Problems

Prove that, if \(a, b\) are real, \[ ab \le \left(\frac{a+b}{2}\right)^2, \] and deduce that, if \(a, b, c, d\) are positive, \[ abcd < \left(\frac{a+b+c+d}{4}\right)^4, \] with equality only when the numbers are all equal. By giving \(d\) a suitable value in terms of \(a, b, c\), or otherwise, prove that, if \(a, b, c\) are positive, \[ abc \le \left(\frac{a+b+c}{3}\right)^3. \]

Prove what you can about the number of real roots of each of the equations

1. [(i)] \((x-a_1)(x-a_2)\dots(x-a_n) + (x-b_1)(x-b_2)\dots(x-b_n) = 0\), where \[ a_1 > b_1 > a_2 > b_2 \dots > a_n > b_n; \]
2. [(ii)] \((x-a)^m+(x-b)^m=0\) where \(a>b\) and \(m\) is a positive integer.

If \(u_n\) denotes the number of ways in which \(n\) men and their wives can pair off at a dance so that no man dances with his wife, prove that \[ u_n = (n-1)(u_{n-1}+u_{n-2}). \] Deduce that \[ \frac{u_n}{n!} - \frac{u_{n-1}}{(n-1)!} = \frac{(-1)^n}{n!}, \] and hence find an expression for \(u_n\).

Prove that, if \((1+x)^n = c_0 + c_1 x + \dots + c_n x^n\), then

1. [(i)] \(\dfrac{c_0}{1} - \dfrac{c_1}{2} + \dfrac{c_2}{3} - \dots + (-1)^n \dfrac{c_n}{n+1} = \dfrac{1}{n+1}\);
2. [(ii)] \(c_0^2 - c_1^2 + c_2^2 - \dots + (-1)^n c_n^2\) is equal to \((-1)^m (2m)!/(m!)^2\) if \(n\) is an even integer \(2m\). Find its value when \(n\) is an odd integer.

Resolve \(x^{2n}+1\) into real quadratic factors, where \(n\) is a positive integer. Express \[ \frac{1}{x^{2n}+1} \] in partial fractions with these factors as denominators.

If three straight lines do not all lie in one plane, prove that, in general, there are infinitely many straight lines which intersect them. Point out any exceptional cases. Of three straight lines \(ABC, DEF, GHK\), no two are in the same plane. They are all met by each of the straight lines \(ADG, BEH, CFK\). Prove that, in general, the lines \(BD, CG, FH\) are concurrent.

A common tangent to two non-intersecting circles \(C_1, C_2\) touches them at \(P_1, P_2\) respectively. \(L\) is one of the limiting points of the coaxal system determined by \(C_1, C_2\). \(P_1L\) meets \(C_1\) again at \(Q_1\) and \(P_2L\) meets \(C_2\) again at \(Q_2\). By inversion with respect to \(L\), or otherwise, prove that \(Q_1Q_2\) is a common tangent to \(C_1, C_2\).

A variable point \(P\) is taken on the parabola \(y^2 = a(x-a)\). The circle on the line joining \(P\) to the origin \(O\) as diameter meets the parabola \(y^2=4ax\) in three points besides \(O\). Prove that the normals at these three points are concurrent and their point of intersection lies on the parabola \[ y^2=4a(x+a). \]

Define carefully what you mean by an asymptote of a curve, and from your definition find the asymptotes of the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \] Find the asymptotes of the curve given in terms of the parameter \(t\) by the equations \[ x = \frac{h}{t^2-1}, \quad y = \frac{kt}{t^2-1}, \] where \(h\) and \(k\) are constants.

Three points \(A, B, C\) are given on a line \(l\). A fourth point \(D_1\) of the line is determined by the following construction. In any plane through \(l\) draw through \(A, B, C\) lines \(AQ_1R_1, BR_1P_1, CP_1Q_1\) intersecting in pairs in the points \(P_1, Q_1, R_1\). Let \(AP_1, BQ_1\) cut in \(U_1\). Then \(D_1\) is the point of intersection of \(R_1U_1\) with \(l\). Prove that, if the same construction is carried out with the lines \(AQ_2R_2, BR_2P_2, CP_2Q_2\), etc., leading to a point \(D_2\), then \(D_1\) and \(D_2\) are the same point.