Problems

Prove that \[ \frac{1}{1.2.3} + \frac{1}{3.4.5} + \frac{1}{5.6.7} + \dots \text{ to infinity} = \log 2 - \frac{1}{2}, \] and that \[ \frac{1}{4} + \frac{1.3}{4.6} + \frac{1.3.5}{4.6.8} + \dots \text{ to infinity} = 1. \]

Prove that if \[ \begin{vmatrix} a & a^3 & a^4-1 \\ b & b^3 & b^4-1 \\ c & c^3 & c^4-1 \end{vmatrix} = 0 \] and \(a,b,c\) are all different, then \[ abc(ab+bc+ca) = a+b+c. \]

Prove that, if the roots of the equation \[ x^3+px+q=0 \] are all real, then \(4p^3+27q^2\) is negative. If the roots are \(\alpha, \beta, \gamma\), prove that the value of \[ \Sigma(\beta-\gamma)^3(\beta+\gamma-2\alpha) \] is \(-27q\).

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 \[ \cos\theta\cos\theta+\cos^2\theta\cos 2\theta+\dots+\cos^n\theta\cos n\theta = \frac{\sin n\theta \cos^{n+1}\theta}{\sin\theta}. \]