Find in how many ways \(mn\) different books can be put in \(m\) boxes, \(n\) books in each box:
Prove that the circumcircles of the four triangles formed by four coplanar lines meet in a point \(O\), and by inverting with respect to this point, or otherwise, shew that the four circumcentres lie on a circle through \(O\).
Four light rods \(AB, BC, CD, DA\) are freely jointed together; \(AB=BC\) and \(CD=DA\). The rod \(AB\) is fixed horizontally and masses \(P, Q\) are suspended from \(C, D\) respectively. Shew that in equilibrium the angles \(DAB, ABC\) will be both acute or both obtuse, and that if \(\alpha, \beta\) are the angles which \(AD, CD\) make with the vertical \[ \frac{\sin 2\beta}{\sin 2\alpha} = 1 + \frac{Q}{P}. \]
A variable tangent \(t\) to a fixed conic meets two fixed tangents in \(A\) and \(B\), and meets any other fixed line \(l\) in \(P\). The harmonic conjugate of \(P\) with respect to \(A, B\) is \(M\). Shew that the locus of \(M\) is a conic which passes through the points in which the fixed tangents meet \(l\).
Denoting by \(x_1, x_2, x_3\) the roots of the equation \(x^3 + px + q = 0\), find the value of the sum \[ x_1 (x_2^3 + x_3^3) + x_2 (x_3^3 + x_1^3) + x_3 (x_1^3 + x_2^3). \]
Prove that, if the lines joining corresponding vertices of two triangles \(ABC, A'B'C'\) are concurrent, the points of intersection of corresponding sides are collinear. Two triangles \(ABC, A'B'C'\) are in perspective. \(BC', B'C\) meet in \(X\); \(CA', C'A\) in \(Y\); \(AB', A'B\) in \(Z\). Shew that the triangle \(XYZ\) is in perspective with \(ABC\) and with \(A'B'C'\).
\(ABCD\) is a uniform lamina, in shape a rhombus with sides of length \(a\) and the angle \(A=2\alpha\). \(P\) and \(Q\) are smooth pegs, \(PQ\) being of length \(l\) and horizontal. Find the angle which \(AC\) makes with the vertical if the lamina can rest with points on the sides \(AB, AD\) in contact with the pegs and with \(AC\) not vertical. Shew that such a position of equilibrium occurs only if \[ a \cos^3\alpha \sin\alpha < l < a\cos^2\alpha. \]
Prove that, if \(n\) be a positive integer, \[ n^n \ge 1 \cdot 3 \cdot 5 \dots (2n-1) \ge (2n-1)^{n/2}, \] and that \[ \left(1+\frac{1}{2}\right)\left(1+\frac{1}{4}\right)\dots\left(1+\frac{1}{2n}\right) < \sqrt{2n+1}. \]
Factorise the expression \[ (bcd + cda + dab + abc)^2 - abcd (a + b + c + d)^2; \] prove that the expression \[ (a - b)^2 (a - c)^2 + (b - c)^2 (b - a)^2 + (c - a)^2 (c - b)^2 \] is a perfect square.
Prove that, if two triangles are inscribed in one conic, then their six sides touch another conic. A triangle is circumscribed about a parabola. Shew that there is a rectangular hyperbola passing through its vertices and having as one asymptote the tangent at the vertex of the parabola.