The inscribed circle of the triangle \(ABC\) touches \(BC\) at \(D\), \(CA\) at \(E\) and \(AB\) at \(F\); \(P\) is the other extremity of the diameter of the circle through \(D\); \(BC\) is cut by \(AP\) at \(M\), by \(PE\) at \(Q\) and by \(PF\) at \(R\). Prove that \(M\) is the middle point of \(QR\).
Having given that \begin{align*} x+y+z &= a, \\ x^2+y^2+z^2 &= b^2, \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} &= \frac{1}{c}, \end{align*} determine \(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\) in terms of \(a, b, c\). \par Solve the equations \[ \left\{ \begin{array}{l} xyz = 1, \\ (x+1)(y+1)(z+1)=9, \\ (y+z)(z+x)(x+y) = 11\frac{1}{4}. \end{array} \right. \]
Find how many different numbers between 1000 and 10,000 can be formed with the digits 0, 1, 2, 3, 4, 5. Shew that the sum of the numbers so formed is 979,920.
The roots of the equation \[ x^3 - ax^2 + bx - c = 0, \] are the lengths of the sides of a triangle. \par Shew that the area of the triangle is \(\frac{1}{4}\{a(4ab-a^3-8c)\}^{\frac{1}{2}}\), and that if the triangle is right-angled then \[ a(4ab-a^3-8c)(a^2-2b)-8c^2=0. \]
\(ABC\) is a triangle, and \begin{align*} \sin A + \sin B + \sin C &= p, \\ \cos A + \cos B + \cos C &= q. \end{align*} Express \[ \sin 2A + \sin 2B + \sin 2C \quad \text{and} \quad \cos 2A + \cos 2B + \cos 2C \] in terms of \(p\) and \(q\).
Solution: Consider \(q+ip = e^{iA}+e^{iB}+e^{iC}\), we would like \begin{align*} e^{i2A} + e^{i2B}+e^{i2C} &= ( e^{iA}+e^{iB}+e^{iC})^2 -2(e^{i(A+B)}+e^{i(B+C)}+e^{i(C+A)}) \\ &= (q+ip)^2-2(e^{i(\pi-C)}+e^{i(\pi-A)} + e^{i(\pi-B)})\\ &= (q+ip)^2+2(e^{-iC}+e^{-iA}+e^{-iB}) \\ &= (q+ip)^2+2(q-ip) \\ &= q^2-p^2+2q+i(2pq-p) \end{align*}
Express \(\log_e(a+b\sqrt{-1})\) in the form \(x+y\sqrt{-1}\). \par Find the value of \(\log_e(-1)\), and point out the fallacy in the following: \[ \log_e(-1) = \tfrac{1}{2}\log_e(-1)^2 = \tfrac{1}{2}\log_e 1=0, \] \[ \therefore -1 = e^0 = 1. \]
Prove that three normals can be drawn to a parabola from a given point. \par The normals at \(P, Q, R\) to a parabola whose vertex is \(A\) intersect in \(T\). A second parabola is drawn through \(P, Q, R\) with its axis perpendicular to the axis of the first parabola. Prove that the second parabola passes through \(A\) and that the length of its latus rectum is equal to half the distance of \(T\) from the axis of the first parabola.
Find the equation of the tangent at any point of the curve \(y^2=x^3\). \par The tangent at \(P\) intersects the curve again at \(Q\), and the lines from the origin to \(P\) and \(Q\) make angles \(\alpha, \beta\) respectively with the axis of \(x\). Shew that \[ \tan\alpha+2\tan\beta=0. \]
A manufacturer's expenses are a fixed sum together with a fixed amount \(c\) for each article sold. The number of articles sold varies as the \(k\)th power of the amount by which the selling price of an article is less than a fixed sum \(p\). Shew that his profits will be greatest if the selling price of an article is \(\frac{p+ck}{1+k}\).
Four forces act at the middle points of the sides of a quadrilateral figure in directions at right angles to the respective sides. Prove that if the forces are proportional to the respective sides they will be in equilibrium. \par Prove also that if the forces are in equilibrium they must be proportional to the sides unless the quadrilateral is such that a circle can be described about it.