A rigid pendulum, mass \(m\), is attached to a point \(A\), which is in turn connected to a fixed point \(O\) by a light elastic spring for which the restoring force is \(\lambda\) times the displacement. The point \(A\) is constrained to remain on a horizontal line containing \(O\). The distance to the centre of mass \(G\) of the pendulum is \(l\) and the moment of inertia about any axis perpendicular to \(AG\) and passing through \(A\), is \(m(l^2 + k^2)\). The displacement of the pendulum from the vertical, in the plane containing \(OA\), is \(\theta\) (positive when the spring is extended). Obtain equations describing the motion of the system and show that \[\ddot{x}\cos\theta + (l^2 + k^2)\ddot{\theta} + gl\sin\theta = 0.\] Show that if \(\lambda(l^2 + k^2) = glm\) then \((l^2 + k^2)\ddot{\theta} - x\) executes simple harmonic motion for small \(\theta\). What is the period of this motion?
A man, whose height can be ignored, stands on a hillside which may be taken as a flat surface making an acute angle \(\alpha\) with the horizontal. He can throw a ball, mass \(M\), with initial velocity \(V\). Derive the complete specification of the direction in which the man should throw the ball in order that its range should be a maximum. Air resistance can be ignored. For a general throw express the range in terms of the angle \(\beta\) between the direction of throw and the horizontal, and the angle \(\gamma\) between the vertical plane containing the direction of throw and that containing the line of greatest slope. Show that, for \(\gamma = \alpha\), the maximum ranges, up and down the slope, are \[\frac{V^2}{g}[(1+\sin\alpha)^{\frac{1}{2}} \pm \sin\alpha](1+\sin\alpha)^{\frac{1}{2}}\] respectively. What is the angle between the directions of throw corresponding to these two ranges?
In the theory of special relativity the kinetic energy of a particle of mass \(m\), moving with velocity \(v\), is given by \(E - mc^2\) where \[E = mc^2(1-v^2/c^2)^{-\frac{1}{2}}\] and \(c\) is the velocity of light. The magnitude \(p\) of the momentum is given by \[p = mv(1-v^2/c^2)^{-\frac{1}{2}}\] and, as in Newtonian mechanics, the momentum is in the direction of the velocity. Show that these definitions reduce to the usual ones when \(v\) is much less than \(c\). A particle with velocity \(v\) makes an elastic collision with an identical particle initially at rest. By using conservation of energy and momentum, as defined by the above equations, show that in the limit of \(v\) nearly equal to \(c\), the angle between the directions of the two particles after the collision is approximately zero. What is the corresponding angle when \(v\) is very small?
Let \(p_i\) (\(1 \leq i \leq n\)) and \(q_i\) (\(1 \leq i \leq n\)) be real numbers such that $$p_1 \geq p_2 \geq \ldots \geq p_n \geq 0$$ and $$q_1 + \ldots + q_i \geq i \quad (1 \leq i \leq n).$$ Show that $$p_1 q_1 + \ldots + p_n q_n \geq p_1 + \ldots + p_n.$$ Hence, or otherwise, show that if \(a_i\) (\(1 \leq i \leq n\)) and \(b_i\) (\(1 \leq i \leq n\)) are real numbers such that $$a_1 \geq a_2 \geq \ldots \geq a_n > 0$$ and $$b_1 b_2 \ldots b_i \geq a_1 a_2 \ldots a_i \quad (1 \leq i \leq n),$$ then $$b_1 + \ldots + b_n \geq a_1 + \ldots + a_n.$$
Show that $$\int_0^{\pi/2} \log(1 + p \tan^2 x) dx = \pi \log(1 + p^t),$$ where \(p\) is any positive real number.
If \(\zeta\), \(\bar{\zeta}\) are conjugate complex numbers, give a geometric description of those numbers \(z\) for which $$|z - \zeta| < |z - \bar{\zeta}|.$$ Let \(z_1, \ldots, z_n\) be \(n\) complex numbers, the imaginary parts of which are strictly positive, and put $$\prod_{j=1}^n (z - z_j) = z^n + (a_1 + ib_1) z^{n-1} + \ldots + (a_n + ib_n),$$ where the \(a_i, \ldots, a_n, b_1, \ldots, b_n\) are real. Show that the roots of $$x^n + a_1 x^{n-1} + \ldots + a_n = 0$$ are all real.
Prove that for each positive integer \(n\) there is a positive integer \(m\) such that the decimal representation of \(nm\) involves all ten digits.
\(ABC\) is an acute-angled scalene triangle, whose incentre is \(I\) and circumcentre is \(O\). Prove that \(IO\), when produced beyond \(O\), meets the longest side of \(ABC\) at an internal point.
\(O\), \(P\), \(P'\) are three distinct collinear points; \(Q\) is another point on the line \(OPP'\). Give a geometrical construction for the point \(Q'\) such that \((P, P'), (Q, Q')\) are pairs of corresponding points in a homography on the line whose self-corresponding points coincide at \(O\). If \(Q\) is at \(P'\), and the corresponding position of \(Q'\) is \(P''\), prove that \(O\) and \(P'\) harmonically separate \(P\) and \(P''\).
A conic \(S\) touches the sides \(BC\), \(CA\), \(AB\) of a triangle \(ABC\) in \(D\), \(E\), \(F\), and \(P\) is a general point of \(S\). A second conic \(S'\) passes through \(A\) and touches \(PB\), \(PC\) at \(B\) and \(C\). Prove that \(S'\) touches \(EF\) at its intersection with \(PD\).