THE OLYMPIC MATHEMATICAL MARATHON · Mathematics Journal, Mathematics Magazine, Crux Mathematicorum. In 2009, he participated as an observer in the World Olympiad which was held in

GEORGE APOSTOLOPOULOS DANIEL SITARU

THE OLYMPIC MATHEMATICAL MARATHON

grades 7-12

VOLUME 1

Dedicated to the International Mathematical Olympiad Romania 2018

GEORGE APOSTOLOPOULOS

George Apostolopoulos was born in Messolonghi – Greece at 23 January 1949. He graduated the University of Ioannina. He is member of "The Mathematical Society of Greece" and member of the tender committee. He is also a member of "The Mathematical Society" of USA, Canada and Romania. He has published proposed mathematical problems and mathematical solutions through the scientific magazines of the "Mathematical Society" of USA and

Canada. There are more than 400 references to his name in mathematical problem-solving situations in famous magazines from the whole world such as American Mathematical Monthly, College Mathematics Journal, Mathematics Magazine, Crux Mathematicorum. In 2009, he participated as an observer in the World Olympiad which was held in Germany, Bremen and in 2010 in the Balkan Olympiad Juniors in Romania. In 2013, he participated as second-in-command at the Balkan Olympiad in Cyprus and in 2014 also as second-in-command at the Balkan Olympiad Juniors held in Ohrid. His book Beauty of proving was very appreciated and became a best-seller in mathematical world.

DANIEL SITARU

Daniel Sitaru, born on 9 August 1963 in Craiova, Romania, is a teacher at National Economic College "Theodor Costescu" in Drobeta-Turnu Severin. He published 33 mathematical books, last two of these, Math Phenomenon and Algebraic Phenomenon, were very appreciated worldwide. He is the founding editor of Romanian Mathematical Magazine, an Interactive Mathematical Journal with 3.200.000 visitors in 2017 (www.ssmrmh.ro). Many problems from his books were published in famous journals such as American Mathematical Monthly, Crux Mathematicorum, Math Problems Journal, The Pentagon Journal, La Gaceta de la RSME, SSMA Magazine. He also published an impressive number of original problems in all mathematical journals from Romania (GMB, Cardinal, Elipsa, Argument, Recreații Matematice). His articles from Crux Mathematicorum and The Pentagon Journal were also very appreciated.

Table of Contents

Chapter I

PROBLEMS .................................................................................................... 8

Chapter II

SOLUTIONS ................................................................................................. 50

Bibliography ............................................................................................ 228

Chapter I PROBLEMS

Notations: − semiperimeter of ∆ , −area of ∆ , − circumradii, − inradii, ℎ , ℎ , ℎ − altitudes, , , − medians, , , − symedians, , , − internal bisectors, , , − exradii

1.Let , , bepositiverealnumbers.Provethat:+ + + + + + + + ≥ .

George Apostolopoulos

2. Prove that in any acute-angled ∆ with length’s sides ≥ ≥ thefollowingrelationshipholds: + 2 + ≤ ( + )( + ) + ( − )( + ). Daniel Sitaru

3.Let , , bepositiverealnumberswith + + = 3.Provethat:( + 1) + ( + 1) + ( + 1) ≥ 6.George Apostolopoulos

4.In ; , , ∈ . Provethat:√ ⋅ ⋅ 1 + 1 + 1 ≤ 53 + 23 + . Daniel Sitaru

5.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:+ + + + + ≥ 32.George Apostolopoulos

6.Provethatif ∈ ( ), ∈ ( ), ∈ ( ) then: + + + + + > 2 1sin . Daniel Sitaru

7.Let , , bepositiverealnumbers.Provethat:( − + )( + ) + ( − + )( + ) + ( − + )( + ) ≥ 316.George Apostolopoulos

VOLUME I

9 8.Letbe , ∈ ( ); , ∈ ( ); , ∈ ( ) in ; ∩ ∩ ≠ ∅. Provethat: 27 ≤ + + .

Daniel Sitaru

9.Let , bepositiverealnumberswith + + = , > 0.Provethat:√ + + √ ≤ √2 + 1√3 ⋅ √ .George Apostolopoulos

10.Provethatinanytriangle thefollowingrelationshipholds:3 ≥ √4 . Daniel Sitaru

11. Let , be positive real numbers such that + + = 9. Find themaximalvalueofexpression:( + ) + ( ) + 2( ) + ( ) − 16. George Apostolopoulos . Find , , , ∈ 0, 2 suchthat:sin sin sin sin + cos cos cos cos = 1.

