436 Index of Problems

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1 Index of Problems Chapter BMO Russia MO 2002 Chapter Viorel Vâjâitu, Alexandra Zaharescu, Gazeta Matematică 2.15 Ireland MO 2000 Chapter IMO, shortlist 1969 (Romania) 3.8 India MO 2003 Chapter France MO South Africa MO Crux Mathematicorum 4.8 Sefket Arslanagic 4.10 Art of problem solving 4.11 Art of problem solving 4.13 Andrei Ciupan, Romania Crux Mathematicorum 4.20 Pham Kim Hung Corollary 4.5: Walther Janous Chapter IMO, shortlist 1974 (Finland) 5.10 Zdravko Cvetkovski 5.13 Zdravko Cvetkovski 5.14 Zdravko Cvetkovski 5.15 Zdravko Cvetkovski Z. Cvetkovski, Inequalities, DOI / , Springer-Verlag Berlin Heidelberg

2 436 Index of Problems Chapter IMO IMO 1964 (Hungary) 6.6 IMO Song Yoon Kim Chapter Darij Grinberg 8.2 Poland MO Calin Popa 8.4 Walther Janous, Crux Mathematicorum 8.6 APMO 2004 Chapter Singapore MO Sefket Arslanagic 9.6 Le Viet Thai 9.7 Pham Kim Hung 9.8 Pham Kim Hung 9.10 Walther Janous, Crux Mathematicorum Chapter Zdravko Cvetkovski 10.6 Zdravko Cvetkovski 10.7 Zdravko Cvetkovski Nguyen Manh Dung Chapter Darij Grinberg 12.7 APMO IMO IMO 1995 Chapter Iran MO United Kingdom 1999 Chapter Vietnam TST Vietnam Darij Grinberg Chapter IMO 2005

3 Index of Problems 437 Chapter Pham Kim Hung 18.5 Nguyen Minh Duc, IMO shortlist 1993 Chapter 20 6 Titu Andreescu, TST 2001 USA 9 Russia Czech and Slovak Republics 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 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, Baltic Way, Gabriel Dospinescu, Marian Tetiva 58 Adrian Zahariuc 68 Vasile Cîrtoaje 71 MOSP Vasile Cîrtoaje, Mircea Lascu, Junior TST 2003 Romania 73 Marian Tetiva 75 Latvia Peru Romania Art of problem solving 86 Japan Art of problem solving 90 Canada Mathlinks contest 97 Kiran Kedlaya 98 Zdravko Cvetkovski, shortlist JBMO Vasile Cîrtoaje, Gazeta Matematică 105 JBMO 2002, shortlist 110 Pham Kim Hung 113 Pham Kim Hung

4 438 Index of Problems 115 Pham Kim Hung 117 Zdravko Cvetkovski 118 Macedonia MO Zdravko Cvetkovski 125 Pham Kim Hung 126 Titu Andreescu, USAMO MYM, Vietnam 136 Vietnam, Zdravko Cvetkovski, BMO shortlist IMO, shortlist 1996 (Slovenia) 154 Nguyen Van Thach 156 Pham Kim Hung 160 Dinu Şerbanescu, Junior TST 2002, Romania 161 Zdravko Cvetkovski, Macedonia MO Pham Kim Hung 163 Romania IMO, shortlist JBMO, Pham Kim Hung 170 Florina Cârlan, Marian Tetiva 173 Darij Grinberg 176 Art of problem solving 179 Art of problem solving 184 Ukraine MO 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 IMO shortlist IMO shortlist 1998 (Russia) 200 IMO shortlist 1993 (USA) 202 IMO, shortlist 1968 (Poland) 207 China MO IMO, shortlist 1992 (Great Britain) 213 A. Teplinsky, Ukraine MO Daniel Campos Salas, Mathematical Reflections Cezar Lupu, Romania MO Titu Andreescu, USAMO Gabriel Dospinescu, Mircea Lascu, Marian Tetiva 237 IMO 1961 (Poland) 250 Zdravko Cvetkovski 257 Calin Popescu, Romania 2004

5 Index of Problems 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 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 Vasile Cîrtoaje, Gazeta Matematică

6 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 / , Springer-Verlag Berlin Heidelberg

7 References 1. Agahanov, H.X., idr.: Vserossi skie Olimpiady Xkolьnikov po Matematike MCNMO, Moskva (2007) (Agakhanov, N.H., et al.: All-Russian Olympic in Mathematics , 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 ; 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: 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: , 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, 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 / , Springer-Verlag Berlin Heidelberg

8 444 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: Matic, I.: Classical Inequalities. The IMO Compendium Group (2007) 25. Nomirovski, D: Neravenstvo Karamaty. Kvant 4, (2000). (Nomirovski, D.: Karamata s inequality. Quant 4, (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, (1997). (Sedrakian, N.: For the usage of recent inequality. Quant 2, (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 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)