Two circles \(A, B\) cut orthogonally in \(X\) and \(Y\). A diameter of \(A\) cuts \(B\) in \(P\) and \(Q\). Prove that the points \(X, P, Y, Q\) subtend a harmonic pencil at any point on the circle.
A quadrilateral is inscribed in one circle and circumscribed about another circle. Prove that the internal diagonals intersect in a limiting point of the coaxal family to which the two circles belong.
If \(O\) be the middle point of a chord \(EF\) of a conic and \(POQ, P'OQ'\) any two chords of the conic, prove that any conic through \(P, P', Q, Q'\) will intersect \(EF\) in points equidistant from \(O\).
If a conic touch the sides of a triangle at points where the perpendiculars from the angular points meet the opposite sides, shew that the distances of its centre from the sides are proportional to the lengths of the sides.
Having given \(n\) points on the circumference of a circle shew that \(\frac{1}{2}(n-1)!\) polygons of \(n\) sides can be formed by joining the points. Also shew that if from any other point on the circumference perpendiculars are drawn to all the sides of one of the polygons, the product of these perpendiculars will be the same for all the polygons.
Sum to \(n\) terms \[ \frac{a}{a^2-1} + \frac{a^2}{a^4-1} + \frac{a^4}{a^8-1} + \dots \] and deduce the sum to \(n\) terms of \[ \operatorname{cosec}\theta + \operatorname{cosec}2\theta + \operatorname{cosec}2^2\theta + \dots. \]
If \(u_n - n(1+k)u_{n-1} + n(n-1)ku_{n-2}=0\), and \(u_2=2u_1k\), shew that \[ \frac{u_2}{2!}+\frac{u_3}{3!}+\frac{u_4}{4!}+\dots = u_1 e^k. \]
If \[ \sin^2\theta = \sin(A-\theta)\sin(B-\theta)\sin(C-\theta), \] and \[ A+B+C=\pi, \] prove that \[ \cot\theta = \cot A + \cot B + \cot C. \]
Two uniform rods \(AB, BC\) of equal weight are hinged at \(B\). The end \(A\) can turn about a fixed point and \(BC\) rests across a smooth horizontal peg. If in equilibrium both rods make angles of 60\(^{\circ}\) with the vertical, prove that the reaction at \(B\) divides the angle \(ABC\) into angles whose tangents are as \(-1:14\).
A circular disc can turn about a smooth pivot through its centre on a rough horizontal table. The pressure of the disc on the table is distributed uniformly. Shew that if \(\mu\) be the coefficient of friction and \(W\) the weight of the disc the least force that will turn the disc round the pivot is \(\frac{2}{3}\mu W\).