Algebraic Inequalities in Mathematical Olympiads: Problems and Solutions

Algebraic Inequalities in Mathematical Olympiads: Problems and Solutions Mohammad Mahdi Taheri July 0, 05 Abstract This is a collection of recent algebraic inequalities proposed in math Olympiads from around the world. mohammadmahdit@gmail.com www.hamsaze.com Problems. (Azerbaijan JBMO TST 05) With the conditions a, b, c R + and a + b + c =, prove that 7 + b + a + 7 + c + b + 7 + a + c 69 4. (Azerbaijan JBMO TST 05) a, b, c R + and a + b + c = 48. Prove that a b 3 + 6 + b c 3 + 6 + c a 3 + 6 4 3. (Azerbaijan JBMO TST 05) a, b, c R + prove that [(3a +) +(+ 3 b ) ][(3b +) +(+ 3 c ) ][(3c +) +(+ 3 a ) ] 48 3 4. (AKMO 05) Let a, b, c be positive real numbers such that abc =. Prove the following inequality: a 3 + b 3 + c 3 + ab a + b + bc b + c + ca c + a 9 5. (Balkan MO 05) If a, b and c are positive real numbers, prove that a 3 b 6 + b 3 c 6 + c 3 a 6 + 3a 3 b 3 c 3 abc ( a 3 b 3 + b 3 c 3 + c 3 a 3) + a b c ( a 3 + b 3 + c 3). 6. (Bosnia Herzegovina TST 05) Determine minimum value of the following expression: a + a(a + ) + b + b(b + ) + c + c(c + ) for positive real numbers such that a + b + c 3

7. (China 05) Let z, z,..., z n be complex numbers satisfying z i r for some r (0, ). Show that z i n ( r ). i= z i= i 8. (China TST 05) Let a, a, a 3,, a n be positive real numbers. For the integers n, prove that ( n j j= n j= a j k= a k ) j n + ( n i= a i) n ( n j ) j= k= a j k n + n 9. (China TST 05) Let x, x,, x n (n ) be a non-decreasing monotonous sequence of positive numbers such that x, x,, xn n is a non-increasing monotonous sequence.prove that n i= x i n ( n i= x i) n n + n n! 0. (Junior Balkan 05) Let a, b, c be positive real numbers such that a + b + c = 3. Find the minimum value of the expression A = a3 a + b3 b + c3. c. (Romania JBMO TST 05) Let x,y,z > 0. Show that : x 3 z 3 + x y + y 3 x 3 + y z + z 3 y 3 + z x 3. (Romania JBMO TST 05) Let a, b, c > 0 such that a bc, b ca and c ab. Find the maximum value that the expression : can acheive. E = abc(a bc )(b ca )(c ab ) 3. (Romania JBMO TST 05) Prove that if a, b, c > 0 and a + b + c =, then bc + a + a + ca + b + + b + ab + c + + c 39 + 0 4. (Kazakhstan 05 ) Prove that + 3 + + (n + ) < n ( ). n

( 5. (Moldova TST 05) Let c 0, π ), ( ) ( a = cos (c) ), b = sin (c) sin(c) cos(c). Prove that at least one of a, b is bigger than 05. 6. (Moldova TST 05) Let a, b, c be positive real numbers such that abc =. Prove the following inequality: a 3 + b 3 + c 3 + ab a + b + bc b + c + ca c + a 9 7. (All-Russian MO 04) Does there exist positive a R, such that for all x R? cos x + cos ax > sin x + sin ax 8. (Balkan 04) Let x, y and z be positive real numbers such that xy + yz + xz = 3xyz. Prove that and determine when equality holds. x y + y z + z x (x + y + z) 3 9. (Baltic Way 04) Positive real numbers a, b, c satisfy a + b + c the inequality a3 + b + b3 + c + c3 + a 3. = 3. Prove 0. (Benelux 04) Find the smallest possible value of the expression a + b + c b + c + d c + d + a d + a + b + + + d a b c in which a, b, c, and d vary over the set of positive integers. (Here x denotes the biggest integer which is smaller than or equal to x.). (Britain 04) Prove that for n the following inequality holds: ( + n + 3 +... + ) > ( n n +... + ). n. (Bosnia Herzegovina TST 04) Let a,b and c be distinct real numbers. a) Determine value of b) Determine value of + ab a b + bc b c + + bc b c + ca c a + + ca c a + ab a b ab a b bc b c + bc b c ca c a + ca c a ab a b 3

c) Prove the following ineqaulity When does quality holds? + a b (a b) + + b c (b c) + + c a (c a) 3 3. (Canada 04) Let a, a,..., a n be positive real numbers whose product is. Show that the sum a +a + a (+a )(+a ) + a 3 (+a )(+a )(+a 3) + + is greater than or equal to n n. a n (+a )(+a ) (+a n) 4. (CentroAmerican 04) Let a, b, c and d be real numbers such that no two of them are equal, a b + b c + c d + d a = 4 and ac = bd. Find the maximum possible value of a c + b d + c a + d b. 5. (China Girls Math Olympiad 04) Let x, x,..., x n be real numbers, where n is a given integer, and let x, x,..., x n be a permutation of,,..., n. Find the maximum and minimum of n x i+ x i i= (here x is the largest integer not greater than x). 6. (China Northern MO 04) Define a positive number sequence sequence {a n } by a =, (n + )a n = (n ) a n. Prove that a + a + + a + n a. n 7. (China Northern MO 04) Let x, y, z, w be real numbers such that x + y + 3z + 4w =. Find the minimum of x + y + z + w + (x + y + z + w) 8. (China TST 04) For any real numbers sequence {x n },suppose that {y n } is a sequence such that: y = x, y n+ = x n+ ( n x i ) (n ). Find the smallest positive number λ such that for any real numbers sequence {x n } and all positive integers m,we have m m m x i λ m i yi. i= i= i= 4

