Problems

Prove that the sum of the first \(r+1\) coefficients in the expansion of \((1-x)^{-n}\) by the binomial theorem, \(n\) being a positive integer, is \[ \frac{(n+r)!}{n!r!}. \] Prove that the number of ways in which \(n\) prizes may be distributed among \(q\) people so that everybody may have one at least is \[ q^n - q(q-1)^n + \frac{q(q-1)}{2!}(q-2)^n - \dots. \]

Prove that \[ 1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha.\cos\beta.\cos\gamma = 4\sin s.\sin(s-\alpha).\sin(s-\beta).\sin(s-\gamma), \] where \(2s = \alpha+\beta+\gamma\).

Solve the system of equations: \begin{align*} yz+by+cz &= a^2-bc, \\ zx+cz+ax &= b^2-ca, \\ xy+ax+by &= c^2-ab. \end{align*}

If \((1+x)^n = c_0+c_1x+\dots+c_nx^n\), where \(n\) is a positive integer, prove that

1. \(c_0^2+c_1^2+\dots+c_n^2 = (2n)!/(n!)^2\),
2. \(c_1+2c_2+\dots+nc_n = n2^{n-1}\),
3. \(\frac{c_0}{1^2}-\frac{c_1}{2^2}+\frac{c_2}{3^2}-\dots+\frac{(-1)^nc_n}{(n+1)^2} = \frac{1}{n+1}\left(1+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{n+1}\right)\).

Prove that the continued fraction \(a-\frac{1}{a-}\,\frac{1}{a-\dots}\) in which \(a\) is equal to \(-1\) and is repeated any number of times, must have one of three values, and that if \(a\) satisfies the equation \(2a^3+3a^2-3a-2=0\), the fraction satisfies this equation.

If \(O\) and \(I\) are the circumcentre and incentre of a triangle \(ABC\), show that \(OI^2=R^2-2Rr\), where \(R, r\) are the radii of the circumcircle and the incircle. If \(OI\) meets the perpendicular from \(A\) to \(BC\) in \(K\), show that \[ OK/OI = \cos(B-C)/\sin\frac{A}{2}. \]

Show that \[ 1+\frac{\cos\theta}{\cos\theta}+\frac{\cos 2\theta}{\cos^2\theta}+\dots+\frac{\cos(n-1)\theta}{\cos^{n-1}\theta} = \frac{\sin n\theta}{\sin\theta\cos^{n-1}\theta}, \] and that \[ \cos\theta\cos\theta+\cos^2\theta\cos 2\theta+\dots+\cos^n\theta\cos n\theta = \frac{\sin n\theta \cos^{n+1}\theta}{\sin\theta}. \]

Find the \(n\) real quadratic factors of \(x^{2n}-2a^nx^n\cos n\phi+a^{2n}\). Show that \(\prod_{r=0}^{r=n-1} \left\{\cos\phi-\cos\frac{2r\pi}{n}\right\} + \prod_{r=0}^{r=n-1} \left\{1-\cos\left(\phi+\frac{2r\pi}{n}\right)\right\} = 0\).

Show that the function \(\sin x + a\sin 3x\) for values of \(x\) between \(0\) and \(\pi\) has two minima with a maximum between, if \(a < -\frac{1}{3}\); one maximum, if \(-\frac{1}{3} < a < \frac{1}{3}\); two maxima with a minimum between, if \(a > \frac{1}{3}\).

Show how to find the asymptotes of an algebraic curve without discussing exceptional cases. Find the asymptotes of the curve \(x^2y+xy^2+xy+y^2+3x=0\). Trace the curve.