Problems

Prove that in general two coplanar conics have a unique self-conjugate triangle. Verify that the two conics, whose equations in any system of homogeneous coordinates (e.g. areal or trilinear coordinates) are \begin{align*} x^2+y^2+z^2+2yz+2zx+6xy&=0, \\ 2x^2+2y^2-z^2-2yz-2zx-4xy&=0, \end{align*} have a common self-conjugate triangle, whose sides are given by \[ x+y=0, \quad x-y=0, \quad x+y+z=0. \]

Find the tangents of the angles that satisfy the equation \[ (m+2)\sin\theta + (2m-1)\cos\theta = 2m+1. \] The sides of a triangle are \(p^2+2pq\), \(p^2+pq+q^2\), and \(p^2-q^2\). Prove that the angles are in arithmetical progression.

If \[ x(1+\sin^2\phi-\cos\phi) = (y\sin\phi+a)(1+\cos\phi) \] and \[ y(1+\cos^2\phi) = (x\cos\phi+a)\sin\phi, \] prove that \[ y^2=ax. \]

A, P, Q, B are four points in order on a straight line. \(AQ=2a, PB=2b\) and \(AB=2c\). Circles are described on AQ, BP and AB as diameters. Prove that the radius of a circle touching all three circles is \[ \frac{c(c-a)(c-b)}{c^2-ab}. \]

Sum to infinity the series \[ 1 - \cos\theta + \frac{\cos 2\theta}{2!} - \frac{\cos 3\theta}{3!} + \dots. \] Prove that the equation whose roots are \(\tan^2 20^\circ, \tan^2 40^\circ\) and \(\tan^2 80^\circ\) is \[ x^3 - 33x^2 + 27x - 3 = 0. \]

A uniform rod rests with one end against a rough vertical wall, the other end being supported by a light string of equal length fastened to a point in the wall; prove that the least angle that the string can make with the wall is \(\tan^{-1}\frac{3}{\mu}\), where \(\mu\) is the coefficient of friction.

A uniform lamina in the form of a parallelogram rests with two adjacent sides on two smooth pegs in the same horizontal line at a distance \(c\) apart. \(2h\) is the length of the diagonal through the intersection of the two sides; \(\alpha, \beta\) and \(\theta\) are the angles which this diagonal makes with the sides and with the vertical (\(\alpha>\beta\)). Prove that \[ h\sin\theta\sin(\alpha+\beta)=c\sin(2\theta-\alpha+\beta). \]

A smooth ring of mass \(M\) is threaded on a light flexible string which is then hung over two smooth fixed pulleys. Masses \(m\) and \(m'\) are tied to the ends of the string, the ring is on the portion of string between the pulleys, and the various portions of the string hang vertically. The system is now released. Find the acceleration of the ring, and prove that it is zero if \[ M = \frac{4mm'}{m+m'}. \]

If a particle is describing a circle of radius \(r\) with uniform speed \(v\), prove that the acceleration is \(\frac{v^2}{r}\) towards the centre. A particle hanging by a light string of length \(l\) from a fixed point \(O\) is projected horizontally from its lowest position with velocity \(\sqrt{\frac{7}{2}gl}\). Prove that the string slackens after swinging through \(120^\circ\).

A particle projected from a point on a smooth inclined plane at the \(r\)th impact strikes the plane normally and at the \(n\)th impact is at the point of projection; if the coefficient of restitution is \(e\), prove that \[ e^n - 2e^r + 1 = 0. \]