If \(u_0=1, u_1=2\) and \[ u_{n+2} = 2(u_{n+1}-u_n) \quad (n=0, 1, 2, \dots), \] show that \(u_{4k}=(-4)^k\) and find \(u_{4k+1}, u_{4k+2}\) and \(u_{4k+3}\). Prove that \[ \sum_{n=1}^{4k} u_n^2 = \frac{2}{3}(16^k-1). \]
Solve completely the system of equations \begin{align*} (b+c)x+a(y+z) &= a, \\ (c+a)y+b(z+x) &= b, \\ (a+b)z+c(x+y) &= c, \end{align*} (i) when \(abc \neq 0\), and (ii) when \(a=0\), but \(b \neq 0\).
If \(a, b\) and \(c\) are the roots of the equation \(x^3=px+q\), express \(a^2+b^2+c^2\), \(a^3+b^3+c^3\) and \(a^4+b^4+c^4\) in terms of \(p\) and \(q\). If \[ \Delta = \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix}, \] show that \(\Delta = (a-b)(b-c)(c-a)\) and that \(\Delta^2 = 4p^3-27q^2\). If \(x, y\) and \(z\) are real numbers, prove that \[ 2(x-y)^2(y-z)^2(z-x)^2 \le [x^2+y^2+z^2-\frac{1}{3}(x+y+z)^2]^3 \] and that the equality sign holds if and only if the three numbers are in arithmetic progression.
Express \(\cos 2\theta\) and \(\sin 2\theta\) in terms of \(\tan \frac{1}{2}\theta\). Find all values of \(\theta\) in the range \(0 \le \theta \le 2\pi\) which satisfy the equation \[ \cos 2\theta + \sin 2\theta = 2 - \cos \theta + 3 \sin \theta. \]
How many \(n\)-digit numbers have no two consecutive digits the same? (The first digit is allowed to be 0.) Show that there are \(9[9^{n-2}-(-1)^n]\) of these numbers for which the first and last digits are the same.
Points \(X, Y, Z\) are taken on the sides \(BC, CA, AB\) of a triangle. Prove that \(AX, BY, CZ\) are concurrent if and only if \[ \frac{BX}{XC} \cdot \frac{CY}{YA} \cdot \frac{AZ}{ZB} = 1. \] The point \(X'\) is chosen on \(BC\) so that the angles \(BAX\) and \(X'AC\) are equal; and points \(Y', Z'\) on \(CA, AB\) are defined similarly. If \(AX, BY, CZ\) are concurrent, show that \(AX', BY', CZ'\) are concurrent.
Show that the locus of the mid-points of chords of constant length \(c\) of the parabola \(y^2+4ax=0\) is given by \[ (y^2+4a^2)(y^2+4ax)+a^2c^2=0. \] Sketch the locus for the case \(a=1, c=4\) and show that its radius of curvature at the point \((-2,0)\) is \(\frac{4}{3}\).
The sides of a triangle when produced divide its plane into seven regions. Prove that it is impossible to draw an ellipse having a piece of its arc in each of these regions. If the sides of the triangle are \(x-5y=2, 2x+y=4, -5x+3y=1\), find in each of the seven regions into which the plane is divided a point which lies on the hyperbola \(xy=1\).
Show that any three collinear points may be inverted to give three points \(P_1, P_2, P_3\) such that \(P_2\) is the mid-point of \(P_1P_3\). Prove that there is a unique circle \(C_i\) through \(P_i\) having the other two points \(P\) as inverse points (\(i=1,2,3\)). (Straight lines count as circles.) Show further that the three circles \(C_1, C_2\) and \(C_3\) have two points in common, and that any two of them are inverses in the third. State a more general result which could be deduced by inverting this figure.
If the four points \(A, B, C, D\) of a hyperbola \(S\) are concyclic, show that \(AB\) and \(CD\) are equally inclined to the axes of \(S\). The normal to the conic \(2x^2-y^2=2a^2-b^2\) at the point \(P\) of coordinates \((a,b)\) meets the conic again in \(Q\), and the circle on \(PQ\) as diameter meets the conic again in \(R\). Find the equation of \(PR\) and show that \(QR\) is the line \[ (8a^2-b^2)(2ax+by)+(8a^2+b^2)(2a^2-b^2)=0. \]