A vessel contains \(p\) gallons of wine, and another contains \(q\) gallons of water. \(c\) gallons are taken out of each and transferred to the other, and this operation is repeated any number of times. Prove that, if \(c = \frac{pq}{p+q}\), the quantity of wine in each vessel will always remain the same after the first operation.
Eliminate \(x, y\) and \(z\) from the equations \begin{align*} \frac{x}{y}+\frac{y}{z}+\frac{z}{x} &= a, \\ \frac{x}{z}+\frac{z}{y}+\frac{y}{x} &= b, \\ \left(\frac{x}{y}+\frac{y}{z}\right)\left(\frac{y}{z}+\frac{z}{x}\right)\left(\frac{z}{x}+\frac{x}{y}\right) &= c. \end{align*} Prove that \[ 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\dots-\frac{1}{2n} = \frac{1}{n+1}+\frac{1}{n+2}+\dots+\frac{1}{2n}. \]
Sum the series:
State and prove a formula for the number of positive integers which are less than a given integer \(N\) and prime to it. Prove also, assuming that unity is prime to \(N\), that the sum of these integers, divided by their number, is equal to \(\frac{1}{2}N\).
In a triangle ABC, D is any point in BC. The angles BAD, CAD, ADC are \(\alpha, \beta\) and \(\theta\) respectively. BD=\(m\), DC=\(n\). Prove that \[ m\cot\alpha - n\cot\beta = (m+n)\cot\theta. \] One diagonal of a parallelogram makes angles \(p, q\) with the sides, and the other makes angles \(r, s\) with them. Show that \(\cot p - \cot r = \cot q - \cot s = 2\cot(p+q)\); and that if \(u\) is the angle between the diagonals, then \[ \cot p - \cot q = \cot r - \cot s = 2\cot u. \]
Sketch very roughly the graph of \(\sin^2 x\), and show that the equation \(x-2\sin^2 x = 0\) has three real roots and only three, \(x\) being measured in radians.
From a house on one side of a street observations were made of the angle subtended by the height of the opposite house; first from the level of the street, in which case the tangent of the angle was 3; and afterwards from two windows one above the other, from each of which the tangent of the angle was found to be \(-3\). The height of the opposite house being 60 feet, find the height of each of the two windows above the street.
Find the sum \(s_n\) of \(n\) terms of the series \[ \sin x + \sin 2x + \sin 3x + \dots \] and prove that the limit, as \(n\) tends to infinity, of \[ \frac{1}{n}(s_1+s_2+s_3+\dots+s_n) = \frac{1}{2}\cot\frac{x}{2}. \]
Prove that if \(n\) is a positive integer, \[ 2\cos n\theta = (2\cos\theta)^n - n(2\cos\theta)^{n-2} + \frac{n(n-3)}{2!}(2\cos\theta)^{n-4} - \dots. \] Form the cubic equation whose roots are \[ \cos\frac{2\pi}{7}, \quad \cos\frac{4\pi}{7}, \quad \cos\frac{6\pi}{7}. \]
Prove