Problems

Find in how many ways \(mn\) different books can be put in \(m\) boxes, \(n\) books in each box:

1. when a distinction is made between different boxes,
2. when no such distinction is made.

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 \[ \left(1+\frac{1}{2}\right)\left(1+\frac{1}{4}\right)\dots\left(1+\frac{1}{2n}\right) < \sqrt{2n+1}. \]

\(ABCD\) 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 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 \(A'B'C'\).

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 the identity \begin{align*} &\cos 2(\beta+\gamma-\alpha-\delta)\sin(\beta-\gamma)\sin(\alpha-\delta) \\ &+ \cos 2(\gamma+\alpha-\beta-\delta)\sin(\gamma-\alpha)\sin(\beta-\delta) \\ &+ \cos 2(\alpha+\beta-\gamma-\delta)\sin(\alpha-\beta)\sin(\gamma-\delta) \\ &= -8\sin(\beta-\gamma)\sin(\gamma-\alpha)\sin(\alpha-\beta)\sin(\alpha-\delta)\sin(\beta-\delta)\sin(\gamma-\delta). \end{align*}

Two small heavy rings connected by a light elastic string can slide without friction one on each of two fixed straight wires \(OA, OB\), which lie in a vertical plane through \(O\), the highest point, and are both inclined to the vertical at \(45^\circ\). Prove that there is only one configuration of equilibrium, and that if the weights of the rings are \(\frac{1}{2}\) and \(\frac{2}{3}\) of the modulus of elasticity of the string, the length of the string is twice its natural length. Investigate the stability of this configuration.

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.

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 the radius of curvature \(\rho\) of a curve \(f(x,y)=0\) is given by the formula \[ \frac{1}{\rho} = (f_{xx}f_y^2 - 2f_{xy}f_xf_y + f_{yy}f_x^2)/(f_x^2+f_y^2)^{3/2}, \] where suffixes denote partial differentiation. The equation \(f(x,y,a)=0\) represents a family of curves. Prove that, if \(\rho, \rho'\) denote the radii of curvature of a particular curve and of the envelope of the family at the point where they touch, then \[ f_{aa}\left(\frac{1}{\rho} - \frac{1}{\rho'}\right) + (f_y f_{ax} - f_x f_{ya})(f_x^2+f_y^2)^{-3/2} = 0. \]