Daniel Sitaru

13.Let , , bepositiverealnumbers.Provethat:(2 + 3 + 3 + 1)(2 + 3 + 3 + 1)(2 + 3 + 3 + 1) ≥≥ + 8 .George Apostolopoulos

14.Provethatif ≤ ≤ in then:( ) ( ) ≥ ( ) .Daniel Sitaru

15.Let , , bepositiverealnumberswith = 1.Provethat:√ + 1 + + 1 + √ + 1+ + ≤ √2.George Apostolopoulos

16.Provethatinanytriangle thefollowingrelationshipholds:√ + √ + √ ≥ √4 √ + √ + √ . Daniel Sitaru

THE OLYMPIC MATHEMATICAL MARATHON 1017.Let , , bepositiverealnumberssuchthat = 1.Provethat:( + 1) + ( + 1) + ( + 1) ⋅ ( + 3)( + 3)( + 3)( + 1)( + 1)( + 1) ≥ 48.


18.Provethatif , , , ∈ ℝ∗then:( − − − ) ≤ 4(1 + )(1 + )(1 + ).Daniel Sitaru

19.Let , , bepositiverealnumbers.Provethat:3 + 5 + 3+ + + 3 + 5 + 3+ + + 3 + 5 + 3+ + ≤ 11.George Apostolopoulos

20. Prove that if ∈ 1, 2 then in any triangle the following relationshipholds: 3 + + ≥ 2√3( ) . Daniel Sitaru

21. Let , be positive real numberswith = 3. Find theminimum value ofexpression:√ + 1 + + 1.George Apostolopoulos

22.Provethatif , , ∈ (0, 1) arethelength’ssidesinanytriangle then: ( − 2) + + 4 − 1( − 1) + + 4 ( − ) ≥ 3(1 − )(1 − )(1 − ). Daniel Sitaru

23.Let , , bepositiverealnumberssuchthat + + = 1.Findthemaximalvalueoftheexpression = − + − + − .George Apostolopoulos

24.Provethatinanytriangle thefollowingrelationshipholds: 6 2 + ≤ 1 . Daniel Sitaru

25.Let , , bepositiverealnumberssuchthat + + = 1.Provethat:+ ( − 1) + + ( − 1) + + ( − 1) + − 1 ( + + ) ≥≥ 13 forallintegers with ≥ 1.George Apostolopoulos

VOLUME I

1126. Provethatif , , , ∈ (0,∞); ( + )( + ) ≠ 0 then: 4(( + ) + ( − ) ) ≥ ( + ) ( + ) .

Daniel Sitaru

27.Let , , bepositiverealnumberssuchthat + + = 2.Provethat:+( + ) + +( + ) + +( + ) ≥ 1.George Apostolopoulos

28.Provethatif0 < < < < π2 then: ( + ) sin + ( − ) sin < ( + ) sin .

Daniel Sitaru . Let , , bepositiverealnumberssuchthat 1 + 1 + 1 = 3.Findthemaximumvalueofexpression:= ( + + ) + ( + + 2) + ( + + 2) .George Apostolopoulos

30.Provethatinanytriangle thefollowingrelationshipholds: sin |cos | ≤ (sin + sin )|cos − sin |. Daniel Sitaru

31.Findalltriples( , , )ofpositiverealnumberssuchthat = 1and1 + 1 + 1 = 3.George Apostolopoulos

32.Provethatinanyacute-angledtriangle thefollowingrelationshipholds:cos 4 − + cos 4 − + cos 4 − > 2 . Daniel Sitaru

33.Let , , and berealnumbers,suchthat + + + = +1or– 1.Provethat: + + + + + + + + + + + ≥ √3.George Apostolopoulos

34.Provethatinanyacute-angledtriangle thefollowingrelationshipholds:(sin + sin ) ≥ 9√2⋅ 2 ; ∈ ℕ∗. Daniel Sitaru

THE OLYMPIC MATHEMATICAL MARATHON 12

. Let , and bepositiverealnumbers, suchthat 8 + 8 + 8 = 1.Provethat: √ + + √ + + √ + ≥ 2.George Apostolopoulos

36.Provethatinanytriangle thefollowingrelationshipholds:( + − ) ≥ 64 (1 − cos − cos − cos ). Daniel Sitaru

37.Let , , and bepositiverealnumberssuchthat:27 + 27 + 27 + 27 = 1.Provethat: √ + + + √ + + + √ + + + √ + + ≥ 3.


