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 \(PD\).
\(ABC\) is a triangle and \(O\) a general point in the plane \(ABC\); \(AO\), \(BO\), \(CO\) meet \(BC\), \(CA\), \(AB\) respectively in \(D\), \(E\), \(F\). A line \(l\) meets \(BC\), \(CA\), \(AB\) in \(L\), \(M\), \(N\). \(P\) is the mate of \(D\) in the involution in which \(B\), \(C\) are corresponding points and \(L\) is a double point; \(Q\) and \(R\) are defined similarly on \(CA\) and \(AB\). Prove that \(AP\), \(BQ\), \(CR\) meet in a point \(S\). Find the locus of \(S\) if \(O\) varies on a general line in the plane, \(l\) remaining fixed.
Given a collection of small pieces of matter, light inextensible strings by which they may be connected, a perfectly smooth horizontal table on which to set the pieces in motion, and instruments for measuring only position and time, describe experiments and measurements from which Newton's laws of motion may be deduced, and independent experiments and measurements to test these deductions. [The classical concepts of mass and force should be supposed unknown in advance.]