# Some random problems in algebra (part b) for RMO and INMO training

1) Solve in real numbers the system of equations:

$y^{2}+u^{2}+v^{2}+w^{2}=4x-1$

$x^{2}+u^{2}+v^{2}+w^{2}=4y-1$

$x^{2}+y^{2}+v^{2}+w^{2}=4u-1$

$x^{2}+y^{2}+u^{2}+w^{2}=4v-1$

$x^{2}+y^{2}+u^{2}+v^{2}=4w-1$

Hints: do you see some quadratics ? Can we reduce the number of variables? …Try such thinking on your own…

2) Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be real numbers such that $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=0$ and $\max_{1 \leq i . Prove that $a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2} \leq 10$.

3) Let a, b, c be positive real numbers. Prove that

$\frac{1}{2a} + \frac{1}{2b} + \frac{1}{2d} \geq \frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a}$

More later

Nalin Pithwa.

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