Old And New Inequalities - Volume 2- Chapter 1(1)

Ve Quec BA CAN COSMIN POHOATA , Old and New Inequa i ies Volume 2 ~GIL Preface ”The last thing one knows when writing a book is what to put first.” Blaise Pascal Mathematics has been called the science of tautology; that is to say, mathematicians have been accused of spending their time proving that things are equal to themselves. This statement is rather inaccurate on two counts. In the first place, mathematics, although the language of science, is not a science. More likely it is a creative art, as G. H. Hardy liked to consider it. Secondly, the fundamental results of mathematics are often inequalities rather than equalities. In the pages that follow, we present a large variety of problems involving such inequal- ities, questions that became famous in (mathematical) competitions or journals because of their beauty. The most important prerequisite for benefiting from this book is the desire to master the craft of discovery and proof. The formal requirements are quite modest. Any- one who knows basic inequalities such as the ones of Cauchy-Schwarz, Ho¨lder, Schur, Chebyshev or Bernoulli is well prepared for almost everything to be found here. The student who is not that experienced will also be exposed in the first part to a wide combi- nation of moderate and easy problems, ideas, techniques, and all the ingredients leading to a good preparation for mathematical contests. Some of the problems we chose to discuss are known, but we have included them here with new solutions which show the diversity of ideas pertaining to inequalities. Nevertheless, the book develops many results which are rarely seen, and even experienced readers are likely to find material that is challenging and informative. To solve a problem is a very human undertaking, and more than a little mystery remains about how we best guide ourselves to the discovery of original solutions. Still, as George Po´lya and the others have taught us, there are principles of problem solving. With practice and good coaching we can all improve our skills. Just like singers, actors, or pianists, we have a path toward a deeper mastery of our craft. The authors September 2008 5 About the authors Vo Quoc Ba Can is a student at the ”Can Tho” University of Medicine and Pharmacy. As a high-school student, he participated in many national contests obtaining several prizes. Though at the moment he is not studying mathematics, his activity in Inequalities has proved to be quite wide lately. Some of his problems were published in specialized jour- nals, but the biggest part of them became popular on the wordwide known MathLinks fo- rum. On the same theme, he (co)authored several manuscripts, which were (unfortunately) published in Vietnamese. Cosmin Pohoat¸a˘ is in present a high-school student at the ”Tudor Vianu” High School in Bucharest, Romania. During his scholar activity he participated in many (mathemati- cal or not) olympiads and contests. Recently, he was awarded with a Gold Medal at the Sharygin International Mathematical Olympiad, which took place in Dubna, Russia from