contests around the world 1999 2000 part 1

... = 0, 1, 2, the last expression equals 22000 1 219 99 −2 219 99 2000 f (3k) · f (2 19 99 +1 k =1 22000 +2 k= f (k) · = k∈S0 = ( 1) f (k) · k∈S1 219 99 +1 (−f (k)) k∈S2 219 99 219 99 f (k) = − k =1 −f ... the sides B1 C1 , C1 A1 and A1 B1 of triangle A1 B1 C1 such that ∠ABC = ∠A1 B1 C1 , ∠BCA = ∠B1 C1 A1 , and ∠CAB = ∠C1 A1 B1 Prove that the orthocenters of the triangle ABC and triangle A1 B1 ... − 1) ci = (an +1 i − 1) ci 1 ⇐⇒ (ai − 1) (bi 1 + bi ) = (an +1 i − 1) (bi−2 + ai 1 bi 1 ) ⇐⇒ (ai − 1) bi 1 + (ai − 1) bi = (an +1 i − 1) bi−2 + (an − ai 1 )bi 1 ⇐⇒ (ai − 1) bi 1 + (an−i − 1) bi 1 = (ai−1...

Ngày tải lên: 15/03/2014, 15:34

... OO1 , ∠G1 OG = ∠CGO = x Therefore, right triangles GOG1 and F1 BH1 are similar Hence BH1 OG1 = F1 H1 GG1 22 Bulgaria Combining the last two equalities yields BH1 · O1 D DH1 · O1 D = OG1 OO1 ... xm = = (n − 1) ! (n − 1) ! n 1 m m =1 n−m n 1 m =1 n−m n 1 n n 1 m m =1 m − m =1 m n 1 m =1 m = n 1 m m =1 n−m n 1 m =1 n−m =n− n 1 + + ··· + n Problem Find all positive integers n such that there exist ... a1 n 1 fn = an + First Solution: Rewrite the recursive relation as 1 an = ( 1) n + nan 1 + n ( 1) n 1 + (n − 1) an−2 2 If ( 1) n 1 + (n − 1) an−2 = an 1 , then we have that an = ( 1) n + 1 n nan−1...

Ngày tải lên: 15/03/2014, 15:35

... OO1 , ∠G1 OG = ∠CGO = x Therefore, right triangles GOG1 and F1 BH1 are similar Hence BH1 OG1 = F1 H1 GG1 22 Bulgaria Combining the last two equalities yields BH1 · O1 D DH1 · O1 D = OG1 OO1 ... xm = = (n − 1) ! (n − 1) ! n 1 m m =1 n−m n 1 m =1 n−m n 1 n n 1 m m =1 m − m =1 m n 1 m =1 m = n 1 m m =1 n−m n 1 m =1 n−m =n− n 1 + + ··· + n Problem Find all positive integers n such that there exist ... a1 n 1 fn = an + First Solution: Rewrite the recursive relation as 1 an = ( 1) n + nan 1 + n ( 1) n 1 + (n − 1) an−2 2 If ( 1) n 1 + (n − 1) an−2 = an 1 , then we have that an = ( 1) n + 1 n nan−1...

Ngày tải lên: 05/06/2014, 18:59

... 13 9 14 0 14 1 14 2 14 4 14 5 14 6 14 7 14 8 14 9 15 0 15 2 15 3 15 5 16 1 16 2 16 3 16 5 16 6 16 7 16 8 16 9 19 97 Regional Contests: Problems 4 .1 Asian Pacific Mathematics Olympiad ... satisfying the condition: a1 + b1 = a1 + b1 a1 + b1 a1 + b1 a1 + b1 a1 + b1 a1 + b1 =4 =5 =6 =7 =8 =9 (6, 3), (12 , 6), (18 , 9), (24, 12 ), (30, 15 ), (36, 18 ), (42, 21) , (48, 24) (12 , 4), (24, 8), (36, 12 ), ... p1 , are the primes in increasing order and n has base representation c0 +2c1 +4c2 +· · ·, then xn = pc0 pc1 · · · In particular, 11 111 1 = · · 11 · 13 · 37 = p1 p3 p4 p5 p10 , so xn = 11 111 1...

Ngày tải lên: 15/03/2014, 15:34

... Let p = 1/ 2 · 3/4 · · 19 97 /19 98 and q = 2/3 · 4/5 · · 19 98 /19 99 Note p < q, so p2 < pq = 1/ 2 · 2/3 · · 19 98 /19 99 = 1 /19 99 Therefore, p < 1 /19 9 91 /2 < 1/ 44 Also, p= 19 98! 19 98 = 2 19 98 , ... square-free Therefore (x1 , y1 ) = (1, 1) or (1, 2) Consider them as separate cases: Case 1: (x1 , y1 ) = (1, 1) Then d2 z = (13 +19 96 )19 97 = 19 97·72 · 41 Since 19 97 is coprime to and 41, d = 1, These ... to 1 a2 > 7b2 As in (a), we obtain 1 sin ∠CB1 M = b1 b1 sin ∠B1 CA1 √ = √ a1 a1 1 a2 + b2 1 4a1 b1 Setting b2 /a2 = x, we get 1 sin ∠CB1 M = √ 14 x − x2 − < √ 2− 1 1= , 49 since x < 1/ 7 Therefore...

