Problems

Two coplanar triangles \(ABC\) and \(A'B'C'\) are in perspective from a point \(O\). Prove that, of the nine points where a side of \(ABC\) meets a side of \(A'B'C'\), three are collinear and the remaining six lie on a conic.

Four equal uniform straight rods \(AB, BC, CD, DE\), each of length \(2a\) and weight \(W\), are smoothly jointed at \(B, C\) and \(D\). The ends \(A\) and \(E\) are freely hinged to fixed points \(2a \operatorname{cosec}(\pi/8)\) apart at the same level. Show that the rods can be made to hang in the form of the lower half of a regular octagon by applying two equal and opposite couples, either to \(AB\) and \(DE\) or to \(BC\) and \(CD\). Prove that the ratio of the couples needed in the two cases is \(-\cot(\pi/8)\).

Two pencils, with vertices \(A\) and \(B\), are homographically related in such a way that the ray \(AB\) of the first corresponds to the ray \(BA\) of the second. Prove that the locus of the points common to two corresponding rays consists of the line \(AB\) together with a second line \(l\). Two homographic ranges lie on the same line. Show that there are two self-corresponding points, which may be either real and distinct, or coincident, or imaginary, and one or both of which may be at infinity. If the self-corresponding points are real, show that the two ranges are in perspective (from different centres) with one and the same range of points on a second suitably chosen line \(l\).

Prove that, if \((1 + x)^n = c_0 + c_1x + \dots + c_nx^n\), then \[c_0 c_2 + c_1 c_3 + \dots + c_{n-2}c_n = \frac{(2n)!}{(n-2)!(n+2)!},\] and \[\frac{c_0}{2} + \frac{c_1}{3} + \dots + \frac{c_n}{n+2} = \frac{2^{n+1}n+1}{(n+1)(n+2)}.\]

Give an account of the application of determinants to the solution of linear algebraic equations. Solve completely the equations \begin{align*} x+y+z &= a+b+c, \\ a^2x+b^2y+c^2z &= a^3+b^3+c^3, \\ a^3x+b^3y+c^3z &= a^4+b^4+c^4, \end{align*} distinguishing between the cases (i) \(bc+ca+ab \neq 0\), (ii) \(bc+ca+ab=0\), it being assumed that no two of the numbers \(a, b, c\) are equal.

Two particles \(A, B\), of the same weight, are joined by a light inextensible string, and placed on a rough horizontal table with the string taut. The coefficient of friction between each particle and the table is \(\mu\). A horizontal force is applied to \(B\) in a direction making an acute angle \(\theta\) with \(AB\) produced, and the magnitude of the force is gradually increased until equilibrium is disturbed. If the initial displacement of \(B\) is in a direction making an angle \(\phi\) with \(AB\) produced, prove that \(\phi=90^\circ\) when \(\theta\ge 45^\circ\), and that \(\phi=2\theta\) when \(\theta < 45^\circ\).

\(X\) and \(Y\) are any points of the line \(AB\), and \(X'\), \(Y'\) are their harmonic conjugates with respect to \(A, B\). Prove that \((ABXY)=(ABX'Y')\). \(P\) and \(Q\) are two points coplanar with a conic \(S\). The tangents from \(P\) to \(S\) have points of contact \(A\) and \(B\). The tangents from \(Q\) to \(S\) have points of contact \(C\) and \(D\). Show that the six points \(P, Q, A, B, C, D\) lie on a conic.

Prove that, if \[ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy\] is the product of two factors linear in \(x, y\) and \(z\), then \[ \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0. \] Prove that, if \(A, B\) and \(C\) are the angles of a triangle, \[ \begin{vmatrix} \sin 2A & \sin C & \sin B \\ \sin C & \sin 2B & \sin A \\ \sin B & \sin A & \sin 2C \end{vmatrix} = 0. \]

Prove that, if \(f(u,v)\) is a homogeneous polynomial in \(u\) and \(v\) of degree \((n-1)\), \[ \frac{f(\sin x, \cos x)}{\sin(x-\alpha_1)\dots\sin(x-\alpha_n)} = \sum_{r=1}^n \frac{f(\sin\alpha_r, \cos\alpha_r)}{\sin(\alpha_r-\alpha_1)\dots\sin(\alpha_r-\alpha_n)} \frac{1}{\sin(x-\alpha_r)}, \] (where the factor \(\sin(\alpha_r - \alpha_r)\) is omitted from the denominator of the \(r\)th term). Prove that, if \(n\) is odd, \[ \frac{1}{\sin(x-\alpha_1)\dots\sin(x-\alpha_n)} = \sum_{r=1}^n \frac{A_r}{\sin(x-\alpha_r)}, \] and find the coefficients \(A_r\). Show what modification is required if \(n\) is even.

A smooth wire bent into the form of a circle of radius \(a\) is fixed in a vertical plane. One end of a light elastic string of modulus \(\lambda\) and natural length \(a\) is attached to the highest point of the wire, and the other end to a bead of weight \(W\) that can slide along the wire. Show that, when the bead rests at the lowest point of the wire, the equilibrium is stable if \(\lambda < 2W\), and unstable if \(\lambda > 2W\). Investigate the stability if \(\lambda = 2W\).