9. (China TST 04) Let n be a given integer which is greater than. Find the greatest constant λ(n) such that for any non-zero complex z, z,, z n,we have z k λ(n) min { z k+ z k }, k n where z n+ = z. k= 30. (China Western MO 04) Let x, y be positive real numbers.find the minimum of x y x + y + + y x. 3. (District Olympiad 04) Prove that for any real numbers a and b the following inequality holds: ( a + ) ( b + ) + 50 (a + ) (3b + ) 3. (ELMO Shortlist 04) Given positive reals a, b, c, p, q satisfying abc = and p q, prove that p ( a + b + c ) ( + q a + b + ) (p + q)(a + b + c). c 33. (ELMO Shortlist 04) Let a, b, c, d, e, f be positive real numbers. Given that def + de + ef + fd = 4, show that ((a + b)de + (b + c)ef + (c + a)fd) (abde + bcef + cafd). 34. (ELMO Shortlist 04) Let a, b, c be positive reals such that a + b + c = ab + bc + ca. Prove that (a + b) ab bc (b + c) bc ca (c + a) ca ab a ca b ab c bc. 35. (ELMO Shortlist 04) Let a, b, c be positive reals with a 04 + b 04 + c 04 + abc = 4. Prove that a 03 + b 03 c c 03 + b03 + c 03 a a 03 + c03 + a 03 b b 03 a 0 +b 0 +c 0. 36. (ELMO Shortlist 04) Let a, b, c be positive reals. Prove that a (bc + a ) b (ca + b b + c + ) c (ab + c c + a + ) a + b a + b + c. 37. (Korea 04) Suppose x, y, z are positive numbers such that x+y +z =. Prove that ( ( + xy + yz + zx)( + 3x 3 + 3y 3 + 3z 3 ) x ) + x 9(x + y)(y + z)(z + x) 4 + y + y + z + z 3 + 9x 4 3 + 9y 4. 3 + 9z 5

38. (France TST 04) Let n be a positive integer and x, x,..., x n be positive reals. Show that there are numbers a, a,..., a n {, } such that the following holds: a x + a x + + a n x n (a x + a x + + a n x n ) 39. (Harvard-MIT Mathematics Tournament 04) Find the largest real number c such that 0 x i cm i= whenever x,..., x 0 are real numbers such that x + + x 0 = 0 and M is the median of x,..., x 0. 40. (India Regional MO 04) Let a, b, c be positive real numbers such that + a + + b + + c. Prove that ( + a )( + b )( + c ) 5. When does equality hold? 4. (India Regional MO 04) Let x, x, x 3... x 04 be positive real numbers such that 04 j= x j =. Determine with proof the smallest constant K such that 04 x j K x j j= 4. (IMO Training Camp 04) Let a, b be positive real numbers.prove that ( + a) 8 + ( + b) 8 8ab(a + b) 43. (Iran 04) Let x, y, z be three non-negative real numbers such that Prove that x + y + z = (xy + yz + zx). x + y + z 3 3 xyz. 44. (Iran 04) For any a, b, c > 0 satisfying a + b + c + ab + ac + bc = 3, prove that a + b + c + abc 3 45. (Iran TST 04) n is a natural number. for every positive real numbers x, x,..., x n+ such that x x...x n+ = prove that: x n +... + x n+ n n n x +... + n n x n+ 46. (Iran TST 04) if x, y, z > 0 are postive real numbers such that x +y + z = x y + y z + z x prove that ((x y)(y z)(z x)) ((x y ) + (y z ) + (z x ) ) 6

47. (Japan 04) Suppose there exist m integers i, i,..., i m and j, j,..., j m, of values in {,,..., 000}. These integers are not necessarily distinct. For any non-negative real numbers a, a,..., a 000 satisfying a +a + + a 000 =, find the maximum positive integer m for which the following inequality holds a i a j + a i a j + + a im a jm.04. 48. (Japan MO Finals 04) Find the maximum value of real number k such that a + 9bc + k(b c) + b + 9ca + k(c a) + c + 9ab + k(a b) holds for all non-negative real numbers a, b, c satisfying a + b + c =. 49. (Turkey JBMO TST 04) Determine the smallest value of (a + 5) + (b ) + (c 9) for all real numbers a, b, c satisfying a + b + c ab bc ca = 3 50. (JBMO 04) For positive real numbers a, b, c with abc = prove that ( a + ) ( + b + ) ( + c + 3(a + b + c + ) b c a) 5. (Korea 04) Let x, y, z be the real numbers that satisfies the following. (x y) + (y z) + (z x) = 8, x 3 + y 3 + z 3 = Find the minimum value of x 4 + y 4 + z 4 5. (Macedonia 04) Let a, b, c be real numbers such that a + b + c = 4 and a, b, c >. Prove that: a + b + c 8 a + b + 8 b + c + 8 c + a 53. (Mediterranean MO 04) Let a,..., a n and b..., b n be n real numbers. Prove that there exists an integer k with k n such that a i a k i= b i a k. i= 54. (Mexico 04) Let a, b, c be positive reals such that a + b + c = 3. Prove: a a + 3 bc + b b + 3 ca + c And determine when equality holds. c + 3 ab 3 7

55. (Middle European MO 04) Determine the lowest possible value of the expression a + x + a + y + b + x + b + y where a, b, x, and y are positive real numbers satisfying the inequalities a + x a + y b + x b + y. 56. (Moldova TST 04) Let a, b R + such that a+b =. Find the minimum value of the following expression: E(a, b) = 3 + a + 40 + 9b. 57. (Moldova TST 04) Consider n positive numbers 0 < x x... x n, such that x + x +... + x n =. Prove that if x n, then there 3 exists a positive integer k n such that 3 x + x +... + x k < 3 58. (Moldova TST 04) Let a, b, c be positive real numbers such that abc =. Determine the minimum value of E(a, b, c) = a 3 + 5 a 3 (b + c) 59. (Romania TST 04) Let a be a real number in the open interval (0, ). Let n be a positive integer and let f n : R R be defined by f n (x) = x + x n. Show that a( a)n + a n + a 3 ( a) n + a( a)n + a < (f an + a n f n )(a) < ( a)n + a where there are n functions in the composition. 60. (Romania TST 04) Determine the smallest real constant c such that k k= k j= x j c k= for all positive integers n and all positive real numbers x,, x n. x k 8

6. (Romania TST 04) Let n a positive integer and let f : [0, ] R an increasing function. Find the value of : max f 0 x x n k= ( x k k n 6. (Southeast MO 04) Let x, x,, x n be non-negative real numbers such that x i x j 4 i j ( i, j n). Prove that x + x + + x n 5 3. ) 63. (Southeast MO 04) Let x, x,, x n be positive real numbers such that x + x + + x n = (n ). Prove that i= x i x i+ x 3 i+ n3 n. here x n+ = x. 64. (Turkey JBMO TST 04) Prove that for positive reals a,b,c such that a + b + c + abc = 4, holds. ( + a ) ( b + ca + b ) ( c + ab + c ) a + bc 7 65. (Turkey TST 04) Prove that for all all non-negative real numbers a, b, c with a + b + c = a + b + a + c + b + c 5abc +. 66. (Tuymaada MO 04 Positive numbers a, b, c satisfy a + b + c = 3. Prove the inequality a3 + + b3 + + c3 + 3. 67. (USAJMO 04) Let a, b, c be real numbers greater than or equal to. Prove that ( 0a ) 5a + min b 5b + 0, 0b 5b + c 5c + 0, 0c 5c + a abc. 5a + 0 68. (USAMO 04) Let a, b, c, d be real numbers such that b d 5 and all zeros x, x, x 3, and x 4 of the polynomial P (x) = x 4 + ax 3 + bx + cx + d are real. Find the smallest value the product can take. (x + )(x + )(x 3 + )(x 4 + ) 9

