Vietnam National Mathematical Olympiad 2012

Problem 1: Define a sequence {x_n} as: \(\left{\begin{aligned}& x_1=3 & x_n=\frac{n+2}{3n}(x_{n-1}+2) \text{for} n\geq 2.\end{aligned}\right.\)
Prove that this sequence has a finite limit as nto+infty. Also determine the limit.

Problem 2:  Let langle a_nrangle and langle b_nrangle be two sequences of numbers, and let m be an integer greater than 2. Define P_k(x)=x^2+a_kx+b_k, k=1,2,cdots, m. Prove that if the quadratic expressions P_1(x), P_m(x) do not have any real roots, then all the remaining polynomials also don’t have real roots.
Problem 3:  Let ABCD be a cyclic quadrilateral with circumcentre O, and the pair of opposite sides not parallel with each other. Let M=ABcap CD and N=ADcap BC. Denote, by P,Q,S,T; the intersection of the angle bisectors of angle MAN and angle MBN; angle MBN and angle MCN; angle MDN and angle MAN. Suppose that the four points P,Q,S,T are distinct.
(a) Show that the four points P,Q,S,T are concyclic. Find the centre of this circle, and denote it as I.
(b) Let E=ACcap BD. Prove that E,O,I are collinear.
Problem 4:  Let n be a natural number. There are n boys and n girls standing in a line, in any arbitrary order. A student X will be eligible for receiving m candies, if we can choose two students of opposite sex with X standing on either side of X in m ways. Show that the total number of candies does not exceed frac 13n(n^2-1).
Problem 5:  For a group of 5 girls, denoted as G_1,G_2,G_3,G_4,G_5 and 12 boys. There are 17 chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met:
(a) Each chair has a proper seat.
(b) The order, from left to right, of the girls seating is G_1; G_2; G_3; G_4; G_5.
(c) Between G_1 and G_2 there are at least three boys.
(d) Between G_4 and G_5 there are at least one boy and most four boys.
How many such arrangements are possible?
Problem 6:  Consider two odd natural numbers a and b where a is a divisor of b^2+2 and b is a divisor of a^2+2. Prove that a and b are the terms of the series of natural numbers langle v_nrangle defined by
v_1 = v_2 = 1; v_n = v_ {n-1}-v_{n-2} text{for} ngeq 3.
Problem 7:  Find all f:R to R such that:
(a) For every real number a there exist real number b:f(b)=a
(b) If x>y then f(x)>f(y)
(c) f(f(x))=f(x)+12x.
 

 

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