Show that, for each pair of positive integers \(m\), \(n\), the number of solutions in non-negative integers \(x_1\), \(x_2\), \(\ldots\), \(x_m\) of the inequality \[x_1 + x_2 + \ldots + x_m \leq n\] is \((m + n)!/m! n!\).
Show that the condition that the two triangles in the Argand plane formed by the two triples of complex numbers \(a_1\), \(a_2\), \(a_3\) and \(b_1\), \(b_2\), \(b_3\) should be similar in the same sense is that \[\frac{a_1 - a_3}{a_1 - a_2} = \frac{b_1 - b_3}{b_1 - b_2}.\] The three triangles \(BCA'\), \(CAB'\), \(ABC'\) are similar in the same sense (although they are not necessarily similar to \(ABC\)). Show that the triangles \(ABC\), \(A'B'C'\) have the same centroid.
Prove, by induction or by using de Moivre's theorem or in any other way, that if \(n\) is a positive integer then \[\tan nx = \frac{U_n(t)}{V_n(t)},\] where \(t = \tan x\) and \(U_n(t)\), \(V_n(t)\) are even polynomials of respective degrees \(n-2\), \(n\) if \(n\) is even and \(n-1\), \(n-1\) if \(n\) is odd. Hence, or otherwise, show that if \(n\) is even, then the product \[\prod_{r=0}^{n-1} \tan \frac{r\pi + c}{n}\] has the same value for all those real numbers \(c\) for which it is defined.
Three points are marked at random on the circumference of a circle. Show that there is probability \(\frac{1}{4}\) that the triangle with these three points as vertices is acute-angled.
Prove that the positive number \(a\) has the property that there exists at least one positive number that is equal to its own logarithm to the base \(a\) if and only if \(a \leq e^{1/e}\).
Prove that, if \(f(x)\) is a polynomial with integral coefficients, then the sum of the infinite series \[f(0) + f(1)/1! + f(2)/2! + \ldots\] is an integral multiple of \(e\).
Prove that, if no two of the real numbers \(a_1\), \(a_2\), \(\ldots\), \(a_n\) are equal, and all the real numbers \(A_1\), \(A_2\), \(\ldots\), \(A_n\) are positive, then the equation \[\frac{A_1}{x - a_1} + \frac{A_2}{x - a_2} + \ldots + \frac{A_n}{x - a_n} = 0\] has exactly \(n - 1\) real roots.
Prove that the quadrilateral of greatest area with sides of prescribed lengths is cyclic. A closed curve has the property that the area that it encloses is greater than that enclosed by any other closed curve of the same length; prove, by considering quadrilaterals inscribed within it, that the curve is a circle. (You may assume that the area enclosed by the curve is convex.)
If \(I(a, b)\) is defined, for all pairs of positive real numbers \(a\), \(b\), by \[I(a, b) = \int_0^{\infty} \frac{x^{a-1}}{(1+x)^{a+b}} dx,\] show, by substituting for \(x\) or otherwise, that \(I(a, b) = I(b, a)\). Prove also that \[I(a+1, b) = \frac{a}{a+b} I(a, b) \text{ and } I(a, b+1) = \frac{b}{a+b} I(a, b).\]
Show that there is a unique pair of real numbers \(a\), \(b\) with the property that \[\int_{-1}^{+1} P(x) dx = P(a) + P(b)\] for all polynomials \(P(x)\) of degree at most three.