69. (Uzbekistan 04) For all x, y, z R\{}, such that xyz =, prove that x (x ) + y (y ) + z (z ) 70. (Vietnam 04) Find the maximum of P = x 3 y 4 z 3 (x 4 + y 4 )(xy + z ) 3 + y 3 z 4 x 3 (y 4 + z 4 )(yz + x ) 3 + z 3 x 4 y 3 (z 4 + x 4 )(zx + y ) 3 where x, y, z are positive real numbers. 7. (Albania TST 03) Let a, b, c, d be positive real numbers such that abcd =.Find with proof that x = 3 is the minimal value for which the following inequality holds : a x + b x + c x + d x a + b + c + d 7. (All-Russian MO 04) Let a, b, c, d be positive real numbers such that (a + b + c + d) abcd. Prove that a + b + c + d abcd. 73. (Baltic Way 03) Prove that the following inequality holds for all positive real numbers x, y, z: x 3 y + z + y3 z + x + z3 x + y x + y + z 74. (Bosnia Herzegovina TST 03) Let x, x,..., x n be nonnegative real numbers of sum equal to. Let. Find: a) min F 3 ; b) min F 4 ; c) min F 5. F n = x + x + + x n (x x + x x 3 + + x n x ) 75. (Canada 03) Let x, y, z be real numbers that are greater than or equal to 0 and less than or equal to (a) Determine the minimum possible value of x + y + z xy yz zx and determine all triples (x, y, z) for which this minimum is obtained. (b) Determine the maximum possible value of x + y + z xy yz zx and determine all triples (x, y, z) for which this maximum is obtained.. 0

76. (China Girls MO 03) For any given positive numbers a, a,..., a n, prove that there exist positive numbers x, x,..., x n satisfying x i = i= such that for any positive numbers y, y,..., y n with y i = i= the inequality holds. i= a i x i x i + y i i= a i 77. (China 03) Find all positive real numbers t with the following property: there exists an infinite set X of real numbers such that the inequality max{ x (a d), y a, z (a + d) } > td holds for all (not necessarily distinct) x, y, z X, all real numbers a and all positive real numbers d. 78. (China Northern MO 03) If a, a,, a 03 [, ] and, find the maximum of a + a + + a 03 = 0. a 3 + a 3 + + a 3 03 79. (China TST 03) Let n and k be two integers which are greater than. Let a, a,..., a n, c, c,..., c m be non-negative real numbers such that i) a a... a n and a + a +... + a n = ; ii) For any integer m {,,..., n}, we have that c + c +... + c m m k. Find the maximum of c a k + c a k +... + c n a k n. 80. (China TST 03) Let n > be an integer and let a 0, a,..., a n be nonnegative real numbers. Definite S k = k ( k i=0 i) ai for k = 0,,..., n. Prove that ( n Sk n ) n n S k 4 45 (S n S 0 ). k=0 k=0 8. (China TST 03) Let k be an integer and let a, a,, a n, b, b,, b n be non-negative real numbers. Prove that ( n ) ( n n n i= a i ) + ( n ) b i i= n (a i + b i ) n. i=

8. (China Western MO 03) Let the integer n, and the real numbers x, x,, x n [0, ].Prove that k<j n kx k x j n 3 kx k. 83. (District Olympiad 03) Let n N and a, a,..., a n R so a + a +... + a k k, ( ) k {,,..., n}.prove that k= a + a +... + a n n + +... + n 84. (District Olympiad 03) Let a, b C. Prove that az + b z, for every z C, with z =, if and only if a + b. 85. (ELMO 03) Let a, a,..., a 9 be nine real numbers, not necessarily distinct, with average m. Let A denote the number of triples i < j < k 9 for which a i + a j + a k 3m. What is the minimum possible value of A? 86. (ELMO 03) Let a, b, c be positive reals satisfying a + b + c = 7 a + 7 b + 7 c. Prove that a a b b c c 87. (ELMO Shortlist 03) Prove that for all positive reals a, b, c, a + b + + b + c + + c + a + 3 3 abc + 3 abc +. 88. (ELMO Shortlist 03) Positive reals a, b, and c obey a +b +c Prove that a + b + c + a b + b c + c a. ab+bc+ca = ab+bc+ca+. 89. (ELMO Shortlist 03) Let a, b, c be positive reals such that a + b + c = 3. Prove that 8 cyc + (ab + bc + ca) 5. (3 c)(4 c) 90. (ELMO Shortlist 03) Let a, b, c be positive reals with a 04 + b 04 + c 04 + abc = 4. Prove that a 03 + b 03 c c 03 + b03 + c 03 a a 03 + c03 + a 03 b b 03 a 0 +b 0 +c 0.

9. (ELMO Shortlist 03) Let a, b, c be positive reals, and let 03 3 a 03 + b 03 + c 03 = P Prove that ( (P + a+b )(P + (P + a+b+c ) cyc a+b ) ) ( (P + 4a+b+c )(P + 3b+3c ) ) (P + cyc 3a+b+c )(P + 3a+b+c ). 9. (Federal Competition for Advanced students 03) For a positive integer n, let a, a,..., a n be nonnegative real numbers such that for all real numbers x > x >... > x n > 0 with x + x +... + x n <, the inequality a k x 3 k < holds. Show that k= na + (n )a +... + (n j + )a j +... + a n n (n + ). 4 93. (Korea 03) For a positive integer n, define set T = {(i, j) i < j n, i j}. For nonnegative real numbers x, x,, x n with x + x + + x n =, find the maximum value of x i x j in terms of n. (i,j) T 94. (Hong Kong 03) Let a, b, c be positive real numbers such that ab + bc + ca =. Prove that 3 3 3 4 a + 6 3b + 4 b + 6 3c + 4 c + 6 3a abc When does inequality hold? 95. (IMC 03) Let z be a complex number with z + >. Prove that z 3 + >. 96. (India Regional MO 03) Given real numbers a, b, c, d, e >. Prove that a c + b d + c e + d a + e b 0 97. (Iran TST 03) Let a, b, c be sides of a triangle such that a b c. prove that: a(a + b ab) + b(a + c ac) + c(b + c bc) a + b + c 3

