Let \(a\) and \(b\) be integers, \(p\) a prime. Use the binomial theorem to show that \((a+b)^p \equiv (a^p+b^p) \pmod{p}\). Show (by induction or otherwise) that \(a^p \equiv a \pmod{p}\). Find all integer solutions of \(x^{p^2}-x^p-x+c \equiv 0 \pmod{p}\), where \(c\) is an integer.
A certain dining club is constituted as follows: There are \(n\) members. The club's dining room seats \(k\) members (\(k \leq n\)) and every dinner is fully attended. No two dinners in one year are attended by the same \(k\) members. Numbers \(s\) and \(t\) are fixed (\(2 \leq s \leq k\), \(t \geq 1\)) and the rules decree that given any \(s\) members, they shall be simultaneously present at precisely \(t\) dinners. Show that, given any \(s - 1\) members, they are simultaneously present at precisely \(\frac{(n-s+1)t}{(k-s+1)}\) dinners, and deduce that every member attends the same number of dinners. Determine how many dinners are held each year, and deduce that \((n/k)^s t \leq 365\).
The quartic equation \(x^4 - s_1 x^3+s_2x^2-s_3x+s_4 = 0\) has roots \(\alpha\), \(\beta\), \(\gamma\), \(\delta\). Find the cubic equation with roots \(\alpha\beta + \gamma\delta\), \(\alpha\gamma + \beta\delta\), \(\alpha\delta + \beta\gamma\). Supposing that methods of solving quadratic and cubic equations are known, describe a procedure for solving a quartic equation.
Solution: Note that:
\(k\) integers are selected from the integers 1, 2, ..., \(n\). In how many ways is it possible if
Describe the path traced out by the point \(w = z+ 1/z\) in the Argand diagram as the point \(z\) traces out the circle \(|z| = r > 0\). The sequence \(w_0\), \(w_1\), \(w_2\), ... is defined by the recurrence relation \(w_n = w_{n-1}^2 - 2\). Show that if \(w_0\) is real and satisfies \(-2 \leq w_0 \leq 2\), then the same is true for all \(w_n\), and that for all other real or complex values of \(w_0\), \(|w_n| \to \infty\).
Show that the set of complex valued \(2 \times 2\) matrices of the form $\begin{pmatrix} z & w\\ -\overline{w} & \overline{z} \end{pmatrix}$ satisfying \(|z|^2+ |w|^2 = 1\) forms a group \(G\) under matrix multiplication. Determine the subsets \(G_2\) consisting of all elements of \(G\) whose square is the identity matrix, and \(G_4\) consisting of all elements of \(G\) whose fourth power is the identity matrix. Do they form subgroups of \(G\)?
State precisely, without proof, the arithmetic-geometric mean inequality. The equation \(f(x) = x^n+a_1x^{n-1}+a_2x^{n-2}+ ... + a_n = 0\) has \(n\) distinct positive roots. Writing \(a_i = (-1)^i\binom{n}{i}b_i\), where \(\binom{n}{i}\) denotes the usual binomial coefficient, prove that \(b_{n-1} > b_n\). By considering \(f'(x)\), or otherwise, prove further that \(b_1 > b_2 > ... > b_{n-1} > b_n\).
Let \(C_1\), \(C_2\) and \(C_3\) be circles in the plane, each pair of which intersect in two points. The common tangents to \(C_2\) and \(C_3\) meet at \(P_1\), and points \(P_2\) and \(P_3\) are defined similarly. Prove that \(P_1\), \(P_2\) and \(P_3\) are collinear. What is the analogous result if the circles are mutually disjoint?
Two adjacent corners \(A\), \(B\) of a rigid rectangular lamina \(ABCD\) slide on the \(x\)-axis and the \(y\)-axis respectively, and all the motion is in one plane. Prove that the locus of \(C\) is an ellipse, and find the area of the ellipse in terms of \(a = AD\) and \(b = AB\). [The area of an ellipse is \(\pi \times\) the product of the lengths of the semi-axes.]
\(P\) and \(Q\) are the intersections of the line \(lx + my + n = 0\) with the parabola \(y^2 = 4ax\). The circle on \(PQ\) as diameter meets the parabola again in \(R\) and \(S\). Find the equation of \(RS\).