Geometric Inequalities: Methods of Proving

454 Pages·2017·17.53 MB·English

by  Hayk Sedrakyan

Problem Books in Mathematics
Hayk Sedrakyan
Nairi Sedrakyan
Geometric
Inequalities
Methods of Proving
Problem Books in Mathematics
SeriesEditor:
PeterWinkler
DepartmentofMathematics
DartmouthCollege
Hanover,NH03755
USA
More information about this series at http://www.springer.com/series/714
Hayk Sedrakyan (cid:129) Nairi Sedrakyan
Geometric Inequalities
Methods of Proving
B
r
1
r
3
r
2
A C
r1 + r2 > r3
HaykSedrakyan NairiSedrakyan
UniversityPierreandMarieCurie Yerevan,Armenia
Paris,France
ISSN0941-3502 ISSN2197-8506 (electronic)
ProblemBooksinMathematics
ISBN978-3-319-55079-4 ISBN978-3-319-55080-0 (eBook)
DOI10.1007/978-3-319-55080-0
LibraryofCongressControlNumber:2017937367
MathematicsSubjectClassification(2010):00A07
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To Margarita,
a wonderful wife and a loving mother
To Ani,
a wonderful daughter and a loving sister
Preface
Geometric inequalities are one of the most interesting sections of elementary
mathematicsandhaveawiderangeofapplicationsingeometryandtheotherfields
ofmathematics,suchasalgebraandtrigonometry.Toprovegeometricinequalities
one often has to use, besides the geometric reasoning, algebraic transformations,
trigonometricrelationsandinequalities,calculusandmathematicalanalysis.
Thisbookisthethirdbookoftheauthorsaboutinequalities.Thefirsttwobooks
werededicatedtoalgebraicinequalitiesandwerepublishedin2015inSouthKorea.
Allthesebooksreflectlongyearsofexperienceoftheauthorsinteaching.Mostof
theproblemswerecreatedorprovedbytheauthorsduringthoseclasses.
Theauthorshavetriednottouse,wheneverpossible,theconceptofaderivative,
therefore making the solutions of many problems understandable to students.
Thebookcontainsmorethan 1000problems. Approximately 800problemsinthe
bookarewiththoroughsolutions.Basically,thesearenon-standardproblems.
The majority of problems are for mathematics competitions and Olympiads.
Manyproblemsinthebookandthemajorityofthesolutionsbelongtotheauthors.
Someofthoseproblemswereusedbytheauthorstoteachtheirstudentsinterested
inmathematicalOlympiads.Infewcases,thesolutionwasproposedbyastudent,
sohis/hernameisindicated.Someproblemsoftheauthors,includedinthisbook,
wereproposedinmathematicalOlympiads(indifferentcountries).Someproblems
were proposed in different mathematical journals, such as the American Mathe-
matical Monthly (MAA), Crux Mathematicorum with Mathematical Mayhem
(Canadian Mathematical Society), Mathematical Reflections (USA), and Kvant
(Russia).
Thebookisdividedintoeightchapters,eachofthechaptersconsistsofone,two,
three,fourorfiveparagraphs.Thebasisoftheclassificationisusuallythemethodof
thesolution.
vii
viii Preface
The authors have tried to find common approaches to different problems. The
goal of the book is to teach the reader new and classical methods for proving
geometricinequalities.
Theauthorswouldliketoexpresstheirgratitudetotheirfamilyforthesupport.
HaykSedrakyan
NairiSedrakyan
Contents
1 TheoremontheLengthoftheBrokenLine. . . . . . . . . . . . . . . . . . 1
1.1 TriangleInequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2 TheoremontheLengthoftheBrokenLine. . . . . . . . . . . . . . . . 24
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 ApplicationofProjectionMethod. . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.1 ConvexPolygonLyingInsideofAnotherPolygon. . . . . . . . . . . 40
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.2 SufficientConditionsforComparisonofLengths
ofTwoBrokenLinesonthePlane. .. . . . . .. . . . . . .. . . . . . .. 59
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.3 InscribedPolygonswiththeLeastPerimeter. . . . . . . . . . . . . . . 66
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4 MethodofProjections. . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . 74
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3 Areas. . . . .. . . .. . . .. . . .. . . . .. . . .. . . .. . . .. . . . .. . . .. . . .. 93
3.1 InequalitieswithAreas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
ix
x Contents
4 ApplicationofVectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.1 ApplicationofVectorsforProvingGeometric
andTrigonometricInequalities. . . . . . . . . . . . . . . . . . . . . . . . . 139
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5 ApplicationofTrigonometricInequalities. . . . . . . . . . . . . . . . . . . 167
5.1 InequalitiesfortheAnglesofaTriangle. . . . . . . . . . . . . . . . . . 168
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
5.2 InequalitiesfortheAnglesofAcute
andObtuseTriangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.3 SomeRelationsforaTriangle. . . . . . . . . . . . . . . . . . . . . . . . . . 198
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
5.4 TrigonometricInequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
5.5 UsingTrigonometricInequalitiesforProving
GeometricInequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
6 InequalitiesforRadiuses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
6.1 InequalitiesforRadiusesofCircles. . . . . . . . . . . . . . . . . . . . . . 262
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
6.2 IntegerLattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
7 MiscellaneousInequalities. . .. . . . .. . . . .. . . .. . . . .. . . . .. . . .. 285
7.1 MiscellaneousInequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
7.2 Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
7.3 ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
8 SomeApplicationsofGeometricInequalities. . . . . . . . . . . . . . . . . 413
8.1 ApplicationofGeometricInequalities
forSolvingGeometricproblems. . . . . . . . . . . . . . . . . . . . . . . . 413
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
Contents xi
8.2 UsingGeometricInequalitiesforProving
AlgebraicInequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
ProblemsforSelf-Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
BasicNotations. .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . 447
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449