2007 Vietnam National Olympiad

Part of Vietnam National Olympiad

Subcontests

(3)

2

Vietnamese national Olympiad 2007, problem 3

Let B,C be fixed points and A be roving point. Let H, G be orthecentre and centroid of triagle ABC. Known midpoint of HG lies on BC, find locus of A

Vietnamese national Olympiad 2007, problem 6

Let ABCD be trapezium that is inscribed in circle (O) with larger edge BC. P is a point lying outer segment BC. PA cut (O) at N(that means PA isn't tangent of (O)), the circle with diameter PD intersect (O) at E, DE meet BC at N. Prove that MN always pass through a fixed point.

2

Vietnamese Olympiad national 2007, problem 2

x,y

be integer number with

x,y\neq-1

\frac{x^{4}-1}{y+1}+\frac{y^{4}-1}{x+1}\in\mathbb{Z}

x^{4}y^{44}-1

x+1

Vietnamese Olympiad national 2007, problem 5

b>0

, find all functions

f: \mathbb{R}\rightarrow\mathbb{R}

f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}

2

Vietnamese Olympiad national 2007, problem 1

Solve the system of equations: \{\begin{array}{l}(1+\frac{12}{3x+y}).\sqrt{x}=2\$1-\frac{12}{3x+y}).\sqrt{y}=6\end{array}$

Vietnamese national Olympiad 2007, problem 4

Given a regular 2007-gon. Find the minimal number

k

such that: Among every

k

vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon.