Problems

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.

A variable circle touches both a given circle and a given straight line. Prove that the chord of contact passes through a fixed point.

Prove that the inverse of a system of non-intersecting coaxal circles with respect to a limiting point is a system of concentric circles having the inverse of the other limiting point as centre. \(S_1, S_2\) are two circles and \(L\) is a limiting point of the system of which they are members. A circle drawn through \(L\) and touching \(S_1\) at \(X\) meets \(S_2\) in \(P\) and \(Q\). Prove that \[ \frac{PX}{QX} = \frac{PL}{QL}. \]