Shew that, if \(x^4 + ax + b\) has a factor \(x^2 + px + q\), then \[ p^6 - 4bp^2 - a^2 = 0 \quad \text{and} \quad q^6 - bq^4 - a^2q^3 - b^2q^2 + b^3 = 0. \] Solve the equation \[ x(x-1)(x-2)(x-3) = a(a-1)(a-2)(a-3), \] and find for what values of \(a\) the roots are all real.
Prove that the circumcentre \(O\), the centroid \(G\), and the orthocentre \(H\), of a triangle \(ABC\) are collinear, and that \(OH=3OG\). A triangle \(ABC\) is inscribed in a fixed circle with centre \(O\), and varies so that \(A\) is fixed and \(BC\) passes through a fixed point \(P\). Prove that the locus of the orthocentre of \(ABC\) is a circle whose radius is equal to \(OP\).
Six equal uniform rods, each of weight \(W\), are smoothly jointed together so as to form a regular hexagon \(ABCDEF\) which hangs from the point \(A\) and is kept in shape by strings \(AC, AD, AE\). Find the tensions in these strings.
Give a geometrical construction for the circle passing through a given point and coaxal with two given circles which do not meet in real points. \(U\) and \(V\) are two fixed circles, \(P\) is a given point; \(Q\) is the inverse of \(P\) with respect to \(U\), \(R\) is the inverse of \(Q\) with respect to \(V\), \(S\) the inverse of \(R\) with respect to \(U\), and so on. Prove that all these points are concyclic.
Shew that, if \(c_r\) is the coefficient of \(x^r\) in the expansion of \((1+x)^n\), where \(n\) is a positive integer, \[ c_0^2 + c_1^2 + c_2^2 + \dots + c_n^2 = \frac{(2n)!}{(n!)^2}. \] Find the value of \(c_0^2 - c_1^2 + c_2^2 - \dots + c_n^2\), where \(n\) is even.
Prove that if a hexagon is inscribed in a conic the points of intersection of pairs of opposite sides are collinear. A conic passes through a given point \(A\) and touches two given lines \(b, c\) at given points \(B, C\). Shew how to construct the tangent at \(A\), and the point where any given line through \(A\) meets the conic again.
A heavy sphere rests on a rough plane inclined at an angle \(\theta\) to the horizontal. The sphere is to be kept from rolling down the plane by a rough cubical block, of negligible weight and edge less than the radius of the sphere, resting on the plane, the edge of the block in contact with the sphere being horizontal. The angles of friction between the block and the plane and between the block and the sphere are \(\lambda_1\) and \(\lambda_2\) respectively, both angles being less than \(45^\circ\). If the points of contact of the sphere with the block and with the plane are \(P_1\) and \(P_2\), shew that the size of the block must be limited by the relation \(\lambda_1 + \lambda_2 \gtreqless \phi \gtreqless \theta\), where \(\phi\) is the angle subtended by \(P_1P_2\) at the centre of the sphere. Explain the significance of the two limits for \(\phi\).
A bag contains \(n\) balls, three red and the rest white. They are drawn out one by one. Find the probability that no two red balls will be drawn consecutively, and shew that this probability is less than \(\frac{1}{2}\) unless \(n\) is greater than 10.
Prove that, if \(n\) is a positive integer, the number of solutions of the equation \(x + 2y + 3z = 6n\), for which \(x, y, z\) are positive integers or zero, is \(3n^2 + 3n + 1\). Find the corresponding number of solutions of the equation \(x + 2y + 3z = 6n + 1\).
Prove that the reciprocal of a circle with respect to a point \(S\) is a conic with one focus at \(S\). Determine what relations between \(S\) and the circle decide whether the conic is an ellipse, parabola, or hyperbola. Given four points \(S, A, B, C\), prove that in general four conics may be drawn through \(A, B, C\) having \(S\) as focus; and that three of them are hyperbolas while the remaining one may be a hyperbola, parabola, or ellipse.