The real numbers \(l_1\), \(l_2\), ..., \(l_n\) satisfy \[l_1 \geq 0, l_1 + l_2 \geq 0, ..., l_1 + l_2 + ... + l_n \geq 0.\] Prove that \[\sum_{i=1}^{n} \alpha_i l_i \geq 0\] for any real numbers \(\alpha_i\) which satisfy \[0 < \alpha_n \leq \alpha_{n-1} \leq ... \leq \alpha_2 \leq \alpha_1.\] Prove that \(y \geq 1 + \log y\) (\(y > 0\)), and deduce that \[\sum_{i=1}^{n} \alpha_i y_i \geq \sum_{i=1}^{n} \alpha_i\] whenever \(y_1 \geq 1\), \(y_1 y_2 \geq 1\), ..., \(y_1 y_2 ... y_n \geq 1\) and the \(\alpha_i\) satisfy (1).
The binary star \(\ast\) \(b\) of two positive integers is defined as follows: Put \(a_0 = a\) and \(b_0 = b\). For \(i \geq 1\), let \[a_i = 2a_{i-1}, \quad b_i = [b_{i-1}/2],\] where \([x]\) is the integer part of \(x\). Let \(n\) be the (unique) index such that \(b_n = 0\). For \(0 \leq i \leq n\) define \[c_i = a_i \quad \text{if} \quad b_i \quad \text{is odd,}\] \[= 0 \quad \text{if} \quad b_i \quad \text{is even.}\] Define \(a\) \(\ast\) \(b\) to be \(c_0 + c_1 + ... + c_n\). Identify \(a\) \(\ast\) \(b\), and justify your answer.
Let \(U\) be a finite set. For subsets \(A\) and \(B\) of \(U\) which are not both empty, define \[d(A, B) = \frac{|A \cup B| - |A \cap B|}{|A \cup B|},\] where \(|T|\) means the number of elements in \(T\). Let \(X\), \(Y\) and \(Z\) be non-empty subsets of \(U\) and let \[W = (X \cap Y) \cup (Y \cap Z) \cup (Z \cap X).\] By considering a Venn diagram, or otherwise, prove that \[d(X, W) + d(Z, W) \geq d(X, Z).\] Prove further that \(d(X, Y) \geq d(X, W)\), and deduce that \[d(X, Y) + d(Y, Z) \geq d(X, Z).\]
Let \(A\) be a finite set having a commutative and associative binary operation * such that \(b = c\) whenever \(a * b = a * c\) for some \(a\). Show that \((A,*)\) is a group. Let \(p\) be a prime. For an integer \(n\), let \([n]\) be the equivalence class of \(n\) under the relation '\(x \sim y\) if and only if \(p\) divides \(x - y\)'. Let \[M = \{[n]; \quad 1 \leq n < p\},\] and define \[[r] * [s] = [rs].\] Prove that \((M,*)\) is a group. Deduce that \(p\) divides \((n^p - n)\) for all integers \(n\). [You may assume that the order of an element of a finite group divides the order of the group.]
The end-points of a variable chord \(l\) of a fixed non-singular conic \(S\) subtend a right angle at a fixed point \(O\). Prove that the foot of the perpendicular from \(O\) to \(l\) lies on a fixed curve \(C\), which in general is a circle, but may exceptionally be a straight line. What is the nature of \(S\) in the latter case? Show further that \(C\) is unaltered if \(S\) is replaced by any conic of the pencil of conics determined by \(S\) and any fixed pair of perpendicular lines through \(O\).
\(A_1 A_2 A_3 A_4\) is a tetrahedron, and the feet of the perpendiculars from a point \(O\) to its faces \(A_2 A_3 A_4\), \(A_1 A_3 A_4\), \(A_1 A_2 A_4\), \(A_1 A_2 A_3\) are the vertices of another tetrahedron \(B_1 B_2 B_3 B_4\). Prove that pairs of lines such as \(A_1 A_2\), \(B_3 B_4\) are mutually perpendicular. The line through \(A_1\) perpendicular to the plane \(B_2 B_3 B_4\) is \(l_1\), and \(l_2\), \(l_3\), \(l_4\) are similarly defined. Prove that any two of the lines \(l_1\), \(l_2\), \(l_3\), \(l_4\) are coplanar, and deduce that all four lines are concurrent.
Prove that through four non-coplanar points \(P_1\), \(P_2\), \(P_3\), \(P_4\) there passes a unique sphere \(S\). Through the mid-point of each pair of the points a plane is taken perpendicular to the line joining the other pair. By the use of vectors with origin at the centre of the sphere, or otherwise, prove that the six planes thus obtained are concurrent, in a point \(Q_5\), say. If \(P_5\) is a fifth point of \(S\), and the points \(Q_1\), \(Q_2\), \(Q_3\), \(Q_4\) are similarly derived from the sets \(\{P_2, P_3, P_4, P_5\}\), \(\{P_1, P_3, P_4, P_5\}\), \(\{P_1, P_2, P_4, P_5\}\), \(\{P_1, P_2, P_3, P_5\}\), prove that the five points \(Q_1\), \(Q_2\), \(Q_3\), \(Q_4\), \(Q_5\) lie on a sphere whose radius is half that of \(S\).
Let \(R\) be a ring in which \(x + x \neq 0\) whenever \(x \neq 0\) (\(x\) in \(R\)). Show that (i) provided \(R \neq \{0\}\), there is at least one element \(x\) in \(R\) with \(x^4 \neq x\); (ii) if \(x^2 = 0\) for every \(x\) in \(R\), then every product of three or more elements of \(R\) is zero. [Warning: \(R\) is not necessarily commutative.]
Let \(a_1\), \(a_2\), ... be an infinite sequence of real numbers. For each positive integer \(n\) let \(k(n)\) be the largest integer \(\leq n\) satisfying \(a_{k(n)} \geq a_j\) for \(j = 1, ..., n\); thus \(a_{k(n)}\) is the largest of the numbers \(a_1, ..., a_n\) and \(k(n)\) is the last place in which it occurs. Prove that either (i) \(k(n)\) tends to \(+\infty\) as \(n \to \infty\) or (ii) \(k(m) = k(m+1) = k(m+2) = ...\) for some integer \(m\). Deduce that there are integers \(m_1\), \(m_2\), ... with \(m_1 < m_2 < ...\) and either \[a_{m_1} \leq a_{m_2} \leq ... \quad \text{or} \quad a_{m_1} \geq a_{m_2} \geq ... \]
Each of \(n\) men attending a dinner leaves his hat in the cloakroom and collects a hat when he departs. It may be assumed that each man's choice of hat end of the dinner is completely random. Let \(P_{n,k}\) be the probability that exactly \(k\) men end up wearing the right hat. Show that \(P_{n,k} = (k+1)P_{n-1, k-1}\), and deduce that if \[F_n(x) \text{ is defined to be } \sum_{k=0}^{n} P_{n,k}x^k, \text{ then}\] \[F_n(x) = \frac{d}{dx}F_{n+1}(x).\] Hence, or otherwise, show that \[P_{n,k} = \frac{1}{k!}\sum_{j=k}^{n}\frac{(-1)^{j-k}}{(j-k)!}.\]