Find a pair of integers \(\alpha\) and \(\beta\) for which \(2^{5n+\alpha} + 4^{3n+\beta}\) is divisible by 29 for all non-negative integers \(n\).
Find a relationship between \(x\), \(y\) and \(z\) which must hold if there are to exist \(p\), \(q\) and \(r\) such that \(x\), \(y\) and \(z\) are respectively the \(p\)th, \(q\)th and \(r\)th terms both of an arithmetical and of a geometrical progression.
Show that 15 distinct pairs of objects can be chosen from six distinct objects. A syntheme is a set of three pairs into which one can partition six such objects. Find the number of distinct synthemes. In how many distinct ways can one select a set of five synthemes which together include all 15 distinct pairs of objects? (Order, whether of objects in a pair, pairs in a syntheme, or synthemes in a set of synthemes, is irrelevant.)
Two elements \(\alpha\), \(\beta\) of a finite group \(G\) are called conjugate if there exists \(\gamma \in G\) such that \(\alpha = \gamma\beta\gamma^{-1}\) Show that conjugacy defines an equivalence relation. The elements \(a\), \(b\) have associative multiplication with unit \(e\) and satisfy \(a^3 = b^2 = (ab)^2 = e\) The set of six elements \(e\), \(a\), \(a^2\), \(b\), \(ab\), \(a^2b\) are distinct. Show that they form a group and separate them into equivalence classes under conjugacy.
The process of representing polynomials by their remainders upon division by \(x^2 + 1\) separates the set of all polynomials with real coefficients into equivalence classes. Denote by \((\alpha, \beta)\) the class containing \(\alpha x + \beta\). If the product \((\alpha_1, \beta_1)(\alpha_2, \beta_2)\) denotes the class containing the products of polynomials from \((\alpha_1, \beta_1)\) and \((\alpha_2, \beta_2)\), obtain it explicitly in the form \((\alpha, \beta)\). Specify the relationship of the field formed by the set of all \((\alpha, \beta)\) and the field of complex numbers. In a like procedure using remainders upon division by \(x^2 + x + 1\), denote by \([\lambda, \mu]\) the class containing \(\lambda x + \mu\). Evaluate the product \([\lambda_1, \mu_1][\lambda_2, \mu_2]\). By relating the set of all \([\lambda, \mu]\) to the complex numbers obtain a 1:1 correspondence between the sets \((\alpha, \beta)\) and \([\lambda, \mu]\) which makes explicit the fact that the corresponding product laws are isomorphic.
Show that the geometric mean of \(n\) positive numbers is less than or equal to their arithmetic mean. Use this result and the binomial theorem to show that \[(n+1)! \leq 2^n\left [1! \cdot 2! \cdots n! \right ]^{2/(n+1)}\]
Solution: \begin{align*} && 2^n &= \sum_{k=0}^n \binom{n}{k} \\ &&& \geq (n+1) \sqrt[n+1]{\prod_{k=0}^n \binom{n}{k}} \\ &&& = (n+1) \sqrt[n+1]{\frac{n!}{0! \cdot n!}\cdot\frac{n!}{1! \cdot (n-1)!}\cdots \frac{n!}{n! \cdot 0!}} \\ &&&=(n+1)^{n+1} \sqrt[n+1]{\frac{(n!)^{n+1}}{\left ( 0! \cdot 1! \cdots n! \right)^2}} \\ &&&= \frac{(n+1)!}{\left [1! \cdot 2! \cdots n! \right ]^{2/(n+1)}} \\ \Rightarrow && (n+1)! & \leq 2^n\left [1! \cdot 2! \cdots n! \right ]^{2/(n+1)} \end{align*}
Show that $\begin{vmatrix} 1+x_1 & x_2 & x_3 & \cdots & x_n \\ x_1 & 1+x_2 & x_3 & \cdots & x_n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_1 & x_2 & x_3 & \cdots & 1+x_n \end{vmatrix} = 1 + x_1 + x_2 + \cdots + x_n$ Hence or otherwise evaluate the \(n\)-rowed determinant $\begin{vmatrix} 0 & 1 & 1 & \cdots & 1 \\ 1 & 0 & 1 & \cdots & 1 \\ 1 & 1 & 0 & \cdots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \cdots & 0 \end{vmatrix}$
Find the two values of \(\lambda\) for which the matrix equation \(\begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \lambda \begin{pmatrix} x \\ y \end{pmatrix}\) has non-trivial solutions for \(x\) and \(y\). For each of these values, find a corresponding solution for \(x\) and \(y\). If \(A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}, \quad M = \begin{pmatrix} x_1 & x_2 \\ y_1 & y_2 \end{pmatrix}\) where \((x_1, y_1)\) and \((x_2, y_2)\) are the two solutions just obtained, find a diagonal matrix \(D\) such that \(AM = MD\)
\(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\) are consecutive vertices of a regular polygon of \(n\) sides (\(n \geq 7\)); \(BE\) meets \(DG\) in \(X\), \(CF\) in \(Y\), and \(AD\) in \(Z\). Prove that \(EX \cdot EZ = EY^2\).
A square \(ABCD\) is such that \(A\) lies on \(y = 0\), \(C\) on \(x = 0\), while \(B\) and \(D\) lie on the circle \(x^2 + y^2 + 2gx + 2fy = 0\) (\(f\), \(g\), \(f \neq g\) all non-zero). Show that three squares satisfy these conditions, and that any pair of these have a common vertex.