Mathematical olympiad challenges

... introduce 2n−1 f (k) < n f (n) ≤ 1/n2 g(n) = k=n Then g(n) − g(n + 1) = f (n) − f (2n) − f (2n + 1) = (f (2n) + f (2n + 1))3 − f (2n) − f (2n + 1) = (f (2n) + f (2n + 1)) f (2n) f (2n + 1) = ... |M \ A| + |F \ A| + |A| = 2( n − l) + l = 2n − l But the degree of any vertex of G is 2k and thus we get 2k ≤ 2n − l, that is, l ≤ 2n − 2k Finally, 2k − n < l ≤ 2n − 2k implies that k < 3n This ... f...

Ngày tải lên: 07/03/2014, 07:20

Ngày tải lên: 07/03/2014, 19:20

Ngày tải lên: 07/03/2014, 19:20

... Questions of Hanoi Open Mathematical Olympiad 1.1 Hanoi Open Mathematical Olympiad 2006 1.1.1 Junior Section, Sunday, April 2006 1.1.2 Senior Section, Sunday, April 2006 1.2 Hanoi Open Mathematical ... = abcd be the 4-digit prime number Prove that the equation ax3 + bx2 + cx + d = has no rational roots 10 1.3 Hanoi Open Mathematical Olympiad 2008 1.3 Hanoi Op...

Ngày tải lên: 07/03/2014, 19:20

... k(k + 2) 1 − k k+2 for any positive integer k, so 1 1 1 + + + + + 15 35 63 99 14 3 1 1 1 + + + + + = × 3 × 5 × 7 × 9 × 11 11 × 13 1 1 1 = − + − + ··· + − 3 11 13 1 = × 1 = 13 13 Example 10 If ab ... 3x 5x − (1 + 3x) 15 − 2x + 16 − 2x =1 = = , 1 15 15 15 10 − 6x x 14 x − (10 − 6x) 10 − 13 x x 2x − − = − = , 2 14 14 x− it follows that 16 − 2x 10 − 13...

Ngày tải lên: 11/03/2014, 16:55

Ngày tải lên: 10/07/2014, 13:57

... (a, b, c) such that the number N= (a − b)(b − c)(c − a) +2 is a power of 2016 (A power of 2016 is an integer of the form 2016n , where n is a non-negative integer.) Solution Let a, b, c be integers ... positive integer such that (a − b)(b − c)(c − a) + = · 2016n We set a − b = −x, b − c = −y and we rewrite the equation as xy(x + y) + = · 2016n (1) If n > 0, then the right hand side is div...

Ngày tải lên: 03/08/2016, 08:32

... What we love to we find time to do! Nguyen Anh Hanoi Mathematical Society Tuan Hanoi Open Mathematical Olympiad 2009 Senior Section Sunday, 29 March 2009 08h45 - 11h45 Important: Answer all 14 questions ... allowed Q1 What is the last two digits of the number 1000.1001 + 1001.1002 + 1002.1003 + … + 2008 .2009? (A) 25; (B) 41; (C) 36; (D) 54; (E) None of the above Q2 Which is l...

Ngày tải lên: 03/12/2016, 05:12

... Challenges in Geometry This page intentionally left blank Challenges in Geometry for Mathematical Olympians Past and Present CHRISTOPHER J BRADLEY Great Clarendon Street, Oxford OX2 6DP Oxford ... (2.1.5) Integral R for integer-sided triangles with integer area Since [ABC] is an integer, R = abc/4[ABC], and a/ sin A = b/ sin B = c/ sin C = 2R, it follows that...

Ngày tải lên: 12/05/2014, 02:23

... HSCOBS adopted the report of the workshop as their strategic plan for bridge engineering The detailed plan follows Grand Challenges: Strategic Plan for Bridge Engineering GRAND CHALLENGES: A STRATEGIC ... Virginia Name Grand Challenges: Strategic Plan for Bridge Engineering 19 Dennis Mertz REPORT WRITER University of Delaware AASHTO Kelley Rehm NCHRP...

Ngày tải lên: 06/09/2012, 14:24

... new health care system increasingly recognized [3] Medical professionals have important role and power in adopting and running health- care systems Therefore, their views on efficiency of such system ... 15 WHO Health system building blocks [http://www.who.int/healthsystems/ EN_HSSkeycomponents.pdf] 16 Ministry of Health: Proceedings of the conference on reform of Iraq...

Ngày tải lên: 25/10/2012, 10:06

... start of an MET program and years later *p

Ngày tải lên: 25/10/2012, 10:35