What conditions must the positive integer \(n\) and the constants \(a\) and \(b\) satisfy in order that the \(n+1\) equations \begin{gather*} x_k - x_{k-1} + x_{k-2}=0, \quad (k=2,3,\dots,n) \\ x_0=a, \quad x_n=b \end{gather*} for the unknowns \(x_0, \dots, x_n\) shall have (i) a solution, (ii) one and only one solution?
Explain briefly how to find the H.C.F. of two integers or two polynomials. If \(m\) and \(n\) are positive integers whose H.C.F. is \(k\), prove that the H.C.F. of the integers \(2^m-1\) and \(2^n-1\) is \(2^k-1\) and that the H.C.F. of the polynomials \(x^{2^m}-x\) and \(x^{2^n}-x\) is \(x^{2^k}-x\).
A pack contains \(n\) cards numbered \(1, 2, \dots, n\). Two cards are drawn from the pack at random and a score is made equal to the product of the numbers on the two cards drawn. What will be the average score for all possible drawings (i) when the two cards are drawn simultaneously, (ii) when the first card is replaced and the pack shuffled before the second card is drawn?
Prove that \[ (1+x)^n - (1-x)^n = 2nx \prod_{k=1}^m \left(1+x^2\cot^2\frac{k\pi}{n}\right), \] where \(n\) is any positive integer and \(m\) is the greatest integer less than \(\frac{1}{2}n\).
If \(0< \theta < \alpha < \phi < 2\pi\) and \(\alpha+\beta=\theta+\phi<2\pi\), show that \[ \sin\alpha + \sin\beta > \sin\theta + \sin\phi. \] Prove that among the \(n\)-sided polygons inscribed in a given circle, the regular ones (those whose sides are all equal) enclose the greatest area.
If \(l,m,p\) and \(q\) are real numbers and \(lm<0\), show that the equations \[ xy=p, \quad (y-lx)(y-mx)=q \] have a real solution. By taking \(\beta x - \alpha y\) and \(\beta x + \alpha y\) as new variables, show that if \(\alpha, \beta, \gamma\) are the sides of a triangle, the equations \[ \frac{a^2+x^2}{\alpha^2} = \frac{b^2+y^2}{\beta^2} = \frac{c^2+(x-y)^2}{\gamma^2} \] always have a real solution. (\(a,b,c\) are real.) Deduce that, given any two triangles, the first may always be regarded as the orthogonal projection of a triangle similar to the second.
The point \(O\) is the centre of the circle \(PQR\) and the tangents at \(O\) to the circles \(OQR\) and \(OPR\) meet the circles \(OPR\) and \(OQR\) again in \(X\) and \(Y\), respectively. Show that, if \(PX\) and \(QY\) meet the circle \(PQR\) again in \(U\) and \(V\), then the parabola which touches \(OX\) and \(OY\) at \(X\) and \(Y\) has focus \(R\) and directrix \(UV\).
Find the coordinates of the centre, the equations of the axes and the lengths of the semi-axes of the ellipse \[ 41x^2 - 24xy + 34y^2 + 106x - 92y + 49 = 0. \]
Show that, as \(t\) varies, the point \[ x = \frac{a_1t^2+2h_1t+b_1}{a_3t^2+2h_3t+b_3}, \quad y = \frac{a_2t^2+2h_2t+b_2}{a_3t^2+2h_3t+b_3} \] in general describes a conic \(S\), and find the condition that the line \(lx+my+n=0\) shall touch \(S\). Prove that the director circle of \(S\) is \[ (a_1-xa_3)(b_1-xb_3) + (a_2-ya_3)(b_2-yb_3) = (h_1-xh_3)^2 + (h_2-yh_3)^2. \]
If \(ABCD\) is a tetrahedron, prove that the lines joining the vertices \(A,B,C,D\) to the centroids of the opposite faces are concurrent. Show further that, if \(AB\) is perpendicular to \(CD\) and \(AC\) is perpendicular to \(BD\), then \(AD\) will be perpendicular to \(BC\).