In the system of equations \begin{align*} -ny+mz &= a, \\ nx - lz &= b, \\ -mx+ly &= c, \\ lx+my+nz &= p, \end{align*} \(l, m, n, a, b, c, p\) are given real numbers, and \(l,m,n\) are not all zero. Prove that a necessary and sufficient condition for the equations to have a solution is that \[ la+mb+nc=0; \] and solve the equations when this condition is satisfied.
If \(x_i\) (\(i=1, 2, 3, \dots n\)) are the \(n\) roots of the equation \(f(x)=0\), when \(f(x)\) is a polynomial of degree \(n\), show that \[ \frac{f'(x)}{f(x)} = \sum_{i=1}^n \frac{1}{x-x_i}. \] If \(S_k\) is the sum of the \(k\)th powers of the roots of the equation \(x^4-4x^3-2x^2+1=0\), prove that, for any integer \(k\) (positive, zero, or negative), \(S_k\) is an integer. Find \(S_3, S_4, S_{-4}\).
Denoting by \(c_\nu\) the coefficient of \(x^\nu y^{n-\nu}\) in the expansion of \((x+y)^n\), where \(n\) is a positive integer, evaluate the sum \[ T = \sum_{\nu=0}^n (2\nu-n)^2 c_\nu. \] Hence, or otherwise, prove that those terms of the sum \[ S = \sum_{\nu=0}^n c_\nu \] for which \[ |2\nu-n| \ge k, \] where \(k\) is a positive integer less than \(n\), contribute less than a fraction \(n/k^2\) of the whole sum. Show that, if \(n=1000\), more than nine-tenths of the sum \(S\) is contributed by fewer than one-tenth of its terms.
Prove by the use of complex numbers, or otherwise, that, if \(n\) is a positive integer, \(\cos n\theta\) can be expressed as a polynomial in \(\cos\theta\) of the form \[ \cos n\theta = p_0 \cos^n\theta - p_1 \cos^{n-2}\theta + p_2\cos^{n-4}\theta - \dots, \] where \(p_0, p_1, p_2, \dots\) are positive integers. (It is to be understood that the summation continues so long as the indices remain non-negative.) Show that \[ p_0+p_1+p_2+\dots = \tfrac{1}{2} \{ (1+\sqrt{2})^n + (1-\sqrt{2})^n \}. \]
Show that a plane is divided by \(n\) straight lines, of which no two are parallel and no three meet in a point, into \(\frac{1}{2}(n^2+n+2)\) regions. Consider the same problem with the plane replaced by the surface of a sphere and the lines by great circles, of which no three meet in a point. Into how many regions is space divided by \(n\) planes, of which no two are parallel, no three meet in a line, and no four meet in a point?
Five points \(A, B, C, D, P\), no three collinear, are given in a plane. Prove that the polars of \(P\) with respect to all conics through \(A, B, C, D\) pass through a point \(P'\). Give a geometrical construction for \(P'\) using straight lines only.
Two confocal central conics \(U\) and \(V\) are given, and a variable point \(P\) in their plane is such that two perpendicular tangents can be drawn from \(P\), one to \(U\) and one to \(V\). Prove that \(P\) lies on a fixed circle \(C\). Show that \(C\) passes through the points of intersection of \(U\) and \(V\).
Prove that the number of (real) circles of a given coaxal system that touch a given line in the plane of the circles is two, one, or none; distinguish the various cases. A line \(l\) is touched at \(P, P'\) by two circles of the given coaxal system, and at \(Q, Q'\) by two circles of the orthogonal system. Show that the point pairs \(P, P'\) and \(Q, Q'\) separate one another harmonically.
A rectangular hyperbola with centre \(O\) and a circle with centre \(C\) meet in four points \(P_1, P_2, P_3, P_4\). Prove that the centroid of these four points is at the mid-point of \(OC\). The line \(P_1O\) meets the hyperbola again at \(Q_1\), and the normal to the hyperbola at \(P_1\) meets the hyperbola again at \(N_1\); and points \(Q_2, Q_3, Q_4\) and \(N_2, N_3, N_4\) are similarly defined. Show that \(Q_1\) is the orthocentre of the triangle \(P_2P_3P_4\). Show also that the four points \(N_1, N_2, N_3, N_4\) lie on a circle. Prove that the four lines \(Q_1N_1, Q_2N_2, Q_3N_3, Q_4N_4\) touch a rectangular hyperbola having its asymptotes parallel to those of the given hyperbola.
Three circles \(A, B, C\) lie in three different planes \(\alpha, \beta, \gamma\). The circles \(B, C\) meet in two distinct points \(P, P'\), the circles \(C, A\) in two distinct points \(Q, Q'\), and the circles \(A, B\) in two distinct points \(R, R'\); but these points are not necessarily all distinct. Prove that, in general, the three circles must lie on a sphere. Specify precisely the implication of "in general", in terms of the number \(n\) of distinct points among \(P, P', Q, Q', R, R'\).