38.Let be a trapezoid. If ∥ ; = ; = ; = ; = ;> then: Area < ( + )( − + + )16( − ) . Daniel Sitaru

39.Let , , bepositiverealnumbers.Provethat:2 + + + 2 + + + 2 + + ≤ 2 + 2 + 2 + 2 + 2 + 2 .George Apostolopoulos

40.Provethatinanyacute-angledtriangle thefollowingrelationshipholds:(2 + 2 + 2 )(2 + 2 + 2 ) < 2 + 3 + 4 . Daniel Sitaru

41.Let , , bepositiverealnumberssuchthat = = = 1.Provethat:( + + − 3 ) ⋅ 1( + ) + 1( + ) + 1( + ) ≥ 1.George Apostolopoulos

42.Provethatinanyacute-angledtriangle thefollowingrelationshipholds:2 + ( + 2 cos( − )) ≥ + 2 cos( − ) + . Daniel Sitaru

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13

43.Let , , bepositiverealnumbers,suchthat + + = 13 .Provethat:+ ++ + + ++ + + ++ ≥ 12.George Apostolopoulos

44.Provethatinanyacute-angledtriangle thefollowingrelationshipholds:2 cos −2 ≤ 3 + 3 + 2 cos( − ).Daniel Sitaru

45.Let , and bepositiverealnumbers.Provethat:( + )( + )( + ) + ( + )( + )( + ) + ( + )( + )( + ) ≥ 6 .George Apostolopoulos

46.Provethatinanynon-isoscelestriangle thefollowingrelationshipholds:2 (ℎ − ℎ )( − ) − (ℎ − ℎ )( − )ℎ + ℎ + ℎ − ℎ ℎ − ℎ ℎ − ℎ ℎ ≤ 3( + + ). Daniel Sitaru

47.Let , , bepositiverealnumbers,suchthat = 1.Provethat:( + )+ + ( + )+ + ( + )+ ≥ 12.George Apostolopoulos

48.Letbe ∈ ( ); = ; = ; = .Provethat: ( + − )( + − )( + − ) ≤ ≤ ( ) . Daniel Sitaru

49.Let , , bepositiverealnumbers.Provethat:+√ + + +√ + + +√ + ≥ 32.George Apostolopoulos

50.Letbe ∈ ( ); = ; = ; = .Provethat: 27( + − )( + − )( + − ) ≤ ( + + ) .Daniel Sitaru

51.Provethat: 1( + 1) + 1( + 1) + 1( + 1) ≥ 2( + + ) + 6+ + − + + forallrealnumbers , , ,eachdifferentfrom– 1andsatisfying = −1.Whentheequalityholds?George Apostolopoulos

THE OLYMPIC MATHEMATICAL MARATHON 14. Provethatif , , ∈ 0, 2 then: 6√ ≤ sin ≤ 3 . Daniel Sitaru

53. Let , and be positive real numbers such that + + = 32 . Find theminimumvalueoftheexpression = + + + + + .George Apostolopoulos

54.Provethatif , , , ∈ ℝ then: + + + (sin + cos + sin cos )( + + ) ≥ 0.Daniel Sitaru

55.Let , , bepositiverealnumberssuchthat = 1.Provethat:2( + 1)( + 1)( + 1)( − + 1)( − + 1)( − + 1) − 12 + + + + + ≤ 1.George Apostolopoulos

56.Letbe ( , ); ∈ 1, 3 theverticesof with , , length’ssides.Provethat: ( ( − ) + ( − ) + ( ( − ) + ( − ) ) >> 3 111 .

Daniel Sitaru 57.Let , , bepositiverealnumberssuchthat = 1.Provethat:( + 1) + ( + 1) + ( + 1) ⋅ ( + 3)( + 3)( + 3)( + 1)( + 1)( + 1) ≥ 48.


58.Provethatinanyacute-angledtriangle thefollowingrelationshipholds:tan + tan + tan tan + tan + tan ≥ + + . Daniel Sitaru

59. Let , , bepositiverealnumberssuchthat + + = 1.Provethat:1+ ( + ) + 1+ ( + ) + 1+ ( + ) ≥ 9 .George Apostolopoulos

VOLUME I

1560.Provethatinanyacute-angledtriangle thefollowingrelationshipholds:(3 + 2 + 3 − 3 ) ≥ 80 .

Daniel Sitaru

61. Let , , bepositive realnumbers such that = 1and let , , be realnumberssuchthat + + ≥ 3.Provethat:(3 + 10) ++ + + (3 + 10) ++ + + (3 + 10) ++ + ≥≥ 26.George Apostolopoulos

62.Provethatif , , ∈ ℝ; + + = 4 then:( sin + sin ( + )) < 5 + cos . Daniel Sitaru

63.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:2 + 1 + 1 + 1 + 1 ≥ + 3 + 2.George Apostolopoulos

64.Provethatinanytriangle thefollowingrelationshipholds: 1 cos cos ≥ 16 sin cos . Daniel Sitaru

65.Let , , , bepositiverealnumberssuchthat = 1.Provethat:+ + + + 4 ≥ 2( + + + ).George Apostolopoulos

