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26 Nov 2015: 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).

21 Oct 2015: a) Let a G. Show that there is a positive integer n such that an = e, the identity element.(The least such positive n is the order of a.). ... b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the

9 Feb 2015: That is, for everya < b and every t there should exist x (a,b) such that f(x) = t. ... 14. Let f : R R be a function that has the intermediate value property: that is, iff(a) < c < f(b) then there exists x (a,b) such that f(x) =

20 Nov 2015: 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 a strategy such that,should they follow it, only finitely many of them will guess wrongly.

10 Feb 2015: Show that for any b C, there exists a sequence of points zn D(a,R) with zn 6= a such thatzn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.

15 Oct 2015: 1. 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.

28 Oct 2015: has no solution with a, b Z. 14. Construct a function f : R R that takes every value on every interval — in other words,for every a < b and every c ... there is an x with a < x < b such that f(x) = c.

4 Mar 2015: Prove that f is. integrable. What is 10f(x)dx? 6. Let a < b and let f : [a,b] R be a Riemann integrable function such that f(x) 0for every ... 1, 1], such that f(x) = g(x) h(x) for every x.

19 Feb 2015: Deducethe more general result that if f ′(a) 6= f ′(b) and z lies between f ′(a) and f ′(b), then thereexists c (a,b) such that f ′(c) = z. ... f(a) < 0 and f(b) > 0 then there exists c (a,b)such that f(c) = 0.

10 Feb 2015: a). n=1. nenz on {z. 0 < r Re(z)}; (b) n=1. ... Can such a map ever be a Möbius trans-formation? 9 Calculateγ z sin z dz when γ is the straight line joining 0 to i.