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11 Oct 2017: ii) Let α be such that |K(α) : K| is odd. ... and let β = α4.Find. rationals a, b, c such that β = a bα cα2.

10 Mar 2017: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... 6. Show that every graph G has a partition of its vertex-set as X Y such that the numberof edges from X to Y is at least 12e(G).

13 Feb 2017: Such a c is called a fixed point of g.Give an example of a bijection h: [0, 1] [0, 1] with no fixed point. ... Thatis, for every a < b and every y there exists x (a,b) such that f(x) = y.Can such a function be continuous at any point?

8 Feb 2017: Show that there exist unique complexnumbers A, B such that for every z C, T(z) = Az Bz̄. ... Can such a map everbe a Möbius transformation? 9. Let f : U C be a holomorphic function where U is an open set (you may assume f isalso C1).

8 Feb 2017: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

10 Mar 2017: Show that for every. closed subinterval I [a,b] of positive length and every ε > 0 there existsa closed subinterval J I of positive length such that supJ f infJ f < ... 0. sinxx. dx = π2. 12. A function f : [a,b] R is said to have bounded variation if

17 Nov 2017: Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T? ... You are asked to supply them with astrategy such that, should they follow it, only finitely many of them will guess wrongly.

8 Nov 2017: Show that there is a divergentseries. n=1 yn with yn > 0 for all n, such that yn/xn 0. ... a < x < b such that f(x) = c.

13 Oct 2017: Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

23 Feb 2017: 7. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of