98. (Macedonia JBMO TST 03) a, b, c > 0 and abc =. Prove that. ( a + b + c ) + + a + + b + + c 3 99. (Turkey JBMO TST 03) For all positive real numbers a, b, c satisfying a + b + c =, prove that a 4 + 5b 4 a(a + b) + b4 + 5c 4 b(b + c) + c4 + 5a 4 ab bc ca c(c + a) 00. (Turkey JBMO TST 03) Let a, b, c, d be real numbers greater than and x, y be real numbers such that Prove that x < y. a x + b y = (a + b ) x and c x + d y = y (cd) y/ 0. (JBMO 03) Show that ( a + b + ) ( b + a + ) 6 a + b + for all positive real numbers a and b such that ab. 0. (Kazakhstan 03) Find maximum value of when a, b, c are reals in [ ; ]. a bc + + b ac + + c ba + 03. (Kazakhstan 03) Consider the following sequence a = ; a n = a [ n ] a [ n ] 3 3 +... + a [ n ] n n Prove that n N a n < a n 04. (Korea 03) Let a, b, c > 0 such that ab + bc + ca = 3. Prove that cyc (a + b) 3 ((a + b)(a + b )) 3 05. (Kosovo 03) For all real numbers a prove that 3(a 4 + a + ) (a + a + ) 06. (Kosovo 03) Which number is bigger 0 0! or 03 03!? + 07. (Macedonia 03) Let a, b, c be positive real numbers such that a 4 + b 4 + c 4 = 3. Prove that 9 a + b 4 + c 6 + 9 a 4 + b 6 + c + 9 a 6 + b + c 4 a6 + b 6 + c 6 + 6 4

08. (Mediterranean MO 03) Let x, y, z be positive reals for which: (xy) = 6xyz Prove that:. x x + yz 3 09. (Middle European MO 03) Let a, b, c be positive real numbers such that Prove that a + b + c = a + b + c. (a + b + c) 3 7a b + + 3 7b c + + 3 7c a +. Find all triples (a, b, c) for which equality holds. 0. (Middle European MO 03) Let x, y, z, w be nonzero real numbers such that x + y 0, z + w 0, and xy + zw 0. Prove that ( x + y z + w + z + w ) + ( x x + y z x) + z ( y + w + w ) y. (Moldova TST 03) For any positive real numbers x, y, z, prove that x y + y z + z z(x + y) x(z + y) y(x + z) + + x y(y + z) z(x + z) x(x + y). (Moldova TST 03) Prove that for any positive real numbers a i, b i, c i with i =,, 3, (a 3 +b 3 +c 3 +)(a 3 +b 3 +c 3 +)(a 3 3+b 3 3+c 3 3+) 3 4 (a +b +c )(a +b +c )(a 3 +b 3 +c 3 ) 3. (Moldova TST 03) Consider real numbers x, y, z such that x, y, z > 0. Prove that ( ) (xy + yz + xz) x + y + x + z + y + z > 5. 4. (Olympic Revenge 03) Let a, b, c, d to be non negative real numbers satisfying ab + ac + ad + bc + bd + cd = 6. Prove that a + + b + + c + + d + 5. (Philippines 03) Let r and s be positive real numbers such that (r + s rs)(r + s + rs) = rs. Find the minimum value of r + s rs and r + s + rs 5

6. (Poland 03) Let k,m and n be three different positive integers. Prove that ( k ) ( m ) ( n ) kmn (k + m + n). k m n 7. (Rioplatense 03) Let a, b, c, d be real positive numbers such that a + b + c + d = Prove that ( a)( b)( c)( d) abcd 8. (Romania 03) To be considered the following complex and distinct a, b, c, d. Prove that the following affirmations are equivalent: i) For every z C this inequality takes place : z a + z b z c + z d ii) There is t (0, ) so that c = ta + ( t) b si d = ( t) a + tb 9. (Romania 03) a)prove that for any m N. + 3 +... + m < m b)let p, p,..., p n be the prime numbers less than 00. Prove that p + p +... + p n < 0 0. (Romania TST 03) Let n be a positive integer and let x,..., x n be positive real numbers. Show that: ( min x, ) + x,, + x n, cos x x n x n ( π n + max x, + x,, x. (Serbia 03) Find the largest constant K R with the following property: if a, a, a 3, a 4 > 0 are numbers satisfying for every i < j < k 4, then a i + a j + a k (a i a j + a j a k + a k a i ) a + a + a 3 + a 4 K(a a + a a 3 + a a 4 + a a 3 + a a 4 + a 3 a 4 ).. (Southeast MO 03) Let a, b be real numbers such that the equation x 3 ax + bx a = 0 has three positive real roots. Find the minimum of a 3 3ab + 3a b + x n + x n, x n ). 6

3. (Southeast MO 03) n 3 is a integer. α, β, γ (0, ). For every a k, b k, c k 0(k =,,..., n) with (k + α)a k α, k= (k + β)b k β, k= (k + γ)c k γ k= we always have Find the minimum of λ (k + λ)a k b k c k λ k= 4. (Today s Calculation of Integrals 03) Let m, n be positive integer such that m < n. () Prove the inequality as follows. n + m m(n + ) < m + (m + ) + + (n ) + n < n + m n(m ) () Prove the inequality as follows. 3 ( lim + n + + n ) (3) Prove the inequality which is made precisely in comparison with the inequality in () as follows. 9 ( 8 lim + n + + n ) 6 36 5. (Tokyo University Entrance Exam 03) Let a, b be real constants. If real numbers x, y satisfy x + y 5, x + y 5, then find the minimum value of z = x + y ax by 6. (Turkey Junior MO 03) Let x, y, z be real numbers satisfying x+y+z = 0 and x + y + z = 6. Find the maximum value of (x y)(y z)(z x) 7. (Turkey 03) Find the maximum value of M for which for all positive real numbers a, b, c we have a 3 + b 3 + c 3 3abc M(ab + bc + ca 3abc) 8. (Turkey TST 03) For all real numbers x, y, z such that x, y, z and x + y + z + xyz = 4, determine the least real number K satisfying z(xz + yz + y) xy + y + z + K. 7

9. (Tuymaada 03) Prove that if x, y, z are positive real numbers and xyz = then x 3 x + y + y3 y + z + z3 z + x 3. 30. (Tuymaada 03) For every positive real numbers a and b prove the inequality a + b ab + 3 3 a +. b 3. (USAMTS 03) An infinite sequence of real numbers a, a, a 3,... is called spooky if a = and for all integers n >, na + (n )a + (n )a 3 +... + a n + a n < 0, n a + (n ) a + (n ) a 3 +... + a n + a n > 0. Given any spooky sequence a, a, a 3,..., prove that 03 3 a + 0 3 a + 0 3 a 3 + + 3 a 0 + a 03 < 345. 3. (Uzbekistan 03) Let real numbers a, b such that a b 0. Prove that a + b + 3 a 3 + b 3 + 4 a 4 + b 4 3a + b. 33. (Uzbekistan 03) Let x and y are real numbers such that x y +yx + = 0. If S = x + + x + y(y + + x ) find (a)maxs (b) mins. 34. (Albania TST 0) Find the greatest value of the expression x 4x + 9 + y 4y + 9 + z 4z + 9 where x, y, z are nonnegative real numbers such that x + y + z =. 35. (All-Russian MO 0) The positive real numbers a,..., a n and k are such that a + + a n = 3k and a + + a n = 3k a 3 + + a 3 n > 3k 3 + k Prove that the difference between some two of a,..., a n is greater than. 8