66.Provethatif , , , ∈ ℝ; ( + )( + ) ≠ 0 then:( + ) − ( − )( + )( + ) ≤ 1 + ( − )( + )( + )( + ) . Daniel Sitaru

67.Let , , , bepositiverealnumberswith = 16.Provethat:+ + + + 4 ≥ + + + .George Apostolopoulos

68.Provethatif , , , > 0; + + + = and= tan + tan tan − tan tan tan

THE OLYMPIC MATHEMATICAL MARATHON 16 = sin ( + ) sin ( + )cos cos cos cos then: 16( − 1) ≤ . Daniel Sitaru

69.Let , , bepositiverealnumberswith + + = 3,Provethat:24 ≤ 8 ≤ ( + )+ ≤ 8 ≤ 8 3 ≤ 8 1 wherethesumsareoverallcyclicpermutationsof( , , ).George Apostolopoulos

70. Let be a convex quadrilater with = = 90∘ and ∈ ( ); ∈ ( ); ∈ ( ); ∈ ( ). Provethat: ⋅ + ⋅ + ⋅ + ⋅ < 2 . Daniel Sitaru

71.Let , , benonnegativerealnumberssuchthat + + = 4.Provethat:3 + + 4 + 3 + + 4 + 3 + + 4 ≤ 12.George Apostolopoulos . Let beatriangle; ∈ ( ); ∈ ( ); ∈ ( ); = = . Provethatif = ; = ; = then:( + + ) ≥ 27( + − )( + − )( + − ).

Daniel Sitaru

73.Let , and bepositiverealnumberssuchthat + + = 3.Provethat:+ 1 + + 1 + + 1 ≥ 3√2.George Apostolopoulos . Intriangle ; ∈ ( ); ∈ ( ); ∈ ( ); = = . Provethat: ( + − ) + ( + − ) + ( + − ) ≥≥ 16 .

Daniel Sitaru

75. Let , = 1, 2, … , bepositiverealnumberswith∏ = 1.Provethat:a. 3 ++ ≥ 2 b. 2 + ≥ 3where = .George Apostolopoulos

VOLUME I

1776.Solvethefollowingequation: (sin ) ⋅ (cos ) = 116.

Daniel Sitaru

77. Let , , bepositiverealnumberssuchthat + + = 1.Provethat:+2 + +2 + +2 ≥ 1.George Apostolopoulos . Provethatif , , ∈ 0, 2 then: 11 − sin sin + 11 − sin sin + 11 − sin sin ≤ 1cos + 1cos + 1cos .

Daniel Sitaru

79. Let , , bepositiverealnumberswith = 1.Provethat:( + ) − −( + ) − − + ( + ) − −( + ) − − + ( + ) − −( + ) − − ≥ 635 .George Apostolopoulos

80.Provethatinany ∆ thefollowingrelationshipholds: (2 − 3 ) + (2 − 3 ) + (2 − 3 ) ≤ 0. Daniel Sitaru

81.Let , , bepositiverealnumberswith + + = 3.Provethat:1 + 3 ⋅ 1 + 3 ⋅ 1 + 3 ≥ 64.George Apostolopoulos

82.In ∆ ; ∈ ( ); ∈ ( ); = . Intheseconditions: ( − )( − ) ≤ 0. Daniel Sitaru

83. Let , = 1, 2, … , be positive real numbers such that ∑ ≤ ,where= .Provethat: 1 + 1⋅ ≥ 2 ∑ .George Apostolopoulos

84.Provethatinany trianglethefollowingrelationshipholds:16( + 3)( + 5)( + 7) ≤ 14 + 1105. Daniel Sitaru

THE OLYMPIC MATHEMATICAL MARATHON 1885.Let , and bepositiverealnumbers.Provethat:(2 + )(2 + )( + 2 )( + 2 ) + (2 + )(2 + )( + 2 )( + 2 ) + (2 + )(2 + )( + 2 )( + 2 ) < 253 .


86.LetHbetheorthocenterof acute-angledtriangle.Provethat:4 ≤ ( + ) . Daniel Sitaru

87.Let , and bepositiverealnumbers.Provethat:a. + 22 + 3 + + + 2+ 2 + 3 + + 23 + + 2 ≤ 3√22 ;b. 2a + ba + 2b + 2 ++ 2 + 2 ++ 2 > 43.


