More Tripos Problems for Practice : RMO and INMO Algebra

If you want to be a Wrangler, you need to tackle the following (so also the problems in previous post :-)):

If $x(2a-y) = y(2a-z) = z(2a-u) = u(2a-x) = b^{2}$ , show that $x = y = z= u$ unless $b^{2}=2a^{2}$ , and that if this condition is satisfied, the equations are not independent.
Out of n straight lines whose lengths are 1, 2, 3, $\ldots , n$ inches respectively, the number of ways in which four may be chosen which will form a quadrilateral in which a circle may be inscribed is $\frac{1}{48} \{ 2n(n-2)(2n-5)-3+3(-1)^{n}\}$ .
For the expansion of $\frac{1+2x}{1-x^{3}}$ , or otherwise, prove that

$1-3n+\frac{(3n-1)(3n-2)}{1.2}-\frac{(3n-2)(3n-3)(3n-4)}{1.2.3}+\frac{(3n-3)(3n-4)(3n-5)(3n-6)}{1.2.3.4} - etc. = (-1)^{n}$ , where n is an integer, and the series stops at the first term that vanishes.

4. Find the real roots of the equations:

$x^{2}+v^{2}+w^{2}=a^{2}$ , $vw + u(y+z)=bc$ ,

$y^{2}+w^{2}+u^{2}=b^{2}$ , $wu + v(z+x)=ca$ ,

$z^{2}+u^{2}+v^{2}=c^{2}$ , $uv + w(x+y) = ab$ .

5. If the equation $\frac{a}{x+a} + \frac{b}{x+b} = \frac{c}{x+c} + \frac{d}{x+d}$ have a pair of equal roots, then either one of the quantities a or b is equal to one of the quantities c or d, or else $\frac{1}{a} + \frac{1}{b} = \frac{1}{c} + \frac{1}{d}$ . Prove also that the roots are then $-a, -a, 0$ ; $-b, -b, 0$ ; or, $0, 0, \frac{-2ab}{a+b}$ .