A bead of mass \(m\) slides on a smooth horizontal rail; a particle, also of mass \(m\), is attached to the bead by a light inelastic string of length \(2a\). The system is released from rest with the string taut, in the vertical plane through the rail, and making an angle \(\alpha\) with the downward vertical. Prove that, if the inclination of the string to the downward vertical at time \(t\) is \(\theta\), then \begin{equation*} \frac{1}{2}\dot{\theta}^2 = \frac{g}{a}\left(\frac{\cos\theta - \cos\alpha}{2-\cos^2\theta}\right). \end{equation*} Hence or otherwise find an expression for the tension in the string at any time in the subsequent motion.
Let \(G\) be a group with identity element \(e\). Prove that the number of solutions of the equation \(x^2 = e\) in \(G\) is either 1, \(\infty\) or even. [Suppose \(a \neq e\) is one solution and consider the solutions satisfying \(ax = xa\).]
Let \(n, p, q\) be integers with \(p, q\) prime, such that \(q\) divides \(n^p - 1\) but not \(n - 1\). Let the relation \(\sim\) on the set \(\{1, 2, \ldots, q - 1\}\) be defined by writing \(x \sim y\) if \(q\) divides \(y - n^x\) for some \(r\). Prove that
Let \(a, b, c\) be integers and let \(f(x, y) = ax^2 + 2bxy + cy^2\). Show that there are integers \(p, q, r, s\) such that \(ps - qr = 1\) and \(f(x, y) = 2(px + qy)(rx + sy)\) if and only if \(a\) and \(c\) are even and \(b^2 - ac = 1\).
\(\Sigma\) is a conic, and \(ABC, A'B'C'\) are triangles such that the lines \(B'C', C'A', A'B'\) are the polars with respect to \(\Sigma\) of \(A, B, C\) respectively. Show that \(AA', BB', CC'\) are concurrent.
If \(A, B\) are points in the plane, the part of the line \(AB\) between \(A\) and \(B\) is the segment \(AB\). Points \(P_1, P_2, \ldots, P_6\) in the plane are such that no three are collinear and no three segments \(P_iP_j, P_kP_l, P_mP_n\) are concurrent. A crossing is a point common to two distinct segments \(P_iP_j, P_kP_l\). Prove that \(P_1, P_2, \ldots, P_6\) always have three crossings, and find six points with exactly three crossings.
Prove that, for any four points \(A, B, C, D\) in a plane, \[\begin{vmatrix} 0 & 2AB^2 & AB^2+AC^2-BC^2 & AB^2+AD^2-BD^2 \\ 2AC^2 & 0 & 2BC^2 & AC^2+AD^2-CD^2 \\ AC^2+AB^2-CB^2 & 2AB^2 & 0 & 2CD^2 \\ AD^2+AB^2-DB^2 & AD^2+AC^2-DC^2 & 2CD^2 & 0 \end{vmatrix} = 0\]
Let \(l_1, l_2, l_3, l_4\) be lines in the plane and let \(C_i\) be the circumcircle of the triangle obtained by omitting \(l_i\). Prove that
Let \[f(x) = \sum_{n=1}^{\infty} \frac{x}{n(n+x)}\] for real positive \(x\). Prove that \[2f(2x) - f(x) - f(x+\frac{1}{2}) = 2\log 2 - \frac{1}{(x+\frac{1}{2})}.\]
Find the most general solution of the 'differential equation' \[f'(x) = \lambda f(1-x),\] where \(\lambda\) is a real constant.