Problems

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

1. [(i)] \(\sim\) is an equivalence relation,
2. [(ii)] each equivalence class has \(p\) elements,
3. [(iii)] \(p\) divides \(q - 1\).

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

1. [(i)] \(C_1, C_2, C_3, C_4\) have a point \(O\) in common.
2. [(ii)] The feet of the perpendiculars from \(O\) to \(l_1, l_2, l_3, l_4\) lie on a line \(l_0\).
3. \(l_0, l_1, l_2, l_3, l_4\) touch a parabola with focus \(O\) and vertex on \(l_0\).

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.

A fair coin is tossed successively until either two heads occur in a row or three tails occur in a row. What is the probability that the sequence ends with two heads?