Shew that a straight tube whose cross-section is a regular hexagon can be completely blocked by a solid cube, and determine the ratio of an edge of the cube to a side of the hexagon.
Two conics inscribed in a triangle \(ABC\) touch \(BC\) at the same point \(P\), touch \(CA\) at \(Q, Q'\) and \(AB\) at \(R, R'\). The conics intersect in \(D, E\). \(PD\) meets \(CA, AB\) in \(L, M\). \(PE\) meets \(CA, AB\) in \(M', L'\). Prove that \(QR, Q'R', LL', MM', DE\) and \(BC\) are concurrent.
(i) Prove that, if \(a,b,c,d\) are real positive numbers not all equal, \[ 64(a^4+b^4+c^4+d^4) > (a+b+c+d)^4. \] (ii) Prove that, if \(m, n\) are positive integers and \(x, y\) and \(z\) real and positive and \(z^m = x^m+y^m\), then \(z^n >\) or \(< x^n+y^n\) according as \(n >\) or \(< m\).
Prove that \[ \tan^2\frac{\pi}{14} + \tan^2\frac{3\pi}{14} + \tan^2\frac{5\pi}{14} = 5, \] and \[ \cos^2\frac{\pi}{14} \cos^2\frac{3\pi}{14} \cos^2\frac{5\pi}{14} = \frac{7}{64}. \]
Prove that, if \(x,y,z\) are functions of two variables \(u,v\) given by the relations \[ x=f(u,v), \quad y=g(u,v), \quad z=h(u,v), \] then \[ \frac{\partial z}{\partial x} (f_u g_v - f_v g_u) = h_u g_v - h_v g_u \] and \[ \frac{\partial^2 z}{\partial x^2} (f_u g_v - f_v g_u)^3 = \begin{vmatrix} f_u & f_v & f_{uu}g_v^2 - 2f_{uv}g_u g_v + f_{vv} g_u^2 \\ g_u & g_v & g_{uu}g_v^2 - 2g_{uv}g_u g_v + g_{vv} g_u^2 \\ h_u & h_v & h_{uu}g_v^2 - 2h_{uv}g_u g_v + h_{vv} g_u^2 \end{vmatrix}, \] where suffixes denote partial differentiation.
Evaluate \[ \int_a^b \sqrt{\{(b-x)/(x-a)\}} dx, \quad a
A hexagonal framework \(ABCDEF\) is formed of six equal uniform rods each of weight \(W\) smoothly jointed at their ends, with two light struts \(BF, CE\). The framework is suspended from \(A\) and \(BF, CE\) are of such length that \(AB, BC\) make angles \(\alpha, \beta\) with the horizontal (\(BF < CE\)). Prove that the thrusts in \(BF, CE\) are \[ \tfrac{1}{2} W(5\cot\alpha-3\cot\beta) \quad \text{and} \quad \tfrac{1}{2} W(3\cot\beta+\cot\gamma), \] where \(\cos\gamma = \cos\alpha+\cos\beta\).
A rigid light rod \(ABC\) has three particles of the same mass \(m\) attached to it at \(A, B, C\), where \(AB=a\) and \(BC=b\) (\(a>b\)). The rod is moving at right angles to its length with velocity \(u\), when its middle point \(O\) is suddenly fixed. Find the impulse at \(O\) and prove that there is a loss of energy \[ 4mu^2(a^2+ab+b^2)/(3a^2+2ab+3b^2). \]
Two particles \(A, B\) of masses \(m_1, m_2\) rest on a smooth horizontal plane connected by an elastic string of modulus \(\lambda\), natural length \(l\) and negligible mass. The particle \(A\) is given a velocity \(u\) in direction \(BA\). Find the velocities of the particles \(t\) seconds after the string begins to stretch, and shew that there is a continuous flux of momentum between the particles amounting to \(2m_1m_2u/(m_1+m_2)\) in \(\pi/n\) seconds, where \[ n^2 = \frac{\lambda}{l} \left( \frac{1}{m_1} + \frac{1}{m_2} \right). \]
The cross-section of a wedge of mass \(M\) is an isosceles triangle of base angles \(\alpha\). It is placed on a rough horizontal plane. Particles of masses \(m,m'\) (\(m>m'\)) are placed one on each of the smooth slant faces and connected by a fine inextensible thread which passes over and at right angles to the top edge of the wedge. Motion is possible in vertical planes through lines of greatest slope. Prove that if the coefficient of friction exceeds \[ (m^2-m'^2)\sin\alpha\cos\alpha / \{(m+m')(M+m+m') - (m-m')^2\sin^2\alpha\} \] the wedge will not move; and write down sufficient equations of motion to determine the acceleration of the wedge and of the particles relative to the wedge when the coefficient of friction has a smaller value.