July 29 to August 1, 2008. In the past few years, he had many important contributions in Euclidean Geometry, distinguishing himself in journals like Forum Geometricorum, Crux Mathematicorum or the American Mathematical Monthly. In Clark Kimberling’s Encyclo- pedia of Triangle Centers, a point appears under his name (X3333 - ”The Pohoata Point”). His main mathematical interests besides Euclidean Geometry are Graph Theory, Combi- natorial Number Theory and, of course, Inequalities. Beyond mathematics, his activities include computer science, philosophy, music, football (soccer) and tennis. 6 Table of contents 1 Problems 8 2 Solutions 22 7 Chapter 1 Problems 1. Prove that for all positive real numbers a, b the following inequality holds√ 2a(a+ b)3 + b √ 2(a2 + b2) ≤ 3(a2 + b2). 2. Consider real numbers a, b, c contained in the interval [ 12 , 1]. Prove that 2 ≤ a+ b 1 + c + b+ c 1 + a + c+ a 1 + b ≤ 3. 3. Let a, b, c be three positive real numbers contained in the interval [0, 1]. Prove that 1 ab+ 1 + 1 bc+ 1 + 1 ca+ 1 ≤ 5 a+ b+ c . 4. Let x, y, z be positive real numbers such that xyz = 1. Show that the following inequality holds: 1 (x+ 1)2 + y2 + 1 + 1 (y + 1)2 + z2 + 1 + 1 (z + 1)2 + x2 + 1 ≤ 1 2 . 5. Let a, b, c be three positive real numbers satisying abc = 8. Prove that a− 2 a+ 1 + b− 2 b+ 1 + c− 2 c+ 1 ≤ 0. 6. Let a, b, c be the sidelengths of an acute-angled triangle. Prove that (a+ b+ c)(a2 + b2 + c2)(a3 + b3 + c3) ≥ 4(a6 + b6 + c6). 7. Let a, b, c be positive real numbers such that ab+ bc+ ca = 1. Prove that a √ b2 + c2 + bc+ b √ c2 + a2 + ca+ c √ a2 + b2 + ab ≥ 3. 8 9 8. Find the maximum value of (x3 + 1)(y3 + 1), for all real numbers x, y, satisfying the condition that x+ y = 1. 9. Let a, b, c be positive real numbers. Prove that a b + b c + c a ≥ a+ b b+ c + b+ c a+ b + 1. 10. If x, y, z are positive real numbers, prove that the following inequality holds: (x+ y + z)2(yz + zx+ xy)2 ≤ 3(y2 + yz + z2)(z2 + zx+ x2)(x2 + xy + y2). 11. Let a, b, c be positive real numbers such that a+ b+ c ≥ 1 a + 1 b + 1 c . Prove that a+ b+ c ≥ 3 a+ b+ c + 2 abc . 12. Let a, b, c be positive real numbers such that 1 a+ b+ 1 + 1 b+ c+ 1 + 1 c+ a+ 1 ≥ 1. Prove that a+ b+ c ≥ ab+ bc+ ca. 13. Let a, b, c be real numbers satisfying a, b, c ≥ 1 and a+ b+ c = 2abc. Prove that 3 √ (a+ b+ c)2 ≥ 3√ab− 1 + 3√bc− 1 + 3√ca− 1. 14. Let a1, a2, . . . , an be positive real numbers satisfying the condition that a1 + a2 + . . .+ an = 1. Prove that n∑ j=1 aj 1 + a1 + . . .+ aj < 1√ 2 . 15. Positive numbers α, β, x1, x2, . . . , xn (n ≥ 1) satisfy the condition x1 + x2 + . . .+ xn = 1. Prove that x31 αx1 + βx2 + x32 αx2 + βx3 + . . .+ x3n αxn + βx1 ≥ 1 n(α+ β) . 16. If three nonnegative real numbers a, b, c satisfy the condition 1 a2 + 1 + 1 b2 + 1 + 1 c2 + 1 = 2, prove that ab+ bc+ ca ≤ 3 2 . 10 17. Let a, b, c be positive real numbers. Prove that a2 b + b2 c + c2 a ≥ a+ b+ c+ 4(a− b) 2 a+ b+ c . 18. If x, y, z are positive numbers satisfying the condition xy + yz + zx = 1, show that 27 4 (x+ y)(y + z)(z + x) ≥ (√x+ y +√y + z +√z + x)2 ≥ 6√3. 19. Let a, b, c be positive real numbers. Prove that a b + b c + c a ≥ 3 + (a− c) 2 ab+ bc+ ca . 