A circular table has radius 1 ft. Five equal circular discs are symmetrically placed so as to cover the table completely. What is their minimum radius? Without detailed calculation, show that it is possible to cover the table by means of five slightly smaller discs.
A table is laid with \(2n\) places in a row. A party of \(2k\) dons, where \(k \leq n\), sit down in such a way that the number of empty spaces between any two of them is even. (It may be zero.) The number of empty spaces at the ends of the row need not be even. In how many ways can this be done?
The sum \(s(m,n)\) is defined by \[ s(m,n) = \sum_{r=1}^n \frac{1}{r}, \] where \(n \geq m \geq 2\). Show that \(s(m,n)\) is never an integer, by proving the following two propositions or otherwise.
The inscribed circle \(\Gamma\) of a triangle \(ABC\) touches the sides of the triangle at \(D\), \(E\), \(F\). Prove that the circumcircle of the triangle \(ABC\) and the nine-point circle of the triangle \(DEF\) are inverse with respect to \(\Gamma\). Show further that the orthocentre and centroid of the triangle \(DEF\) lie on the line joining the circumcentre and incentre of the triangle \(ABC\).
Show that with \(n\) rods of lengths \(1, 2, 3, \ldots, n\) it is possible to form exactly \(\frac{1}{24}n(n-2)(2n-5)\) triangles if \(n\) is even, and find the corresponding number if \(n\) is odd.
In a euclidean plane a point \(P'\) is said to be the reflection of a point \(P\) in a point \(A\) if \(A\) is the mid-point of \(PP'\); \(P'\) is the reflection of \(P\) in a line \(l\) if the line \(l\) bisects \(PP'\) at right angles. The operation of reflection in a point or a line will be denoted by the same symbol as the point or line itself. A symbol such as \(\ln B/l\) denotes the operation of successive reflection in \(l\), then in \(B\), then in \(A\) etc., taken in this order. If \(A\) is any operation of this type, \(R\) is denoted by \(R^{-1}\) and so on. The identity operation, in which every point of the plane is left unaltered, is denoted by \(I\). Show that every point of the plane is left unaltered if and only if \(R = I\) and only if
A uniform solid hemisphere is balanced in equilibrium with its curved surface in contact with a sufficiently rough inclined plane. Find the greatest possible value of the inclination of the plane to the horizontal, and less inclination there may be two positions of equilibrium, one stable and one unstable. Show also that the coefficient of friction has to be greater than \(3/\sqrt{(55)}\) for the plane to be sufficiently rough.
A uniform rod of length \(l\) lies horizontally on a rough plane inclined to the horizontal at an angle \(\alpha\). The coefficient of friction \(\mu\) is greater than \(\tan \alpha\). A gradually increasing force is applied upwards along the line of greatest slope at one end of the rod. Show that when the rod begins to move, the length of rod which moves upwards is less than \(l/\sqrt{2}\).
List clearly and concisely the main dynamical principles and problems involved in designing (i) an earth satellite, and (ii) a moon rocket. Where relevant express your answers also in mathematical form.
A photon of momentum \(k_0\) is absorbed by an electron initially at rest which instantly recoils and emits a second photon of momentum \(k\) in a direction making an angle \(\theta\) with the direction of \(k_0\). The electron at rest has an energy \(m_0\), and when moving with momentum \(p\) has an energy \(\sqrt{(m_0^2 + p^2)}\). The photons have energies \(k_0\) and \(k\) respectively. Prove that \[ \frac{1}{k} - \frac{1}{k_0} = \frac{1}{m_0}(1 - \cos \theta). \] [It is to be assumed that total energy is conserved, and that total momentum is conserved in the direction of \(k_0\) and perpendicular to this direction. Gravitational effects can be neglected.]