If \(a\), \(b\), \(c\) and \(d\) are all positive, prove that there is a positive value of \(t\) such that the equations $$ax + by = tx,$$ $$cx + dy = ty$$ have solutions other than \(x = y = 0\), and that there are solutions corresponding to this value of \(t\) in which both \(x\) and \(y\) are positive.
State and prove the Remainder Theorem for polynomials. What is the remainder when the polynomial \(f(x)\) is divided by \((x-a)(x-b)\) (where \(a \neq b\)), in terms of \(f(a)\) and \(f(b)\)?
Writing \(C(n,r)\) for \(\frac{n!}{r!(n-r)!}\) (and taking \(C(n,0) = C(n,n) = 1\)), prove that, if \(0 \leq r \leq n\), $$\sum_{r=0}^{s} (-1)^r C(n,r)C(n,s-r) = 0 \quad \text{if } s \text{ is odd}$$ $$= (-1)^{s/2} C(n,\frac{s}{2}) \quad \text{if } s \text{ is even}.$$ What can you say about $$\sum_{r=0}^{s} (-1)^r C(n+r-1,r)C(n+s-r-1,s-r)?$$ Justify your answer.
State the relations between the roots \(\alpha\), \(\beta\), \(\gamma\) of the equation \(ax^3 + bx^2 + cx + d = 0\) and the coefficients \(a\), \(b\), \(c\), \(d\). Prove that \(\{0, i, -i\}\) (where \(i^2 = -1\)) is the only set of three distinct numbers (real or complex) such that each is equal to the sum of the cubes of the other two.
Prove that (if all the numbers involved are positive) $$(ab)^{\frac{1}{2}} \leq \frac{1}{2}(a+b) \quad \text{and} \quad (abcd)^{\frac{1}{4}} \leq \frac{1}{4}(a+b+c+d).$$ By taking a special value for \(d\), or otherwise, prove that $$(abc)^{\frac{1}{3}} \leq \frac{1}{3}(a+b+c).$$ Prove also that $$8abc \leq (b+c)(c+a)(a+b).$$
Imagine that you are provided with a straight-edge and a parallel ruler (which is a device by means of which a straight line parallel to a given line may be drawn through a given point), but not a graduated ruler or a pair of compasses. Describe, with adequate detail, constructions for (a) the bisection, (b) the trisection of a given straight line segment.
\(ABC\) is a triangle, \(O\) a point inside it. Prove that $$\lambda(BC + CA + AB) > OA + OB + OC > \mu(BC + CA + AB),$$ where \(\lambda = 1\), \(\mu = \frac{1}{2}\). Prove also that if \(\lambda < 1\), \(\mu > \frac{1}{2}\) there is a triangle such that each of the above inequalities is false for a suitably chosen point \(O\) (which will not in general be the same for the two inequalities).
\(A\) is a fixed point, \(C\) a circle passing through two given fixed points. Prove that in general the polar of \(A\) with respect to \(C\) passes through a fixed point. Are there any exceptional cases? Consider the similar problem when \(C\) passes through one fixed point and touches a fixed line.
Two rectangular hyperbolas are such that the asymptotes of one are the axes of the other. Prove that they intersect at right angles, and that each common tangent subtends a right angle at the centre.
Write a short essay on that aspect of the theory of conics which you find most interesting.