Problems

Find for what values of the constant \(a\) the equation \(x^3 - 3x + a = 0\) has three distinct real roots. Shew that if \(h>0\) the equation \(x^3 - 3x - 2 - 27h = 0\) has just one real root, and that, if this is denoted by \(2+3\xi\), then \(0 < \xi < h\); and with the aid of this result obtain the narrower limits \[ \frac{h}{(1+h)^2} < \xi < h. \]

Prove that \[ \sum_{r=0}^{n} {}^nC_r \left(r-\frac{1}{2}n\right)^2 = 2^{n-2}n, \] where \(n\) is a positive integer and \({}^nC_r\) is the coefficient of \(x^r\) in the expansion of \((1+x)^n\). Hence, or otherwise, prove that, if \(S(n)\) is the sum of all the coefficients \({}^nC_r\) and \(S_\delta(n)\) the sum of those coefficients \({}^nC_r\) for which \(r/n\) lies outside the interval \((\frac{1}{2}-\delta, \frac{1}{2}+\delta)\), where \(\delta\) is a number satisfying \(0<\delta<\frac{1}{2}\), then \[ \frac{S_\delta(n)}{S(n)} < \frac{1}{4\delta^2n}. \]

A circular disc of radius \(a\) rests in a vertical plane upon two rough pegs which are at a distance \(\sqrt{2}a\) apart in a horizontal line. If the centre of gravity of the disc is at a distance \(c\) from its centre, shew that the disc can rest in any position provided that \[ a \sin 2\lambda > \sqrt{2}c, \] where \(\lambda\) is the angle of friction at either peg. (Only the case \(\lambda < \frac{\pi}{4}\) need be considered.)

State and prove the harmonic property of the complete quadrilateral. Points \(P\) and \(Q\) and a line \(p\) are given. Construct by means of a ruler and compass the circle through \(Q\) with respect to which \(P\) and \(p\) are pole and polar.

A square \(ABCD\) is divided into twenty-five squares by two sets each of four equidistant lines. Shew that the number of paths of length \(2AB\) connecting \(A\) to \(C\) and made up of segments of the dividing lines or the edges of the square is 252.

Two polynomials \(P(x)\) and \(Q(x)\) satisfy the identity \[ 1 - \{P(x)\}^2 = \{Q(x)\}^2(1-x^2). \] Prove that \(P'(x)\) is divisible without remainder by \(Q(x)\) and that \[ \frac{P'(x)}{\sqrt{1-\{P(x)\}^2}} = \frac{n}{\sqrt{1-x^2}}, \] where \(n\) is a constant, and interpret its value.

A uniform circular ring of radius \(a\) and weight \(2\pi aw\) hangs in equilibrium under gravity over a fixed peg. If the ring is cut through at an end of its horizontal diameter, find expressions for the tension, shearing force and bending moment at any point of the ring.

Two curves \(C_1\), \(C_2\) and a point \(P\) common to them are inverted with respect to any circle whose centre is \(O\) into the curves \(C_1'\), \(C_2'\) and the point \(P'\). Prove that the angle of intersection at \(P\) is equal to the angle of intersection at \(P'\). Hence prove that if a circle \(C\) is inverted into the circle \(C'\) the centre of \(C\) is inverted into the point which is the inverse of \(O\) in \(C'\). \(L\) and \(M\) are inverse points with respect to a circle \(C\), \(L\) being outside the circle. Prove that the circles through \(L\) which touch \(C\) meet the perpendicular bisector of \(LM\) at a constant angle.

If \(O\) is a point inside a triangle \(ABC\), and \(A'\), \(B'\), \(C'\) are the feet of the perpendiculars from \(O\) to the sides \(BC\), \(CA\), \(AB\), respectively, prove that the radius of the circumcircle of the triangle \(A'B'C'\) is \[ \frac{xyz \sin A \sin B \sin C}{2 (y'z' \sin A + z'x' \sin B + x'y' \sin C)}, \] where \(x=OA\), \(y=OB\), \(z=OC\), and \(x'=OA'\), \(y'=OB'\), \(z'=OC'\).

\(C_1, C_2\) are two circles in a plane. A direct common tangent touches them at \(A_1, A_2\), and a circle \(C\) touches them externally at \(B_1, B_2\). Shew that \(A_1B_1, A_2B_2\) meet in a point \(X\) of \(C\) which lies on the radical axis of \(C_1, C_2\).