20. Let a, b, c be nonnegative real numbers satisfying ab+ bc+ ca = 3. Prove that 1 1 + a2(b+ c) + 1 1 + b2(c+ a) + 1 1 + c2(a+ b) ≤ 3 1 + 2abc . 21. Let a, b, c be positive real numbers such that 2a+ b = 1. Prove that 5a3 bc + 4b3 ca + 3c3 ab ≥ 4. 22. i) If x, y and z are three real numbers, all different from 1, such that xyz = 1, then prove that x2 (x− 1)2 + y2 (y − 1)2 + z2 (z − 1)2 ≥ 1. ii) Prove that equality is achieved for infinitely many triples of rational numbers x, y and z. 23. Let a, b, c be positive real numbers. Prove that a b(b+ c)2 + b c(c+ a)2 + c a(a+ b)2 ≥ 9 4(ab+ bc+ ca) . 24. Let a, b, c be nonnegative real numbers such that a+ b+ c = 1. Prove that√ a+ (b− c)2 4 + √ b+ √ c ≤ √ 3. 25. Let a, b, c be the sidelengths of a triangle. Prove that∑ cyc a3 a3 + (b+ c)3 + 1 ≥ 2 ∑ cyc a2 a2 + (b+ c)2 . 26. Prove that for any real numbers a, b, c the following inequality holds (b+ c− a)2(c+ a− b)2(a+ b− c)2 ≥ (b2 + c2− a2)(c2 + a2− b2)(a2 + b2− c2). 11 27. Let a, b, c be the sidelengths of a given triangle. Prove that (a+ b)(b+ c)(c+ a) + (−a+ b+ c)(a− b+ c)(a+ b− c) ≥ 9abc. 28. Let a, b, c be positive real numbers. Prove that( 1 a + 1 b + 1 c )( 1 a+ 1 + 1 b+ 1 + 1 c+ 1 ) ≥ 9 abc+ 1 . 29. Let a, b, c be positive real numbers contained in the interval [0, 1]. Prove that 2a 1 + bc + 2b 1 + ca + 2c 1 + ab + abc ≤ 4. 30. Let a, b, c be nonnegative real numbers a, b, c satisfying max {b+ c− a, c+ a− b, a+ b− c} ≤ 1. Prove that a2 + b2 + c2 ≤ 1 + 2abc. 31. If x, y, z are real numbers satisfying xyz = −1, prove that x4 + y4 + z4 + 3(x+ y + z) ≥ y 2 + z2 x + z2 + x2 y + x2 + y2 z . 32. Let a, b, c, d be positive real numbers satisfying the condition a + b + c + d = abc+ bcd+ cda+ dab. Prove that a+ b+ c+ d+ 2a a+ 1 + 2b b+ 1 + 2c c+ 1 + 2d d+ 1 ≥ 8. 33. Let a, b, c be nonnegative real numbers. Prove that a2 + 2bc b2 + c2 + b2 + 2ca c2 + a2 + c2 + 2ab a2 + b2 ≥ 3. 34. Let a, b, c be positive real numbers. Prove that ab c(c+ a) + bc a(a+ b) + ca b(b+ c) ≥ a c+ a + b a+ b + c b+ c . 35. Let a, b, c be positive real numbers such that ab+ bc+ ca ≥ 3. Prove that a√ a+ b + b√ b+ c + c√ c+ a ≥ 3√ 2 . 12 36. Let x, y, z, t be positive real numbers such that 1 x+ 1 + 1 y + 1 + 1 z + 1 + 1 t+ 1 = 1. Prove that min { 1 x + 1 y + 1 z , 1 y + 1 z + 1 t , 1 z + 1 t + 1 x , 1 t + 1 x + 1 y } ≤ 1 ≤ max { 1 x + 1 y + 1 z , 1 y + 1 z + 1 t , 1 z + 1 t + 1 x , 1 t + 1 x + 1 y } . 37. Let a1, a2, . . . , an be positive real numbers. Prove that n∏ k=1  n∑ j=1 aTkj  ≥ ( n∑ k=1 a Tn+1 3 k )n , where Tk = k(k+1)2 is the k-th triangular number. 38. Let a, b, c, d be positive numbers. Prove that 3(a2 − ab+ b2)(c2 − cd+ d2) ≥ (a2c2 − abcd+ b2d2). 39. Let a, b, c be real numbers such that a+ b+ c = 1. Prove that a a2 + 1 + b b2 + 1 + c c2 + 1 ≤ 9 10 . 40. Let n be a positive integer, and let x and y be positive real numbers such that xn + yn = 1. Prove that( n∑ k=1 1 + x2k 1 + x4k )( n∑ k=1 1 + y2k 1 + y4k ) < 1 (1− x)(1− y) . 41. Let a, b, c be positive real numbers such that a+ b+ c+ 1 = 4abc. Prove that 1 a + 1 b + 1 c ≥ 3 ≥ 1√ ab + 1√ bc + 1√ ca . 