\(ABC\) 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.]
A uniform sphere, of radius \(a\), is projected with velocity \(V\) down a rough plane of inclination \(\alpha\). The sphere also has an angular velocity \(\Omega\) about a horizontal diameter, in such a sense as to tend to cause rolling up the plane. Prove that it will never stop slipping unless the coefficient of friction \(\mu\) is greater than \(\frac{2}{5} \tan \alpha\), and that, if \(bV < 2a\Omega\), it will move back up the plane if $$\mu > \frac{\tan \alpha}{1 - \frac{5V}{2a\Omega}}.$$
Solve the simultaneous equations \begin{align} x + y + z &= 6, \\ (y + z)(z + x)(x + y) &= 60, \\ \begin{vmatrix} x & y & z \\ y & z & x \\ z & x & y \end{vmatrix} &= -18. \end{align}
Show that if an integer of the form \(4n + 3\) is expressed as a product of integers, then one at least of these integers is also of the form \(4n + 3\). Show that each pair of integers \(x_i\), \(x_j\) \((i \neq j)\) chosen from the sequence \(x_1\), \(x_2\), \(\ldots\), defined by \[x_1 = 1, \quad x_{n+1} = 4x_1x_2\ldots x_n + 3 \quad (n \geq 1)\] are coprime (that is, have highest common factor 1). Deduce that there are an infinity of prime numbers of the form \(4n + 3\).
The numbers \(c_0\), \(c_1\), \(\ldots\), \(c_n\) are defined by the identity \[(1 + x)^n = c_0 + c_1x + \ldots + c_nx^n;\] prove that \(\sum c_ic_j\) summed over all integer pairs \(i\), \(j\) such that \(0 \leq i < j \leq n\) is equal to \[2^{2n-1} - \frac{(2n-1)!}{n!(n-1)!}.\]
\(n\) different names are placed in a hat. One name is drawn at random, read out, and replaced in the hat. This is repeated until \(m\) names in all have been read out. What is the probability that no name has been read out twice? If \(r\) is a given integer, how large must \(m\) be in order to ensure that one name at least is read out \(r\) times?
Obtain a reduction formula for \[I_n = \int x^n \cos rx\,dx \quad (r \neq 0).\] If \[u_n = \int_0^{1\pi} x^n \sin^2 x\,dx,\] prove that, for \(n \geq 2\), \[u_n = \frac{(\tfrac{1}{2}\pi)^{n+1}}{2n+2} + \frac{n(\tfrac{1}{2}\pi)^{n-1}}{4} - \tfrac{1}{4}n(n-1)u_{n-2}.\]
Obtain a series expansion of \(\log_e\{1 + (1/x)\}\) in ascending powers of \(1/(2x+1)\). For what ranges of values of \(x\) is this expansion valid? Prove that if \(x\) is strictly positive, for what \[\frac{2x+1}{2x(x+1)} > \log_e\left(1 + \frac{1}{x}\right) > \frac{2}{2x+1}.\]
If \[f(x) = (\sin x - \sin a)^{-1} - (x - a)^{-1}\sec a\] evaluate \[\frac{d}{da}\left[\text{Lt}_{x \to a} f(x)\right] - \text{Lt}_{x \to a} f'(x).\]