Prove that there are four plane sections of a cube which are regular hexagons. Shew that a flexible elastic ring stretched tight round a smooth cube along one of these plane sections would be in equilibrium.
Prove that any triangle inscribed in a rectangular hyperbola has the orthocentre as another point on the curve. What theorems arise from the cases of coincidence of (i) two vertices of the triangle, (ii) all three vertices?
Prove that by a suitable choice of rectangular axes the equations of any two circles take the forms \[ x^2+y^2+2gx+c=0, \quad x^2+y^2+2g'x+c=0. \] Prove that the signs of \(g\) and \(g'\) will be different if, and only if, neither circle surrounds the diameter of the other which is perpendicular to the line joining the centres.
From any point \(P\) on the parabola \(y^2=ax\) perpendiculars \(PM, PN\) are drawn to the coordinate axes. Prove that the line through \(P\) perpendicular to \(MN\) is a normal of the parabola \(y^2=4a(x+3a)\).
Shew that four normals can be drawn from a given point to the conic \(ax^2+by^2=1\), and that the feet of these normals lie on a rectangular hyperbola whose asymptotes are parallel to the coordinate axes. Prove that the normals to the conic at its intersections with \(lx+my=1\) meet at the point \[ \left( \frac{l(a-b)(m^2-b)}{b(am^2+bl^2)}, \frac{m(b-a)(l^2-a)}{a(am^2+bl^2)} \right). \]
Find the condition that the lines \(ax^2+2hxy+by^2=0\) should be harmonic conjugates with respect to the lines \(a'x^2+2h'xy+b'y^2=0\). Prove that the locus of a point such that the tangents from it to two parabolas, which have a common vertex and axes at right angles, form a harmonic pencil, is a rectangular hyperbola, having the axes as asymptotes, and a transverse axis which is a mean proportional to the latera recta of the parabolas.
Shew that there is one hyperbola which has asymptotes parallel to the lines \(3x^2-8xy+3y^2=0\), and has a focus at \((0,1)\) corresponding to a directrix which passes through \((1,0)\). Draw a rough sketch of the hyperbola, and find the coordinates of the second focus.
Through any point \(P\) lines are drawn parallel to the internal bisectors of the angles of a triangle \(ABC\) to meet the opposite sides in \(D, E, F\). Prove that if \(D, E, F\) are collinear \(P\) lies on the conic \[ (b+c)\beta\gamma + (a+c)\alpha\gamma + (a+b)\alpha\beta = 0, \] where the coordinates are trilinear and \(ABC\) is the triangle of reference. Prove that the centre of the conic is the centre of the inscribed circle of the triangle whose vertices are the mid-points of the sides of \(ABC\).
If a set of numbers is added together, shew that the sum of the digits in them is equal to the sum of the digits in the answer or exceeds the latter by a multiple of nine.
If \(a, b, c, d\) are in ascending order of magnitude, the equation \[ (x-a)(x-c) = k(x-b)(x-d) \] has real roots for all values of \(k\).