42. Let a, b, c be nonnegative real numbers such that a+b+c = 3. Set x = √ a2 − a+ 1, y = √ b2 − b+ 1 and z = √c2 − c+ 1. Prove that: xy+yz+zx ≥ 3 and x+y+z ≤ 2 + √ 7. 43. Let n ≥ 2 be a given integer. Determine (a) the largest real cn such that 1 1 + a1 + 1 1 + a2 + . . .+ 1 1 + an ≥ cn 13 holds for any positive numbers a1, a2, . . . , an with a1a2 . . . an = 1. (b) the largest real dn such that 1 1 + 2a1 + 1 1 + 2a2 + . . .+ 1 1 + 2an ≥ dn holds for any positive numbers a1, a2, . . . , an with a1a2 . . . an = 1. 44. Let a, b, c be positive real numbers. Prove that bc a2 + bc + ca b2 + ca + ab c2 + ab ≤ a b+ c + b c+ a + c a+ b . 45. Real numbers a1, a2, . . . , an are given. For each i (1 ≤ i ≤ n) define di = max {aj | 1 ≤ j ≤ i} −min {aj | i ≤ j ≤ n} and let d = max {di | 1 ≤ i ≤ n}. (a) Prove that for any real numbers x1 ≤ x2 ≤ . . . ≤ xn, we have max {|xi − ai| | 1 ≤ i ≤ n} ≥ d2 . (b) Show that there are real numbers x1 ≤ x2 ≤ . . . ≤ xn such that we have equality in (a). 46. Let a, b, c be nonzero positive numbers. Prove that√ a2 4a2 + ab+ 4b2 + √ b2 4b2 + bc+ 4c2 + √ c2 4c2 + ca+ 4a2 ≤ 1. 47. Let a, b, c be positive numbers such that 4abc = a+ b+ c+ 1. Prove that b2 + c2 a + c2 + a2 b + a2 + b2 c ≥ 2(ab+ bc+ ca). 48. Let a, b, c be positive real numbers. Prove that a3 (a+ b)3 + b3 (b+ c)3 + c3 (c+ a)3 ≥ 3 8 . 49. Let a, b, c, x, y, z be positive real numbers. Prove that (a2 + x2)(b2 + y2)(c2 + z2) ≥ (ayz + bzx+ cxy − xyz)2. 50. Let x, y, z be positive real numbers. Prove that √ y + z x + √ z + x y + √ x+ y z ≥ 4 (x+ y + z)√ (y + z) (z + x) (x+ y) . 14 51. Let a, b, c be nonnegative real numbers such that abc = 4 and a, b, c > 1. Prove that (a− 1)(b− 1)(c− 1) ( a+ b+ c 3 − 1 ) ≤ ( 3 √ 4− 1 )4 . 52. Let a, b, c be positive real numbers satisfying abc = 1. Prove that 1 b(a+ b) + 1 c(b+ c) + 1 a(c+ a) ≥ 3 2 . 53. Prove that for all positive real numbers a, b, c the following inequality holds: 1 a+ b+ c ( 1 b+ c + 1 c+ a + 1 a+ b ) ≥ 1 ab+ bc+ ca + 1 2(a2 + b2 + c2) . 54. Let a, b, c be the sidelengths of a triangle. Prove that √ b+ c− a√ b+ √ c−√a + √ c+ a− b√ c+ √ a−√b + √ a+ b− c√ a+ √ b−√c ≤ 3. 55. Let a, b, c be the sidelengths of a triangle with perimeter 1. Prove that 1 < b√ a+ b2 + c√ b+ c2 + a√ c+ a2 < 2. 56. Prove that for any positive real numbers a, b and c, we have that√ b+ c a + √ c+ a b + √ a+ b c ≥ √ 6 · a+ b+ c 3 √ abc . 57. Let a, b, c be positive real numbers. Prove that a√ ab+ b2 + b√ bc+ c2 + c√ ca+ a2 ≥ 3√ 2 . 58. Let a1 ≤ a2 ≤ . . . ≤ an be positive real numbers such that a21 + a 2 2 + . . .+ a 2 n n = 1, a1 + a2 + . . .+ an n = m, where 1 ≥ m > 0. Prove that for all i satisfying ai ≤ m, we have n− i ≥ n(m− ai)2. 59. Let x, y, z be positive real numbers. Prove that 3 √ 3 2 ≤ √x+ y + z · ( √ x y + z + √ y z + x + √ z x+ y ) . 15 60. Let a, b, c be positive real numbers. Prove that a√ a2 + 2bc + b√ b2 + 2ca + c√ c2 + 2ab ≤ a+ b+ c√ ab+ bc+ ca . 61. Let a, b, c be distinct positive real numbers. Prove the following inequality: a2b+ a2c+ b2a+ b2c+ c2a+ c2b− 6abc a2 + b2 + c2 − ab− bc− ca ≥ 16abc (a+ b+ c)2 . 