The roots of the equation \[ x^3+3qx+r=0 \] are \(\alpha, \beta, \gamma\). Express \(P^2\) as a polynomial in \(q\) and \(r\), where \(P=(\beta-\gamma)(\gamma-\alpha)(\alpha-\beta)\). Explain why \(P\) cannot be expressed in this form. From your expression for \(P^2\), or otherwise, obtain necessary and sufficient conditions for the given equation to have
Solution: Note that \(\alpha+\beta+\gamma = 0\) and let \(p_k = \alpha^k+\beta^k+\gamma^k\). \begin{align*} && P^2 &= \det \begin{pmatrix} 1 & \alpha & \alpha^2 \\ 1 & \beta & \beta^2 \\ 1 & \gamma & \gamma^2 \end{pmatrix}^2 \\ &&&= \det \begin{pmatrix} 1 & \alpha & \alpha^2 \\ 1 & \beta & \beta^2 \\ 1 & \gamma & \gamma^2 \end{pmatrix} \det \begin{pmatrix} 1 & \alpha & \alpha^2 \\ 1 & \beta & \beta^2 \\ 1 & \gamma & \gamma^2 \end{pmatrix}^T\\ &&&= \det \begin{pmatrix} 1 & \alpha & \alpha^2 \\ 1 & \beta & \beta^2 \\ 1 & \gamma & \gamma^2 \end{pmatrix} \det \begin{pmatrix} 1 &1 &1 \\ \alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2 \end{pmatrix}\\ &&&= \det \begin{pmatrix} p_0 & p_1 &p_2 \\ p_1 & p_2 & p_3\\ p_2 & p_3 & p_4 \end{pmatrix}\\ \end{align*} Node that \(p_0 = 3, p_1 = 0, p_2 = p_1^2 - 6q = -6q\) and \(p_3 = -3qp_1 - 3r = -3r\) and \(p_4 = -3qp_2-rp_1 = 18q^2\) So \begin{align*} P^2 &= \det \begin{pmatrix}3 & 0 & -6q \\ 0 & -6q & -3r \\ -6q & -3r & 18q^2 \end{pmatrix} \\ &= 3 \left \lvert \begin{matrix} -6q & -3r \\ -3r & 18q^2 \end{matrix} \right \rvert - 0 + (-6q) \left \lvert \begin{matrix} 0& -6q \\ -6q & -3r \end{matrix} \right \rvert \\ &= 3(-6 \cdot 18 q^3-9r^2) +6^3 q^3 \\ &= 27(q^3(-12+8) -r^2) \\ &= 27(-4q^3-r^2)\\ &= -27(4q^3+r^2) \end{align*} \(P\) cannot be written as a combination of elementary symmetric polynomials since it isn't symmetric under the transposition \(\alpha \leftrightarrow \beta\).
By putting the expression \[ \frac{(x+1)(x+2)\dots(x+n)}{x(x-1)(x-2)\dots(x-n)} \] into partial fractions, or otherwise, prove that the system of \(n\) equations \[ \sum_{r=0}^n \frac{X_r}{r+s} = 0 \quad (s=1, 2, \dots, n) \] in the \(n+1\) unknowns \(X_0, X_1, \dots, X_n\) is satisfied by \[ X_r = \frac{(-1)^{n-r}(n+r)!}{(r!)^2(n-r)!} \quad (r=0, 1, \dots, n). \] Show also that, with these values, \[ \sum_{r=0}^n X_r = 1. \]
Prove the identity \[ \sum_{s=0}^{N-1} \frac{1}{z-e^{is\theta}} = \frac{N}{z^N-1} - \frac{1}{2}\frac{z^{N-1}+1}{z^N-1}\cot\frac{\theta}{2} + \frac{i}{2}\frac{z^{N-1}+1}{z^N-1}. \] Hence, or otherwise, evaluate the sums \[ \sum_{s=0}^{N-1} \tan(s\theta), \quad \sum_{s=0}^{N-1} \tan^2(s\theta). \]
If \[ D_N = \begin{vmatrix} 1 & a & a^N \\ 1 & b & b^N \\ 1 & c & c^N \end{vmatrix} \quad (N=0, 1, 2, \dots), \] prove that \[ D_N = D_2 P_{N-2}, \] where \(P_n\) is the coefficient of \(t^n\) in \[ (1+at+\dots+a^Nt^N)(1+bt+\dots+b^Nt^N)(1+ct+\dots+c^Nt^N) \] for any \(N \ge n\). Write out \(P_n\) explicitly, as a polynomial in \(a, b, c\), in the cases \(n=0, 1, 2, 3\).
A certain odd integer \(n\) is expressed as a sum of two squares in two different ways, \[ n = x^2+y^2=X^2+Y^2, \] where \(x, X\) are even positive integers, \(y, Y\) are odd positive integers, and \(x < X\). Prove that positive integers \(a, r, s, b\) can be found so that \[ X-x=2ar, \quad y-Y=2as, \] \(r,s\) are co-prime (i.e. have no common factor greater than 1), and \[ (x+ar)r = (y-as)s = brs. \] (It may be assumed that, if an integer is divisible by each of two co-prime integers, it is divisible by their product.) Express \(x,y,n\) in terms of \(a, r, s, b\), and deduce that \(n\) is not a prime number.
A general point \(O\) is taken in the plane of a triangle \(ABC\); the lines \(AO, BO, CO\) meet \(BC, CA, AB\) respectively in \(L, M, N\); and \(MN, NL, LM\) meet \(BC, CA, AB\) respectively in \(P, Q, R\). Prove that \(P, Q, R\) are collinear. Prove also that, if \(BM, CN\) meet \(NL, LM\) respectively in \(Y, Z\), then \(P, Y, Z\) are collinear.
A parallelogram \(PQRS\) circumscribes an ellipse. Prove that, if \(P\) lies on a directrix, then \(Q\) and \(S\) lie on the auxiliary circle.
A variable chord \(PQ\) of a given central conic \(S\) passes through a fixed point \(O\). Prove that the mid-point of \(PQ\) lies on a fixed conic \(S'\). Show that, if \(O\) now varies along a fixed line, \(S'\) passes through two fixed points.
Prove that four normals can be drawn to a rectangular hyperbola from a general point \(N\) in its plane. If the feet of these normals are four distinct real points \(A, B, C, D\), establish the following facts:
The inside of a box, with lid closed, has the form of a cube of edge \(2a\). A circular ring of radius \(b\), made of wire of negligible thickness, is to be placed in the box and the lid closed. How would you suggest placing the ring so as to allow \(b/a\) to be as large as possible, and what is the largest value of \(b/a\) with the suggested arrangement?