Problems

\(O\) is a fixed point in the plane of a given conic \(S\). Prove that chords of \(S\) subtending a right angle at \(O\) in general envelope a conic \(\Sigma\). If \(\Sigma\) is a parabola, what type of conic is \(S\)?

Find in radians, correct to two places of decimals, the solutions of:

1. \(3\sin\theta = 2\theta\);
2. \(\cos\theta = \theta\).

If \(I\) is the incentre of a triangle \(ABC\), prove that the circumcentre of the triangle \(BIC\) is collinear with \(A\) and \(I\). If \(R, R_1, R_2, R_3\) are the circumradii of the triangles \(ABC, BIC, CIA, AIB\), prove that \(\frac{R_1 R_2 R_3}{2R^2}\) is the radius of the inscribed circle of \(ABC\).

The inscribed circle of a triangle \(ABC\) touches the side \(BC\) at \(X\) and the inscribed circles of the triangles \(ABX, ACX\) touch \(BC\) at \(Y,Z\); prove that \(X\) bisects \(YZ\).

If \(P, Q\) are inverse points with respect to a circle \(\gamma\) and \(P', Q', \gamma'\) are the inverses of \(P, Q, \gamma\) with respect to any coplanar circle, prove that \(P', Q'\) are inverse points with respect to \(\gamma'\). Determine the locus of the inverse of a given point with respect to a variable circle of a given coaxal system, when the system has real limiting points.

A variable obtuse-angled triangle inscribed in a fixed circle with centre \(O\) has a fixed orthocentre \(H\); prove that the triangle is self-polar with respect to another fixed circle and that its sides touch a fixed conic with foci at \(O\) and \(H\).

Shew that, if there is a 1-1 correspondence between points \(P, P'\) on a straight line, there are in general two distinct points \(A, B\) on the line, such that the cross-ratio of \(A, B, P, P'\) is the same for all corresponding points \(P, P'\). State the nature of the correspondence when (i) \(A\) or \(B\) is at infinity, (ii) \(A\) and \(B\) are both at infinity.

If \(P\) is a variable point on a fixed circle and \(O\) is any point not in the plane of the circle, prove that the plane through \(P\) perpendicular to \(OP\) passes through a fixed point.

The sides of a triangle lie along the lines \(u \equiv x\cos\alpha+y\sin\alpha-p=0\), \(v \equiv x\cos\beta+y\sin\beta-q=0\), \(w \equiv x\cos\gamma+y\sin\gamma-r=0\); prove that (i) the orthocentre, (ii) the circumcentre, (iii) the centroid of the triangle are determined by the equations:

1. \(u \cos(\beta-\gamma) = v \cos(\gamma-\alpha) = w \cos(\alpha-\beta)\),
2. \(u \sec(\beta-\gamma) = v \sec(\gamma-\alpha) = w \sec(\alpha-\beta)\),
3. \(u \sin(\beta-\gamma) = v \sin(\gamma-\alpha) = w \sin(\alpha-\beta)\).

A variable line \(\lambda\) cuts the fixed conics \[ ax^2+by^2+k(a-b)=0, \quad a'x^2+b'y^2-k(a'-b')=0 \] in two pairs of points which are harmonic conjugates; prove that \(\lambda\) touches a fixed circle.