88.Provethatinany acute-angledtrianglewehave: 27 cos cos cos ≤ ( + ) Daniel Sitaru

89.Let , and bepositiverealnumbers.Provethat:a. + 22 + 3 + + + 2+ 2 + 3 + + 23 + + 2 ≤ 32 ;b. 2 ++ 2 + 2 ++ 2 + 2 ++ > 2.George Apostolopoulos

90. Solvethefollowingsystem: = 6arctan + arctan + arctan = .+ + = 11 Daniel Sitaru

91.Let , and bepositiverealnumberssuchthat + + = 3√2.Provethat: 1+ + 4 + 1+ + 4 + 1+ + 4 ≤ 38.George Apostolopoulos

92. Let , be the orthocenter and the centroid of acute-angled triangle.Provethat: ≥ 108+ + . Daniel Sitaru

VOLUME I

1993.Let , , bepositiverealnumberssuchthat + + = 3.a.Findthemaximumvalueoftheexpression= + +( + + ) ⋅ ( + + ).b.Findtheminimumvalueoftheexpression= ( − + 1) + ( − + 1) + ( − + 1).


94.Provethatif , , ∈ 0, 2π and + + = 1 then: 2(tan + tan + tan ) ≥ 11 − ( + + ).

Daniel Sitaru . Let , , bepositiverealnumberswith 1 + 1 + 1 = 3. Provethat:( + ) + ( + ) + ( + ) ≥ 23 ( + + ) + 4 .George Apostolopoulos

96.Provethat: 1sin + 12 sin + 13 sin + ⋯+ 12015 sin > 20152016.Daniel Sitaru

97. Provethat: (sin ) + cos(2 − cos ) ≥ 1, ∈ ℝ.George Apostolopoulos

98.Provethat: 12015 sin 12 + 12014 sin 13 +⋯+ 12 sin 12015 < 20142015

Daniel Sitaru

99.Let , , bepositiverealnumberssuchthat + + = 1.Provethat:+ + + + + ≤ 12( + + ).


100.Provethat (∀) ∈ ℝ: sin + cos + sin cos ≤ 12 + √2. Daniel Sitaru

THE OLYMPIC MATHEMATICAL MARATHON 20101. Let , , , … , be real numbers such that > > > ⋯ > .Provethat: 1− + 1− +⋯+ 1− + − ≥ 2( − 1).


102. Prove that in any acute-angled triangle the following relationshipholds: 2 cot 2 + cot 2 + cot 2 + tan tan tan ≥ 9√3. Daniel Sitaru

103.Let , , bepositiverealnumberssuchthat + + = 1.Provethat:( ) + ( ) + ( ) < 2 + √33 .George Apostolopoulos

104.Provethat: 1 − cos 235 + 1 − cos 263 > 1 − cos 445. Daniel Sitaru

105.Let , , bepositiverealnumberssuchthat = .Provethat:1+ 2 + 1+ 2 + 1+ 2 ≤ 1 .George Apostolopoulos

106.Provethatinany trianglethefollowingrelationshipholds: + > 32.Daniel Sitaru

107.Iftherootsoftheequation + + + = 0; , , , ∈ ℝwith > 0,areallnonnegative.Provethat: ≤ 4 −9 .George Apostolopoulos

108.Provethatinany trianglethefollowingrelationshipholds: ∑ √∑ √ ≥ ∑ √∑ √ . Daniel Sitaru

109.Let , , bepositiverealnumberssuchthat + + = .Finda.theminimumvalueofexpression

VOLUME I

21

= 13 + 1 + 13 + 1 + 13 + 1;b.themaximumvalueofexpression = − 2( + + ).George Apostolopoulos

110.Provethatinany trianglethefollowingrelationshipholds: sin + sin + sin ≤ 2 1 + cos( − ) cos . Daniel Sitaru

111. Let , , be positive real numbers such that + + = . Find themaximumvalueofexpression = 2( + + ) − − − .George Apostolopoulos

112.Provethatinany trianglethefollowingrelationshipholds: ( + ) cos ≥ 1. Daniel Sitaru

113.Let , bedistinctrealnumberssuchthat:+ − 3( + ) + 8 ≤ 2( + )(2 − ).Findthevalueoftheexpression = ( ) + ( + 1) + ( + 2) ,where isapositiveinteger.George Apostolopoulos

114.Provethat (∀) , ∈ ℝ: sin + sin + sin( + ) ≤ 2 1 − cos( + ) cos( − ). Daniel Sitaru

115.Let , , bepositiverealnumberssuchthat = 1.Provethat:+ ( − 1) + + ( − 1) + + ( − 1) ≥ 3foreachpositiveinteger .George Apostolopoulos

116.Provethat: 13 + 16 sin 16 + 13 cos > 518. Daniel Sitaru

117.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:+ ++ + + + ++ + + + ++ + ≥ 3.George Apostolopoulos

THE OLYMPIC MATHEMATICAL MARATHON 22118. Prove that in any acute-angled triangle we have the followingrelationship: tan + tan + tan > sin + sin + sin .

