Problems

Prove that

1. If \(n\) is an integer greater than 2, then \[ \left(\frac{n+1}{2}\right)^n > n! > n^{n/2}, \] \[ n! \left(2-\frac{1}{n}\right)\left(2-\frac{3}{n}\right)\dots\left(2-\frac{2n-1}{n}\right) > 1; \]
2. If \(p > m > 0\), then \[ \frac{p+m}{p-m} \ge \frac{x^2-2mx+p^2}{x^2+2mx+p^2} \ge \frac{p-m}{p+m} \] for all real \(x\).

If \(a,b,c,d\) are roots of \(x^4+px^3+qx^2+rx+s=0\):

1. find the value of \(\sum \frac{a^3}{b^2}, \sum a^3b^2c^2\);
2. find the equation whose roots are \[ b^2+c^2+d^2-a^2, \dots. \]

1. If \[ y = \tan^{-1} \frac{x\sin\alpha}{1+x\cos\alpha}, \] prove \[ \frac{d^n y}{dx^n} = \frac{(n-1)!}{\sin^n\alpha} \sin n(\alpha-y)\sin^n(\alpha-y). \] (The OCR is slightly different here, but this form seems more likely based on context. Original OCR: `sin n(y-a) sin^n(y-a)` and `sin^n a`. The image is clearer showing \(\alpha - y\)).
2. In the equation \[ f(x+h) = f(x) + hf'(x) + \frac{h^2}{2}f''(x+\theta h), \] find the first three terms of the expansion of \(\theta\) in ascending powers of \(h\), and calculate them for \(f(x) = \sin x\).

Evaluate the integrals \[ \int \frac{dx}{(1+x)(4+6x+4x^2+x^3)}, \quad \int \frac{\sin^2 x \, dx}{\cos^3 x}, \quad \int_0^{\pi/2} \log(\sin x \cos x) \, dx. \]

1. Show that if \(a_1+a_2+\dots\) be a divergent series of positive terms, then the series \[ \frac{a_1}{a_1+1} + \frac{a_2}{(a_1+1)(a_2+1)} + \frac{a_3}{(a_1+1)(a_2+1)(a_3+1)} + \dots \] converges to \(+1\).
2. Find the sum of the series \[ 1 + \cos\theta\tan\theta + \frac{1}{2!}\cos 2\theta \tan^2\theta + \frac{1}{3!}\cos 3\theta\tan^3\theta + \dots. \]

Normals are drawn to an ellipse at the ends of two conjugate diameters. Find all maxima and minima distances of their common point from the centre.

Tangents at right angles are drawn to the four-cusped hypocycloid \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\). Show that the bisectors of the angles between them either pass through the origin or envelop the curve \[ (x+y)^{\frac{2}{3}} + (x-y)^{\frac{2}{3}} = (2a)^{\frac{2}{3}}. \]

The diagonals of a quadrilateral inscribed in a circle subtend acute angles \(\theta\) and \(\phi\) at the circumference. Prove that if a circle can be inscribed in this quadrilateral the acute angle between its diagonals has its tangent equal to \[ \frac{\sin\theta + \sin\phi}{\cos\theta\cos\phi}. \]

Four variables are connected by two independent relations. Show that \[ \left(\frac{\partial y}{\partial z}\right)_u \left(\frac{\partial z}{\partial x}\right)_u \left(\frac{\partial x}{\partial y}\right)_u = -1; \quad \left(\frac{\partial y}{\partial z}\right)_u \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial y}\right)_z = 1; \] and \[ \left(\frac{\partial z}{\partial x}\right)_u \left(\frac{\partial x}{\partial y}\right)_u = -\left(\frac{\partial z}{\partial y}\right)_u; \quad \left(\frac{\partial z}{\partial x}\right)_u \left(\frac{\partial u}{\partial y}\right)_x + \left(\frac{\partial z}{\partial y}\right)_x = 0, \] (The OCR is hard to read here; the provided transcription is based on standard identities, but the scanned text appears to show different equations. For instance, the first appears as \(\left(\frac{\partial y}{\partial z}\right)_u \left(\frac{\partial z}{\partial x}\right)_u \left(\frac{\partial x}{\partial y}\right)_u = - \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial y}\right)_z = 1\). The second equation in the prompt is transcribed from the second line of the original which seems to be \(\left(\frac{\partial z}{\partial x}\right)_u \left(\frac{\partial x}{\partial y}\right)_u = \left(\frac{\partial z}{\partial y}\right)_u + \left(\frac{\partial u}{\partial z}\right)_x \left(\frac{\partial z}{\partial y}\right)_x = 1\). This is very unusual. Let's transcribe the visual from the scan) \[ \left(\frac{\partial y}{\partial z}\right)_u \left(\frac{\partial z}{\partial x}\right)_u \left(\frac{\partial x}{\partial y}\right)_u = - \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y \left(\frac{\partial x}{\partial y}\right)_z = 1; \] and \[ \left(\frac{\partial z}{\partial x}\right)_u \left(\frac{\partial x}{\partial y}\right)_u + \left(\frac{\partial z}{\partial y}\right)_u \left(\frac{\partial u}{\partial y}\right)_x = 1, \] where the suffix in each case denotes the second independent variable in the differential coefficients to which it is attached.

Trace the curve \[ y^2+2(x^2-2)xy+x^4=0, \] and find the areas of the loops.