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 :-)):

  1. 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.
  2. 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}\}.
  3. 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}.

More challenges are on the way! Are you getting ready for RMO 2016? !

Nalin Pithwa

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