VN-TST 2008

Find all values of the positive integer $m$ such that there exists polynomials $P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $a,b$ which satisfying $a^m-b^2=0$ , we always have that $P(R(a,b))=a$ and $Q(R(a,b))=b$ .

Let an integer $n > 3$ . Denote the set $T=\{1,2, \ldots,n\}.$ A subset S of T is called wanting set if S has the property: There exists a positive integer $c$ which is not greater than $\frac {n}{2}$ such that $|s_1 - s_2|\ne c$ for every pairs of arbitrary elements $s_1,s_2\in S$ . How many does a wanting set have at most are there ?

Let $m$ and $n$ be positive integers. Prove that $6m | (2m + 3)^n + 1$ if and only if $4m | 3^n + 1$

Let $k$ be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that $\frac {AL}{AD} = \frac {BM}{BE} = \frac {CN}{CF} = k$ . Denote $(O_1),(O_2),(O_3)$ be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C.

1. Prove that when $k = \frac{1}{2}$ , three circles $(O_1),(O_2),(O_3)$ have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles.
2. Find all values of k such that three circles $(O_1),(O_2),(O_3)$ have exactly two common points

Consider the set $M = \{1,2, \ldots ,2008\}$ . Paint every number in the set $M$ with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets:

$S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}$ ; $S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}$ .

Prove that $2|S_1| > |S_2|$ (where $|X|$ denotes the number of elements in a set $X$ ).