Problems

In the system of equations \begin{align*} ax+by+cz &= 0, \\ cx+ay+bz &= 0, \\ bx+cy+az &= 0, \end{align*} the numbers \(a,b,c\) are real and not all equal. Prove the following facts:

1. [(i)] If \(a+b+c \ne 0\), the system has no solution other than \(x=y=z=0\).
2. [(ii)] If \(a+b+c=0\), the system has other solutions, but every solution satisfies \(x=y=z\).

Express each of the polynomials \(x^m-1, x^n-1, x^{mn}-1\) as a product of linear factors involving the complex number \[ z = \cos \frac{2\pi}{mn} + i \sin \frac{2\pi}{mn}. \] Hence, or otherwise, prove that, if \(m, n\) are prime to one another (i.e. have no common factor greater than 1), then \[ (x-1)(x^{mn}-1) = (x^m-1)(x^n-1)P(x), \] \[ \frac{x^{mn}-1}{(x^m-1)(x^n-1)} = \frac{1}{x-1} + p(x), \] where \(P(x)\) and \(p(x)\) are polynomials.

Show that, if \(n\) is a positive integer or zero, then \[ (1+\sqrt{2})^n = u_n+v_n\sqrt{2}, \quad (1-\sqrt{2})^n=u_n-v_n\sqrt{2}, \] where \(u_n\) and \(v_n\) are integers. A sequence \(\{x_n\}\) satisfies the recurrence relation \[ x_{n+2} = 2x_{n+1}+x_n \quad (n=0,1,2,\dots). \] Express \(x_n\) in terms of \(u_n, v_n, x_0, x_1\). Prove that, if \(x_0=x_1=1\), then \(x_n\) is, for \(n \ge 1\), the integer nearest to \(\frac{1}{2}(1+\sqrt{2})^n\).

The numbers \(a_1, a_2, \dots, a_n\) are positive and not all equal, and their arithmetic and geometric means are \(A\) and \(G\), respectively. Prove that \(A>G\). Prove that, if \(x>0\), the geometric mean of the numbers \(x+a_r\) (\(r=1,2,\dots,n\)) is greater than \(x+G\).

A man has a balance with two pans \(P\) and \(Q\), and a supply of weights of \(1, 2, \dots, k\) pounds, the weights of any one kind being unlimited in number and indistinguishable from each other. The symbols \(a(n), b(n), c(n)\) denote, respectively, the number of ways in which an article \(A\) of \(n\) pounds can be weighed by the following methods:

1. [(a)] \(A\) is placed in \(P\), and weights in \(Q\), but not more than one weight of any one kind;
2. [(b)] as in (a), but without restriction on the number of weights of any one kind;
3. [(c)] \(A\) is placed in \(P\), and weights in one or both of \(P\) and \(Q\), but not more than one of any one kind altogether.

Show that \(a(n), b(n)\) are, respectively, the coefficients of \(x^n\) in the expansions of the "generating functions" \[ A(x) = \prod_{r=1}^k (1+x^r), \] \[ B(x) = \prod_{r=1}^k (1+x^r+x^{2r}+\dots), \] where the number of terms in each bracket in \(B(x)\) may be any number greater than \(n\). Write down a generating function \(C(x)\) for \(c(n)\). Prove that \begin{align*} a(n) &= b_1(n) \quad (1 \le n \le k), \\ c(n) &= b_2(n+K), \\ a(n)-a(n-1) &= a_3(n) \quad (1 \le n \le k), \end{align*} where \(K\) is the greatest number of pounds that can be weighed by the (a) method, \(a(0)\) is defined to be 1, and the suffixes 1, 2, 3 attached to a letter indicate, respectively, that the method corresponding to that letter is to be modified by (1) the exclusion of weights \(2, 4, 6, \dots\) pounds, (2) a restriction to not more than two weights of any one kind, (3) the exclusion of weights \(1, 2, 2^2, 2^3, \dots\) pounds.

Four points \(A,B,C,D\) lie on a circle. The orthocentres of the triangles \(BCD, ACD, ABD, ABC\) are \(P, Q, R, S\), respectively. Prove that the lines \(AP, BQ, CR, DS\) bisect each other.

Three circles \(S_1, S_2, S_3\) are in general position in a plane, and their centres are \(O_1, O_2, O_3\), respectively. The radical axis of \(S_i\) and \(S_j\) is \(p_{ij}\). Prove that \(p_{23}, p_{31}, p_{12}\) meet in a point \(R\). Prove that, if \(p_{23}\) passes through \(O_1\) and \(p_{31}\) passes through \(O_2\), then \(p_{12}\) passes through \(O_3\). Prove also that, in this case, the mid-points of \(O_2O_3, O_3O_1, O_1O_2, O_1R, O_2R, O_3R\) all lie on one circle.

The feet of the three normals from a general point \(P\) to a given parabola are \(L, M, N\). Show that the circumcircle of the triangle \(LMN\) passes through the vertex \(A\) of the parabola. Prove also that the conic through \(L, M, N, A, P\) is an ellipse having \(AP\) for a diameter.

A variable chord \(PQ\) of a given ellipse \(S\) subtends a right angle at the centre of the ellipse. Show that \(PQ\) touches a fixed circle \(\Sigma\). The figure is projected orthogonally so that \(S\) becomes a circle \(S'\). Prove that \(\Sigma\) becomes an ellipse \(\Sigma'\) having \(S'\) for its director circle.

By considering the points where the curve \[ x^3+y^3=3axy \] is met by the line \(y=px\), or otherwise, express the co-ordinates of a general point \(P\) of the curve as rational functions of a parameter \(p\). Obtain a necessary and sufficient condition, in terms of the parameters \(p_1, p_2, p_3\), for three points \(P_1, P_2, P_3\) on the curve to be collinear. A straight line meets the curve in three points \(P, Q, R\) (real or imaginary), and the tangents at \(P, Q, R\) meet the curve again in \(U, V, W\), respectively. Prove that \(U, V, W\) are collinear. Prove also that a given set of three collinear points \(U, V, W\) on the curve can be derived in this way from any one of four lines, which may be denoted by \(PQR, PQ'R', P'QR', P'Q'R\), where \(P, P', Q, Q', R, R'\) are suitable points on the curve.