Sum the series \[ \sum_{n=1}^N \frac{3n-1}{n(n+1)(n+3)}. \]
Prove that, if \(a>0\) and \(ac-b^2>0\), then \(ax^2+2bx+c > 0\) for all real values of \(x\). Examine whether it is possible to find real values of \(x, y\) and \(z\) which give a negative value to the expression \[ 7x^2+10y^2+z^2-6yz+zx-8xy. \]
Given that \(u_0=1\), \(u_1=\frac{3}{2}\), and that \[ 2u_n - 3u_{n-1} + u_{n-2} = 0 \quad (n\ge2), \] find \(u_n\) in terms of \(n\). Prove that, if \(-1 < x < 1\), \[ \sum_{n=0}^\infty u_n x^n = \frac{2}{(1-x)(2-x)}, \] \[ \sum_{n=0}^\infty u_n u_{n+1} x^n = \frac{12}{(1-x)(2-x)(4-x)}. \]
Prove that, in general, \[ \frac{\sin x}{\sin(x-a)\sin(x-b)} \] can be expressed in the form \[ \frac{A}{\sin(x-a)} + \frac{B}{\sin(x-b)}, \] when \(A\) and \(B\) are trigonometrical functions (to be found) independent of \(x\). Extend your result to \[ \frac{\sin^2 x}{\sin(x-a)\sin(x-b)\sin(x-c)}. \]
Prove that, if \(\cos\theta=c\), and the \(a\)'s are constants, \[ \cos n\theta = a_n c^n + a_{n-2}c^{n-2} + \dots, \] the last term being \(a_1 c\) or \(a_0\) according as \(n\) is odd or even. Prove that \(a_n = 2^{n-1}\). Prove that a polynomial of degree \(n-2\) can be found which differs from \(x^n\) for \(-1\le x \le 1\) by at most \(1/2^{n-1}\), and find such a polynomial when \(n=6\).
Points \(D, E, F\) are given on the respective sides \(BC, CA, AB\) of a triangle \(ABC\) such that \[ \frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = -1. \] Prove that \(D, E, F\) are collinear. If \[ \frac{BD}{DC} = \frac{m}{n'}, \quad \frac{CE}{EA} = \frac{n}{l'}, \quad \frac{AF}{FB} = \frac{l}{m'}, \] prove that \[ \frac{EF}{l(m-n)} = \frac{FD}{m(n-l)} = \frac{DE}{n(l-m)}. \]
Spheres are described to touch two given planes and to pass through a given point. Prove that, in general, they all pass through a second fixed point and that the locus of their points of contact with either plane is a circle.
Prove that chords of an ellipse which subtend a right angle at the centre touch a fixed circle.
A conic touches the sides \(BC, CA, AB\) of a triangle \(ABC\) at \(D, E, F\) respectively. Prove that \(AD, BE, CF\) meet at a point \(P\). Prove that, if the conic varies so that it always touches a fourth fixed line \(l\), then the locus of \(P\) is a conic \(S\) passing through \(A, B, C\). Prove that the tangent to \(S\) at \(A\) meets \(l\) on \(BC\).
Prove that the feet of the normals from the point \((h, k)\) to the rectangular hyperbola \(xy=c^2\) lie on the rectangular hyperbola \[ x^2 - y^2 - hx + ky = 0. \] Prove that, if \(hk=4c^2\), the same four points are the feet of the normals to the second hyperbola from the point \((-\frac{1}{2}h, -\frac{1}{2}k)\).