Some random assorted (part A) problems in algebra for RMO and INMO training

You might want to take a serious shot at each of these. In the first stage of attack, apportion 15 minutes of time for each problem. Do whatever you can, but write down your steps in minute detail. In the last 5 minutes, check why the method or approach does not work. You can even ask — or observe, for example, that if surds are there in an equation, the equation becomes inherently tough. So, as a child we are tempted to think — how to get rid of the surds ?…and so on, thinking in math requires patience and introversion…

So, here are the exercises for your math gym today:

1) Prove that if x, y, z are non-zero real numbers with $x+y+z=0$ , then

$\frac{x^{2}+y^{2}}{x+y} + \frac{y^{2}+z^{2}}{y+z} + \frac{z^{2}+x^{2}}{x+z} = \frac{x^{3}}{yz} + \frac{y^{3}}{zx} + \frac{z^{3}}{xy}$

2) Let a b, c, d be complex numbers with $a+b+c+d=0$ . Prove that

$a^{3}+b^{3}+c^{3}+d^{3}=3(abc+bcd+adb+acd)$

3) Let a, b, c, d be integers. Prove that $a+b+c+d$ divides

$2(a^{4}+b^{4}+c^{4}+d^{4})-(a^{2}+b^{2}+c^{2}+d^{2})^{2}+8abcd$

4) Solve in complex numbers the equation:

$(x+1)(x+2)(x+3)^{2}(x+4)(x+5)=360$

5) Solve in real numbers the equation:

$\sqrt{x} + \sqrt{y} + 2\sqrt{z-2} + \sqrt{u} + \sqrt{v} = x+y+z+u+v$

6) Find the real solutions to the equation:

$(x+y)^{2}=(x+1)(y-1)$

7) Solve the equation:

$\sqrt{x + \sqrt{4x + \sqrt{16x + \sqrt{\ldots + \sqrt{4^{n}x+3}}}}} - \sqrt{x}=1$

8) Prove that if x, y, z are real numbers such that $x^{3}+y^{3}+z^{3} \neq 0$ , then the ratio $\frac{2xyz - (x+y+z)}{x^{3}+y^{3}+z^{3}}$ equals $2/3$ if and only if $x+y+z=0$ .