Shew that, if by inversion in a plane three given points are inverted into three points forming the vertices of an equilateral triangle, the origin of inversion has two possible positions. Extend the result to the case of possible origins not in the plane of the original triangle and shew that they lie on a circle of which the line, joining the two origins in the plane, is the diameter.
The asymptotes of each of two rectangular hyperbolas are parallel to the axes of the other, and each hyperbola passes through the centre of the other. Prove that the normals to each hyperbola at its points of intersection with the other are concurrent, and that the centres of the hyperbolas are the points of trisection of the line joining the two points of concurrence.
Prove that the two conics, which pass through the four corners of a given square and touch a given line, are real unless the given line separates one corner of the square from the other three corners. State the analogous theorem in regard to the two conics touching the sides of a given square and passing through a given point.
Two equal uniform smooth cylinders, of radius \(a\), rest in a horizontal cylindrical groove of radius \(b\). A third cylinder, equal in all respects to the first two, is placed upon them. Shew that the two lower cylinders will just separate if \((b-a)^2 = 28a^2\).
A tetrahedron \(ABCD\) is formed of light rods smoothly jointed at their extremities and \(X, Y\), the middle points of \(AB, CD\), are joined by a string in which there is a tension \(T\). Prove that the tension in \(AB\) is \(\frac{1}{4}T \cdot \frac{AB}{XY}\) and write down the stresses in the other rods, stating in each case whether the stress is a tension or a thrust.
Two imperfectly elastic particles of equal mass, whose coefficient of restitution is \(e\), are suspended from the same point by light strings of equal length. One particle is drawn aside a small distance \(x_0\), and then released. Shew that, between the \(n\)th and \((n+1)\)th impacts, the particle originally drawn aside swings through a distance \(\frac{1}{2}\{1+(-e)^n\}x_0\) on one side of the vertical through the point of suspension.
Two particles, masses \(M\) and \(m\) (\(M>m\)), are attached to the ends of a string, length \(2l\), which passes over a smooth peg at a height \(l\) above a smooth plane inclined at an angle \(\alpha\) to the vertical. The particles are initially held at rest on the plane at the point vertically below the peg, \(M\) being below \(m\). Prove that, if the particles are released, \(m\) will oscillate through a vertical distance \(2M(M-m)l/(m^2 \sec^2\alpha - M^2)\), provided that \(\tan^2\alpha\) is greater than \((3M+m)(M-m)/m^2\).
Find the remainder when a polynomial \(f(x)\) is divided by (i) \(x-a\), (ii) \((x-a)(x-b)\). Find all the factors of \((y-z)^5+(z-x)^5+(x-y)^5\). Prove that \((y-z)^7+(z-x)^7+(x-y)^7\) is divisible by \((x^2+y^2+z^2-yz-zx-xy)^2\).
Having given that a quadratic function of \(x\) assumes the values \(V_1, V_2, V_3\) for the values \(x=a, x=b, x=c\) prove that the function must be \[ V_1 \frac{(x-b)(x-c)}{(a-b)(a-c)} + V_2 \frac{(x-c)(x-a)}{(b-c)(b-a)} + V_3 \frac{(x-a)(x-b)}{(c-a)(c-b)}. \] A variable quantity which can be represented by a quadratic function of the time assumes the values 144, 15.6, 18 at 10 a.m., 1 p.m. and 2 p.m. on a certain day. Find the value at noon, and find at what time the quantity assumes its least value.
Find the number of homogeneous products of \(n\) dimensions formed from \(r\) letters \(a,b,c,\dots,k\); and shew that the sum of such products is equal to \(\sum \frac{a^{n+r-1}}{(a-b)(a-c)\dots(a-k)}\).