36. (All-Russian MO 0) Any two of the real numbers a, a, a 3, a 4, a 5 differ by no less than. There exists some real number k satisfying Prove that k 5 3. a + a + a 3 + a 4 + a 5 = k a + a + a 3 + a 4 + a 5 = k 37. (APMO 0) Let n be an integer greater than or equal to. Prove that if the real numbers a, a,, a n satisfy a + a + + a n = n, then must hold. i<j n n a i a j n 38. (Balkan 0) Prove that (x + y) (z + x)(z + y) 4(xy + yz + zx), cyc for all positive real numbers x, y and z. 39. (Baltic Way 0) Let a, b, c be real numbers. Prove that ab + bc + ca + max{ a b, b c, c a } + 3 (a + b + c). 40. (Bosnia Herzegovina TST 0) Prove for all positive real numbers a, b, c, such that a + b + c = : a 3 b + c + b3 c + a + c3 a + b 3 + 3. 4. (Canada 0) Let x, y and z be positive real numbers. Show that x + xy + xyz 4xyz 4 4. (CentroAmerican 0) Let a, b, c be real numbers that satisfy = and ab + bc + ac > 0. a+c Show that a + b + c abc ab + bc + ac 4 a+b + b+c + 43. (China Girls Math Olympiad 0) Let a, a,..., a n be non-negative real numbers. Prove that a + + a ( + a )( + a ) + a a ( + a )( + a )( + a 3 ) + + a a a n ( + a )( + a ) ( + a n ). 44. (China 0) Let f(x) = (x + a)(x + b) where a, b > 0. For any reals x, x,..., x n 0 satisfying x + x +... + x n =, find the maximum of F = min {f(x i ), f(x j )} i<j n 9

45. (China 0) Suppose that x, y, z [0, ]. Find the maximal value of the expression x y + y z + z x. 46. (china TST 0) Complex numbers x i, y i satisfy x i = y i = for i =,,..., n. Let x = n x i, y = n y i and z i = xy i + yx i x i y i. Prove that. i= i= z i n i= 47. (China TST 0) Given two integers m, n which are greater than. r, s are two given positive real numbers such that r < s. For all a ij 0 which are not all zeroes,find the maximal value of the expression f = ( n j= ( m i= as ij ) r s ) r ( m i= ) n j= ar ij ). s r ) s 48. (China TST 0) Given an integer k. Prove that there exist k pairwise distinct positive integers a, a,..., a k such that for any nonnegative integers b, b,..., b k, c, c,..., c k satisfying a b i a i, i =,,..., k and k i= bci i < k i= b i, we have k k i= b ci i < k b i. 49. (Czech-Polish-Slovak MAtch 0) Let a, b, c, d be positive real numbers such that abcd = 4, a + b + c + d = 0 Find the maximum possible value of i= ab + bc + cd + da 50. (ELMO Shortlist 0) Let x, x, x 3, y, y, y 3 be nonzero real numbers satisfying x + x + x 3 = 0, y + y + y 3 = 0. Prove that x x + y y (x + y )(x + y )+ x x 3 + y y 3 (x + y )(x 3 + y 3 )+ x 3 x + y 3 y (x 3 + y 3 )(x + y ) 3. 5. (ELMO Shortlist 0) Let a, b, c be three positive real numbers such that a b c and a + b + c =. Prove that a + c a + c + b + c b + c + a + b a + b 3 6(b + c) (a + b )(b + c )(c + a ). 5. (ELMO Shortlist 0) Let a, b, c 0. Show that (a +bc) 0 +(b +ca) 0 +(c +ab) 0 (a +b +c ) 0 +(ab+bc+ca) 0 0

53. (ELMO Shortlist 0) Let a, b, c be distinct positive real numbers, and let k be a positive integer greater than 3. Show that a k+ (b c) + b k+ (c a) + c k+ (a b) a k (b c) + b k (c a) + c k (a b) k + (a + b + c) 3(k ) and a k+ (b c) + b k+ (c a) + c k+ (a b) a k (b c) + b k (c a) + c k (a b) (k + )(k + ) (a + b + c ). 3k(k ) 54. (Federal competition for advanced students 0) Determine the maximum value of m, such that the inequality (a + 4(b + c ))(b + 4(a + c ))(c + 4(a + b )) m holds for every a, b, c R \ {0} with a + b + c 3. When does equality occur? 55. (Korea 0) Let x, y, z be positive real numbers. Prove that x + xy (y + zx + z) + y + yz (z + xy + x) + z + zx (x + yz + y) 56. (Finnish National High School Math Competition 0) Let k, n N, 0 < k < n. Prove that k j= ( ) n = j ( ) n + ( ) n +... + ( ) n n k. k 57. (IMO 0) Let n 3 be an integer, and let a, a 3,..., a n be positive real numbers such that a a 3 a n =. Prove that ( + a ) ( + a 3 ) 3 ( + a n ) n > n n. 58. (India Regional MO 0) Let a and b be positive real numbers such that a + b =. Prove that a a b b + a b b a 59. (India Regional MO 0) Let a, b, c be positive real numbers such that abc(a + b + c) = 3. Prove that we have Also determine the case of equality. (a + b)(b + c)(c + a) 8. 60. (Iran TST 0) For positive reals a, b and c with ab + bc + ca =, show that a a 3( a + b + c) bc + b b ca + c c ab.