62. Let a, b, c be nonzero positive real numbers. Prove that a3 + abc b+ c + b3 + abc c+ a + c3 + abc a+ b ≥ a(b 3 + c3) a2 + bc + b(c3 + a3) b2 + ca + c(a3 + b3) c2 + ab . 63. Let a, b, c, d be real numbers with sum 0. Prove the inequality: (ab+ ac+ ad+ bc+ bd+ cd)2 + 12 ≥ 6(abc+ abd+ acd+ bcd). 64. Let a, b, c be positive real numbers satisfying a+ b+ c = 1. Prove that( 1 a − 2 )2 + ( 1 b − 2 )2 + ( 1 c − 2 )2 ≥ 8(a 2 + b2 + c2)2 (1− a)(1− b)(1− c) . 65. Let x1, x2, . . . , xn be real numbers from the interval [0, 1] satisfying x1x2 . . . xn = (1− x1)2(1− x2)2 . . . (1− xn)2. Find the maximum value of x1x2 . . . xn. 66. Let a, b, c be three positive real numbers with sum 3. Prove that 1 a2 + 1 b2 + 1 c2 ≥ a2 + b2 + c2. 67. Let a, b, c be positive real numbers satisfying a+ b+ c = 3. Prove that a 2b+ 1 + b 2c+ 1 + c 2a+ 1 ≤ 1 abc . 68. For any three positive numbers a, b, c, prove the inequality (1 + abc) ( 1 a (1 + b) + 1 b (1 + c) + 1 c (1 + a) ) ≥ 3. 69. Let a, b, c, d be real numbers such that a2 + b2 + c2 + d2 = 1. Prove that 1 1− ab + 1 1− bc + 1 1− cd + 1 1− da ≤ 16 3 . 16 70. Let x1, x2, . . . , xn be positive real numbers such that x1 + x2 + . . . + xn = 1. Prove that ( n∑ i=1 √ xi )( n∑ i=1 1√ 1 + xi ) ≤ n 2 √ n+ 1 . 71. Let a, b, c be positive real numbers. Prove that a4 + b4 + c4 ab+ bc+ ca + 3abc a+ b+ c ≥ 2 3 (a2 + b2 + c2). 72. Let a, b, c be nonnegative real numbers, from which at least two are nonzero. Prove that 3 √ a2 + bc b2 + c2 + 3 √ b2 + ca c2 + a2 + 3 √ c2 + ab a2 + b2 ≥ 9 3 √ abc a+ b+ c . 73. Let a1, a2, . . . , a100 be nonnegative eral numbers such that a21+a22+. . .+a2100 = 1. Prove that a21a2 + a 2 2a3 + . . .+ a 2 100a1 < 12 25 . 74. Let a, b, c be nonnegative real numbers, no two of which are zero. Prove that ab+ ac+ 4bc b2 + c2 + bc+ ba+ 4ca c2 + a2 + ca+ cb+ 4ab a2 + b2 ≥ 4. 75. Let a, b, c be positive real numbers. Prove that a+ b2 + c3 ab+ c2 + b+ c2 + a3 bc+ a2 + c+ a2 + b3 ca+ b2 ≥ 9 2 . 76. Let a, b and c be positive real numbers satisfying a+ b+ c = 2. Prove that 1 2 + ∑ cyc a b+ c ≤ a 2 + bc b+ c + b2 + ca c+ a + c2 + ab a+ b ≤ 1 2 + ∑ cyc a2 b2 + c2 . 77. Real numbers ai, bi (1 ≤ i ≤ n) satisfy ∑n i=1 a 2 i = ∑n i=1 b 2 i = 1 and ∑n i=1 aibi = 0. Prove that ( n∑ i=1 ai )2 + ( n∑ i=1 bi )2 ≤ n. 78. Let a, b, c be positive real numbers satisfying a+ b+ c = 1. Prove that ab√ ab+ bc + bc√ bc+ ca + ca√ ca+ ab ≤ √ 2 2 . 79. Let a, b, c be nonnegative real numbers satisfying a2 + b2 + c2 = 1. Prove that 1 ≤ a 1 + bc + b 1 + ca + c 1 + ab ≤ √ 2. 17 80. Let a, b, c be positive real numbers such that a ≤ b ≤ c and abc = 1. Prove that a+ b2 + c3 ≥ 1 a + 1 b2 + 1 c3 . 81. Given k + 1 positive real numbers x0, . . . , xk and a positive integer n, show that∑ σ (xσ1 + . . .+ xσk) −n ≤ k−n k∑ i=0 x−ni , where the sum on the left is taken of the k + 1 distinct k-element subsets of {x0, . . . , xk}. 82. Let a, b, c be nonnegative real numbers, such that at least two are nonzero and which satisfy the condition a+ b+ c = 1. Prove that a√ a+ 2b + b√ b+ 2c + c√ c+ 2a ≤ 4 √ 27 (√ 3− 1)√ 2 . 