Daniel Sitaru

119.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:++ + + ++ + + ++ + ≥ 2.George Apostolopoulos

120.Provethatif 0 < ≤ ≤ < π2 then: tan − tan tan tan < tan − tan .Daniel Sitaru

121.Let , , bepositiverealnumberssuchthat + + = 18 .Provethat:+ 4 + + + 4 + + + 4 + ≤ + + .George Apostolopoulos

122.Find , , , ∈ (0, ) suchthat: sin cos + sin cos + sin cos = 12sin cos + sin cos + sin cos = 32. Daniel Sitaru

123.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:+ 1+ 1 + + 1+ 1 + + 1+ 1 ≥ 3 .George Apostolopoulos

124.Provethatinanyacute-angled triangle: 27 ≤ 1( − ) + 1( − ) + 1( − ) . Daniel Sitaru

125.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:+ 1+ 1 + + 1+ 1 + + 1+ 1 ≥ + + .George Apostolopoulos

VOLUME I

23126.Provethatinany trianglethefollowingrelationshipholds: √tan tan √tan + √tan ≤ 2 tan tan tan .

Daniel Sitaru

127.Let , , bepositiverealnumberssuchthat = 1.Provethat:+ 1 + + 1 + + 1 − + +3 < 72.George Apostolopoulos

128.Provethatinanytriangle thefollowingrelationshipholds:(2 + ) + + + < ( + )( + + ).Daniel Sitaru

129.Let , , bepositiverealnumberssuchthat = 1.Provethat:+ 1+ 1 + + 1+ 1 + + 1+ 1 ≤ + + .George Apostolopoulos

130.Provethatinany trianglethefollowingrelationshipholds: 1 + 1 + 1 + > . Daniel Sitaru

131.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:+ + 1+ + 1 + + + 1+ + 1 + + + 1+ + 1 ≥ 3 .George Apostolopoulos

132.Letbe , , , ∈ ℝ∗ suchthatarctan + arctan + arctan + arctan = . Provethat: 1 + 1 + 1 + 1 ≥ 4( ) . Daniel Sitaru

133.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:+ + ≤ 3 + 3 + 3 .George Apostolopoulos

134.Provethatinany trianglethefollowingrelationshipholds:+ + ≥ 24√2( ) . Daniel Sitaru

THE OLYMPIC MATHEMATICAL MARATHON 24135. Findtheminimumvalueofexpression= 1+ + 1 + 1 + 1 ,when , , > 0and + + = 1.( , , arerealnumbers).


136.Provethatinany trianglethefollowingrelationshipholds: ( + )( + ) ≥ 8√ √ + √ + √ . Daniel Sitaru

137.Findallpossiblepairs( , )ofintegerssatisfying+ (4 − 3 )( − − 1) = 2.George Apostolopoulos

138.Provethatinany trianglethefollowingrelationshipholds: + ++ + ≤ + ++ + . Daniel Sitaru

139. Findallpairsofpositiveintegers , satisfyingtheequation4 + 3 − 7 − 6 + 5 = 0.George Apostolopoulos

140.Provethatinany trianglethefollowingrelationshipholds:(1 + sin ) ≥ 4030 + 2 sin 2 .Daniel Sitaru

141.Findallpossiblepairs( , )ofintegerssatisfying− 3 + 3 − 2 − 9 + 19 + 10 + 23 = 0.George Apostolopoulos

142.Provethatinanytriangle thefollowingrelationshipholds: sin + cos ≤ 4 . Daniel Sitaru

143.Findallpairsofintegers( , )suchthat+ ( + − 1)(4 − 3 ) = 2.George Apostolopoulos

144.Provethatinany trianglethefollowingrelationshipholds:+ + + + + ≤ . Daniel Sitaru

145.Findallpossiblepairs( , )ofintegerssatisfying+ (3 + 3 − 4)(1 − ) = 1.George Apostolopoulos

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146.Provethatif , , ∈ 1,

2π then: 1 ++ tan ≥ 2+ tan tan tan . Daniel Sitaru

147.Theequation3 − 3 + 1 = 0,hasrootsthenumbers , , .Let = + + and = + + . Find ⋅ .George Apostolopoulos

148. Provethatinany trianglethefollowingrelationshipholds: 3( + + ) < 2 + + + 6( + + ). Daniel Sitaru

149.Let , , bepositiverealnumberssuchthat + + = 1k .Provethat:+ + + + + ≤ 32 + + .


