[Toán THCS] SOLVE VIA MAIL COMPETITION QUESTIONS (Translated by Nam Vũ Thành)

1. Compare $\dfrac{A}{B}$ and 3, where A = 1 + $2^{2014}$ + $3^{2013}$ + $4^{2012}$ + ... + $2014{2}$ + 2015 , and B = 1 + $2^{2013}$ + $3^{2012}$ + $4^{2011}$ + ... + $2013^2$ + 2014.

2. Given that x, y, z are integers such that ( $x^3$ + $y^3$ + $z^3$ ) $\vdots$ 27 , prove that either x, y, z are all divisible by 3 or the sum of two numbers among them are divisible by 9.
3. Solve the equation
$\sqrt{\dfrac{2 \sqrt{x}+1}{x+ \sqrt{x+ \sqrt{x}}}}$ = $\sqrt{2 \sqrt{x}+1}$ - $\sqrt{x+ \sqrt{x}}+1$ .

4. Let a, b and c be positive real numbers such that $\dfrac{1}{2a+1}$ + $\dfrac{1}{2b+1}$ + $\dfrac{1}{2c+1}$ $\geq$ 1 . Prove that $\dfrac{1}{6a+1}$ + $\dfrac{1}{6b+1}$ + $\dfrac{1}{6c+1}$ $\geq$ $\dfrac{3}{7}$ .

5. Given the set P={red, green, black, white} . Determine if each of the following is a partition of P.
a) $P_{1}$ = [{red}, {green, black}]
b) $P_{2}$ = [{white, black, red, green}]
c) $P_{3}$ = [\phi, {red, green}, {black, white}].

6. Let O be a point on the equilateral triangle ABC. Given that the area of the shaded region is half the area of the triangle. Prove that the point O lies on one of the median of the triangle.