Discover all the real roots of each of the equations
(i) If \(k = 9^9\), use the information given in four-figure tables to prove that \(9^k\) is a number of more than 368,000,000 figures. (ii) Prove that, if \(m\), \(n\), \(p\) are positive integers such that $$(m^n)^p = m^{(n,p)},$$ then the only possibilities are that either \(m = 1\) or \(p = 1\) or \(n = p = 2\).
Obtain conditions on the positive integer \(n\) and the constants \(a\), \(b\) in order that the \(n+1\) equations for \(x_0\), \(\ldots\), \(x_n\) $$x_k - x_{k-1} + x_{k-2} = 0 \quad (k = 2, 3, \ldots, n), \quad x_0 = a, \quad x_n = b,$$ shall have (i) exactly one solution, (ii) no solution, (iii) more than one solution.
Prove that, if \(h(x)\) is the H.C.F. of two polynomials \(p(x)\), \(q(x)\), then polynomials \(A(x)\), \(B(x)\) exist such that $$A(x)p(x) + B(x)q(x) \equiv h(x).$$ Obtain an identity of this form when $$p(x) = x^{10} - 1, \quad q(x) = x^6 - 1.$$
Prove that, if \(n > 1\), $$\sum_{r=1}^n \left(1 + \frac{1}{r}\right) > n(n+1)^{1/n}.$$ If you appeal to any general inequality, prove it. Prove that $$n\{(n+1)^{1/n} - 1\} < \sum_{r=1}^n \frac{1}{r} < n\left\{1 + \frac{1}{n+1} - \frac{1}{(n+1)^{1/n}}\right\}.$$
On the sides of a triangle \(Z_1Z_2Z_3\) are constructed isosceles triangles \(Z_2Z_3W_1\), \(Z_3Z_1W_2\), \(Z_1Z_2W_3\) lying outside the triangle \(Z_1Z_2Z_3\). The angles at \(W_1\), \(W_2\), \(W_3\) are all \(\frac{2\pi}{13}\). By assuming complex numbers \(z_1\), \(z_2\), \(z_3\) to \(Z_1\), \(Z_2\), \(Z_3\) and calculating the numbers representing \(W_1\), \(W_2\), \(W_3\), or otherwise, prove that \(W_1W_2W_3\) is equilateral.
A regular polygon \(\Pi\) of \(n\) sides is given. A variable regular polygon of \(n\) sides is inscribed in \(\Pi\), having one vertex on each side of \(\Pi\). Prove that the sides of the variable polygon envelop parabolas. When \(n = 4\), identify the foci and latera recta of the parabolas.
A circle cuts the conic \(Ax^2 + By^2 = 1\) in four points \(P_1\), \(P_2\), \(P_3\), \(P_4\). Establish a result about the directions of the lines \(P_1P_2\), \(P_3P_4\). If the conic is an ellipse and the eccentric angle of \(P_k\) is \(\alpha_k\) (\(k = 1, 2, 3, 4\)), prove that \(\alpha_1 + \alpha_2 + \alpha_3 + \alpha_4\) is an integral multiple of \(2\pi\). Investigate the analogous result if the conic is a hyperbola and the coordinates of \(P_k\) are $$x = a\cosh u_k, \quad y = b\sinh u_k.$$
A point moves in space so that its distance from each of two intersecting straight lines is a given length \(l\). Prove that it lies on one of two ellipses which have a common minor axis.
Prove that the locus of the point $$\frac{x}{a_1t^2 + 2b_1t + c_1} = \frac{y}{a_2t^2 + 2b_2t + c_2} = \frac{1}{a_3t^2 + 2b_3t + c_3},$$ where the coefficients \(a_1\), \(\ldots\), \(c_3\) are real and \(t\) is a parameter, is, in general, a conic.