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Hanoi Open Mathematical Olympiad - Problems and solutions

Hanoi Open Mathematical Olympiad 2006 Junior Section, Sunday, 9 April 2006 Q1. What is the last two digits of the number (11 + 12 + 13 + · · · + 2006)2 ? Q2. Find the last two digits of the sum 200511 + 200512 + · · · + 20052006 . Q3. Find the number of different positive integer triples (x, y, z) satisfying the equations x2 + y − z = 100 and x + y 2 − z = 124. Q4. Suppose x and y are two real numbers such that x + y − xy = 155 Find. | HANOI MATHEMATICAL SOCIETY NGUYEN VAN MAU HANOI OPEN MATHEMATICAL OLYMPIAD PROBLEMS AND SOLUTIONS Hanoi 2009 Contents Questions of Hanoi Open Mathematical Olympiad 3 1.1 Hanoi Open Mathematical Olympiad 2006 . 3 1.1.1 Junior Section Sunday 9 April 2006 . 3 1.1.2 Senior Section Sunday 9 April 2006 . 4 1.2 Hanoi Open Mathematical Olympiad 2007 . 5 1.2.1 Junior Section Sunday 15 April 2007 . 5 1.2.2 Senior Section Sunday 15 April 2007 . 7 1.3 Hanoi Open Mathematical Olympiad 2008 . 10 1.3.1 Junior Section Sunday 30 March 2008 . 10 1.3.2 Senior Section Sunday 30 March 2008 . 11 1.4 Hanoi Open Mathematical Olympiad 2009 . 12 1.4.1 Junior Section Sunday 29 March 2009 . 12 1.4.2 Senior Section Sunday 29 March 2009 . 14 2 Questions of Hanoi Open Mathematical Olympiad 1.1 Hanoi Open Mathematical Olympiad 2006 1.1.1 Junior Section Sunday 9 April 2006 Q1. What is the last two digits of the number 11 12 13 2006 2 Q2. Find the last two digits of the sum 200511 200512 20 0 52006. Q3. Find the number of different positive integer triples x y z satisfying the equations x2 y z 100 and x y2 z 124. Q4. Suppose x and y are two real numbers such that x y xy 155 and x2 y2 325. Find the value of x3 y3 . Q5. Suppose n is a positive integer and 3 arbitrary numbers are choosen from the set 1 2 3 . 3n 1 with their sum equal to 3n 1. What is the largest possible product of those 3 numbers