6. (JBMO 0) Let a, b, c be positive real numbers such that a + b + c =. Prove that ( a b + a c + c b + c a + b c + b ) a b c a + 6 + +. a b c When does equality hold? 6. (JBMO shortlist 0) Let a, b, c be positive real numbers such that abc =. Show that : a 3 + bc + b 3 + ca + (ab + bc + ca) c 3 + ab 6 63. (JBMO shortlist 0) Let a, b, c be positive real numbers such that a + b + c = a + b + c. Prove that : a a + ab + b b + bc + c c + ca a + b + c 64. (JBMO shortlist 0) Find the largest positive integer n for which the inequality a + b + c abc + + n abc 5 holds true for all a, b, c [0, ]. Here we make the convention abc = abc. 65. (Macedonia JBMO TST 0) Let a,b,c be positive real numbers and a + b + c + = abc. Prove that a b + + b c + + c a +. 66. (Turkey JBMO TST 0) Find the greatest real number M for which a + b + c + 3abc M(ab + bc + ca) for all non-negative real numbers a, b, c satisfying a + b + c = 4. 67. (Turkey JBMO TST 0) Show that for all real numbers x, y satisfying x + y 0 (x + y ) 3 3(x 3 + y 3 )(xy x y) 68. (Moldova JBMO TST 0) Let a, b, c, d, e, f, g, h, k 9 and a, b, c, d, e, f, g, h, k are different integers, find the minimum value of the expression and prove that it is minimum. E = abc + def + ghk 69. (Moldova JBMO TST 0) Let a, b, c be positive real numbers, prove the inequality: (a + b + c) + ab + bc + ac 6 abc(a + b + c)

70. (Kazakhstan 0) Let a, b, c, d > 0 for which the following conditions: a) (a c)(b d) = 4 b) a+c a +b +c +d a+b+c+d Find the minimum of expression a + c 7. (Kazakhstan 0) For a positive reals x,..., x n prove inequality: x + +... + x n + n n + x +...+ xn 7. (Korea 0) a, b, c are positive numbers such that a + b + c = abc +. Find the maximum value of (a bc)(b ca)(c ab) 73. (Korea 0) Let {a, a,, a 0 } = {,,, 0}. Find the maximum value of 0 (na n n a n ) n= 74. (Kyoto University Entry Examination 0) When real numbers x, y moves in the constraint with x + xy + y = 6. Find the range of x y + xy x xy y + x + y. 75. (Kyrgyzstan 0) Given positive real numbers a, a,..., a n with a +a +... + a n =. Prove that ( ) ( ) ( )... a (n ) n n a a 76. (Macedonia 0) If a, b, c, d are positive real numbers such that abcd = then prove that the following inequality holds bc + cd + da + ab + cd + da + ab + bc + da + ab + bc + cd. When does inequality hold? 77. (Middle European MO 0) Let a, b and c be positive real numbers with abc =. Prove that 9 + 6a + 9 + 6b + 9 + 6c 3 + 4(a + b + c) 78. (Olympic Revenge 0) Let x, x,..., x n positive real numbers. Prove that: x 3 cyc i + x i x i x i+ x cyc i x i+ (x i + x i+ ) 3

79. (Pre-Vietnam MO 0) For a, b, c > 0 : abc = prove that a 3 + b 3 + c 3 + 6 (a + b + c) 80. (Puerto rico TST 0) Let x, y and z be consecutive integers such that Find the maximum value of x + y + z > 45. x + y + z 8. (Regional competition for advanced students 0) Prove that the inequality a + a 3 a 4 a 6 < holds for all real numbers a. 8. (Romania 0) Prove that if n is a natural number and x, x,..., x n are positive real numbers, then: ( x 3 4 x 3 + x3 x 3 3 +... + x3 n x 3 n + x3 n x 3 ) x + x x + x 3 x n + x n x n + x (x x ) + (x x 3 ) +... + (x n x n ) + (x n x ) 83. (Romania 0) Let a, b and c be three complex numbers such that a + b + c = 0 and a = b = c =. Prove that: for any z C, z. 3 z a + z b + z c 4, 84. (Romania 0) Let a, b R with 0 < a < b. Prove that: a) for x, y, z [a, b]. b) { x + y + z 3 ab x + y + z 3 + ab a + b 3 xyz + ab x, y, z [a, b]} = [ ab, a + b]. 3 xyz 85. (Romania TST 0) Let k be a positive integer. Find the maximum value of a 3k b + b 3k c + c 3k a + k a k b k c k, where a, b, c are non-negative reals such that a + b + c = 3k. 4

86. (Romania TST 0) Let f, g : Z [0, ) be two functions such that f(n) = g(n) = 0 with the exception of finitely many integers n. Define h : Z [0, ) by h(n) = max{f(n k)g(k) : k Z}. Let p and q be two positive reals such that /p + /q =. Prove that ( h(n) n Z n Z f(n) p)/p( n Z g(n) q ) /q. 87. (South East MO 0) Let a, b, c, d be real numbers satisfying inequality a cos x + b cos x + c cos 3x + d cos 4x holds for any real number x. Find the maximal value of a + b c + d and determine the values of a, b, c, d when that maximum is attained. 88. (South East MO 0) Find the least natural number n, such that the following inequality holds: n 0 n 0 n 03 n 0 < 3 3 0 0 0 03 89. (Stanford Mathematics Tournament 0) Compute the minimum possible value of For real values x (x ) + (x ) + (x 3) + (x 4) + (x 5) 90. (Stanford Mathematics Tournament 0) Find the minimum value of xy, given that x + y + z = 7, and x, y, z are real numbers xz + xy + yz = 4 9. (TSTST 0) Positive real numbers x, y, z satisfy xyz + xy + yz + zx = x + y + z +. Prove that ( ) ( ) + x 3 + x + + y + z + y + 5/8 x + y + z. + z 3 9. (Turkey Junior MO 0) Let a, b, c be positive real numbers satisfying a 3 + b 3 + c 3 = a 4 + b 4 + c 4. Show that a a + b 3 + c 3 + b a 3 + b + c 3 + c a 3 + b 3 + c 5

93. (Turkey 0) For all positive real numbers x, y, z, show that x(x y) y(y z) z(z x) + + y(z + x) z(x + y) x(y + z) 94. (Turkey TST 0) For all positive real numbers a, b, c satisfying ab+bc+ ca, prove that a + b + c + ( 3 8abc a + + b + + ) c + 95. (Tuymaada 0) Prove that for any real numbers a, b, c satisfying abc = the following inequality holds a + b + 3 + b + c + 3 + c + a + 3. 96. (USAJMO 0) Let a, b, c be positive real numbers. Prove that a 3 + 3b 3 5a + b + b3 + 3c 3 5b + c + c3 + 3a 3 5c + a 3 (a + b + c ) 97. (Uzbekistan 0) Given a, b and c positive real numbers with ab+bc+ca =. Then prove that a 3 + 9b ac + b 3 + 9c ab + c 3 (a + b + c)3 + 9a bc 8 98. (Vietnam TST 0) Prove that c = 0 4 is the largest constant such that if there exist positive numbers a, a,..., a 7 satisfying: 7 i= a i = 4, 7 7 a 3 i + a i < c i= i= then for every i, j, k such that < j < k 7, we have that x i, x j, x k are sides of a triangle. Solutions. http://www.artofproblemsolving.com/community/c6h08444p4785586. http://www.artofproblemsolving.com/community/c6h084465p478607 3. http://www.artofproblemsolving.com/community/c6h084477p4786093 4. http://www.artofproblemsolving.com/community/c6h07850p46768 5. http://www.artofproblemsolving.com/community/c6h08543p479493 6. http://www.artofproblemsolving.com/community/c6h09044p48488 7. http://www.artofproblemsolving.com/community/c6h687p36859 6

