Problems

Prove that the Arithmetic Mean of a number of positive quantities is not less than their Geometric Mean. If \(a_1, a_2 \dots a_N\) be \(N\) given positive constants and \(n_1, n_2 \dots n_N\) be any \(N\) positive integers, shew that \[ a_1^{n_1}a_2^{n_2}\dots a_N^{n_N} \text{ is not greater than } n_1^{n_1}n_2^{n_2}\dots n_N^{n_N}\left(\frac{a_1+a_2+\dots+a_N}{n_1+n_2+\dots+n_N}\right)^{n_1+n_2+\dots+n_N}. \] (Note: This inequality appears to be transcribed correctly from the paper, but may contain a misprint in the original document.)

If \(a,b,c,d\) are four real quantities whose sum is zero, shew that \[ \frac{a^5+b^5+c^5+d^5}{5} = \frac{a^3+b^3+c^3+d^3}{3}\frac{a^2+b^2+c^2+d^2}{2}. \] If \(d\) is zero, shew further that \[ \frac{a^7+b^7+c^7}{7} = \frac{a^5+b^5+c^5}{5}\frac{a^2+b^2+c^2}{2} = \frac{a^4+b^4+c^4}{2}\frac{a^3+b^3+c^3}{3}. \]

If \(\omega\) is one of the imaginary \(n\)th roots of unity, shew that \[ \sum_{r=1}^{n-1}\frac{1-\omega^r}{y-\omega^r} = \frac{n(y^{n-1}-1)}{y^n-1}. \] By the use of the calculus, or otherwise, prove that if \(x>1\), then \[ (n+1)^2(x+3)(x-1) > 4n^2\{x^{n+1}+(n+1)x^n-n-2\} - 4n(n+1)\sum_{r=1}^{n-1}(1-\omega^r)\log\frac{1}{1-\omega^r x^{-1}} \] \[ > 4(n+1)^2(x-1), \] where \(n-1\) is a positive integer, and \(x^n\) is real.

State and prove Leibnitz's theorem on the \(n\)th differential coefficient of the product of two functions. If \(y=(x+a)\cot^{-1}\frac{x}{a}\) satisfies identically the equation \[ P\frac{dy}{dx}+Qy+R=0, \] where \(P, Q, R\) are polynomials in \(x\), find the simplest possible forms for \(P, Q,\) and \(R\), and shew that if \(\frac{p_n}{n!}\) is the coefficient of \(x^n\) in the Maclaurin expansion of \(y\) in a series of ascending powers of \(x\) then \[ a^3p_n+(n-3)a^2p_{n-1}+(n-1)(n-2)ap_{n-2}+(n-1)(n-2)(n-5)p_{n-3}=0, \] provided \(n>4\).

Write an account of the theory of rectilinear asymptotes of a plane curve whose equation is given either in rectangular cartesian form or in polar form. The lines whose equations are \(x=y, x+y=0, x=2y\) are the asymptotes of a cubic curve which touches the axis of \(x\) at the origin and which passes through the point \((0,b)\). What is the equation of the curve?

Sketch the curve whose polar equation is \(r^2(\sec n\theta+\tan n\theta)=a^2\), where \(n\) is a positive integer and \(a\) is a constant. In the case \(n=1\) shew that the only real point at which the circle of curvature passes through the pole is given by \(\theta=\tan^{-1}\sqrt{1+\sqrt{\frac{28}{3}}}\).

Explain how to find the intrinsic \((s, \psi)\) form of the equation of a plane curve whose pedal \((p,r)\) equation is known. Shew that the \((s, \psi)\) equation of the curve \(p^3=ar^2\) is \(s=3a\tan\frac{\psi}{2}+a\tan^3\frac{\psi}{2}\), where \(s\) and \(\psi\) are measured from the apse.

Prove that the centre of mass of a uniform lamina bounded by part of the parabola \(y^2=2lx\) and a focal chord of the parabola always lies on the parabola \[ y^2 = \frac{5l}{4}\left(x-\frac{3l}{10}\right) \] whatever the inclination of the focal chord to the axis of the parabola.

On two fixed straight lines, \(p, p'\), fixed points \(A, B, C, A', B', C'\) are taken. Variable points \(P, P'\) are taken on \(p, p'\), respectively, such that the cross ratios \((ABCP), (A'B'C'P')\) are equal. Prove that the line \(PP'\) envelops a conic, and discuss what happens if \(AA', BB', CC'\) meet in a point. A variable conic is drawn through four fixed points, \(A, B, C, D\). A fixed line through \(D\) cuts the conic again in \(P\). Prove that the tangent to the conic at \(P\) envelops a fixed conic inscribed in the triangle \(ABC\), and touching the fixed line at \(D\).

(i) Shew that with a suitable choice of the triangle of reference, the equations of any two coplanar conics may be taken in the form \[ ax^2+by^2+cz^2=0 \] \[ a'x^2+b'y^2+c'z^2=0. \] Deduce that if two conics have four real common tangents, their points of intersection are either all real or all imaginary. Hence or otherwise, shew that the four conics which can be drawn through two given points to touch the sides of a given triangle are either all real, or all imaginary. (ii) A conic touches the sides \(QR, RP, PQ\) of a triangle in the points \(A, B, C\). From any point \(O\) the lines \(OP, OQ, OR\) are drawn to meet the lines \(QR, RP, PQ\), in points \(X, Y, Z\). From \(X, Y, Z\) further tangents are drawn to the conic, meeting each other in points \(A', B', C'\). Prove that the lines \(AA', BB', CC'\) meet in the point \(O\).