150.Provethatinany trianglethefollowingrelationshipholds: + + ≥ 16 . Daniel Sitaru

151.Let , , , = 1, 2, 3, 4bepositiverealnumbers.Provethat:3 ⋅ + + + 1+ + ≥ 23 .George Apostolopoulos

152.Provethatif ∆ isanacute-angledtrianglethen:(tan )√tan tan ≥ 3 tan , ∈ ℕ∗. Daniel Sitaru

153.Let , berealnumberswith ≠ , ≥ −1suchthat+ − 3( + ) + 8 ≤ 2( + )(2 − ).Provethat:32 √ + 1 + √ + 2 +⋯+ √ + + √ + + 1 > , ∈ ℕ∗.George Apostolopoulos

THE OLYMPIC MATHEMATICAL MARATHON 26154. Provethatinany trianglethefollowingrelationshipholds: ≥ 27 .

Daniel Sitaru 155.Let , , berealnumberssuchthat + + = 3.Provethat:− + 1+ 1 + − + 1+ 1 + − + 1+ 1 ≥ 92( + + ).


156.Provethatinany trianglethefollowingrelationshipholds: 64 ≥ 27 . Daniel Sitaru

157.Let , , bepositiverealnumbers.Provethat:− + + + − +− + + ≤ 3.George Apostolopoulos

158.Provethatinany trianglethefollowingrelationshipholds: 16 ≥ 9 ( + + ). Daniel Sitaru

159.Let , , bepositiverealnumbers.Provethat:+ + + 3 ≥ 3 ( + 1) ⋅ ( + 1)( + 1) .George Apostolopoulos

160.Provethatinany trianglethefollowingrelationshipholds:ℎ + ℎ + ℎ ≥ 92 √ .Daniel Sitaru

161.Let , , bepositiverealnumbers.Provethat:+ + + + + + + + ≥ ( ) foreachpositiveinteger .George Apostolopoulos

162.Provethatinany trianglethefollowingrelationshipholds: + + ≤ . Daniel Sitaru

163.Let , , bepositiverealnumberswith + + = 1.Provethat:1 + 12 + ⋅ 1 + 12 + ⋅ 1 + 12 + ≥ 2.George Apostolopoulos

VOLUME I

27164.Provethatinany trianglethefollowingrelationshipholds: ≤ √3 where – isthelengthofsimediancorrespondingtothe vertex.

Daniel Sitaru

165.Foranytrianglewithsidesoflengths , and ,provethat:+ − + + − + + − ≥ 3, ∈ ℕ.George Apostolopoulos

166.Provethatin ∆ with + ≠ wehave:1( + − ) + +4 ≥ 8116( + + ). Daniel Sitaru

167.Let , , bepositiverealnumberssatisfying + + = 3.Provethat:2√2 + + + + ≤ 92.George Apostolopoulos

168.Provethatinany trianglethefollowingrelationshipholds: ( + )cos ≥ 8. Daniel Sitaru

169.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:(3 + 2) ++ + + (3 + 2) ++ + + (3 + 2) ++ + ≥≥ 10 .George Apostolopoulos

170.Provethatinany trianglethefollowingrelationshipholds:1 + 2 + 1 5 + + + > 64. Daniel Sitaru

171. Let , , bepositiverealnumberssuchthat + + = 1,and ∈ ℕ.Provethat:⋅ +√ + + ⋅ +√ + + ⋅ +√ + ≥ 3 2 .George Apostolopoulos

THE OLYMPIC MATHEMATICAL MARATHON 28172. Provethatinany trianglethefollowingrelationshipholds:8 ( − ) + 2 ≥ 3√3 .

Daniel Sitaru

173. Let , , be nonnegative real numbers, of which either two are notsimultaneouslyzero.Provethat:+√ + + +√ + + +√ + ≥ + + + + + .George Apostolopoulos

174.Provethatinany ∆ thefollowingrelationshipholds:+ 2 < + + 2 cos + 2 cos . Daniel Sitaru

175.Let , , bepositiverealnumbers.Provethat:1+ + ≤ √62 + .George Apostolopoulos

176.Provethatinany trianglethefollowingrelationshipholds: √ sin 4 + cos 4 ≤ 3√ . Daniel Sitaru

177.Let , and benonnegativerealnumberssuchthat + + = 1.Provethat: ≤ + +3 + + +3 + + +3 ≤ + + .George Apostolopoulos

178.Provethatinany trianglethefollowingrelationshipholds: √sin sin ≤ cos 2 + cos 2 + cos 2. Daniel Sitaru

179.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:+ +2 ≥ + + + + + .George Apostolopoulos

180.Provethatinany ∆ thefollowingrelationshipholds:cos 2 cos 2 ≤ cos −4 + cos −4 + cos −4 . Daniel Sitaru

