Prove that, if \(x, y, z\) are positive numbers such that \begin{align*} x+y+z &= 6, \\ x^2+y^2+z^2 &= 18, \end{align*} then none of \(x, y, z\) can exceed 4.
Prove that, if \(h(x)\) is the H.C.F. of two polynomials \(f(x), g(x)\), then polynomials \(A(x), B(x)\) exist such that \[ A(x)f(x) + B(x)g(x) = h(x). \] Obtain an identity of this form when \[ f(x) = x^{10} + 1, \quad g(x) = x^6 + 1. \]
If \(y = (kx+d)/(x+k)\), evaluate \(y-x\) and \(y^2-d\) in terms of \(d, x, k\). Suppose now that \(d\) is a positive integer which is not a perfect square, and that \(l, m\) are positive integers such that \(l/m\) is an approximation to \(\sqrt{d}\). Prove that, if \(k\) is the integer next greater than \(\sqrt{d}\), a better approximation to \(\sqrt{d}\) is given by \((kl+dm)/(l+km)\). Prove also that the two approximations are both greater or both less than \(\sqrt{d}\).
The sequence \(u_0, u_1, u_2, \dots\) is defined by \[ u_0 = 1, \quad u_1 = 2, \quad u_n = 2u_{n-1} - 5u_{n-2} \quad (n=2, 3, \dots). \] Obtain the general expression for \(u_n\). \[ u_n = \frac{1}{2} 5^{\frac{1}{2}(n+1)} \sin(n+1)\theta, \] where \(\theta\) is the acute angle defined by \(\tan\theta=2\).
In order to locate a thin plane stratum of rock beneath the surface of a horizontal plain, borings are made at three stations \(A, B, C\) on the plain, and the rock is reached at depths \(h_1, h_2, h_3\) respectively. The station \(B\) is due E. of \(A\), and \(C\) is due S. of \(B\); also \(AB=BC=a\). Taking the case \(h_1 < h_2 < h_3\), find the angle between the North and the direction of the line in which the stratum would meet the ground, and show that a line of greatest slope of the stratum makes with the horizontal an angle whose tangent is \[ \{(h_2 - h_1)^2 + (h_3-h_2)^2\}^{\frac{1}{2}}/a. \]
\(P, A, B, C\) are four points in space. Through the mid-points of \(BC, CA, AB\), lines are drawn parallel to \(PA, PB, PC\) respectively. Prove that these lines meet in a point.
The points \(O\) and \(P\) are inverse with respect to a circle \(\Sigma\), \(O\) being outside the circle. Prove that the circles through \(O\) which touch \(\Sigma\) cut the perpendicular bisector of \(OP\) at a constant angle.
If \(A, B, C\) are points on a rectangular hyperbola, prove that the circle through the mid-points of \(BC, CA, AB\) passes through the centre of the hyperbola.
The circle of curvature at a point \(P\) of the parabola \(y^2 = 4ax\) cuts the parabola again at \(Q\). Prove that the chord \(PQ\) touches the parabola \(y^2+12ax = 0\).
Prove that the locus of the point \[ \frac{x}{a_1t^2+2b_1t+c_1} = \frac{y}{a_2t^2+2b_2t+c_2} = \frac{1}{a_3t^2+2b_3t+c_3}, \] as the parameter \(t\) varies, is, in general, a conic.