Prove that, if \[ a+b+c=0, \] then \[ a^3+b^3+c^3=3abc, \] and \[ a^6+b^6+c^6 = \frac{1}{4}(a^2+b^2+c^2)^3+3a^2b^2c^2. \] Solve the equations \begin{align*} x+y+z &= 4, \\ x^2+y^2+z^2 &= 66, \\ x^3+y^3+z^3 &= 280. \end{align*}
Find the number of combinations of \(n\) unlike things (1) \(r\) at a time, (2) any number at a time. Prove that the number of sentences of \(r\) words and \(n\) letters which can be formed with \(n\) different letters is \[ \frac{n!(n-1)!}{(n-r)!(r-1)!}, \] any arrangement of letters being regarded as a word, and any arrangement of words as a sentence.
Explain how to find the \(n\)th term \(u_n\) of a series, whose terms satisfy for all values of \(n\) the relation \[ u_n+ku_{n-1}+lu_{n-2}=0, \] \(k\) and \(l\) being given numbers. A plant produces \(p\) seeds at the beginning of the second year of its life, \(p\) more at the beginning of the third year, and dies at the end of its third year. Prove that if all the seeds come to maturity the number produced from one seed in the \(n\)th year from planting that seed is \[ \frac{1}{2^n\cdot q}\{(p+q)^n-(p-q)^n\}, \] where \[ q^2=p^2+4p. \]
If \(\frac{p_{n-1}}{q_{n-1}}\) and \(\frac{p_n}{q_n}\) are the \((n-1)\)th and \(n\)th convergents of the continued fraction \[ \frac{1}{a_1+} \frac{1}{a_2+} \frac{1}{a_3+} \dots, \] prove that \[ p_n q_{n-1} - p_{n-1}q_n = (-1)^n. \] Prove also that if \(w_n \left(=\frac{p_n}{q_n}\right)\) is the \(n\)th convergent, \[ \frac{(w_{n+1}-w_n)(w_{n-1}-w_{n-2})}{(w_{n+1}-w_{n-1})(w_n-w_{n-2})} + \frac{1}{a_n a_{n+1}} = 0. \]
Prove that if \(x\) is numerically less than unity \[ \log_e(1+x)=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\dots. \] Prove also that, if \(n>1\), the coefficient of \(x^n\) in the expansion of \[ \{\log_e(1+x)\}^2 \] is \[ (-1)^n \frac{2}{n} \left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n-1}\right). \]
If \(\tan 4\theta = \tan 4\alpha\), express in terms of the trigonometrical ratios of \(\alpha\) the possible values of \(\sin\theta\) and \(\cos\theta\). If \(\beta, \gamma\) are values of \(x\) satisfying the equation \[ \sin(x+\lambda) = m\sin 2\lambda, \] and if \((\beta\sim\gamma)\) is not a multiple of \(\pi\), prove that \[ \cos\frac{\beta-\gamma}{2} = \pm m\sin(\beta+\gamma). \]
An aeroplane is travelling in a straight line with constant velocity \(v\) feet per second at a constant height. At one moment a man observes it due north of him, at an angle of elevation \(\alpha\); \(t\) seconds later he sees it in a direction \(\beta\) east of north, and the angle of elevation is \(\gamma\). Prove that the aeroplane's course is \(\delta\) south of east, where \[ \tan\delta = \cot\alpha . \text{cosec}\beta . \tan\gamma - \cot\beta, \] and its height is \[ vt \text{ cosec}\beta . \tan\gamma . \cos\delta \text{ feet}. \]
Prove that in any triangle, with the usual notation,
Resolve \(x^{2n}-2x^n\cos n\theta+1\) into \(n\) real quadratic factors. Express \((x+iy)^{a+ib}\) in the form \(X+iY\), and show that one of the values is real if \(\frac{1}{2}b\log(x^2+y^2)+a\tan^{-1}\frac{y}{x}\) is a multiple of \(\pi\).
By means of De Moivre's theorem, or otherwise, express \(\tan n\theta\) in terms of \(\tan\theta\). Prove that \[ \tan 10^\circ . \tan 50^\circ . \tan 70^\circ = \frac{1}{\sqrt{3}}. \] Prove that \[ \tan\theta + \frac{1}{2}\tan\frac{\theta}{2} + \frac{1}{4}\tan\frac{\theta}{4} + \dots \text{to infinity} = \frac{1}{\theta} - 2\cot 2\theta. \]