41

b c (014 14 ) = c b = (c b)(c + b) 014 14 c b c + b c b c + b (014 14 ) = 8 19 8 5 8 c b 7 9 c b = c + b c b > c + b 1 (7 9 1) x = c b y = c + b y > x x y c = y+x b = y x 1 (7 9 1) 4 = 4! +! +! +! +! + 1! + 1! n a 1 a a k k! n = [a k, a k 1,..., a, a 1 ] i + 1 i! (i + 1)! a i i 4 = [1,, 0, 0] [k, k 1,...,, 1]+1 = [1, 0,..., 0] [, 4,,, 1] = 1! +! +! +! +! +! + 4! + 4! + 4! + 4! + 5! + 5! = = (5! 1) + 5! + 5! = 59. n n

n = 8,5 A B A B n 7 n = 7 8 S i (x, y) x y x y i S 9 S 1 S S 1 S = {(, ), (4, 9), (6, 6)} S S 1 S S 1 S

S S 1 S = {(1, 4)} S 4 S 4 = {(, )} S 5 = {(1, 6)}, S 6 = {(, 4)}, S 7 = {(, 6)}, S 8 = {(4, 4)}, S 9 = {(, 8)}. 00 9 r O T V S P O X OP T Y V T Y SP Z T V O T X S Z Y P

V Y = 10 V T T O T XO V SY T X = 1 1 r OX = 5 1 r T Z = V S r OX = 10 18 1 r, V SY T ZY ( ZY = 10 18 ) 1 r 5 15 = 50 1 6 r. SZ = SY ZY = 15 6r ZP = T X SP = SZ + ZP = 4 6 r + 15 6 r = 9 6 r = r. P S 100 SP = 100 r = 100 r = 00 9 7 7 1,,..., 7 90 c irc

A B C D E F G A B C D E F G 1 4 5 6 7 1 4 5 6 7 1 A B C D E F G 1 A B C D E F G 4 4 5 5 6 6 7 7 A B C AB D E F AC EF BF CD G H DEG 18 F GH C F G H A D E B

1,8 [XY Z] XY Z AEC G EG GF = DG = 1 DC A B C A + 1B + C A+C = 4F BF + 1 = 4 1 A + B = D CD +1 = H DH = 1 DG 5 DC GH = 15 DC GH = 5 [DEG] DG EG = [F GH] F G HG = 5 = 10, [F GH] = 1,8 x 1 x 1 x 5 9 x = 1 1 6 = k k k 6 k = 9

ABC DEF D E F D E F AD BE CF AB = 1 AD = AD = h + AD = h + 1, h F D E A D B C h = / AD = 1 6 h = 9 / = 18 AD = h + 1 5 = = 5 1 = 5 9.