83. For real numbers xi > 1, 1 ≤ i ≤ n, n ≥ 2, such that x2i xi − 1 ≥ S = n∑ j=1 xj , for all i = 1, 2, . . . , n find, with proof, sup S. 84. Let a, b, c, d be positive real numbers such that a+b+c+d = abc+bcd+cda+dab. Prove that(√ a2 + 1 + √ b2 + 1 )2 + (√ c2 + 1 + √ d2 + 1 )2 ≤ (a+ b+ c+ d)2. 85. Let a, b, c be the sidelengths of a triangle. Prove that a2 ( b c − 1 ) + b2 ( c a − 1 ) + c2 (a b − 1 ) ≥ 0. 86. Let x, y, z be positive real numbers such that x2 + y2 + z2 ≥ 3. Prove that x5 − x2 x5 + y2 + z2 + y5 − y2 y5 + z2 + x2 + z5 − z2 z5 + x2 + y2 ≥ 0. 87. Let a, b, c be the sidelengths of a given triangle. Prove that a b + b c + c a + 3 ≥ 2 ( a+ b b+ c + b+ c c+ a + c+ a a+ b ) . 88. Let x, y, z be positive real numbers. Prove that( x+ y z + y + z x + z + x y ) − 4 ( x y + z + y z + x + z x+ y ) ≥ 1− 8xyz (x+ y)(y + z)(z + x) . 18 89. Let a, b, c be real numbers satisfying a2 + b2 + c2 = 9. Prove that 3 ·min {a, b, c} ≤ 1 + abc. 90. Let x1, x2, . . . , x3n be positive real numbers. Prove that 2n · 3n∏ k=1 1 + x2k 1 + xk ≥ ( 1 + 3n∏ k=1 x 1/n k )n . 91. Given an integer n ≥ 2, find the largest constant C(n) for which the inequality n∑ i=1 xi ≥ C(n) ∑ 1≤j 0, show that( n∏ i=0 (s+ i) ) n∑ j=0 1 s+ j  < (n+ 1) n∏ k=1 ( s+ k − 1 2 ) . 96. Let a, b, c be real numbers contained in the interval [ 0, 35 ] and also satisfying the condition a+b+c = 1. Determine the maximum value that the following expression can reach: P (a, b, c) = a3 + b3 + c3 + 3 4 abc. 97. Let a, b, c, d be positive real numbers such that a ≥ b ≥ c ≥ d and abcd = 1. Prove that 1 a3 + 1 + 1 b3 + 1 + 1 c3 + 1 ≥ 3 abc+ 1 . 19 98. Let a, b, c be nonnegative real numbers, from which at least are nonzero. Prove that a2(b+ c)2 b2 + c2 + b2(c+ a)2 c2 + a2 + c2(a+ b)2 a2 + b2 ≥ 2(ab+ bc+ ca). 99. Let a, b, c be nonnegative real numbers, from which at least two are nonzero. Prove that (a+ b+ c)2 ab+ bc+ ca ≥ a(b+ c) a2 + bc + b(c+ a) b2 + ca + c(a+ b) c2 + ab ≥ (a+ b+ c) 2 a2 + b2 + c2 . 100. Let a, b, c be positive real numbers. Prove that a2 − bc 4a2 + 4b2 + c2 + b2 − ca 4b2 + 4c2 + a2 + c2 − ab 4c2 + 4a2 + b2 ≥ 0. 101. Let a, b, c be nonnegative real numbers, from which at least two are nonzero. Prove that a(b+ c) b2 + c2 + b(c+ a) c2 + a2 + c(a+ b) a2 + b2 ≥ a(b+ c) a2 + bc + b(c+ a) b2 + ca + c(a+ b) c2 + ab . 102. Let n be a positive integer. Find the minimum value of (a− b)2n+1 + (b− c)2n+1 + (c− a)2n+1 (a− b)(b− c)(c− a) for distinct real numbers a, b, c with bc+ ca ≥ 1 + ab+ c2. 103. Let a, b, c be positive real numbers. Prove that ab+c (b+ c)2 + bc+a (c+ a)2 + ca+b (a+ b)2 ≥ 3 4 . 104. Find the least real number c such that if n ≥ 1 and a1, . . . , an > 0 then n∑ k=1 k∑k j=1 1/aj ≤ c n∑ k=1 ak. 105. Let a, b, c be nonnegative real numbers satisfying a+b+c = 1, and moreover from which at least two are nonzero. Prove that a √ 4b2 + c2 + b √ 4c2 + a2 + c √ 4a2 + b2 ≤ 3 4 . 106. Let a, b be real numbers such that a + b 6= 0 and let x, y > 1 be some given constants. Determine the minimum value of the following expression: f(a, b) = (a2 + 1)x(b2 + 1)y (a+ b)2 . 20 107. It is given that real numbers x1, x2, . . . , xn (n > 2) satisfy∣∣∣∣∣ n∑ i=1 xi ∣∣∣∣∣ > 1, |xi| ≤ 1 (i = 1, 2, . . . , n). Prove that ther