By the methods of abridged notation, or otherwise, prove that if three conics have one chord common to all, their three other common chords are concurrent. Prove also that if two conics have each double contact with a third, their chords of contact with the third conic, and a pair of their chords of intersection with each other, will all meet in a point and form a harmonic pencil.
Shew how to construct a circle to touch a given straight line and pass through two given points on the same side of the straight line. Two circles cut in \(A\) and \(B\); \(P\) is a point on one of the circles; the tangent at \(A\) to that circle cuts the other circle in \(Q\) and \(BP\) cuts it in \(R\). Prove that \(QR\) is parallel to \(AP\).
From a point \(P\) on the circumscribing circle of the triangle \(ABC\) perpendiculars \(PL, PM\) and \(PN\) are drawn to the sides \(BC, CA, AB,\) respectively. Prove that \(LMN\) is a straight line. Prove also that, if \(PL\) produced cut the circle again in \(K\), the straight line \(LMN\) is parallel to \(AK\).
\(ABC\) and \(A'B'C'\) are two triangles such that \(AA', BB'\) and \(CC'\) meet in a point. Prove that the points of intersection of \(BC\) and \(B'C'\), \(CA\) and \(C'A'\), \(AB\) and \(A'B'\) are collinear.
\(PQ\) is a straight line, \(AB\) is a straight line through \(P\) at right angles to \(PQ\), and \(CD\) is a straight line through \(Q\) at right angles to the plane containing \(PQ\) and \(AB\). Prove that the volume of the tetrahedron \(ABCD\) is \(\frac{1}{6}AB \cdot CD \cdot PQ\).
Prove that, if the sides of a triangle touch a parabola the focus of the parabola is a point on the circle circumscribing the triangle. \(P\) is a point on a parabola whose focus is \(S\); the perpendicular bisector of \(SP\) cuts the tangent at \(P\) in \(T\) and the diameter through \(T\) cuts the parabola in \(Q\). Prove that the other tangent from \(T\) is equal to \(4TQ\).
Prove that the rectangle contained by the perpendiculars from the foci of an ellipse on any tangent is constant. Tangents \(TP, TP'\) are drawn to an ellipse, whose foci are \(S\) and \(S'\), such that \(SP\) is parallel to \(S'P'\). Determine the locus of \(T\).
Shew that the equations \[ x^2+\lambda x + \mu = 0 \] and \[ x^3 + \lambda' x + \mu' = 0 \] have one root the same if \((\lambda\mu' - \lambda'\mu)(\lambda - \lambda') + (\mu - \mu')^2 = 0\); and that the other two roots are given by the quadratic \[ (\mu - \mu')^2 x^2 + (\lambda\mu' - \lambda'\mu)(\lambda'^2 - \lambda^2 + 2\mu - 2\mu')x + \mu\mu'(\lambda-\lambda')^2 = 0. \]
Prove that \(\left(1+\frac{1}{x}\right)^x\) is never greater than 3, however large \(x\) is. Prove that \[ \frac{1}{1 \cdot 2} - \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} - \frac{1}{4 \cdot 5} + \dots = 2\log 2 - 1. \]
Prove the rule for the formation of successive convergents of a continued fraction. If \(\frac{p_n}{q_n}\) is the \(n\)th convergent of the continued fraction \(\frac{1}{1+}\frac{1}{2+}\frac{1}{1+}\frac{1}{2+}\dots\), prove that \(p_n + p_{n-4} = 4p_{n-2}\).