Cheenta Ganit Kendra: Vietnam National Mathematical Olympiad 2012

Thursday, 9 February 2012

Vietnam National Mathematical Olympiad 2012

Problem 1: Define a sequence ${x_n}$ as: $left{begin{aligned}& x_1=3 \ & x_n=frac{n+2}{3n}(x_{n-1}+2) text{for} ngeq 2.end{aligned}right.$
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

be an integer greater than

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

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

(b) Let

E=ACcap BD.

Prove that

E,O,I

are collinear.

Problem 4: Let

be a natural number. There are

boys and

girls standing in a line, in any arbitrary order. A student

will be eligible for receiving

candies, if we can choose two students of opposite sex with

standing on either side of

in

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

boys. There are

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

and

where

is a divisor of

b^2+2

and

is a divisor of

a^2+2.

Prove that

and

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

there exist real number

:

f(b)=a

(b) If

x>y

then

f(x)>f(y)

(c)

f(f(x))=f(x)+12x.

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