Prove that \(\displaystyle \binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}.\) Hence prove that for \(n > r\) \(\displaystyle \binom{n}{r} = \sum_{i=0}^r \binom{n-i-1}{r-i}.\) [The binomial coefficient \(\displaystyle \binom{n}{r}\) is defined by \(\displaystyle \binom{n}{r} = \frac{n!}{r!(n-r)!}\).]
Show that the sum of the first \(n\) odd positive integers is a perfect square. The odd positive integers are arranged in blocks with \(n\) integers in the \(n\)th block and \(a_n\) denotes the sum of the numbers in the \(n\)th block, so that \(a_1 = 1\), \(a_2 = 3+5\), \(a_3 = 7+9+11\), etc. Find an expression for \(a_n\) and deduce that \(\displaystyle \sum_{r=1}^n r^3 = \frac{1}{4}n^2(n+1)^2\). Show that \(\displaystyle \sum_{r=1}^n r^5 = \frac{1}{12}n^2(n+1)^2(2n^2+2n-1)\) by considering the situation in which the \(n\)th block has \(n^2\) integers.
Show that a number is divisible by 3 if and only if the sum of its digits is divisible by 3. All possible numbers between 1,000 and 10,000 are formed from the digits 0, 1, 2, 3, 5 and 7, no digit being repeated in any one number. What proportion of these numbers is divisible by 3 and what proportion by 6?
Find necessary and sufficient conditions on the coefficients of the quartic equation \[x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0\] which ensure that whenever \(z\) is a root so is \(1/z\). Hence show that the roots of a quartic equation of this type may be found by solving several appropriate quadratic equations.
Let \[a_n = \frac{1}{2\sqrt{2}}\{(1+\sqrt{2})^n - (1-\sqrt{2})^n\}.\] Establish a linear relationship between \(a_n\), \(a_{n+1}\) and \(a_{n+2}\), and deduce that \(a_n\) is an integer for all positive integers \(n\). Show also that the greatest integer less than or equal to \((1+\sqrt{2})^n/\sqrt{2}\) is always even.
Let \(z = \cos\theta + i\sin\theta\) (\(\theta \neq \pi\)) and \(w = (z-1)(z+1)^{-1}\). Show that \(w\) is purely imaginary, and hence show that the angle in a semi-circle is a right angle.
A matrix \(B\) satisfies \(B^2 = B\) and is known to be of the following form: \[B = \begin{pmatrix} a & 0 & a \\ -b & b & -a \\ -b & 0 & -b \end{pmatrix},\] where \(a\) and \(b\) are non-zero real numbers. Find the matrix \(B\). Find a non-zero column matrix \(Z\) such that \(BZ = 0\), and determine the condition for a column matrix \(X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\) to satisfy \(BX = X\). Hence, by defining its columns suitably, find an invertible matrix \(P\) such that \[BP = P\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}.\]
Let \(G\) be a group and let \(g \in G\). Let \[C(g) = \{x \in G : xg = gx\}.\] Show that \(C(g)\) is a subgroup of \(G\). Now let \(G\) be the group of symmetries of the square \(ABCD\). Let \(a\) be the rotation through \(\pi /2\) about an axis through the centre and perpendicular to the square. Let \(b\) be the rotation through \(\pi\) about an axis through the mid-points of \(AB\) and \(CD\). Show that every element of \(G\) can be written in one of the forms \(a^i\) or \(ba^i\) for \(i = 0, 1, 2, 3\). Determine those elements whose square is the identity. Show further that \(C(a^2) = G\) and that \(C(b) \neq G\).
A triangle inscribed in the parabola \(y^2 = x\) has fixed centroid \((\xi, \eta)\) (where \(\eta^2 < \xi\)). Show that the area of the triangle is greatest when one vertex is at the point \((\eta^2, \eta)\). [The area of the triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) is given by the absolute value of \[\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}.\] The product of the differences of the roots of the cubic equation \(t^3 + \alpha t^2 + \beta t + \gamma = 0\) is the square root of \(\alpha^2\beta^2 - 4\beta^3 + (18\alpha\beta - 4\alpha^3)\gamma - 27\gamma^2\).]
Show that there is an infinite number of rectangles circumscribing a given ellipse and that their vertices lie on a circle. Hence find the circumscribing rectangle of greatest area.