Four equal spheres of radius \(r\) all touch one another. Find the radius of the smallest sphere that could enclose them all.
Prove that the two straight lines \[ x^2 \sin^2\alpha \cos^2\theta + 4xy \sin\alpha \sin\theta + y^2\{4\cos\alpha-(1+\cos\alpha)^2\cos^2\theta\} = 0 \] meet at an angle \(\alpha\).
An ellipse of given eccentricity \(\sin 2\beta\) passes through the focus of the parabola \(y^2 = 4ax\) and has its own foci on the parabola. Prove that the major axes of all such ellipses touch the parabola \[ y^2 = 4a(1-\tan^2\beta)(x-a\tan^2\beta). \]
The circle of curvature of the rectangular hyperbola \(x^2-y^2=a^2\) at the point \((a\operatorname{cosec}\theta, a\cot\theta)\) meets the curve again at \((a\operatorname{cosec}\phi, a\cot\phi)\). Prove that \[ \tan^4\frac{\theta}{2} \tan\frac{\phi}{2} = 1. \]
\(V\) is a given point on a given conic. Any chords \(VP, VQ\) are drawn, equally inclined to a given straight line. Prove that \(PQ\) passes through a fixed point.
If two conics have each double contact with a third, prove that 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.
Prove that if the equations \[ cy^2-2fyz+bz^2=0, \quad az^2-2gzx+cx^2=0, \quad bx^2-2hxy+ay^2=0 \] are satisfied by values of \(x,y,z\) all different from zero, then \[ abc+2fgh-af^2-bg^2-ch^2=0. \] Shew that when this condition is satisfied the equations are, in general, satisfied by two sets of values of \(x,y,z\) not proportional to one another.
If \(u_0, u_1, u_2, \dots\) are numbers connected by the recurrence formula \[ u_n-4u_{n-1}+5u_{n-2}-2u_{n-3}=0, \] find an expression for \(u_n\), given \(u_0=0, u_1=2, u_2=5\).
The roots of the equation \[ x^3+3px+q=0 \] are \(\alpha, \beta, \gamma\). Find the equation whose roots are \((\beta-\gamma)^2, (\gamma-\alpha)^2, (\alpha-\beta)^2\). Deduce that if \[ 4p^3+q^2 > 0, \] the original equation has one real and two imaginary roots. Prove also that if \(a,b,c\) are the roots of the above equation of squared differences, \[ a^2+b^2+c^2 = 2(bc+ca+ab). \]
(a) Prove that \[ \sin nx = 2^{n-1} \sin x \prod_{m=1}^{n-1} \left\{\cos x - \cos\frac{m\pi}{n}\right\}. \] (b) Prove that if \(x_1, x_2, \dots, x_n\) are the roots of the equation \[ x^{n-1}(x-1)+(x^2+1)(a_0x^{n-2}+a_1x^{n-3}+\dots+a_{n-2})=0, \] then \[ \sum_{r=1}^n \tan^{-1} x_r = (m+\tfrac{1}{2})\pi, \] where \(m\) is an integer.