\(a, b, c, d\) 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:
\(p\) 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 \(m_0\).
Two particles \(A\) and \(B\), of equal mass, are joined by a light inextensible string. \(A\) moves on a rough horizontal table (coefficient of friction \(\mu\)) and the string passes through a small smooth hole \(O\), so that \(B\) hangs below the table. Show that, if \((r, \theta)\) are polar coordinates of the position of \(A\) relative to \(O\), $$\frac{d}{dt}(r^2\dot{\theta}) = -\mu gr^2\dot{\theta}/v,$$ $$2\ddot{r} - r\dot{\theta}^2 = -g(1 + \mu^2/v),$$ where \(v^2 = \dot{r}^2 + r^2\dot{\theta}^2\) and the dot denotes differentiation with respect to the time \(t\). Initially \(r = R\) and the velocity of \(A\) is at right angles to the string and of magnitude \(\sqrt{(gR)}\). If \(\mu\) is small an approximate solution to the equations of motion is $$r = R + \mu\rho(t),$$ $$r\theta = \sqrt{(gR)} + \mu h(t),$$ where the functions \(\rho(t)\) and \(h(t)\) are independent of \(\mu\). Show that $$h = -gt - \sqrt{(g/R)}\rho$$ and $$\ddot{\rho} + \frac{3g}{2R^2}\rho + g\sqrt{\frac{g}{R}}t = 0.$$