\(TP\) and \(TQ\) are the tangents at \(P\) and \(Q\) to an ellipse whose centre is \(C\). Prove that \(CT\) bisects \(PQ\). If a pair of conjugate diameters of an ellipse is produced to meet either directrix, prove that the orthocentre of the triangle so formed is the corresponding focus of the curve.
\(P, R\) and \(S\) are points on a conic, and the normal at \(P\) bisects the angle \(RPS\) and cuts the conic again in \(Q\). Prove that \(QR\) and \(QS\) are harmonic conjugates to \(QP\) and the tangent at \(Q\).
The vertex \(A\) of a triangle \(ABC\) is fixed, the magnitude of the vertical angle \(BAC\) is given, and the base \(BC\) lies on a given straight line. Prove that the locus of the circumscribing circle is a hyperbola with \(A\) as focus. Prove also that this circle touches a fixed circle whose centre is the other focus of the hyperbola.
A given conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] is cut by a line \(lx+my=1\) in \(P\) and \(Q\), such that \(OP\) and \(OQ\) are at right angles, where \(O\) is the origin of coordinates. The perpendicular from \(O\) to \(PQ\) meets \(PQ\) in \(N\). Prove that \[ a+b+2(gl+fm)+c(l^2+m^2)=0, \] and that the locus of \(P\) is the circle \[ (a+b)(x^2+y^2)+2gx+2fy+c=0. \]
Prove that in general three normals can be drawn from a point \(P\) to a parabola. Prove also that if \(S\) is the focus, the sum of the angles the three normals make with the axis of the parabola differs from the angle between \(SP\) and the axis by a multiple of two right angles.
Prove that any point from which the tangents to the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) are equally inclined to the line \(y=x\tan\alpha\) lies on the rectangular hyperbola \[ x^2-2xy\cot 2\alpha - y^2 = a^2-b^2. \]
A sphere rolls or slides on a fixed parabolic wire, always touching it at two points. Prove that the centre of the sphere describes part of an equal parabola.
\(S\) is a conic inscribed in the triangle \(ABC\). \(T\) is a conic that touches \(AB, AC\) at \(B\) and \(C\) respectively, and cuts the conic \(S\) in \(P\) and \(Q\). Prove that the tangents to \(S\) at \(P\) and \(Q\) intersect on \(T\).
Find the linear factors of \[ a^3(b-c) + b^3(c-a) + c^3(a-b). \] Show that if \(x^3+y^3+z^3 = 3mxyz\), and \begin{align*} ax^2+by^2+cz^2&=0 \\ ayz+bzx+cxy&=0 \end{align*} then \[ a^3+b^3+c^3 = 3mabc. \]
Thirty balls of which twelve are alike and black, and eighteen are alike and white, are dropped into three boxes, so that there are ten balls in each box. Taking into account permutations among the boxes, find the number of ways in which the balls can be distributed between the boxes so that no box contains more than five black balls.