If \(x\), \(y\), \(z\) are all different and \(x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x},\) prove that the common value of these three expressions is \(\pm 1\).
For any fixed angle \(\theta\) with \(\sin \frac{1}{2}\theta \neq 0\), write \(S_N = \sum_{n=1}^{N} \sin n\theta.\) Prove that there exist numbers \(\varepsilon_M\) (which may depend on \(\theta\)) such that \(\varepsilon_M \to 0\) as \(M \to \infty\) and \(\left| \frac{1}{M+1} \sum_{n=N}^{M+N} S_n - \frac{1}{2}\cot \frac{1}{2}\theta \right| < \varepsilon_M\) for all positive \(M\), \(N\). For what values of \(\theta\) does the series \(\sum_{n=1}^{\infty} \sin n\theta\) converge?
If \(p\), \(q\), \(r\), \(s\) are positive integers with \(qr - ps = 1\), prove that any fraction which lies between \(p/q\) and \(r/s\) must have denominator at least \(q + s\).
If \(a_r = r!(n-r)!\) for \(0 < r < n\) and \(a_0 = a_n = n!\), prove that \(\frac{1}{a_0^2} + \frac{1}{a_1^2} + \cdots + \frac{1}{a_n^2} = \frac{(2n)!}{(n!)^4}.\)
Explain the principle of mathematical induction, and use it to prove that the \(n\)th derivative of the function \(\frac{1}{x^2 + 1}\) is \((-1)^{n+1} n! \cos^{n+1}\theta \sin(n+1)(\theta - \frac{1}{2}\pi),\) where \(-\frac{1}{2}\pi < \theta < \frac{1}{2}\pi\) and \(\tan \theta = x\).
(i) Evaluate \(\int_0^{\infty} e^{-\alpha x} \cos \beta x \cos \gamma x \, dx, \quad \text{where } \alpha > 0.\) (ii) Prove that \(\int_0^{\pi} xf(\sin x) \, dx = \pi \int_0^{\pi/2} f(\sin x) \, dx,\) and hence evaluate \(\int_0^{\pi} \frac{x \sin x \, dx}{2 - \sin^2 x}.\) (iii) Prove that, for \(x > 0\), \(\int_0^x [t] \, dt = (x - \frac{1}{2})[x] - \frac{1}{2}[x]^2,\) where \([t]\) is the greatest integer \(\leq t\).
For \(a \leq x \leq b\) the function \(f(x)\) is positive and decreasing, and the graph of \(y = f(x)\) is concave upwards. Prove that \((b-a)f(b) < \int_a^b f(x) \, dx < \frac{1}{2}(b-a)\{f(a) + f(b)\}.\) If \(S(n,k) = \frac{1}{k+1} + \frac{1}{k+2} + \cdots + \frac{1}{nk},\) where \(k\) and \(n\) are positive integers, show, by splitting the range of integration of \(\int_1^x \frac{dx}{x}\) into suitable parts, that \(0 < \log n - S(n,k) < \frac{n-1}{2nk}.\) Deduce that \(\log n - \left( \frac{1}{n+1} + \frac{1}{n+2} + \cdots + \frac{1}{n^2} \right) \to 0\) as \(n \to \infty\).
The function \(f(x)\) is continuous in the range \(a \leq x \leq b\). Show that a value of \(\theta\) can be found with \(0 < \theta < 1\) such that \(\int_a^b f(x) \, dx = (b-a)f\{a + \theta(b-a)\}.\) The coefficients in the equation \(a_0 x^n + a_1 x^{n-1} + \cdots + a_n = 0\) are connected by the relation \(\frac{a_0}{n+1} + \frac{a_1}{n} + \cdots + \frac{a_{n-1}}{2} + a_n = 0.\) Show that it has at least one root between 0 and 1.
Two numbers \(x\) and \(y\) are chosen at random between 0 and 2. Find the chance that \(x^m y^n \leq 1\) in the three cases:
A function \(f(r, \theta)\) is transformed into \(g(u, s)\) by means of the relations \(r \cos \theta = 1/u\), \(\tan \theta = s\). Prove that