Through any point \(P\) lines are drawn parallel to the internal bisectors of the angles of a triangle \(ABC\) to meet the opposite sides in \(D, E, F\). Prove that if \(D, E, F\) are collinear \(P\) lies on the conic \[ (b+c)\beta\gamma + (a+c)\alpha\gamma + (a+b)\alpha\beta = 0, \] where the coordinates are trilinear and \(ABC\) is the triangle of reference. Prove that the centre of the conic is the centre of the inscribed circle of the triangle whose vertices are the mid-points of the sides of \(ABC\).
If a set of numbers is added together, shew that the sum of the digits in them is equal to the sum of the digits in the answer or exceeds the latter by a multiple of nine.
If \(a, b, c, d\) are in ascending order of magnitude, the equation \[ (x-a)(x-c) = k(x-b)(x-d) \] has real roots for all values of \(k\).
Prove by induction or otherwise that if \(r\) is a positive integer then the sum of the infinite series \[ \frac{1^r}{1!} + \frac{2^r}{2!} + \frac{3^r}{3!} + \dots \] is an integral multiple of \(e\).
If three angles be such that the sum of their cosines is zero and the sum of their sines is zero, prove that any two of them differ by \(2r\pi \pm \frac{2\pi}{3}\), where \(r\) is an integer, and that the sum of the squares of their cosines is equal to the sum of the squares of their sines.
Two isosceles triangles have the same inscribed circle and the same circumscribed circle: prove that their vertical angles \(V_1\) and \(V_2\) satisfy the equation \[ \sin\frac{1}{2}V_1 + \sin\frac{1}{2}V_2 = 1. \]
From the focus \(S\) of an ellipse a perpendicular \(SY\) is drawn to a tangent and produced to \(Z\) so that \(SZ=2SY\); shew that the square of the tangent from \(Z\) to the director circle is equal to \(2SY^2\).
From a point \(T\) a perpendicular \(TL\) is drawn on its polar with respect to a parabola; prove that when \(T\) moves on a line parallel to the axis the locus of \(L\) is a line through the focus.
Given two vertices of a triangle and its area, shew that the locus of its orthocentre is two parabolas.
On the sides of a triangle \(ABC\) equilateral triangles \(BPC, CQA,\) and \(ARB\) are described externally; shew that the lines \(AP, BQ, CR\) are of equal length.