Problems

Find the smallest positive integer which, when divided by 28, leaves a remainder 21, and when divided by 19, leaves a remainder 17.

1. [(i)] Solve the equations \begin{align*} \tan x + \tan y &= 4, \\ \tan 2x + \tan 2y &= 0. \end{align*}
2. [(ii)] If \[ \sin\theta = \frac{\sin\alpha+\sin\beta}{1+\sin\alpha\sin\beta}, \] prove that \[ \tan\left(\frac{\pi}{4}-\frac{\theta}{2}\right) = \tan\left(\frac{\pi}{4}-\frac{\alpha}{2}\right)\tan\left(\frac{\pi}{4}-\frac{\beta}{2}\right). \]

If \(D,E,F\) are the feet of the perpendiculars from the vertices \(A,B,C\) of a triangle \(ABC\) on the opposite sides, prove that the radius of the circle inscribed in the triangle \(DEF\) is \[ \frac{r \cos A \cos B \cos C}{2\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}}. \]

A bowl is in the shape of a segment of a sphere, greater than a hemisphere. The diameter of the horizontal circular rim is \(2c\), and the depth of the bowl is \(a\). Three equal balls float in the bowl, which is full of water, the balls being only just completely immersed. Prove that the greatest possible radius of the balls under these conditions is \[ \frac{c}{4a}\{\sqrt{9c^2+12a^2}-3c\}. \]

1. [(i)] Prove that the coefficient of \(x^n\) in the expansion of \(e^x\cos x\) in ascending powers of \(x\) is \[ \frac{2^{\frac{1}{2}n}}{n!}\cos\frac{n\pi}{4}. \]
2. [(ii)] If \(\alpha = \cos^2\theta - \frac{1}{2}\cos^3\theta\cos 3\theta + \frac{1}{3}\cos^4\theta\cos 5\theta - \dots\) to infinity, prove that \[ \tan 2\alpha = 2\cot^2\theta. \]

Two regular polygons of \(m\) and \(n\) sides have equal perimeters \(l\). Prove that, if \(m\) and \(n\) are large, the areas of the polygons differ by \(\dfrac{\pi l^2}{12}\left(\dfrac{1}{m^2}\sim\dfrac{1}{n^2}\right)\) approximately.

Points \(L, M, N\) are taken in the sides \(BC, CA, AB\) of a triangle. Prove that the normals to the sides at these points are concurrent if, and only if, \[ BL^2 + CM^2 + AN^2 = CL^2 + AM^2 + BN^2. \] At each of any three points \(P, Q, R\) taken on a straight line perpendiculars \(PP', QQ', RR'\) of arbitrary lengths are drawn. Show that the perpendiculars from \(P, Q, R\) upon \(Q'R', R'P', P'Q'\) are concurrent.

Three circles touch one another in pairs. Show that the circle through their points of contact cuts each of the three orthogonally.

Show how to reciprocate confocal conics into coaxal circles. If a conic \(C\) has one focus \(S\) in common with a family of confocal conics, prove that pairs of common tangents to \(C\) and members of the confocal family meet in points on a line through \(S'\), the second focus of the confocal family, and cut any other tangent to \(C\) in points in involution.

It is required to inscribe a triangle in a conic so that the sides pass respectively, through three given points \(A, B, C\). Prove that there are in general two solutions, and that if there are more than two, there is an infinite number. Shew that if \(ABC\) is self-conjugate with respect to the conic, then there is an infinite number of solutions.