The tangents at \(B, C\) to the circumcircle of a triangle \(ABC\) meet in \(L\); \(AL\) cuts the circle in \(P\); and \(Q\) is the mid-point of \(AP\). Prove that \(AQ\) bisects the angle \(BQC\), and that the triangles \(QAB, QCA\) are similar.
Prove that, if \(SY, HZ\) are the perpendiculars from the foci \(S, H\) on the tangent to an ellipse at any point \(P\), then \(Y\) and \(Z\) lie on the circle whose diameter is the major axis \(AA'\). Prove also that, if the line drawn from the centre \(C\) perpendicular to the tangent at \(P\) cuts \(SP, HP\) in \(Q\) and \(R\), then (1) \(SQ=HR=CA\); and (2) \(YQZR\) is a rhombus having its sides equal to \(SC\).
Solve the equations \[ x^2+y^2-3x+1=0, \quad 3y^2-xy+2x-2y-3=0. \]
Prove that \[ \frac{3^3}{1.2} + \frac{5^3}{1.2.3} + \frac{7^3}{1.2.3.4} + \dots = 21e. \]
Prove that, if \(A+B+C+D=\pi\), \[ \cos 2A+\cos 2B - \cos 2C - \cos 2D = 4(\cos A\cos B\sin C\sin D - \sin A\sin B\cos C\cos D). \]
\(O\) is the circumcentre of a triangle \(ABC\) and \(AO, BO, CO\) cut the sides \(BC, CA, AB\) in \(X, Y, Z\). Prove that the ratio of the area of the triangle \(XYZ\) to the area of the triangle \(ABC\) \[ = 2OX.OY.OZ : R^3. \]
Through a point \(P(\alpha,\beta)\) a pair of lines are drawn parallel to the lines \[ ax^2+2hxy+by^2=0 \] which cut the axis \(Ox\) in \(X, X'\) and the axis \(Oy\) in \(Y, Y'\). Find the equation of the line joining the mid-point of \(XX'\) to the mid-point of \(YY'\); and prove that, if this line is perpendicular to the line joining \(P\) to the origin, the locus of \(P\) is a pair of perpendicular lines passing through the origin.
Prove that the tangents to a parabola at any three points \(P, Q, R\) form a triangle whose area is half the area of the triangle \(PQR\).
Prove that, if a rhombus is inscribed in the conic \(ax^2+by^2=1\), its sides must touch the circle \((a+b)(x^2+y^2)=1\).
\(P\) and \(Q\) are two points on the curve \(ay^2=x^3\) such that \(PQ\) subtends a right angle at the cusp. Prove that the locus of the point of intersection of the tangents at \(P\) and \(Q\) is the parabola \[ 4y^2 = a(3x-a). \]