Ngày tải lên: 15/03/2014, 15:34

... Ireland (10 0); Italy (10 2); the Netherlands (10 5); Norway (10 8); Portugal (11 1); Spain (11 4); Sweden (11 7); Switzerland (12 0); and the United Kingdom (12 3) vii 16 747$ CNTS 02 -12 -08 14 :29:26 ... Latin America 15 3 Negotiating Primers for Argentina (15 7); Brazil (15 9); Chile (16 3); Colombia (16 5); Costa Rica (16 8); Cuba (17 1); Ecuador (17 4); Guatemala (17 7); Mexico (18 0); Peru (18 3); and Venezuela ... otherwise, without the prior written permission of AMACOM, a division of American Management Association, 16 01 Broadway, New York, NY 10 019 Printing number 10 16 747$ $$FM 02 -12 -08 14 :29 :17 ...

Ngày tải lên: 17/08/2013, 09:49

Ngày tải lên: 01/11/2013, 16:20

... the marketplace: Sub-Saharan Africa 6.2 More than just colonial traces: South and South-East Asia 6.3 Thousand Islands: The Pacific 6.4 Future aspirations: East Asia 11 1 11 2 12 2 13 4 13 5 14 9 16 4 ... differences between the North and the South 84 Printed Jamaican creole in a newspaper cartoon 10 1 10 Treaty of Waitangi 11 4 11 Regional syntax: transitive farewell in New Zealand English 11 6 12 Hong Kong ... considerations 210 220 222 224 Conclusion vi 18 9 229 18 9 19 7 211 213 215 218 Contents Appendix 1: Phonetic characters Appendix 2: A list of guiding questions on English in any specific region 2 31 Glossary...

Ngày tải lên: 26/12/2013, 19:29

... Love 61 The Family Circle 75 Clocking On 87 Time O◊ 10 1 Eating and Drinking 11 3 Below Par 12 5 From Cradle to Grave 13 1 Otherworldly 14 3 All Creatures Great and Small 14 9 Whatever the Weather 16 3 ... China, whilst other forms of wordless communication include the talking drums (ntumpane) of the Kele in Congo, the xylophones used by the Northern Chin of Burma, the banging on the roots of trees ... Things 17 1 Seeing Things 17 9 Number Crunching 18 5 What’s in a Name? 2 01 Foreword My interest in the quirkiness of foreign words was triggered when one day, working as a researcher for the BBC...

Ngày tải lên: 15/01/2014, 10:51

... as they placed themselves at the whist-table, they continued to argue the matter Stuart and Flanagan played together, while Phileas 16 Around the World in 80 Days Fogg had Fallentin for his partner ... country.’ 42 Around the World in 80 Days ‘I regulate my watch? Never!’ ‘Well, then, it will not agree with the sun.’ ‘So much the worse for the sun, monsieur The sun will be wrong, then!’ And the worthy ... man then descended, the street-door was double-locked, and at the end of Saville Row they took a cab and drove rapidly to Charing Cross The cab stopped 24 Around the World in 80 Days before the...

Ngày tải lên: 19/01/2014, 16:20

... airtelbroadband.in # Zombies 34,480 33, 714 33 ,10 5 27,4 51 22, 714 21, 609 21, 146 20,793 16 ,088 14 ,857 Source: Commtouch Labs Page 14 www.commtouch.com Top Spam Topics Topics of Spam Email Q1 2009 Loans – 28% Dating ... productivity After the fall of McColo in Q4 2008 and the subsequent drop in the amount of spam being transmitted, the levels have slowly returned to the levels they were before the incident Spammers ... that the message was a legitimate technology article recommended by someone in the community; both the “Sender” and the closing line refer to the Tech Update service The hyperlink within the email...

Ngày tải lên: 18/02/2014, 02:20

... 316 10 .11 Polish 19 93 317 10 .12 Polish 19 94 318 10 .13 Polish 19 95 319 10 .14 Polish 19 96 320 10 .15 Polish 19 97 3 21 10 .16 Polish ... 13 9 3 .14 BMO 19 78 14 0 3 .15 BMO 19 79 14 1 3 .16 BMO 19 80 14 2 3 .17 BMO 19 81 14 3 3 .18 BMO 19 82 14 4 3 .19 BMO 19 83 14 5 ... 338 11 .11 Spanish 2000 339 11 .12 Spanish 20 01 340 11 .13 Spanish 2002 3 41 11. 14 Spanish 2003 342 12 Swedish (19 61 – 2003) 343 12 .1 Swedish...

