Problems

If \begin{align*} x^2 - yz + (a-\lambda)x &= 0, \\ y^2 - zx + (b-\lambda)y &= 0, \\ z^2 - xy + (c-\lambda)z &= 0, \end{align*} and \[ x^2y^2+y^2z^2+z^2x^2 = xyz(x+y+z), \] prove that \[ 3\lambda = a+b+c, \] and that \[ a^2+b^2+c^2 = bc+ca+ab. \]

State and prove the Binomial Theorem for a positive integral exponent. If \[ (1+x)^{4m} = 1+c_1x+c_2x^2+\dots, \] where \(m\) is a positive integer, prove that \[ 1+c_1+c_2+c_3+\dots = 2^{4m} \] and that \[ 1-c_2+c_4-c_6+\dots = (-1)^m \cdot 2^{2m}. \]

Find a number of six digits, such that if another number is formed by taking its last three digits and placing them in the same relative order in front of the first three, the second number shall be six times the first. Prove that there is only one such number.

Establish a formula for the number of combinations of \(n\) things taken \(r\) at a time. Find in how many ways sixty similar articles can be divided among three men so that no man has less than ten or more than twenty-five.

Solve the equations

1. [(i)] \(x+y+z=x^2+y^2+z^2 = \frac{1}{3}(x^3+y^3+z^3)=3\).
2. [(ii)] \(\frac{(x+a)(x+b)}{(x-a)(x-b)} + \frac{(x-a)(x-b)}{(x+a)(x+b)} = \frac{(x+c)(x+d)}{(x-c)(x-d)} + \frac{(x-c)(x-d)}{(x+c)(x+d)}\).
3. [(iii)] \(2\cot 2x - \tan 2x = 3\cot 3x\).

Sum the series

1. [(i)] \(1+\frac{2^3}{\lfloor 2} + \frac{3^3}{\lfloor 3} + \frac{4^3}{\lfloor 4} + \dots\) to infinity.
2. [(ii)] \(\tan\alpha.\tan 2\alpha + \tan 2\alpha.\tan 3\alpha + \tan 3\alpha.\tan 4\alpha + \dots\) to \(n\) terms.
3. [(iii)] \(x\cos\alpha + \frac{1}{2}x^2\cos 2\alpha + \frac{1}{3}x^3\cos 3\alpha + \dots\) to infinity, when \(x<1\).

Find a formula for the radius of the inscribed circle of a triangle. The circle inscribed in the triangle \(ABC\) touches \(BC, CA, AB\) at \(D, E, F\); and circles are inscribed in the triangles \(AEF, BFD, CDE\). If \(r, \rho_1, \rho_2, \rho_3\) are the radii of these circles, show that \[ r(2r-\rho_1-\rho_2-\rho_3)^2 = 2\rho_1\rho_2\rho_3. \]

Prove that, if \(\alpha, \beta, \gamma\) do not differ by a multiple of \(\pi\), and if \[ \frac{\cos(\alpha+\theta)}{\sin(\beta+\gamma)} = \frac{\cos(\beta+\theta)}{\sin(\gamma+\alpha)}, \] then each fraction is equal to \[ \frac{\cos(\gamma+\theta)}{\sin(\alpha+\beta)}, \] and is also equal to \(\pm 1\).

A person standing between two towers observes that they subtend angles each equal to \(\alpha\), and on walking \(a\) feet along a straight horizontal path inclined at an angle \(\gamma\) to the line joining the towers, he finds that they subtend angles each equal to \(\beta\). Prove that the heights \(h\) and \(h'\) of the towers are given by \[ hh'(\cot^2\beta - \cot^2\alpha)=a^2, \] \[ (h'-h)(\cot^2\beta-\cot^2\alpha) = 2a\cot\alpha.\cos\gamma. \]

Prove that \((\cos m\theta+i\sin m\theta)\) is one of the values of \[ (\cos\theta+i\sin\theta)^m, \] where \(m\) is a real rational number, and \(i^2=-1\). Prove that \[ x^2-2x\cos\theta+1 \] is a factor of \[ x^{2n}-2x^n\cos n\theta+1; \] and resolve this last expression into its \(n\) real quadratic factors.