Problems

Discuss the solution of the system of linear equations \begin{align} x + y + z + t &= 4,\\ x + 2y + 3z + 4t &= 10,\\ x + 4y + az + bt &= c, \end{align} with due regard to the special cases which may arise for particular values of \(a\), \(b\) and \(c\).

1. Solve $$x^4 - x^3 - 4x^2 - x + 1 = 0.$$
2. Solve $$2^x + 8^x = 4^{x+1}.$$
3. Find the positive root of $$\log_4(x + 1) = \frac{3}{4}(\log_2 x + \log_3 x).$$

Let \(J_1\) be the operation of taking the inverse (reciprocal) of a number, and \(J_2\) the operation of subtracting a number from 1. Prove that the operations \(J_1\) and \(J_2\), applied repeatedly in any order to a number \(\lambda\) (\(\lambda \neq 0, \lambda \neq 1\)) can only lead to one of a finite set of numbers. Express each of these numbers in terms of \(\lambda\). If \(J_r\) is the operation by which the \(r\)th member of the set (excluding \(\lambda\) itself) is obtained from \(\lambda\), show that either \(J_r\) applied twice reproduces the original number, or \(J_r\) does so when applied three times.

\(x_1, \ldots, x_n\) are distinct numbers and, for \(1 \leq r \leq n\), \(p_r(x)\) is written for $$(x - x_1) \ldots (x - x_{r-1})(x - x_{r+1}) \ldots (x - x_n).$$ By considering $$\sum_{r=1}^{n} \alpha_r p_r(x),$$ for suitably chosen \(\alpha_r\), show that it is possible to find a polynomial of degree not exceeding \(n-1\) which takes given values at \(x_1, \ldots, x_n\). Similarly, by considering $$\sum_{r=1}^{n} (\beta_r x + \gamma_r)\{p_r(x)\}^3,$$ show that it is possible to find a polynomial of degree not exceeding \(2n-1\) which takes given values at \(x_1, \ldots, x_n\) and whose first derivative also takes given values at these points.

1. [(i)] By considering the series expansion of \(e^{-x}(e^x - 1)^{n+1}\), or otherwise, show that $$n^n - (n+1)(n-1)^n + \frac{(n+1)n}{2!}(n-2)^n - \ldots(\text{to } n \text{ terms}) = 1.$$
2. [(ii)] Show that $$1^n - n2^n + \frac{n(n-1)}{2!}3^n - \ldots(\text{to } n+1 \text{ terms}) = (-1)^n n!.$$

1. [(i)] Prove, by induction or otherwise, that \(3^{2n+1} + 2^{n+2}\) is divisible by 7 for any positive integer \(n\).
2. [(ii)] Let $$u(m, n) = \{(n+1)!\}^2 + (n-1)!^{3m} + \{(n+1)! - (n-1)!\}^{2m},$$ where \(m\) and \(n\) are positive integers. Show that, for each pair of values of \(m\) and \(n\), \(u(m, n)\) is an integer. Show also (by finding a relation between \(u(m, n)\), \(u(m+1, n)\) and \(u(m+2, n)\), or otherwise) that \(u(m, n)\) is divisible by \(2^{m+1}\).

\(C_1\) and \(C_2\) are two circles; the polars of a point \(A\) with respect to \(C_1\) and \(C_2\) meet at \(B\). Prove that \(B\) is on the radical axis of \(C_1\) and \(C_2\) if and only if \(A\) is also.

A point moves so that its least distances from each of two fixed circles are equal; describe its locus in each of the various cases that may arise, and justify your statements.

Three tangents are drawn to a parabola so that the sum of the angles which they make with the axis of the parabola is \(\pi\). Prove that the circumcircle of the triangle formed by the tangents touches the axis of the parabola at the focus.

Take any two of the standard concurrence theorems for the triangle (medians, altitudes, bisectors of angles, perpendicular bisectors of sides), formulate analogous theorems in three dimensions, with `triangle' replaced by `tetrahedron', and discuss whether or not the three-dimensional theorems are true.