\(A, B, C, D\) are the corners of a square of side \(a\) on level ground. Inside the square is a flagstaff, and the angles of elevation of the top of the flagstaff, as seen from \(A, B, C\) are \(\alpha, \beta, \gamma\) respectively. Find an equation giving the height of the flagstaff in terms of \(\alpha, \beta, \gamma, a\), and deduce that there are two possible values, \(h_1\) and \(h_2\). Show that \[ \frac{1}{h_1^2} + \frac{1}{h_2^2} = \frac{1}{a^2}(\cot^2\alpha+\cot^2\gamma), \] and that \[ 4\cot^2\beta(\cot^2\alpha+\cot^2\gamma-\cot^2\beta) > (\cot^2\alpha-\cot^2\gamma)^2. \]
If \(l, m, l', m', l''\) and \(m''\) are integers, and if \(\alpha/\beta\) is not rational, and if \[ l\alpha + m\beta = l'\alpha + m'\beta, \] shew that \(l=l'\), and \(m=m'\). Also shew that no two of the numbers \[ (2l+5m)\alpha+l\beta, \quad (2l'+5m'+1)\alpha+l'\beta, \quad (2l''+5m'')\alpha+(l''+1)\beta \] are equal.
Shew that, if \(n\) is a positive integer, the number of solutions of the equation \[ n = 2n_1 + 3n_2, \] where \(n_1\) and \(n_2\) are positive integers or zero, is equal to \(N\) or \(N+1\) according as \(r\) is or is not equal to 1; here \(N\) denotes the quotient and \(r\) the remainder when \(n\) is divided by 6.
Shew that \[ \sum_{m=0}^{N} \frac{\cos m\phi}{\cos^m \theta} = \frac{\cos^2 \theta - \cos\theta\cos\phi}{\cos^2\theta-2\cos\theta\cos\phi+1} - \frac{\cos(N+1)\phi}{\cos^{N-1}\theta} + \frac{\cos N\phi}{\cos^N\theta}. \] Indicate how the value of \(\sum_{m=0}^{N} \frac{m\sin(m-1)\phi}{\cos^{m+1}\phi}\) could be found from the above equation.
(i) If \(I\) be the in-centre and \(O\) the circumcentre of a triangle \(ABC\), shew that \[ OI^2 = R^2 - 2Rr, \] where \(r\) is the radius of the in-circle, and \(R\) is the radius of the circumcircle. (ii) If \(H\) be the orthocentre of the triangle \(ABC\), shew that \[ OH^2 = R^2 - 8R^2\cos A \cos B \cos C. \]
Express \(bc+ca+ab\) and \(abc\) in terms of \(s, p\) and \(q\), where \[ 2s=a+b+c, \quad 2p=a^2+b^2+c^2, \quad 3q=a^3+b^3+c^3. \] In a triangle the sum of the lengths of the sides, \(a, b\) and \(c\), is constant; shew that the small variation, \(\delta R\), in the radius of the circumcircle when the triangle is slightly changed, is given by \[ 24\Delta^2 \delta R = s[3s(3q-2sp)\delta p + (2s^3-3q)\delta q] \] approximately, where \(\Delta\) is the area of the triangle.
(i) If \[ \frac{d^2y}{dx^2} + \frac{1}{x}\frac{dy}{dx} + y = 0, \] shew that \[ x^2\frac{d^3y}{dy^3} - \left(\frac{dx}{dy}\right)^2 - xy\left(\frac{dx}{dy}\right)^3 = 0. \] (ii) If \[ \frac{d^2y}{dx^2} + \left(\frac{l}{x-a}+\frac{m}{x-b}+\frac{n}{x-c}\right)\frac{dy}{dx} = 0, \] where \(l+m+n=2\), shew that \[ \frac{d^2y}{dX^2} + \left(\frac{l}{X-A}+\frac{m}{X-B}+\frac{n}{X-C}\right)\frac{dY}{dX} = 0, \] where \[ x = \frac{\alpha X + \beta}{\gamma X + \delta}, \quad a = \frac{\alpha A + \beta}{\gamma A + \delta} \text{ etc., and } \alpha\delta - \beta\gamma = 1. \]
Assuming that \(\pi[ab-h^2]^{-\frac{1}{2}}\) is the area of the ellipse \(ax^2+2hxy+by^2=1\), shew that the minimum area of an ellipse which passes through the points \((\pm p, 0), (q, r)\) and \((-q, -r)\) is equal to \(\pi pr\).
Shew that if \(m\) and \(n\) are integers \[ \int_0^{\frac{\pi}{2}} \sin^n\theta \cos^m\theta d\theta \] is decreased when either \(m\) or \(n\) is increased. Hence shew that \[ \frac{(n!)^2 2^{2n}}{(2n+1)(2n-1)^2 \dots 3^2} < \frac{\pi}{2} < \frac{(n+1)(n!)^2 2^{2n+1}}{(2n+1)^2 \dots 3^2}. \]
Shew that \[ \int_0^{\frac{\pi}{4}} \sec^3 x dx = \frac{1}{2}\sqrt{2} + \frac{1}{2}\log(1+\sqrt{2}). \] Evaluate \[ \int_0^1 \frac{dx}{(1+x)^2(2x+1)}. \]