Problems

For each positive integer \(n\), let \(u_n\) be the number of finite sequences \(a_1, a_2, \ldots, a_r\) satisfying the following conditions:

1. [(i)] \(r \geq 1\);
2. [(ii)] \(a_i = 1\) or \(2\) \quad \((i = 1, 2, \ldots, r)\);
3. [(iii)] \(a_1 + a_2 + \ldots + a_r = n\).

Show that \begin{equation*} u_n = \sum_{r=m}^{n} \binom{r}{n-r}, \end{equation*} where \(m\) is the least integer greater than or equal to \(\frac{1}{2}n\). Prove also that \(u_n = u_{n-1} + u_{n-2}\) \((n \geq 3)\), and that \begin{equation*} u_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1} - \left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right] \quad (n \geq 1). \end{equation*}

Suppose \(H_1\), \(H_2\), \(H_3\) are subgroups of a group \(G\), such that \(H_i \neq G\) \((i = 1, 2, 3)\). Of the following two statements, show that (i) is always false, and find an example where (ii) is false:

1. [(i)] Every element of \(G\) is in \(H_1\) or \(H_2\).
2. [(ii)] Every element of \(G\) is in \(H_1\), \(H_2\) or \(H_3\).

Let \(z_1\), \(z_2\), \(z_3\) be complex numbers, and suppose that \(z_1^k+z_2^k+z_3^k\) is real for \(k = 1, 2, 3\). Show that at least one of the numbers \(z_1\), \(z_2\), \(z_3\) is also real.

Let \(z_1\), \(z_2\), \(z_3\), \(z_4\) be real numbers, and suppose that \(z_1^2 + z_2^2 + z_3^2 + z_4^2 = 0\) for \(i = 1, 2, 3\). Show that the notation for the four numbers can be chosen in such a way that \(z_1 + z_2 + z_3 + z_4 = 0\).

Show that if \(y = \sum_{r=0}^{\infty} e^{rx}\), then \begin{equation*} (-1)^m\frac{d^m y}{dx^m} + e^x \sum_{k=0}^{m} \binom{m}{k} \frac{d^k y}{dx^k} = (n+1)^m e^{(n+1)x} \end{equation*} for all \(m > 0\). Deduce that if \(s_k = \sum_{r=0}^{\infty} r^k\), then \begin{equation*} \sum_{k=0}^{m-1} \binom{m}{k} s_k = (n+1)^m \quad (m > 0). \end{equation*} Prove that \(s_2 = \frac{1}{6}n(n+1)^2\).

Evaluate the indefinite integral \begin{equation*} \int \frac{px + q}{rx^2 + 2sx + t} dx \end{equation*} where \(p, q, r, s, t\) are real constants.

\(P\) is a variable point on a plane curve \(\Gamma\), and \(R\) is the centre of curvature of \(\Gamma\) at \(P\). Let \(\Delta\) be the locus of \(Q\), where \(Q\) is the mid-point of \(PR\). Show that if \(\phi\) is the angle between the tangent to \(\Gamma\) at \(P\) and the tangent to \(\Delta\) at \(Q\) then \begin{equation*} \tan\phi = \frac{d\rho}{ds}, \end{equation*} where \(\rho = PR\) and \(s\) is the arc length of \(\Gamma\). Prove that if \(\Gamma\) is defined by the equation \(\rho^2 + s^2 = a^2\), then \(\Delta\) is a straight line.

A triangle \(ABC\) is said to be self-conjugate with respect to a circle if \(A\) is the pole of \(BC\), \(B\) is the pole of \(CA\), and \(C\) is the pole of \(AB\). Show that if the triangle \(ABC\) has an obtuse angle there is just one circle with respect to which it is self-conjugate, but that otherwise there is no such circle.

The points \(O\), \(A\), \(B\), \(C\) are not coplanar, and the position vectors of \(A\), \(B\), \(C\) with respect to \(O\) as origin are \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) respectively. If \(\mathbf{p}\) is any vector, show that \begin{equation*} [\mathbf{a}, \mathbf{b}, \mathbf{c}]\mathbf{p} = (\mathbf{a} \cdot \mathbf{p})\mathbf{b} \times \mathbf{c} + (\mathbf{b} \cdot \mathbf{p})\mathbf{c} \times \mathbf{a} + (\mathbf{c} \cdot \mathbf{p})\mathbf{a} \times \mathbf{b}. \end{equation*} \(X\), \(Y\), \(Z\) are such that \(X\) is the centre of the sphere through \(O\), \(A\), \(B\), \(C\); \(Y\) is the centre of a sphere which touches the lines \(OA\), \(OB\), \(OC\); and \(Z\) is the second common point of the spheres through \(O\) with centres \(A\), \(B\) and \(C\). Show that the position vectors of \(X\), \(Y\), \(Z\) are of the form \(\mathbf{x}\), \(\lambda\mathbf{y}\), \(\mu\mathbf{z}\) respectively, where \begin{align*} 2[\mathbf{a}, \mathbf{b}, \mathbf{c}]\mathbf{x} &= |\mathbf{a}|^2 \mathbf{b} \times \mathbf{c} + |\mathbf{b}|^2 \mathbf{c} \times \mathbf{a} + |\mathbf{c}|^2 \mathbf{a} \times \mathbf{b}\\ \mathbf{y} &= |\mathbf{a}| \mathbf{b} \times \mathbf{c} + |\mathbf{b}| \mathbf{c} \times \mathbf{a} + |\mathbf{c}| \mathbf{a} \times \mathbf{b}\\ \mathbf{z} &= \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} + \mathbf{a} \times \mathbf{b} \end{align*} and \begin{equation*} \mu = \frac{2[\mathbf{a}, \mathbf{b}, \mathbf{c}]}{|\mathbf{z}|^2}. \end{equation*}

Let \(f_N(a) = \underset{R}{\textrm{Max}}(x_1 x_2 \ldots x_N)\), where \(a \geq 0\), and \(R\) is the region of values determined by \begin{equation*} x_1 + x_2 + \ldots + x_N = a \end{equation*} and \(x_i \geq 0\) for all \(i\). Show that \begin{equation*} f_N(a) = \underset{0 \leq z \leq a}{\textrm{Max}} \{zf_{N-1}(a-z)\} \end{equation*} \((N > 1)\), with \(f_1(a) = a\). Hence show that \begin{equation*} f_N(a) = \frac{a^N}{N^N}. \end{equation*}