Problems

Show that three distinct points in the plane with integral coordinates (in the usual Cartesian system) cannot form an equilateral triangle.

Let \(a_1, a_2, \ldots, a_k, \ldots\) be a sequence of real numbers which is periodic modulo a positive integer \(k\), that is \[a_{n+k} = a_n \quad (n = 1, 2, \ldots).\] Show that there is a positive integer \(N\) such that \[a_{N+1} + a_{N+2} + \ldots + a_{N+n} > nA\] for every positive integer \(n\), where \(A\) is defined by \[kA = a_1 + a_2 + \ldots + a_k.\]

A tripod \(VA\), \(VB\), \(VC\) is made of three uniform rods of length \(2l\) and weight \(w\). If freely pivoted together at \(V\), it stands symmetrically making a regular tetrahedron, one face of which is flat on the ground. A wind is blowing and the projection of \(VA\) on the ground points exactly into the wind. The feet \(B\) and \(C\) are gently dug into the ground and may be taken as freely pivoted but the foot \(A\) rests on hard ground whose coefficient of friction is \(\mu\). If the force due to the wind on each rod is directed exactly downwind and of magnitude \(\frac{1}{2}w^2l\) per unit length, show that the least wind speed \(v\) that will topple the tripod is given by \[2v^2 = \frac{2\sqrt{2}\mu + 1}{2\sqrt{2}\mu + 3}\sqrt{2}.\]

A small cork of density \(\rho\) and mass \(M\) is inside a large bottle filled with water of density \(\rho'\). The cork is in equilibrium completely immersed at a height \(x_0\) above the bottom of the bottle to which it is attached by a light spring of natural length \(a\). If the cork is moved through the water it experiences a resisting force whose magnitude is \(2\lambda\) times the speed of the cork through the water. The bottle and cork are initially at rest but they are then dropped. Show that the height \(x\) of the cork in the bottle at all times until the bottle hits the ground is given by \[x = a + (x_0 - a)e^{-\lambda t}[(z/m)\sin mt + \cos mt],\] where \[m^2 = \left[\frac{\rho' - \rho}{\rho} \cdot \frac{g}{x_0 - a} - \lambda^2\right].\]

\(a, b, c, d\) are integers lying between 1 and 9, inclusive, and $$n = 10^4a + 10^3b + 10^2b + 10c + d.$$ $$n = b^2(10a + b)(10^2d + 10d + b)$$ is the decomposition of \(n\) into prime factors. Prove that there is exactly one \(n\) with this property, and find \(n\).

\(a_1, a_2, \ldots, a_n\) are distinct numbers, and \(b_1 > b_2 > \cdots > b_n\). If \(\rho\) is a permutation of \((1, 2, \ldots, n)\), so that \(i\) becomes \(\rho(i)\), the number \(F(\rho)\) is defined by $$F(\rho) = a_{\rho(1)}b_1 + a_{\rho(2)}b_2 + \cdots + a_{\rho(n)}b_n.$$ Show that \(F(\rho)\) attains a maximum value (as \(\rho\) varies) when \(\rho\) is chosen so that $$a_{\rho(1)} > a_{\rho(2)} > \cdots > a_{\rho(n)}.$$ \((z_1, z_2, \ldots, z_{n1}, \ldots)\) is a sequence of positive integers, and \(x_i = x_j\) if and only if \(i = j\). Show that $$\sum_{i=1}^{\infty} \frac{1}{x_i(i + 1)}$$ is convergent. What is the largest possible value that this sum can take?

Describe the following transformations of the complex \(z\)-plane geometrically:

1. [(i)] \(R(z) = iz - i + 1\);
2. [(ii)] \(S(z) = z/(z - 1)\).

Show that \(R^4 = S^2 = I\), where \(I\) is the identity transformation \(I(z) = z\), and also that \(RS = SR^3\). Deduce that there are eight, and only eight, distinct transformations which can be obtained by arbitrary (finite) combinations of the transformations \(R\) and \(S\).

\(p\) is a parabola, with axis \(a\). \(X\) is a fixed point of \(p\), not on \(a\), and \(l\) is the line from \(X\) parallel to \(a\) and lying outside \(p\). Let \(P\) be a general point of \(p\). The tangents to \(p\) from \(P\) touch \(p\) at \(T_1\) and \(T_2\). The lines through \(T_1\) and \(T_2\) perpendicular to \(T_1T_2\) meet \(p\) again in \(S_1\) and \(S_2\), and the tangents to \(p\) at \(S_1\) and \(S_2\) meet in \(P'\). What is the locus of \(P'\) as \(P\) varies on \(l\)?

\(a, b, c, d\) and \(l\) are five coplanar lines, no three of which are concurrent, and \(E, F, G\) are the joins of the points \((ab)\) and \((cd)\), \((ac)\) and \((bd)\), and \((ad)\) and \((bc)\), respectively. \(E, F\) and \(G\) are the harmonic conjugate of \((cd)\) with respect to \((ab)\) and \((cd)\), and \(F\) and \(G\) are similarly defined. Show that \(E, F\) and \(G\) are collinear.

A cloud of stationary droplets has mean density \(k\rho\). A raindrop falls through the cloud under the influence of gravity and those droplets of the cloud that adhere to it. The raindrop remains spherical and of constant density. Find the speed \(v\) of the raindrop when its mass is \(m\), if it starts from rest with mass \(m_0\).