Index of Problems Chapter 1 1.8 BMO 2001 1.17 Russia MO 2002 Chapter 2 2.4 Viorel Vâjâitu, Alexandra Zaharescu, Gazeta Matematică 2.15 Ireland MO 2000 Chapter 3 3.6 IMO, shortlist 1969 (Romania) 3.8 India MO 2003 Chapter 4 4.1 France MO 1996 4.3 South Africa MO 1995 4.7 Crux Mathematicorum 4.8 Sefket Arslanagic 4.10 Art of problem solving 4.11 Art of problem solving 4.13 Andrei Ciupan, Romania 2007 4.15 Crux Mathematicorum 4.20 Pham Kim Hung Corollary 4.5: Walther Janous Chapter 5 5.3 IMO, shortlist 1974 (Finland) 5.10 Zdravko Cvetkovski 5.13 Zdravko Cvetkovski 5.14 Zdravko Cvetkovski 5.15 Zdravko Cvetkovski Z. Cvetkovski, Inequalities, DOI 10.1007/978-3-642-23792-8, Springer-Verlag Berlin Heidelberg 2012 435
436 Index of Problems Chapter 6 6.2 IMO 1975 6.4 IMO 1964 (Hungary) 6.6 IMO 1995 6.10 Song Yoon Kim Chapter 8 8.1 Darij Grinberg 8.2 Poland MO 1999 8.3 Calin Popa 8.4 Walther Janous, Crux Mathematicorum 8.6 APMO 2004 Chapter 9 9.1 Singapore MO 2002 9.2 Sefket Arslanagic 9.6 Le Viet Thai 9.7 Pham Kim Hung 9.8 Pham Kim Hung 9.10 Walther Janous, Crux Mathematicorum Chapter 10 10.5 Zdravko Cvetkovski 10.6 Zdravko Cvetkovski 10.7 Zdravko Cvetkovski 10.11 Nguyen Manh Dung Chapter 12 12.5 Darij Grinberg 12.7 APMO 2004 12.10 IMO 1984 12.11 IMO 1995 Chapter 14 14.2 Iran MO 1996 14.6 United Kingdom 1999 Chapter 15 15.2 Vietnam TST 1996 15.3 Vietnam 2002 15.4 Darij Grinberg Chapter 17 17.2 IMO 2005
Index of Problems 437 Chapter 18 18.3 Pham Kim Hung 18.5 Nguyen Minh Duc, IMO shortlist 1993 Chapter 20 6 Titu Andreescu, TST 2001 USA 9 Russia 2002 12 Czech and Slovak Republics 2005 14 Walther Janous, Crux Mathematicorum 16 Vasile Cîrtoaje, Gazeta Matematică 20 Titu Andreescu, Gabriel Dospinescu 22 Art of problem solving 23 Vasile Cîrtoaje 28 Art of problem solving 33 Art of problem solving 36 Belarus 1996 37 Art of problem solving 40 Zdravko Cvetkovski 41 Zdravko Cvetkovski 42 Zdravko Cvetkovski 45 Mircea Lascu, Gazeta Matematică 48 Art of problem solving 49 IMO 2000, Titu Andreescu 50 Bulgaria, 1997 56 Baltic Way, 2005 57 Gabriel Dospinescu, Marian Tetiva 58 Adrian Zahariuc 68 Vasile Cîrtoaje 71 MOSP 2001 72 Vasile Cîrtoaje, Mircea Lascu, Junior TST 2003 Romania 73 Marian Tetiva 75 Latvia 2002 81 Peru 2007 82 Romania 2003 84 Art of problem solving 86 Japan 2005 88 Art of problem solving 90 Canada 2008 91 Mathlinks contest 97 Kiran Kedlaya 98 Zdravko Cvetkovski, shortlist JBMO 2010 99 Vasile Cîrtoaje, Gazeta Matematică 105 JBMO 2002, shortlist 110 Pham Kim Hung 113 Pham Kim Hung
