Two triangles \(ABC, A'B'C'\) are such that lines through \(A,B,C\) parallel respectively to \(B'C', C'A', A'B'\) are concurrent. Shew that the same is true of lines through \(A', B', C'\) parallel to \(BC, CA, AB\).
Given four points and one line, shew that there is in general one and only one conic through the four points which has an axis parallel to the given line; and give a geometrical construction for any number of points upon it. Indicate the case in which there are an infinite number of conics satisfying the given conditions.
Shew that there are in general two triangles whose sides pass through three given points and whose vertices lie on a given conic.
Two circles intersect orthogonally in two fixed points. Shew that their common tangent envelopes an ellipse of eccentricity \(1/\sqrt{2}\).
Two of the normals from a point \(P\) to a given parabola make equal angles with a given straight line. Prove that the locus of \(P\) is a parabola.
Four suits of cards, each suit consisting of thirteen cards numbered from 1 to 13, are dealt to four persons. Find the chance that each person's cards contain all the numbers from 1 to 13.
Eliminate \(x, y, z\) from the equations \[ ax^2+by^2+cz^2 = ax+by+cz = yz+zx+xy=0 \] and reduce the result to a symmetrical form.
Prove that, if \(u_n = (\alpha+\beta)u_{n-1} - \alpha\beta u_{n-2}\) and \(u_2=\alpha\beta u_1\), then \[ \frac{u_n}{u_1} = \frac{\alpha\beta}{\beta-\alpha} \{\alpha^{n-2} - \alpha^{-1}\beta^{n-1} - \beta^{n-2} + \beta^{-1}\alpha^{n-1} \}. \]
If \(\theta=t^n e^{-(x^2+y^2)/4t}\), find what value of \(n\) will make \[ \frac{\partial^2\theta}{\partial x^2} + \frac{\partial^2\theta}{\partial y^2} = \frac{\partial\theta}{\partial t}. \]
By finding the fourth differential coefficient of \((\sin^2 x)/x^2\), or otherwise, shew that as \(x\) tends to zero the limit of \[ \frac{15}{x^5} - \frac{2x^4-18x^2+15}{x^6}\cos 2x + \frac{8x^2-24}{x^5}\sin 2x \] is \(\frac{4}{15}\).