8. http://www.artofproblemsolving.com/community/c6h064458p469006 9. http://www.artofproblemsolving.com/community/c6h06957p464545 0. http://www.artofproblemsolving.com/community/c6h0699p50884. http://www.artofproblemsolving.com/community/c6h089033p483839. http://www.artofproblemsolving.com/community/c6h08904p48386 3. http://www.artofproblemsolving.com/community/c6h095599p49078 4. http://www.artofproblemsolving.com/community/c6h6040p370678 5. http://www.artofproblemsolving.com/community/c6h07847p46767 6. http://www.artofproblemsolving.com/community/c6h07850p46768 7. http://www.artofproblemsolving.com/community/c6h587594p347899 8. http://www.artofproblemsolving.com/community/c6h5885p34849 9. http://www.artofproblemsolving.com/community/c6h63430p36497 0. http://www.artofproblemsolving.com/community/c6h598349p3550637. http://www.artofproblemsolving.com/community/c6h586449p3469333. http://www.artofproblemsolving.com/community/c6h588854p3486360 3. http://www.artofproblemsolving.com/community/c6h589037p3487565 4. http://www.artofproblemsolving.com/community/c6h593098p356794 5. http://www.artofproblemsolving.com/community/c6h6050p357550 6. http://www.artofproblemsolving.com/community/c6h603p357465 7. http://www.artofproblemsolving.com/community/c6h6090p35736 8. http://www.artofproblemsolving.com/community/c6h580475p34734 9. http://www.artofproblemsolving.com/community/c6h5865p344049 30. http://www.artofproblemsolving.com/community/c6h6057p357744 3. http://www.artofproblemsolving.com/community/c6h59373p350937 3. http://www.artofproblemsolving.com/community/c6h59936p3557460 33. http://www.artofproblemsolving.com/community/c6h59937p355747 34. http://www.artofproblemsolving.com/community/c6h59937p355747 35. http://www.artofproblemsolving.com/community/c6h599343p355747 36. http://www.artofproblemsolving.com/community/c6h599363p355746 37. http://www.artofproblemsolving.com/community/c6h58893p3438505 7

38. http://www.artofproblemsolving.com/community/c6h593544p35999 39. http://www.artofproblemsolving.com/community/c9h596746p35405 40. http://www.artofproblemsolving.com/community/c6h66647p367349 4. http://www.artofproblemsolving.com/community/c6h66665p3673483 4. http://www.artofproblemsolving.com/community/c6h597388p35450 43. http://www.artofproblemsolving.com/community/c6h587736p34796 44. http://www.artofproblemsolving.com/community/c6h60466p3590688 45. http://www.artofproblemsolving.com/community/c6h58589p346585 46. http://www.artofproblemsolving.com/community/c6h590573p3497489 47. http://www.artofproblemsolving.com/community/c6h5704p3354443 48. http://www.artofproblemsolving.com/community/c6h576463p3397493 49. http://www.artofproblemsolving.com/community/c6h594596p356776 50. http://www.artofproblemsolving.com/community/c6h59489p35906 5. http://www.artofproblemsolving.com/community/c6h6845p37766 5. http://www.artofproblemsolving.com/community/c6h58584p346073 53. http://www.artofproblemsolving.com/community/c6h59838p35504 54. http://www.artofproblemsolving.com/community/c6h6406p365343 55. http://www.artofproblemsolving.com/community/c6h60704p3607066 56. http://www.artofproblemsolving.com/community/c6h57900p346647 57. http://www.artofproblemsolving.com/community/c6h5830p3447545 58. http://www.artofproblemsolving.com/community/c6h58304p344755 59. http://www.artofproblemsolving.com/community/c6h6879p377794 60. http://www.artofproblemsolving.com/community/c6h6884p377806 6. http://www.artofproblemsolving.com/community/c6h6896p377848 6. http://www.artofproblemsolving.com/community/c6h599769p35608 63. http://www.artofproblemsolving.com/community/c6h59999p35660 64. http://www.artofproblemsolving.com/community/c6h6463p365436 65. http://www.artofproblemsolving.com/community/c6h58030p34670 66. http://www.artofproblemsolving.com/community/c6h59756p35458 67. http://www.artofproblemsolving.com/community/c5h58759p347768 8

68. http://www.artofproblemsolving.com/community/c5h58753p3477753 69. http://www.artofproblemsolving.com/community/c6h59765p35458 70. http://www.artofproblemsolving.com/community/c6h5697p334430 7. http://www.artofproblemsolving.com/community/c6h535703p307475 7. http://www.artofproblemsolving.com/community/c6h5353p3070765 73. http://www.artofproblemsolving.com/community/c6h569073p3338904 74. http://www.artofproblemsolving.com/community/c6h53500p3068370 75. http://www.artofproblemsolving.com/community/c6h59049p3496307 76. http://www.artofproblemsolving.com/community/c6h54868p38684 77. http://www.artofproblemsolving.com/community/c6h5604p90650 78. http://www.artofproblemsolving.com/community/c6h5573p399584 79. http://www.artofproblemsolving.com/community/c6h5507p969643 80. http://www.artofproblemsolving.com/community/c6h55603p97604 8. http://www.artofproblemsolving.com/community/c6h56394p984683 8. http://www.artofproblemsolving.com/community/c6h549479p387785 83. http://www.artofproblemsolving.com/community/c6h5483p9685 84. http://www.artofproblemsolving.com/community/c6h54790p96808 85. http://www.artofproblemsolving.com/community/c6h53954p304304 86. http://www.artofproblemsolving.com/community/c6h539538p304300 87. http://www.artofproblemsolving.com/community/c6h545067p35935 88. http://www.artofproblemsolving.com/community/c6h545069p35937 89. http://www.artofproblemsolving.com/community/c6h545070p35938 90. http://www.artofproblemsolving.com/community/c6h599343p355747 9. http://www.artofproblemsolving.com/community/c6h54507p35939 9. http://www.artofproblemsolving.com/community/c6h539383p30399 93. http://www.artofproblemsolving.com/community/c6h5660p98347 94. http://www.artofproblemsolving.com/community/c6h567387p3346 95. http://www.artofproblemsolving.com/community/c7h54803p376380 96. http://www.artofproblemsolving.com/community/c6h56697p33065 97. http://www.artofproblemsolving.com/community/c6h53375p303549 9

