The number \(n\) whose digits in the scale of 10 are \(a\), \(b\), \(c\), \(d\) in that order is the same as the number whose digits in the scale of 9 are \(d\), \(b\), \(c\), \(a\) in that order; in other words, we have \[10^3a + 10^2b + 10c + d = 9^3d + 9^2b + 9c + a,\] and the digits \(a\), \(b\), \(c\), \(d\) all lie between 0 and 8 inclusive. Prove that there is exactly one number \(n\) (\(\neq 0\)) with this property, and find \(n\).
Given a plane curve \(C\) and a fixed point \(O\), the pedal curve of \(C\) with respect to \(O\) is defined to be the locus of the foot of the perpendicular from \(O\) to a variable tangent of \(C\). Let \(P\) be an arbitrary point on \(C\), \(PZ\) the tangent at \(P\) and \(OZ\) the perpendicular from \(O\) to \(PZ\). Prove (i) that the angle between \(OP\) and \(PZ\) is equal to the angle between \(OZ\) and the tangent to the pedal curve at \(Z\), and (ii) that, if we write \(p\) for \(OZ\) and \(q\) for \(PZ\), then \[q = \frac{dp}{d\psi},\] where \(\psi\) is the angle between \(OZ\) and a fixed line \(OX\). Find, in rectangular coordinates, the equation of the curve whose pedal curve with respect to the point \((0, 0)\) in the \((x, y)\) plane is the parabola \(y^2 = 4ax\). Give a rough sketch of the two curves.
Define the function \(f(x)\) for positive values of \(x\) by the equation \[f(x) = \int_x^{\infty} \frac{e^{x-t}}{t}dt.\] Prove that, for each positive integer \(n\), \[f(x) = \frac{1}{x} - \frac{1}{x^2} + \frac{2!}{x^3} - \frac{3!}{x^4} + \ldots + \frac{(-1)^{n-1}(n-1)!}{x^n} + (-1)^n n! \int_x^{\infty} \frac{e^{x-t}}{t^{n+1}} dt.\] Show that there is no value of \(x\) for which the infinite series \[\frac{1}{x} - \frac{1}{x^2} + \frac{2!}{x^3} - \ldots + \frac{(-1)^{n-1}(n-1)!}{x^n} + \ldots\] is convergent. Writing \[S_n(x) = \frac{1}{x} - \frac{1}{x^2} + \frac{2!}{x^3} - \frac{3!}{x^4} + \ldots + \frac{(-1)^{n-1}(n-1)!}{x^n},\] prove that \(f(x)\) always lies between \(S_n(x)\) and \(S_{n+1}(x)\). Use the series to obtain a value for \(f(10)\) with an error smaller than 1 part in 4000.
Let \(A\), \(B\), \(C\), \(D\) be four given points in a plane, no three of them being collinear. Suppose that a given line meets \(AB\), \(CD\), \(BC\), \(AD\) in \(E\), \(F\), \(H\), \(K\) respectively. Let \(O\), \(P\), \(Q\) be respectively the intersections \((AC, BD)\), \((BF, CE)\), \((CK, DH)\). Prove that \(OP\) and \(OQ\) are harmonically conjugate with respect to \(OA\) and \(OB\).
Let \(O\), \(A\), \(B\) be three points on a conic \(S\) and let \(D\) be the pole of \(AB\). Prove that two points \(U\) and \(V\) lying on \(OA\) and \(OB\) respectively are conjugate with respect to \(S\) if and only if \(UV\) passes through \(D\). Now let \(P\), \(Q\), \(R\) be three further points on \(S\), and suppose that \(DP\), \(DQ\), \(DR\) meet \(S\) again in \(L\), \(M\), \(N\) respectively; denote the intersections \((OL, QR)\), \((OM, RP)\), \((ON, PQ)\) by \(P_1\), \(Q_1\), \(R_1\) respectively. Prove that \(D\), \(P_1\), \(Q_1\), \(R_1\) are collinear.
