A polygon \(P\) has vertices \(A_1, \dots, A_n\) where the coordinates \(x_r, y_r\) of \(A_r\) are both integers for \(r=1, \dots, n\). The number of points inside \(P\) (and not on the sides) whose co-ordinates \(x, y\) are both integers is denoted by \(m(P)\) and the number of points on the sides whose coordinates are both integers is denoted by \(m'(P)\); this including the vertices \(A_1, \dots, A_n\). Prove that the area of \(P\) is given by the formula \[ \text{Area} = m(P) + \tfrac{1}{2}m'(P) - 1 \] in the following cases:
The real number \(a\) is greater than 1 and an approximation \(x\) to the square root of \(a\) is given which is also greater than 1. A new approximation \(y\) to the square root of \(a\) is defined by the formula \[ y = \tfrac{1}{2}\left(x+\frac{a}{x}\right). \] Prove that the square root of \(a\) lies between \(y\) and \(y - \frac{1}{8y}(x^2-a)^2\). By taking \(x=1 \cdot 4\), obtain an approximation to \(\sqrt{2}\) which is correct to 3 places of decimals, proving that this is the case.
Prove that \[ \int_1^a \frac{f(x)}{x} dx = \int_{1/a}^1 \frac{f(1/x)}{x} dx \] where \(a>0\). Evaluate the integral \[ \int_{1/a}^a \frac{(\log x)^2 dx}{x(1+x^n)}. \]
The triangle \(ABC\) is acute-angled; \(P\) is a point that can vary on \(BC\) (but not outside the segment \(BC\)). The mirror images of \(B, P\) in \(AC\) are \(B', P'\) and of \(C, P'\) in \(A B'\) are \(C'', P''\). Find for what position of \(P\) on \(BC\) the distance \(PP''\) is least. The points \(Q, R\) vary on the segments \(AC, AB\); the image of \(R\) in \(AC\) is \(R'\). By comparing \(PQ+QR'+R'P''\) with \(PP''\), or otherwise, determine for any given position of \(P\) the positions of \(Q\) and \(R\) for which the perimeter of the triangle \(PQR\) is least; hence determine the positions of \(P, Q\) and \(R\) for which this perimeter is least. [It may be assumed without proof that, for any position of \(P\) in \(BC\), the segment from \(P\) to \(P''\) cuts \(AC, AB'\) internally in that order.]
A regular dodecahedron is bounded by twelve regular pentagons each with side of unit length. Prove that the obtuse angle between two adjacent faces is \(116\frac{1}{2}^\circ\) approximately and find the radii of the inscribed and circumscribed spheres of the dodecahedron.
A uniform solid cylinder of mass \(m\) and radius \(a\) rolls down a rough plane inclined at an angle \(\alpha\) to the horizontal, the coefficient of friction being \(\mu\). A retarding couple \(G\) is then applied to the cylinder for the purpose of stopping the translational motion as quickly as possible. Show that the cylinder will not be stopped unless \[ \mu > \tan\alpha, \quad G > mga\sin\alpha, \] but that there is no advantage to be gained by applying a couple of greater magnitude than \[ \tfrac{1}{2}mga(3\mu\cos\alpha - \sin\alpha). \]
A smooth and perfectly elastic ball is dropped on to a smooth plane which is inclined at an angle \(\beta\) to the horizontal. If \(\theta_n\) denotes the angle between the plane and the direction of motion immediately after the \(n\)th bounce, show that \(\theta_n \to 0\) as \(n\to\infty\), and that \[ \theta_{n+1} = \theta_n - 2\theta_n^2 \tan\beta \] for large values of \(n\), neglecting terms of the order of \(\theta_n^3\).
A cloud of water vapour moves vertically upwards with velocity \(V\), and a spherical drop of water in the cloud moves in a vertical line under the action of gravity. The mass of the drop increases, by condensation from the cloud, at a rate \(k_1S\), where \(S\) is the surface area of the drop. Relative motion between the drop and the cloud is opposed by a force of magnitude \(k_2S\) times the relative velocity. If the drop is initially very small and moving with the cloud, show that it will begin to fall after a time \[ \frac{V}{g}\left(4+3\frac{k_2}{k_1}\right). \]
A plank rests across a cylindrical barrel on flat ground and initially has one end on the ground. A man walks up the plank. DIscuss qualitatively the apossible resulting motions on the assumption that no slipping occurs. [For example, consider different ratios of masses and dimensions and consider the man taking short or long steps and moving irregularly.]
If \(a, b, c\) are three constants, all different, show that the equations \begin{align*} x+y+z &= 3(a+b+c), \\ ax+by+cz &= (a+b+c)^2, \\ xyz &= ayz+bzx+cxy, \end{align*} have in general only one solution in which \(x, y, z\) are unequal, and find this solution.