98. http://www.artofproblemsolving.com/community/c6h535756p307549 99. http://www.artofproblemsolving.com/community/c6h5366p30897 00. http://www.artofproblemsolving.com/community/c6h53664p3080 0. http://www.artofproblemsolving.com/community/c6h54049p30966 0. http://www.artofproblemsolving.com/community/c6h55649p9769 03. http://www.artofproblemsolving.com/community/c6h54677p96784 04. http://www.artofproblemsolving.com/community/c6h569p375975 05. http://www.artofproblemsolving.com/community/c4h5340p95533 06. http://www.artofproblemsolving.com/community/c6h53393p95579 07. http://www.artofproblemsolving.com/community/c6h5855p3006896 08. http://www.artofproblemsolving.com/community/c6h538745p3097790 09. http://www.artofproblemsolving.com/community/c6h589868p34933 0. http://www.artofproblemsolving.com/community/c6h589873p349338. http://www.artofproblemsolving.com/community/c6h59888p3064. http://www.artofproblemsolving.com/community/c6h5999p3079 3. http://www.artofproblemsolving.com/community/c6h59936p30835 4. http://www.artofproblemsolving.com/community/c6h5884p9645 5. http://www.artofproblemsolving.com/community/c6h597p93775 6. http://www.artofproblemsolving.com/community/c6h534767p306575 7. http://www.artofproblemsolving.com/community/c6h60354p3583779 8. http://www.artofproblemsolving.com/community/c6h57865p3000 9. http://www.artofproblemsolving.com/community/c6h57866p30004 0. http://www.artofproblemsolving.com/community/c6h6906p377903. http://www.artofproblemsolving.com/community/c6h58744p3009475. http://www.artofproblemsolving.com/community/c6h5467p36497 3. http://www.artofproblemsolving.com/community/c6h54854p377403 4. http://www.artofproblemsolving.com/community/c7h5867p9085 5. http://www.artofproblemsolving.com/community/c6h5345p94446 6. http://www.artofproblemsolving.com/community/c6h56489p330604 7. http://www.artofproblemsolving.com/community/c6h564744p330778 30

8. http://www.artofproblemsolving.com/community/c6h5785p3000065 9. http://www.artofproblemsolving.com/community/c6h545390p353978 30. http://www.artofproblemsolving.com/community/c6h544756p349358 3. http://www.artofproblemsolving.com/community/c3h56384p393988 3. http://www.artofproblemsolving.com/community/c6h535868p3076 33. http://www.artofproblemsolving.com/community/c6h535869p30767 34. http://www.artofproblemsolving.com/community/c6h534833p306669 35. http://www.artofproblemsolving.com/community/c6h4899p699660 36. http://www.artofproblemsolving.com/community/c6h48890p69946 37. http://www.artofproblemsolving.com/community/c6h47956p648 38. http://www.artofproblemsolving.com/community/c6h4773p6763 39. http://www.artofproblemsolving.com/community/c6h50834p855937 40. http://www.artofproblemsolving.com/community/c6h480359p68993 4. http://www.artofproblemsolving.com/community/c6h4785p67739 4. http://www.artofproblemsolving.com/community/c6h4847p7599 43. http://www.artofproblemsolving.com/community/c6h4937p7665 44. http://www.artofproblemsolving.com/community/c6h457944p57078 45. http://www.artofproblemsolving.com/community/c6h50330p8569 46. http://www.artofproblemsolving.com/community/c6h469500p6848 47. http://www.artofproblemsolving.com/community/c6h47050p634653 48. http://www.artofproblemsolving.com/community/c6h4756p639960 49. http://www.artofproblemsolving.com/community/c6h5938p3067 50. http://www.artofproblemsolving.com/community/c6h48697p78454 5. http://www.artofproblemsolving.com/community/c6h48696p78453 5. http://www.artofproblemsolving.com/community/c6h48699p78457 53. http://www.artofproblemsolving.com/community/c6h48698p78456 54. http://www.artofproblemsolving.com/community/c6h483098p706843 55. http://www.artofproblemsolving.com/community/c6h47575p640006 56. http://www.artofproblemsolving.com/community/c6h53979p3049598 57. http://www.artofproblemsolving.com/community/c6h48834p736375 3

58. http://www.artofproblemsolving.com/community/c6h50977p86508 59. http://www.artofproblemsolving.com/community/c6h5098p86598 60. http://www.artofproblemsolving.com/community/c6h479380p68478 6. http://www.artofproblemsolving.com/community/c6h48674p749 6. http://www.artofproblemsolving.com/community/c6h63849p3734374 63. http://www.artofproblemsolving.com/community/c6h6385p3734377 64. http://www.artofproblemsolving.com/community/c6h63854p3734384 65. http://www.artofproblemsolving.com/community/c6h4959p78474 66. http://www.artofproblemsolving.com/community/c6h4864p697895 67. http://www.artofproblemsolving.com/community/c6h48773p698639 68. http://www.artofproblemsolving.com/community/c6h467886p69744 69. http://www.artofproblemsolving.com/community/c6h49308p765684 70. http://www.artofproblemsolving.com/community/c6h478833p68337 7. http://www.artofproblemsolving.com/community/c6h480480p690898 7. http://www.artofproblemsolving.com/community/c6h494547p77588 73. http://www.artofproblemsolving.com/community/c6h49455p77584 74. http://www.artofproblemsolving.com/community/c6h466336p6546 75. http://www.artofproblemsolving.com/community/c6h535p3045 76. http://www.artofproblemsolving.com/community/c6h47398p653838 77. http://www.artofproblemsolving.com/community/c6h498389p800446 78. http://www.artofproblemsolving.com/community/c6h48056p70040 79. http://www.artofproblemsolving.com/community/c6h45403p5505 80. http://www.artofproblemsolving.com/community/c6h476867p67058 8. http://www.artofproblemsolving.com/community/c6h48086p693 8. http://www.artofproblemsolving.com/community/c6h473475p65093 83. http://www.artofproblemsolving.com/community/c6h473478p65096 84. http://www.artofproblemsolving.com/community/c6h473479p65097 85. http://www.artofproblemsolving.com/community/c6h47906p68 86. http://www.artofproblemsolving.com/community/c6h479953p687057 87. http://www.artofproblemsolving.com/community/c6h5449p34434 3

88. http://www.artofproblemsolving.com/community/c6h544p344356 89. http://www.artofproblemsolving.com/community/c383h46504p605906 90. http://www.artofproblemsolving.com/community/c383h46637p6056 9. http://www.artofproblemsolving.com/community/c6h489749p74586 9. http://www.artofproblemsolving.com/community/c6h5070p875068 93. http://www.artofproblemsolving.com/community/c6h508908p859975 94. http://www.artofproblemsolving.com/community/c6h4778p6433 95. http://www.artofproblemsolving.com/community/c6h49000p747508 96. http://www.artofproblemsolving.com/community/c5h4767p6694 97. http://www.artofproblemsolving.com/community/c6h4863p697894 98. http://www.artofproblemsolving.com/community/c6h475453p66737 33