Prove that, if \(p/q\) is a fraction in its lowest terms, then integers \(r\) and \(s\) can be found such that \(qr-ps=1\). Prove that, if \(p/q\) and \(r/s\) are fractions such that \(qr-ps=1\), then the denominator of any fraction whose value lies between \(p/q\) and \(r/s\) is at least \(q+s\).
Prove that \[ a^2+b^2+c^2-bc-ca-ab = (a+\omega b+\omega^2 c)(a+\omega^2 b + \omega c), \] where \(\omega\) is a complex cube root of 1. Prove that, if \[ (b-c)^n+(c-a)^n+(a-b)^n \] is divisible by \(\Sigma a^2 - \Sigma bc\), then \(n\) is an integer not a multiple of 3. Prove that, if the same expression is divisible by \((\Sigma a^2 - \Sigma bc)^2\), then \(n\) is greater by one than a multiple of 3.
Prove that, if \((1+x)^n = c_0+c_1x+\dots+c_nx^n\), then \[ c_0c_2+c_1c_3+\dots+c_{n-2}c_n = \frac{(2n)!}{(n-2)!(n+2)!}, \] \[ \frac{c_0}{1} - \frac{c_1}{2} + \frac{c_2}{3} - \dots + (-1)^n\frac{c_n}{n+1} = \frac{1}{n+1}, \] \[ \frac{c_0}{1^2} - \frac{c_1}{2^2} + \frac{c_2}{3^2} - \dots + (-1)^n\frac{c_n}{(n+1)^2} = \frac{1}{n+1}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n+1}\right). \]
Find an equation connecting the expressions \[ \cos A + \cos B + \cos C, \] \[ \sin A \sin B \sin C, \] where \(A, B\) and \(C\) are the angles of a triangle. Prove that the sum of the cosines of the angles of a triangle is greater than 1 and not greater than \(\frac{3}{2}\).
Prove that, if \(H\) and \(O\) are the orthocentre and circumcentre of a triangle \(ABC\), \[ OH^2=R^2(1-8\cos A \cos B \cos C), \] where \(R\) is the radius of the circumcircle. Prove that if \(K\) is the middle point of \(OH\), \[ AK^2+BK^2+CK^2 = 3R^2-\frac{1}{4}OH^2. \]
Prove that the function \[ \frac{\sin^2 x}{\sin(x-\alpha)}, \] where \(0 < \alpha < \pi\), has infinitely many maxima equal to 0 and minima equal to \(\sin\alpha\). Sketch the graph of the function.
If \(x, y, z\) are connected by an equation \(\phi(x,y,z)=0\), explain the meaning of the partial differential coefficient \(\partial z/\partial x\), and express it in terms of the partial differential coefficients of the function \(\phi(x,y,z)\). If \[ \frac{x^2}{a^2+z} + \frac{y^2}{b^2+z} = 1, \] where \(a\) and \(b\) are constants, prove that \[ \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2 = 2\left(x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y}\right). \]
Find the limiting values of the expression \[ \frac{\sin x \sin y}{\cos x - \cos y} \] as the point \((x,y)\) approaches the origin along curves of the form (i) \(y=xk\), where \(k\) is positive, (ii) \(y=ax+bx^2\), where \(a\) and \(b\) have various constant values. Point out any cases in which the limits are infinite.
Integrate \[ (1+x^2)^{\frac{3}{2}}, \quad \frac{1-\tan x}{1+\tan x}. \] Prove that, if \(n\) is a positive integer, \[ \int_{-\infty}^{\infty} \frac{dx}{(1+x^2)^{n+1}} = \frac{1 \cdot 3 \dots (2n-1)}{2 \cdot 4 \dots 2n} \pi. \]
Prove that \[ \int_0^\pi \frac{\sin n\theta}{\sin\theta} d\theta = 0 \text{ or } \pi, \text{ according as } n \text{ is an even or odd positive integer.} \] Evaluate \(\displaystyle\int_0^\pi \frac{\sin^2 n\theta}{\sin^2\theta} d\theta\), where \(n\) is a positive integer.