If \(f(x) = (x+1)(2x^2-x+1)^{1/2}(x-1)^{-1/2}\) prove that \(f(x) = f(\{1-x\}/\{1+x\})\). Show that the equation \(f(x) = f(3)\) has two real and two imaginary roots, giving the values of each.
Prove that the product of any set of integers, each of which can be expressed as the sum of the squares of two integers, is equal to the sum of the squares of two integers. Express 7540 as the sum of the squares of two integers.
Show that, if \(a_1, a_2, \dots, a_m\) are distinct prime numbers other than unity, the number of solutions in integers (including unity) of the equation \(x_1 x_2 x_3 \dots x_n = a_1 a_2 a_3 \dots a_m\) is \(n^m\). Show also that the number of solutions in which at least one of the \(x\)'s is unity is \[ n! \left\{ \frac{(n-1)^m}{(n-1)! 1!} - \frac{(n-2)^m}{(n-2)! 2!} + \dots + (-)^{n-2} \frac{1^m}{1!(n-1)!} \right\}. \]
Find the sum of the terms after the \(n\)th in the expansion of \((1+x)/(1-x)^2\) in ascending powers of \(x\). Prove that the ratio of this sum to the sum of the corresponding terms in the expansion of \((1-x)^{-2}\) can be made equal to any given number \(\lambda\), which is greater than \(2n\), by suitable choice of \(x\). Explain clearly why the restriction upon \(\lambda\) is necessary.
In any triangle \(ABC\) prove that the sum of the squares of the distances of the centre of the inscribed circle from the vertices is \(bc+ca+ab - 6abc/(a+b+c)\). Investigate the corresponding result for the sum of the squares of the distances of the centre of an escribed circle from the vertices.
Prove that the least value of \(a\cos\theta + b\sin\theta\) is the negative square root of \(a^2+b^2\). Prove also that the least value of \[ x^2 + 2x(a\cos\theta+b\sin\theta) + c\cos 2\theta + d\sin 2\theta \] is \[ -\frac{1}{2}(a^2+b^2) - \{c^2+d^2+\frac{1}{4}(a^2+b^2)^2+c(b^2-a^2)-2abd\}^{1/2}. \]
Three spheres, each of radius 3 inches, rest in mutual contact on a horizontal table, and a fourth sphere, of radius 2 inches, rests upon them. Find (i) the height above the table of the highest point of the smaller sphere, and (ii) the inclination to the horizontal of a plane which touches the smaller sphere and two of the larger ones.
Find values of \(a, b, c, d\) such that the curve \(y=ax^3+bx^2+cx+d\) touches the lines \(3x-y-6=0, 3x+3y+2=0\) at their points of intersection with the axes of \(x\) and \(y\) respectively. Prove that the curve touches the axis of \(x\), and that the curvature at the point of contact is 2.
Prove that a function \(f(x)\) has a minimum for \(x=a\), if \(f'(a)=0\) and \(f''(a)>0\). A thin closed rectangular box is to have one edge \(n\) times the length of another edge, and the volume is to be \(V\). Prove that the least surface \(S\) is given by \(nS^3=54(n+1)^2V^2\).
Sketch the curve \(a^2y^2 = x^2(a^2-x^2)\). Find the area of a loop of the curve, and prove that the volume generated by revolution of a loop about the \(y\)-axis is \(\pi a^3/4\).