VOLUME I

29181. Let , , be the exradii of a triangle with inradius and circum-radius .Provethat: 89 ≤ 1 + 1 + 1 ≤ 81 − 576648 .


182. Let be a tetrahedron and ∈ ( ); ∈ ( ); ∈ ( ); ∈ ( ) suchthat: = ; = 1+ 2 ; = + 13 ; = 6+ 2, , ∈ (0,∞). Provethatif , , , arecoplanarthen ( + 1) ≥ 4 .Daniel Sitaru

183. Let , = 1, 2, … , be positive real numbers such that∑ = . Provethat: + 1+ 1 ≥ .George Apostolopoulos

184.Provethatifin ∆ : cos + cos = 2 sin sin then: > 8( − ) . Daniel Sitaru

185.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:(3 − ) + 1+ 1 ≥ 6 .George Apostolopoulos

186.Provethatinany ∆ thefollowingrelationshipholds:(5 + 7 + 3 )(7 + 3 + 5 ) ≥ 71( + + ). Daniel Sitaru

187.Let , , bepositiverealnumbers.Provethat:a. ( + )+ + + ( + )+ + + ( + )+ + ≤ 4;b. + + = 3, then + + + + + + + + ≤ 1.George Apostolopoulos

188.Provethatinanytriangle thefollowingrelationshipholds:sin −4 + sin −4 + sin −4 > 1 − ( − )( − )( − )32 .Daniel Sitaru

THE OLYMPIC MATHEMATICAL MARATHON 30189. Real numbers , , satisfy √ + 5 + 4 + 2√ − 5 + 4 = √5 + 40.Provethat + ≥ .


190.Provethatinany ∆ thefollowingrelationshipholds:6 2 + ≤ 1 . Daniel Sitaru

191.Let , , bepositiverealnumbers.Findthemaximalvalueofexpression:= + 22 + 3 + + + 22 + 3 + + + 22 + 3 + .George Apostolopoulos

192.Provethatinany ∆ thefollowingrelationshipholds:tan −2 tan 2 < 1.Daniel Sitaru

193.Let , , bepositiverealnumberssuchthat + + = 1.Provethat:+ + + + + ≥ 9 6 ⋅ ( ) forall ∈ ℕ.George Apostolopoulos

194.Provethatinany ∆ thefollowingrelationshipholds: sin − sinsin + sin < 1. Daniel Sitaru

195.Let , , bepositiverealnumberswith + + = 1.Provethat:+ + ≥ .George Apostolopoulos

196.Let beatetrahedronwhere:= √11; = 3; = √14; = √3; = 2; = √13 Provethatm ∢( , ) > 90°. Daniel Sitaru

197. Let , , be positive real numbers such that + + = 3. Find themaximumvalueofexpression= ( + ) ⋅ ( + + + 5 )( + ) ⋅ ( + + − 2 ).George Apostolopoulos

198.Provethatif , , ∈ ℝthen:(2 − − − + ) ≤ ( + 2)( + 2)( + 2).Daniel Sitaru

VOLUME I

31199.Let beatrianglewithcircumradius andinradius .Provethat:27 ≤ sin + sin + sin ≤ 278 1 − .


200.Provethatinanyacute-angled triangle,thefollowingrelationship(sin 2 + sin 2 ) 1sin 2 + 1sin 2 ≤ (tan + tan )(cot + cot ). Daniel Sitaru

201.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:( + )( + )( + ) − 2 ≤ 6.George Apostolopoulos

202.Provethatif , , ∈ 0,∞) thenin thefollowingrelationshipholds:√2 ( + ) + ( + ) + ( + ) ≥ √sin + √sin + √sin . Daniel Sitaru

203.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:( + )( + )( + ) + 4 ≤ 12.George Apostolopoulos

204.Provethatif , ∈ ℝ; ∈ 0,∞) then: sin( − ) cos( + ) + cos cos ≤ cos( + ) + cos( + ) + 2√2 . Daniel Sitaru

205. Let , , , bepositiverealnumberssuchthat + + + = 4.Provethat:( + + + ) 1( + ) + 1( + ) + 1( + ) + 1( + ) ≥ 8.George Apostolopoulos

206.Provethatif , , arethesidesofatrianglethen:sin + sin + sin ≤ 4 sin sin( − ) sin( − ) sin( − ). Daniel Sitaru

207.Let , , bepositiverealnumberssuchthat + + = 3.Provethat:3 + + + + + + + + + 1 ≤ 4 .George Apostolopoulos

208.Provethatin trianglewehave: 8 tan −2 tan 2 ≤ 1 − ⋅ 1 − ⋅ 1 − . Daniel Sitaru

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THE OLYMPIC MATHEMATICAL MARATHON · Mathematics Journal, Mathematics Magazine, Crux Mathematicorum. In 2009, he participated as an observer in the World Olympiad which was held in