A uniform circular disc of mass \(m\) and radius \(a\) lies flat on a smooth horizontal plane, with its centre at the origin of rectangular axes \(Oxy\) located in the plane. A particle of mass \(m\) is projected along the plane with speed \((u, 0)\) and strikes the rim of the disc, to which it adheres, at the point \((-a\cos\theta, -a\sin\theta)\). Immediately after impact the velocity of the centre of the disc is \((v, w)\) and the angular velocity of the disc is \(\Omega\). State carefully the momentum laws appropriate to the problem, and use them to find \(v\), \(w\) and \(\Omega\) in terms of \(u\) and \(\theta\). Find the kinetic energy lost in the collision.
A uniform thin hollow right circular cylinder, of mass \(m\) and radius \(a\), rolls perfectly rough horizontal table. The surface of the cylinder experiences a force which at each point \(P\) acts along the tangent \(l\) to the normal cross-section of the cylinder. The magnitude of the drag at the generator through \(P\) is \(kv^2\) per unit length, where \(v\) is the component along \(l\) of the velocity of \(P\). Show that, if the initial speed of the axis of the cylinder is \(V\), then its speed at subsequent time \(t\) is \[2mV/(2m + 5\pi ka Vt).\]
A crate of mass \(m\) rests on the floor of a truck of mass \(M\), at a distance \(a\) from the vertical front end of the truck. The truck is travelling along straight horizontal rails with uniform speed \(V\), when its wheels are locked and it is slowed down by friction between the wheels and the rails, for which the coefficient of friction is \(\mu\). The coefficient of friction between the crate and the floor of the truck is \(\mu'\). Show that, if \(\mu' < \mu\), the crate will strike the front end of the truck when the truck is still moving provided that \[V^2 > 2ga\left[\frac{\mu M + (\mu - \mu') m}{(\mu - \mu') M(M + m)}\right].\] In the case \(m = M\), \(\mu' = \frac{1}{2}\mu\), \(V = 2\sqrt{(2\mu ga)}\), and on the assumption that the impact between the crate and the end of the truck is inelastic, find the total distance travelled by the truck after its wheels are locked. What would happen if \(\mu'\) were greater than \(\mu\)?
A smooth straight narrow tube \(AB\), of length \(b\) and closed at \(B\), is kept in rotation about \(A\) in a horizontal plane with constant angular velocity \(\omega\). A particle inside the tube is released from rest relative to the tube at a distance \(a\) from \(A\). Show that, if impact of the particle on the end of the tube at \(B\) is perfectly elastic, then the motion of the particle relative to the tube is periodic with period \(T = (2/\omega)\cosh^{-1}(b/a)\). Show also that, if the impact is not perfectly elastic, but has coefficient of restitution \(1-\epsilon\) where \(\epsilon\) is small, then the particle first comes to relative rest again near its point of release after a time \[T - \frac{b\sqrt{(b^2-a^2)}\epsilon}{a^2\omega}\] approximately.
Two particles, each of mass \(m\), hang at the ends \(A\), \(B\) of two light inextensible strings, each of length \(a\), the other ends of which are fixed at the same level at a distance \(b\) apart. The particles are joined by a light spring of natural length \(b\) and modulus \(\lambda\) and initially the system is at rest in its equilibrium position. The particle at \(A\) is then struck by an impulse \(I\) directed towards \(B\). In the subsequent motion the angles \(\theta\), \(\phi\) which the strings make with the vertical (measured in the same sense) remain small. Show that \[\theta + \phi = -\frac{(g/a)(\theta + \phi)}{x}\] \[\theta - \phi = -\frac{(g/a)(1 + \epsilon)(\theta - \phi)}{x}\] where \(\epsilon = 2\lambda a/(mga)\), and hence find \(\theta\) and \(\phi\) as functions of \(t\). Defend the statement that, if \(\epsilon\) is small, the motion can be described as the repeated transfer, from particle \(A\) to particle and \(B\) back, of an oscillatory motion, with a repetition time approximately \(4\pi\sqrt{(a/g)\epsilon}\).