Ngày tải lên: 22/02/2014, 16:09

... 23 10 15 25 20 13 14 26 19 12 11 22 23 13 17 14 20 16 18 11 21 9.9 10 .25 10 .35 12 .55 14 14 .05 14 .15 15 .1 15.25 15 .35 15 .75 15 .9 18 .55 20.05 21. 8 27 24 23 26 28 30 29 26 21 25 28 24 29 30 16 27 ... per 10 0, OECD 17 Household penetration, OECD 10 3G penetration, GC 18 17 Wi-Fi hotspots per 10 0000, Jwire 27 10 15 29 12 19 11 16 12 14 13 10 15 20 19 18 22 21 25 11 13 17 12 15 18 14 19 16 20 ... 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Per inhabitant rank 17 10 11 12 15 14 18 16 13 20 19 25 21 24 27 23 28 22 26 30 29 Change in rank 40% 30% 20% 10 % 0% Denmark Netherlands...

Ngày tải lên: 06/03/2014, 21:20

... 2 010 n AMFm US$ (millions) 2,000 n Others 1, 500 n World Bank 1, 000 n DFID 500 n PMI 2000 20 01 2002 2003 2004 2005 2006 2007 2008 2009 2 010 2 011 2 012 2 013 2 014 2 015 n Global Fund www.mdghealthalliance.org ... eliminate the transmission of HIV from mother-to-child by the end of 2 015 In collaboration with UNAIDS, PEPFAR, the countries suffering the greatest burden and other partners, virtually eliminate the ... by 2 015 and Keeping Their Mothers Alive: 2 011 -2 015 This Global Plan provides the foundation for country-led progress towards the elimination of new HIV infections among children and keeping their...

Ngày tải lên: 14/03/2014, 11:20

... HIV-AIDS Based on research findings, 13 2 participants with savings accounts saved 3,729 USD, not including the 2 :1 match With the match, participants saved a total of 11 ,18 7 USD (Curley, Ssewamala, & ... their biological father present in the household than did youths in the control group (Curley, Ssewamala & Han, 2009; Ssewamala, Han, & Neilands, 2009) 11 Participants’ caregivers are the mother ... YOUTH SAVINGS AROUND THE WORLD Youth Savings around the World: Youth Characteristics, Savings Performance, and Potential Effects The global population of youth is increasing rapidly, particularly...

Ngày tải lên: 15/03/2014, 10:20

... in 2000, 1, 159 (8 .1 percent) were born at low birth weight; 257 (1. 8 percent) were born at very low birth weight; and more than 10 0 infants died (Allegheny County Health Department, 19 99, 2000a,b) ... is needed because the county fails to meet six other HP 2 010 objectives, as shown in Table 1. 2 The result of the county’s failure to meet these objectives means that of the 14 ,249 infants born ... for all mothers as a percentage of live births Allegheny County 16 .6 14 7.5 3.9 1. 8 82.7 HP 2 010 Objective 11 0.9 0.9 90 Improving Maternal and Child Health Care Casey Foundation, 19 99) ; in comparison,...

Ngày tải lên: 22/03/2014, 09:20

... About These Women and Organizations Afterword 15 3 15 6 16 3 17 5 18 0 18 5 18 9 19 1 19 5 Dr Helene Gayle, President and CEO, CARE Acknowledgments About the Author 203 205 Index 206 Foreword ; A s a ... Saving the Environment One Plate at a Time Building for the Future Safeguarding the Environment in Your Own Life 11 7 12 2 12 9 13 9 14 5 14 9 Part Five addressing discrimination Overcoming Discrimination ... publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 11 1 River Street, Hoboken, NJ 07030, 2 01- 748-6 011 , fax 2 01- 748-6008, or online at www.wiley.com/go/...

Ngày tải lên: 28/03/2014, 19:20

... collaborations around the world By the end of FASD 2 013 , participants can expect to: - recognize the implications of cutting-edge and emerging evidence-based research and knowledge of practice around the world ... Discussion regarding the need for further research and potential opportunities for research will be facilitated in the session 1: 10pm D1 Concurrent Session D (Repeated G1) Oral Papers (15 -Min Presentation ... Session #1 (12 :00pm - 1: 00pm) Concurrent Breakout Session D (D1 - D8) Break Plenary Plenary Lunch (Provided) Optional Lunch Session #2 (12 :20pm - 1: 20pm) Concurrent Breakout Session E (E1 - E8)...

Ngày tải lên: 31/03/2014, 15:20

... 13 and 18 Article 12 of the Convention on the Rights of the Child recognizes the right of the children to express their own views in matters affecting them Article 13 enshrines the right to freedom ... Children’s/Young People’s Use of the Media Television Television is the dominant medium for young people—and adults around the world From the mid -19 80s to the mid -19 90s, the number of television channels, ... 31% 16 %16 % issues in the Brazilian media and the model is 5.7% now being applied in eight other Latin American countries TROC The Young Reporter of Albania Dozens of young people, from 13 to 18 ,...

Ngày tải lên: 31/03/2014, 18:20