Find all the solutions of the equations \begin{align} Bx + 7y + 8z &= A \\ 4x - 2y - 3z + 9 &= 0 \\ x + y + z &= 11 \end{align} for all possible values of the constants \(A\) and \(B\).
Express in partial fractions $$\frac{(x+1)(x+2)\ldots(x+n+1)-(n+1)!}{x(x+1)(x+2)\ldots(x+n)}.$$ Hence, or otherwise, prove that $$\frac{C_1}{1} - \frac{C_2}{2} + \frac{C_3}{3} - \ldots + (-1)^{n-1} \frac{C_n}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n},$$ where \(C_r = n!/r!(n-r)!\).
If \(a\), \(b\) and \(c\) are the roots of the equation $$x^3 + px^2 + qx + r = 0,$$ form the equation with the roots \(a^3 - bc\), \(b^3 - ca\), and \(c^3 - ab\). Prove that one of the roots is the geometric mean of the other two if, and only if, \(rp^3 = q^3\). Find in a similar way a condition for one root to be the arithmetic mean of the other two. What can be said about \(a\), \(b\) and \(c\) if both these conditions hold?
By considering the solution of the recurrence relation $$U_{n+2} - 6U_{n+1} + 4U_n = 0 \quad \text{with} \quad U_0 = 2, U_1 = 6,$$ or otherwise, prove that the whole number next greater than \((3+\sqrt{5})^n\) is, for \(n \geq 1\), divisible by \(2^n\). Is it possible for this number to be divisible by \(2^{n+1}\) for all \(n \geq N\) (for some integer \(N\))?
If \(a_1, \ldots, a_n\) and \(b_1, \ldots, b_n\) are real numbers prove, by considering the minimum value as \(x\) varies of \(\sum_{r=1}^n (xa_r + b_r)^2\), or otherwise, that $$\left(\sum_{r=1}^n a_r^2\right)\left(\sum_{r=1}^n b_r^2\right) > \left(\sum_{r=1}^n a_r b_r\right)^2.$$ Hence prove by analytical geometry that, if \(ABC\) is a triangle, \(AB + BC > AC\). Two circles, \(C_1\) and \(C_2\), touch at \(T\). A variable circle \(C\) goes through \(T\) and cuts \(C_1\) and \(C_2\) again orthogonally in \(X\) and \(Y\). Prove that in general \(XY\) passes through a fixed point. Also discuss the exceptional case.
A variable chord \(AB\) of a conic subtends a right angle at a fixed point \(O\). Show that in general the foot of the perpendicular from \(O\) to \(PQ\) lies on a circle. What is the exceptional case?
\(ABCD\) is a tetrahedron. \(O\) is a point not lying on any of its faces. The line through \(O\) and \(A\) cuts \(BCD\) in \(P\) and \(P'\), respectively. Similarly the line through \(O\) meeting \(BC\) and \(AD\) cuts them in \(Q\) and \(Q'\) respectively, and that meeting \(AB\) and \(CD\) cuts them in \(R\) and \(R'\), respectively. Prove that \(AP\), \(BQ\), \(CR\) and \(DO\) are concurrent.
\(T\) is a point on a parabola of which \(S\) is the focus. A circle through \(S\) and \(T\) cuts the tangent to the parabola at \(T\) again in \(U\). Prove that the tangent to the circle at \(U\) is also a tangent to the parabola.
tangent to the parabola.
Given an ellipse, describe how to find its centre, axes and foci using ruler and compasses only, and justify your constructions.