If \[x_1x_2 + y_1y_2 = a_1, \quad x_2x_3 + y_2y_3 = a_2, \quad x_3x_1 + y_3y_1 = a_3, \quad x_1^2 + y_1^2 = x_2^2 + y_2^2 = x_3^2 + y_3^2 = b,\] prove that \[b^3 - (a_1^2 + a_2^2 + a_3^2)b + 2a_1a_2a_3 = 0.\] Deduce that, if the \(2n\) equations \[x_1x_2 + y_1y_2 = a_1, \quad x_2x_3 + y_2y_3 = a_2, \ldots, x_{n-1}x_n + y_{n-1}y_n = a_{n-1}, \quad x_nx_1 + y_ny_1 = a_n,\] \[x_1^2 + y_1^2 = x_2^2 + y_2^2 = \ldots = x_n^2 + y_n^2 = b,\] for the \(2n\) unknowns \(x_1, \ldots, x_n, y_1, \ldots, y_n\), are consistent, there must be an algebraic relation connecting \(b\) and \(a_1, a_2, \ldots, a_n\).
(i) Prove that \[\left(\sum_{i=1}^n a_ib_i\right)^2 \leq \sum_{i=1}^n a_i^2 \sum_{i=1}^n b_i^2,\] determining when equality arises. (ii) If \(g\) is the geometrical mean of \(n\) positive numbers \(a_1, \ldots, a_n\), prove that \[(1+a_1)(1+a_2)\ldots(1+a_n) > (1+g)^n,\] unless \(a_1 = a_2 = \ldots = a_n\).
\(O\) is a point inside a convex polygon \(ABC\ldots N\), of \(n\) sides; \(A_1, B_1, C_1, \ldots, N_1\) are the feet of the perpendiculars from \(O\) on to the sides \(AB, BC, CD, \ldots, NA\) respectively. The process is repeated for the same point \(O\) and the polygon \(A_1B_1C_1\ldots N_1\), to get a polygon \(A_2B_2C_2\ldots N_2\), and is continued, to get in succession polygons \(A_iB_iC_i\ldots N_i\), \(i = 3, 4, \ldots\). Prove that the polygon \(A_nB_nC_n\ldots N_n\) is similar to the original polygon \(ABC\ldots N\). Examine whether, in the case \(n = 3\), the result remains true if \(O\) is exterior to the triangle, mentioning any special cases that may arise.
Establish the existence of the nine-point circle of a triangle, and prove that its centre is the mid-point of the join of the circumcentre and the orthocentre. The feet of the perpendiculars from the vertices of a triangle \(ABC\) on to the opposite sides are \(P, Q, R\) respectively; prove that the corresponding sides of the triangles \(ABC, PQR\) meet in points lying on the radical axis of the circumcircle and nine-point circle of the triangle \(ABC\).
A tetrahedron \(ABCD\) is such that there is a sphere which touches its six edges. Prove also that the three sums of pairs of opposite edges are the same. Prove also that the three loci of points of contact of opposite edges are concurrent.
Determine \(\theta\) so that the line \[lx + my + n = \theta(l'x + m'y + n')\] is perpendicular to the line \[\lambda x + \mu y + \nu = 0.\] Prove that the perpendiculars from the vertices \(A_1, A_2, A_3\) of the triangle formed by the lines \[l_ix + m_iy + n_i = 0 \quad (i = 1, 2, 3)\] on to the sides \(B_2B_3, B_3B_1, B_1B_2\), respectively, of the triangle formed by the lines \[\lambda_ix + \mu_iy + \nu_i = 0 \quad (i = 1, 2, 3)\] are concurrent if \[(l_1\lambda_1 + m_1\mu_1)(l_2\lambda_2 + m_2\mu_2)(l_3\lambda_3 + m_3\mu_3) = (l_1\lambda_2 + m_1\mu_2)(l_2\lambda_3 + m_2\mu_3)(l_3\lambda_1 + m_3\mu_1).\] Deduce that in this case the perpendiculars from the vertices of \(B_1B_2B_3\) on to the respective sides of \(A_1A_2A_3\) are also concurrent.
Prove that four normals can be drawn from a point \(O\), whose rectangular cartesian coordinates are \(f, g\), to the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.\] Prove also that the equation of the parabola which touches the tangents to the ellipse at the feet of the four normals is \[(fx + gy)^2 - 2(a^2 - b^2)(fx - gy) + (a^2 - b^2)^2 = 0.\]
Tangents are drawn to the parabola \(y^2 = 4ax\) from a point \(P\), and the normals at the points of contact meet in \(Q\). If the join \(PQ\) passes through a fixed point \((z, \beta)\), prove that \(P\) lies on the curve \[y(x^2 + y^2) - (a + z)xy - \beta y^2 + a(2a - z)y + 2a\beta x - 2a^2\beta = 0.\]
(i) Given four distinct points \(A, B, C, D\) on a line \(l\), prove that there is a projectivity (a one-one algebraic correspondence) which sends the points \(A, B, C, D\) respectively into \(B, A, D, C\). (ii) A projectivity on \(l\) sends a point \(P\) into a different point \(Q\), and also sends \(Q\) into \(P\); prove that the projectivity is involutory, i.e. if any point \(R\) goes into \(S\), then \(S\) goes into \(R\). (iii) Any two points \(S, S'\) are taken on a line which meets \(l\) in \(M\); \(A, A'\) and \(B, B'\) are two pairs of points on \(l\): \(SA, S'A'\) meet in \(V\) and \(SB, S'B'\) in \(W\); \(VW\) meets \(l\) in \(N\). Prove that the projectivity which sends \(M\) into itself, \(A\) into \(A'\) and \(B\) into \(B'\) also sends \(N\) into itself; and that \(A, B'; B, A', M, N\) are pairs of an involution.
Find the coordinates of the point of intersection of the tangents to the conic whose equation in general homogeneous coordinates is \[ax^2 + by^2 + cz^2 = 0,\] at the points in which it is met by the line \[lx + my + nz = 0.\] Prove that, if the tangents to a conic at the points \(P, Q\) meet in \(R\), and the tangents at \(P', Q'\) meet in \(R'\), then the six points \(P, Q, R, P', Q', R'\) lie on a conic. If \(RR'\) meets \(PQ\) in \(L\), \(P'Q'\) in \(L'\), and the conic in \(M, M'\), prove that \(L, L'; M, M'; R, R'\) are pairs of an involution.