The opposite sides of a quadrilateral inscribed in a circle meet in \(P\) and \(Q\). Prove that the bisectors of the angles at \(P\) and \(Q\) form a rectangle.
Three circles \(OBC, OCA, OAB\) are cut by a fourth circle through \(O\) in points \(P, Q, R\) respectively. Prove that \(BP \cdot CQ \cdot AR = CP \cdot AQ \cdot BR\).
A cylindrical barrel of radius \(a\) rests with its curved surface on a horizontal floor. A uniform straight plank of length \(2l\) lies symmetrically across it in such a position that its centre is in contact with the barrel, and its lower end rests on the floor. The coefficients of friction at all three points of contact are equal. There is limiting friction at one of the three points of contact. Shew that this point is never the point of contact of the barrel with the floor, but is the point of contact of the plank with the floor or with the barrel according as \(3a^2 \lesseqgtr l^2\).
Give an account of the principal properties of a system of confocal ellipses and hyperbolas. Establish a connection with (1) coaxal circles, and (2) conics touching four given straight lines; and deduce some properties of these systems from properties of confocals.
A variable circle passes through a fixed point \(A\) and cuts at right angles a given circle whose centre is \(O\). Prove that the locus of the centre of the first circle is a straight line perpendicular to \(OA\).
Given two points \(A, B\) on a rectangular hyperbola, and the tangents \(AT, BT'\) at these points, obtain a geometrical construction for the centre.
Four equal uniform rods of length \(a\) are jointed so as to form a square. Two adjacent sides rest in contact with two smooth pegs which are on the same level and at a distance \(2c\) apart. The square is kept rigid by two strings stretched between opposite corners and in the position of equilibrium these two strings are horizontal and vertical respectively. If \(T_1\) and \(T_2\) are the tensions of the two strings and \(W\) is the weight of each rod, shew that \(T_1 - T_2 = 4W(c\sqrt{2}/a - 1)\).
Give an account of the methods employed for the solution of triangles, giving as many alternative methods as possible, with the advantages or disadvantages of each, stating the formulae employed. Treat carefully alternative methods of solving the ambiguous case.
Shew that an approximate solution of \(x \log x + x - 1 = \epsilon\), where \(\epsilon\) is small, is \[ x = 1 + \frac{\epsilon}{2} - \frac{\epsilon^2}{16}. \]
Prove that, if \(u_1, u_2, \dots, u_n, \dots\) are connected by the relation \[ u_n = u_{n-1} + n^2 u_{n-2} \] for all positive integral values of \(n \ge 3\), and \(u_1=1, u_2=5\), then \[ u_n/(n+1)! = 1 - \frac{1}{2} + \frac{1}{3} - \dots + \frac{(-1)^n}{n+1}. \] (Note: The condition on \(n\) for the recurrence is inferred from mathematical consistency, as the original text is ambiguous.)