Problems

If, for \(n = 1, 2, 3, \ldots\), the polynomial $$\frac{1}{n!}x(x-1)(x-2)\cdots(x-n+1)$$ is denoted by \(P_n(x)\), show that $$P_{n+1}(x+1) - P_{n+1}(x) = P_n(x) \quad (n = 1, 2, 3, \ldots).$$ Hence, or otherwise, prove that each of the polynomials \(P_n(x)\) takes integral values for all integral values of \(x\).

If, for each real number \(x\), \(\{x\}\) denotes the distance of \(x\) from the nearest integer (so that, for example, \(\{\pi\} = \pi - 3\) and \(\{2\frac{3}{4}\} = \frac{1}{4}\)), show that, if \(n\) is any integer \(\geq 1\), $$\{(\sqrt{2} + 1)^n\} = (\sqrt{2} - 1)^n$$

Each of the following rules defines a map (or transformation) from the set \(Z\) of all integers (positive, negative, or zero) into the same set \(Z\):

1. [(a)] \(x \rightarrow \cos\pi x\);
2. [(b)] \(x \rightarrow 3x - 1\).

(For `\(x \rightarrow \cos\pi x\)' read `\(x\) is mapped into, or transformed into, \(\cos\pi x\)'.) If \(M\) is one of these maps and \(S\) is a subset of \(Z\) (which may be \(Z\) itself) then we denote by \(M(S)\) the subset of \(Z\) consisting of all integers expressible in the form \(M(s)\) for some integer \(s\) in \(S\). Thus in case (b) if \(S\) is the set of all positive integers divisible by 3 then \(M(S)\) is the set \(M(Z)\) in each of the cases (a) and (b). Furthermore, in case (a) find a non-empty subset \(S\) of \(Z\) such that \(M(S) = S\); and in case (b) show that there is no such non-empty subset.

A circular disc of radius \(r\) is thrown at random onto a large board divided into squares of side \(a\) (where \(a > 2r\)). Show that the probability that the disc comes to rest entirely within one square is \(\left(1 - \frac{2r}{a}\right)^2\). If, instead of a disc, a thin pencil of length \(l\) (where \(l \leq a\)) is thrown on to the board, show that the probability that the pencil does not come to rest entirely within one square is \(\frac{(4a-l)l}{\pi a^2}\).

If \(x_0\) and \(x_1\) are two given positive real numbers and \(x_2, x_3, \ldots\) are determined successively by the formula $$x_n = \sqrt{(x_{n-1} \cdot x_{n-2})} \quad (n = 2, 3, \ldots),$$ show that \(x_n \rightarrow x_0^{1/3} x_1^{2/3}\) as \(n \rightarrow \infty\).

Either by showing that \(n!e\) is never an integer (for \(n = 1, 2, \ldots\)), or in any way, prove that $$e = 1 + \frac{1}{1!} + \frac{1}{2!} + \cdots$$ is irrational (that is, it cannot be expressed in the form \(p/q\) with \(p\) and \(q\) integral).

By using the identity $$\frac{1}{y+1} = \frac{1}{y-1} - \frac{2}{y^2-1},$$ or otherwise, determine for what real values of \(x\) the series $$\sum_{n=1}^{\infty} \frac{2n}{x^{2n} + 1}$$ is convergent and show that when it is convergent the series has sum \(\frac{2}{x^2-1}\).

Show that for each integer \(n \geq 1\) there is a polynomial \(T_n(x)\) of degree \(n\) such that $$T_n(\cos t) = \cos nt$$ for all real \(t\). Show furthermore that, for each such integer \(n\), \(|T_n(x)| \leq 1\) if \(-1 \leq x \leq 1\) but \(|T_n(x)| > 1\) for all real \(x\) outside this range.

Show that the function $$f(x) = \int_x^{2x} \frac{\sin t}{t} dt$$ is bounded for \(x > 0\), and find the points \(x\) at which it attains its greatest and least values in this range. (A function \(f(x)\) is said to be bounded over a certain range if a real number \(C\) can be found such that \(|f(x)| \leq C\) for all \(x\) in that range.)

If \(\Gamma\) is a circle with centre \(C\), and \(A, B\) are two points in the same plane as \(\Gamma\) (but not on \(\Gamma\)), show that \(AX + BX\) attains its minimum as \(X\) varies on \(\Gamma\) at a point at which \(AX\) and \(BX\) are equally inclined to \(CX\). Hence or otherwise show that if \(A, B, C\) are three fixed points and \(P\) is a point (in the plane of \(A, B\) and \(C\)) at which \(AP + BP + CP\) is a minimum, then either \(P\) is one of the points \(A, B\), or \(C\), or else $$\widehat{BPC} = \widehat{CPA} = \widehat{APB} = \frac{2\pi}{3}.$$