Problems

S is a given conic and A, B are two fixed points not lying on S. P is a variable point on S, PA meets S again in Q, and QB meets S again in R. Show that the pairs P, R belong to an involution on S if, and only if, A and B are conjugate with respect to S. Prove also that, if ABC is a self-polar triangle with respect to S, there is an infinite number of triangles LMN inscribed in S whose sides MN, NL, LM pass through A, B, C respectively.

P, Q and R are corresponding points of homographic ranges on three lines p, q and r which do not lie in a plane.

1. [(i)] If p, q and r all pass through a point O which is a self-corresponding point of the three ranges, prove that the planes PQR pass through a fixed line.
2. [(ii)] If p and q meet in a point O, and r meets the plane containing p and q in a point H distinct from O, and if O is a self-corresponding point of the ranges on p and q, prove that the planes PQR pass through a fixed point. If also the points of p and q which correspond to H lie on a line through H, prove that the planes PQR pass through a fixed line.

Show that the circles with respect to which a fixed line \[ ax+by+c=0 \] is the polar of the origin form a coaxal system, and find the line of centres, the radical axis, and the limiting points.

P and Q are the intersections of the line \[ lx+my+n=0 \] with the parabola \[ y^2-x=0. \] The circle on PQ as diameter meets the parabola again in R and S. Find the equation of the line RS.

Find the condition that the line \[ lx+my+n=0 \] should touch the conic \[ S_\lambda \equiv ax^2 + by^2 + c + \lambda(px+qy+r)^2 = 0. \] Show that if \[ \lambda\left(\frac{p^2}{a} + \frac{q^2}{b} + \frac{r^2}{c}\right) + 2 = 0, \] the polar reciprocal of \(S_\lambda\) with respect to the conic \[ S \equiv ax^2 + by^2 + c = 0 \] is the conic \(S_\lambda\) itself, and that the polar reciprocal of \(S\) with respect to \(S_\lambda\) is the conic \(S\) itself.

XYZ is the triangle of reference and H, K are the points \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\). The line HK meets YZ, ZX, XY in P, Q, R respectively, and the harmonic conjugates of these points with respect to H and K are P', Q', R' respectively. Find the coordinates of P', Q', R' and show that XP', YQ', ZR' are concurrent.

Show that the eight points of contact of the common tangents of the conics \begin{align*} ax^2 + by^2 + cz^2 &= 0, \\ a'x^2 + b'y^2 + c'z^2 &= 0, \end{align*} lie on a conic, and find its equation.

Shew that the Arithmetic mean of a number of positive quantities is never less than their Geometric mean. If \(x, y, z\) are positive quantities such that \(x+y+z=1\), prove that

1. [(i)] \(\displaystyle \frac{x}{2-x} + \frac{y}{2-y} + \frac{z}{2-z} \le \frac{3}{5}\).
2. [(ii)] \(x^2 y^2 z \le \frac{4^3}{7^7}\).

(i) Denoting the roots of the equation \(x^4-x+1=0\) by \(x_1, x_2, x_3, x_4\), shew that, if \(y_r = x_r^3+x_r\), [\(r=1,2,3,4\)], then \(y_r\) satisfies the equation \(y^4-3y^3+7y^2-7y+5=0\). (ii) Given that the sum of two of its roots is zero, solve completely the equation \[ 4x^4+8x^3+13x^2+2x+3=0. \]

(i) Find the sum to \(n\) terms of the series: \[ \frac{1.2.12}{4.5.6} + \frac{2.3.13}{5.6.7} + \frac{3.4.14}{6.7.8} + \dots. \] (ii) By expanding \(\log_e(1+x^2+x^4)\), or otherwise, shew that \[ \sum_{r=\frac{n}{2}}^n \frac{(-1)^r|r-1|}{(n-r)|2r-n|} = 2\sum_{r=n}^{2n} \frac{(-1)^r|r-1|}{(2n-r)|2r-2n|}, \] where \(n\) and \(r\) are positive integers.