MathDB

3

Part of 2009 IMO Shortlist

Problems(3)

IMO Shortlist 2009 - Problem C3

Source:

7/6/2010
Let nn be a positive integer. Given a sequence ε1\varepsilon_1, …\dots, εn−1\varepsilon_{n - 1} with εi=0\varepsilon_i = 0 or εi=1\varepsilon_i = 1 for each i=1i = 1, …\dots, n−1n - 1, the sequences a0a_0, …\dots, ana_n and b0b_0, …\dots, bnb_n are constructed by the following rules: a_0 = b_0 = 1,   a_1 = b_1 = 7, \begin{array}{lll} a_{i+1} = \begin{cases} 2a_{i-1} + 3a_i, \\ 3a_{i-1} + a_i, \end{cases} & \begin{array}{l} \text{if } \varepsilon_i = 0, \\ \text{if } \varepsilon_i = 1, \end{array} & \text{for each } i = 1, \dots, n - 1, \$$15pt] b_{i+1}= \begin{cases} 2b_{i-1} + 3b_i, \\ 3b_{i-1} + b_i, \end{cases} & \begin{array}{l} \text{if } \varepsilon_{n-i} = 0, \\ \text{if } \varepsilon_{n-i} = 1, \end{array} & \text{for each } i = 1, \dots, n - 1. \end{array} Prove that an=bna_n = b_n.
Proposed by Ilya Bogdanov, Russia
linear algebracombinatoricsrecursionSequenceIMO Shortlist
IMO Shortlist 2009 - Problem G3

Source:

7/5/2010
Let ABCABC be a triangle. The incircle of ABCABC touches the sides ABAB and ACAC at the points ZZ and YY, respectively. Let GG be the point where the lines BYBY and CZCZ meet, and let RR and SS be points such that the two quadrilaterals BCYRBCYR and BCSZBCSZ are parallelogram. Prove that GR=GSGR=GS.
Proposed by Hossein Karke Abadi, Iran
geometrysymmetryIMO ShortlistTriangleparallelogram
IMO Shortlist 2009 - Problem N3

Source:

7/5/2010
Let ff be a non-constant function from the set of positive integers into the set of positive integer, such that a−ba-b divides f(a)−f(b)f(a)-f(b) for all distinct positive integers aa, bb. Prove that there exist infinitely many primes pp such that pp divides f(c)f(c) for some positive integer cc.
Proposed by Juhan Aru, Estonia
functionalgebramodular arithmeticnumber theoryDivisibilityIMO Shortlist