Show that if \(\omega\) is one of the imaginary cube roots of unity, then the other is \(\omega^2\); and that \[ x^2+y^2+z^2-yz-zx-xy = (x+\omega y+\omega^2 z)(x+\omega^2 y+\omega z). \] Prove that if \((y-z)^n+(z-x)^n+(x-y)^n\) is divisible by \(\Sigma x^2 - \Sigma yz\), \(n\) must be an integer which is not a multiple of 3. Prove also that if it is divisible by \((\Sigma x^2 - \Sigma yz)^2\), \(n\) must exceed by unity a multiple of 3.
Find the number of homogeneous products of degree \(r\) in \(n\) letters, and show that if there are three letters \(a, b, c\), the sum of these products is \[ \Sigma a^{r+2}(b-c)/\Sigma a^2(b-c). \]
Prove that the infinite series whose \(n\)th terms are (i) \(\frac{n^2}{2^n}\), (ii) \(\frac{n+2}{n(n+1)(n+3)}\), are convergent, and find their sums.
Find the sides of the pedal triangle of a triangle \(ABC\) in terms of the sides of \(ABC\). \(L, M, N\) are points on the sides \(BC, CA, AB\) of a triangle such that the angles \(ALC, BMA, CNB\) are each equal to \(\alpha\). If \(BM, CN\) meet in \(A'\), \(CN, AL\) in \(B'\), and \(AL, BM\) in \(C'\), show that the triangle \(A'B'C'\) is similar to \(ABC\) with corresponding sides in the ratio \(2\cos\alpha:1\). If the circle \(A'B'C'\) touches the side \(BC\), show that \(\cos\alpha = \cos B \cos C\).
The hemispherical dome of a building is surmounted by a cross. The elevation of the top of the cross is \(\alpha\) and of the highest visible point of the dome \(\beta\) as seen from a point of a level road leading straight to the building. From a point of the road distant \(a\) nearer, the cross is just disappearing from view behind the dome and the elevation is \(\gamma\). Prove that the radius of the dome is \[ \frac{a \sin\gamma \cos\gamma \sin(\alpha-\beta)}{\sin(\gamma-\alpha)(\cos\beta-\cos\gamma)}. \]
Sum the infinite series \[ \cos\theta - \frac{1}{3}\cos 3\theta + \frac{1}{5}\cos 5\theta - \dots. \] Obtain the values of \((1+e^{i\theta})^{\frac{1}{2}}\), each in the form \(a+ib\), where \(a,b\) are real.
Show that the function \[ \frac{\sin^2 x}{\sin(x-a)\sin(x-b)} \] where \(a, b\) lie between \(0\) and \(\pi\), has an infinity of minima equal to \(0\) and of maxima equal to \(-4\sin a \sin b / \sin^2(a-b)\). Sketch the graph of the function.
Prove the formula \(\rho = r \frac{dr}{dp}\) for the radius of curvature of a curve \(f(r,p)=0\). If \(t\) is the length of the tangent from the pole to the circle of curvature at any point of the curve, show that \(t^2 = \frac{d}{dr}(r^3 v)\), where \(v=\frac{1}{p}\). Deduce that if all the circles of curvature of a curve pass through a fixed point, the curve must be a circle.
Integrate \[ \int \frac{dx}{x\sqrt{1+x^2}}, \quad \int \frac{dx}{(x+1)^2(x^2+x+1)}, \quad \int x^2 \sec^2 x \tan x \,dx. \] If \(\int f(x) \,dx = \log[1+f(x)]\), determine \(f(x)\).
Trace the curve \(r=a(1+2\cos\theta)\), and show in the figure the area represented by \[ \frac{1}{2} \int_0^{2\pi} r^2 \,d\theta. \] Find separately the areas bounded by each loop of the curve.