438 Index of Problems 115 Pham Kim Hung 117 Zdravko Cvetkovski 118 Macedonia MO 1999 119 Zdravko Cvetkovski 125 Pham Kim Hung 126 Titu Andreescu, USAMO 2004 129 MYM, 2001 131 Vietnam 136 Vietnam, 1998 139 Zdravko Cvetkovski, BMO shortlist 2010 144 IMO, shortlist 1996 (Slovenia) 154 Nguyen Van Thach 156 Pham Kim Hung 160 Dinu Şerbanescu, Junior TST 2002, Romania 161 Zdravko Cvetkovski, Macedonia MO 2010 162 Pham Kim Hung 163 Romania 2008 165 IMO, shortlist 1987 166 JBMO, 2003 167 Pham Kim Hung 170 Florina Cârlan, Marian Tetiva 173 Darij Grinberg 176 Art of problem solving 179 Art of problem solving 184 Ukraine MO 2004 188 JBMO 2002, shortlist 190 IMO, shortlist 1990 (Thailand) 191 Titu Andreescu, Mircea Lascu 195 Poru Loh, Crux Mathematicorum 196 Pham Kim Hung 197 Moldova TST 2005 198 IMO shortlist 1994 199 IMO shortlist 1998 (Russia) 200 IMO shortlist 1993 (USA) 202 IMO, shortlist 1968 (Poland) 207 China MO 1996 212 IMO, shortlist 1992 (Great Britain) 213 A. Teplinsky, Ukraine MO 2005 214 Daniel Campos Salas, Mathematical Reflections 2007 217 Cezar Lupu, Romania MO 2005 218 Titu Andreescu, USAMO 2001 219 Gabriel Dospinescu, Mircea Lascu, Marian Tetiva 237 IMO 1961 (Poland) 250 Zdravko Cvetkovski 257 Calin Popescu, Romania 2004
Index of Problems 439 264 Pham Kim Hung 265 Mihai Piticari, Dan Popescu 271 Art of problem solving 280 Tran Nam Dung 281 Le Trung Kien 287 Zdravko Cvetkovski 288 Vasile Cîrtoaje, Gazeta Matematică 292 Serbia 2008 294 Vasile Cîrtoaje, Gazeta Matematică 298 Pham Kim Hung 300 Art of problem solving 302 Art of problem solving 303 Art of problem solving 304 India 2002 305 Vasile Cîrtoaje, Gazeta Matematică
Abbreviations APMO Asian-Pacific Mathematical Olympiad BMO Balkan Mathematical Olympiad IMO International Mathematical Olympiad JBMO Junior Balkan Mathematical Olympiad MYM Mathematics and Youth magazine, Vietnam MO Mathematical Olympiad MOSP Mathematical Olympiad Summer Program USAMO United States of America Mathematical Olympiad Z. Cvetkovski, Inequalities, DOI 10.1007/978-3-642-23792-8, Springer-Verlag Berlin Heidelberg 2012 441
References 1. Agahanov, H.X., idr.: Vserossi skie Olimpiady Xkolьnikov po Matematike 1993 2006. MCNMO, Moskva (2007) (Agakhanov, N.H., et al.: All-Russian Olympic in Mathematics 1993 2006, MCNMO, Moscow, 2007) 2. Alfutova, B.N., Ustinov, A.B.: Algebra i Teori Qisel dl Matematiqeskih Xkol. MCNMO, Moskva (2003) (Alfutova, B.N., Ustinov, A.V.: Algebra and Number Theory for Mathematical Schools, MCNMO, Moscow, 2003) 3. Andreescu, T., Enescu, B.: Mathematical Olympiad Treasures. Birkhäuser, Basel (2003) 4. Andreescu, T., Cirtoaje, V., Dospinescu, G., Lascu, M.: Old and New Inequalities. Gil Publishing House, Zalau (2004) 5. Andreescu, T., Feng, Z., Lee, G.: Mathematical Olympiads 2000 2001; Problems and Solutions from Around the World. Math. Assoc. of America, Washington (2001) 6. Arslanagič, Š.: Matematika za Nadarene. Bosanska Riječ, Sarajevo (2005) (Mathematics for Gifted Students, Bosanska Riječ, Sarajevo, 2005) 7. Arslanagič, Š.: Metodička Zbirka Zadataka sa Osnovama Teorije iz Elementarne Matematike. Grafičar promet, Sarajevo (2006) (Methodical Collection of Problems with the Basic Theory of Elementary Mathematics, Grafičar promet, Sarajevo, 2006) 8. Art of Problem solving: http://www.artofproblrmsolving.com 9. Bottema, O., et al.: Geometric Inequalities. Wolters-Noordhoff Publishing, Groningen (1969) 10. Dimovski, D., i dr.: Praktikum po Elementarna Matematika. Prosvetno delo, Skopje (1993). (Dimovski, D., et al.: Practicum for Elementary Mathematics. Prosvetno delo, Skopje (1993)) 11. Djukić, D., Janković, V., Matić, I., Petrović, N.: The IMO Compendium, a Collection of Problems Suggested for the International Mathematical Olympiad: 1959 2004, 1st edn. Springer, Berlin (2006) 12. Engel, A.: Problem-Solving Strategies. Springer, New York (1997) 13. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1967) 14. Iжboldin, O., Kurl andqik, L.: Neravenstvo Iencena. Kvant 4, 7 10 (2000). (Izboldin, O., Kurlandcik, L.: Jensen s inequality. Quant 4, 7 10 (2000)) 15. Kedlaya, K.S.: A<B, http://www.unl.edu.com 16. Kim Hung, P.: The stronger mixing variables method. Math. Reflect. 6, 1 8 (2006) 17. Kim Hung, P.: Secrets in Inequalities, vol. 1. Gil Publishing House, Zalau (2007) 18. Larson, C.L.: Problem Solving Through Problems. Springer, New York (1983) 19. Lee, H.: Topics in Inequalities Theorems and Techniques. The IMO Compendium Group (2007) 20. Li, Y.K.: Math Problem Book I. Hong Kong Mathematical Society (2001) 21. Lozanski, E., Rousseau, C.: Wining Solutions. Springer, New York (1996) Z. Cvetkovski, Inequalities, DOI 10.1007/978-3-642-23792-8, Springer-Verlag Berlin Heidelberg 2012 443
444 References 22. Malqeski, P.,Malqeski, A.:Izbrani sodrжini od elementarna matematika. SDMI, Skopje (1994) (Malcheski, R., Malcheski, A.: Selected Content from Elementary Mathematics, SDMI, Skopje 1994) 23. MathLinks: http://www.mathlinks.ro 24. Matic, I.: Classical Inequalities. The IMO Compendium Group (2007) 25. Nomirovski, D: Neravenstvo Karamaty. Kvant 4, 43 45 (2000). (Nomirovski, D.: Karamata s inequality. Quant 4, 43 45 (2000)) 26. Prasolov, B.B.: Zadaqi po algebre, arifmetike i analizu (2005) (Prasolov, V.V.: Problems in Algebra, Arithmetic and Analysis, 2005) 27. Puong, T.: Diamonds in Mathematical Inequalities. HaNoi Publishing House (2007) 28. Sedrak n, N.: O primenenii odnogo neravenstva. Kvant 2, 42 44 (1997). (Sedrakian, N.: For the usage of recent inequality. Quant 2, 42 44 (1997)) 29. Shklarsky, D.O., Chentzov, N.N., Yaglom, I.M.: The USSR Olympiad Problem Book. Dover, New York (1994) 30. Slinko, A.: USSR Mathematical Olympiads 1989 1992. Australian Mathematical Trust, Belconnen (1997) 31. Verdiyan, V., Salas, D.C.: Simple trigonometric substitutions with broad result. Math. Reflect. 6, 1 12 (2007) 32. Vo Quoc, B.: On a class of three-variable inequalities. Math. Reflect. 2, 1 8 (2007)