(i) In the equation \[\frac{k_1}{x-a_1} + \frac{k_2}{x-a_2} + \ldots + \frac{k_n}{x-a_n} = 0\] the numbers \(k_i\) are positive and the \(a_i\) are distinct real numbers. Prove that the roots of the equation are all real. (ii) Find necessary and sufficient conditions for the equation \[2x^5 - 5px^2 + 3q = 0,\] where \(p\) and \(q\) are positive, to have (a) one, (b) three real roots.
Two regular polygons of \(n_1\) and \(n_2\) sides are inscribed in two concentric circles of radii \(r_1\) and \(r_2\) respectively. Prove that the sum of the squares on all the lines joining the vertices of one to the vertices of the other is \[n_1n_2(r_1^2 + r_2^2).\]
The function \(\log x\), where \(x\) is real and positive, is defined by the formula \[\log x = \int_1^x \frac{dt}{t}.\] From the definition prove that
A conic \(S\) is inscribed in a triangle \(ABC\), its point of contact with \(BC\) being \(D\). \(O\) is a general point of \(S\), and the lines \(OA\), \(OB\), \(OC\) meet \(S\) again in \(A'\), \(B'\), \(C'\) respectively. Prove that the tangent at \(O\) passes through the intersection of the lines \(B'C'\), \(A'D\).
Prove Desargues' theorem that, if the lines joining corresponding vertices of two coplanar triangles are concurrent, the intersections of corresponding sides are collinear. The sides \(A_1A_2\), \(A_2A_3\), \(A_3A_4\), \(A_4A_1\), \(A_1A_3\) of a quadrangle \(A_1A_2A_3A_4\) pass successively through the vertices \(b_1b_2\), \(b_2b_3\), \(b_3b_4\), \(b_4b_1\), \(b_1b_3\) of a quadrilateral \(b_1b_2b_3b_4\), so that \(A_1A_3\) passes through \(b_1b_3\).
Prove Pascal's theorem that the three intersections of pairs of opposite sides of a hexagon inscribed in a conic are collinear. \(A\), \(B\), \(C\), \(D\), \(E\) are five points of a conic \(S\). Show how to construct (i) the tangent at \(A\), (ii) the second intersection with \(S\) of a given line through \(A\). Hence or otherwise, sketch a method of constructing the polar line of a general point \(P\) with respect to \(S\).
Show that three distinct points in the plane with integral coordinates (in the usual Cartesian system) cannot form an equilateral triangle.
Let \(a_1, a_2, \ldots, a_k, \ldots\) be a sequence of real numbers which is periodic modulo a positive integer \(k\), that is \[a_{n+k} = a_n \quad (n = 1, 2, \ldots).\] Show that there is a positive integer \(N\) such that \[a_{N+1} + a_{N+2} + \ldots + a_{N+n} > nA\] for every positive integer \(n\), where \(A\) is defined by \[kA = a_1 + a_2 + \ldots + a_k.\]
A tripod \(VA\), \(VB\), \(VC\) is made of three uniform rods of length \(2l\) and weight \(w\). If freely pivoted together at \(V\), it stands symmetrically making a regular tetrahedron, one face of which is flat on the ground. A wind is blowing and the projection of \(VA\) on the ground points exactly into the wind. The feet \(B\) and \(C\) are gently dug into the ground and may be taken as freely pivoted but the foot \(A\) rests on hard ground whose coefficient of friction is \(\mu\). If the force due to the wind on each rod is directed exactly downwind and of magnitude \(\frac{1}{2}w^2l\) per unit length, show that the least wind speed \(v\) that will topple the tripod is given by \[2v^2 = \frac{2\sqrt{2}\mu + 1}{2\sqrt{2}\mu + 3}\sqrt{2}.\]
A small cork of density \(\rho\) and mass \(M\) is inside a large bottle filled with water of density \(\rho'\). The cork is in equilibrium completely immersed at a height \(x_0\) above the bottom of the bottle to which it is attached by a light spring of natural length \(a\). If the cork is moved through the water it experiences a resisting force whose magnitude is \(2\lambda\) times the speed of the cork through the water. The bottle and cork are initially at rest but they are then dropped. Show that the height \(x\) of the cork in the bottle at all times until the bottle hits the ground is given by \[x = a + (x_0 - a)e^{-\lambda t}[(z/m)\sin mt + \cos mt],\] where \[m^2 = \left[\frac{\rho' - \rho}{\rho} \cdot \frac{g}{x_0 - a} - \lambda^2\right].\]