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23 Mar 2007: Fix some subset B of [n]. How large a family A of subsets of [n] can we find such that the intersection of any two sets in A contains a cyclic ... Chung, Graham, Frankl and Shearer have proved that, in the case where B is a block of length t, we can do

20 Jan 2007: Show that the curve joining A and B with length and such that together with AB bounds the largest possible area is an arc of a circle passingthrough A and B. ... Compute the Gaussian curvature and the mean curvature. 10. (i) Determine an equation for the

21 Nov 2007: 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 1.

12 Oct 2007: On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a, b] such that fn(xn) 6 f ... b) Give an example of a sequence of uniformly continuous, real-valued functions (fn) on R such thatfn

12 Mar 2007: b) Explicitly construct such a representation ρ : SU2 SO3(R) by showing that SU2acts on the vector space of matrices of the form. ... by conjugation. Show that this subspace is isomorphic to R3, that (A,B) 7 tr(AB)is a positive definite non-degenerate

2 Mar 2007: Prove that there exists such a reparametrization Γ̃ = Γffor which ‖dΓ̃/dt‖ is constant, namely l. ... 9) For a surface of revolution S, corresponding to a curve η : (a,b) R3 given byη(u) = (f(u),0,g(u)), where η is parametrized in such a way

23 Feb 2007: Show that for anyb C, there exists a sequence of pointszn D(a, R) with zn 6= a such thatzn a andf (zn) b asn. ... 16 Let f be a meromorphic function onC such thatf (1/z) is also meromorphic.

16 Nov 2007: Can we find such a bijection f which isorder-preserving (that is, satisfies f(x) < f(y) whenever x < y)? ... Let S be a collection of subsets of N such that for every A, B S we have eitherA B or B A.

17 Oct 2007: But such a numberwould clearly be very interesting indeed. Therefore, all positive integers are interesting. ... 12. Find a positive integer k such that, for all positive integers n, the number n4 k iscomposite.

5 Feb 2007: b) Let N Zn be a submodule of rank n. Show that there is a matrix A Mn(Z) such thatA(Zn) = N and the index [Zn : N] of N in ... 8) An integral domain R is called a euclidian domain if there is a map N : R{0} Z>0 such thatfor all a,b R, b 6= 0, there are

15 Mar 2007: Assume there is θ OK such that 1,θ,. ,θn1 isan integral basis of OK. ... dK| (nn. n! )2 (π4. )n. Deduce that |dK| > 1 for every number field K 6= Q.(b) Show that there are constants A > 1, c > 1, such that for every

21 Nov 2007: If A is the matrix of the first of these (say), find a non-singular matrix P such that P T AP is diagonalwith entries 1. ... Show that r(P Q) is odd. 19. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

7 Nov 2007: a linear func-tional θ V such that θ(x) 6= θ(y).(b) Suppose that V is finite dimensional. ... If no such subspace exists,show that there exists a basis v, α(v),.

8 May 2007: b) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... b) (x, y, z) (x′, y′, z′) x = x′, y = y′, z = 1 z′.

14 May 2007: Is there always an orientable such surface, i.e. one in which“turning clockwise” makes sense globally on the surface? ... b) What is the smallest genus (number of holes) of such a surface with boundary thetrefoil knot, which is the knot which lies on

8 Nov 2007: Moreover, the Kretchmann scalar RαβγδRαβγδblows up along such curves, thus B can be viewed as a singular boundary. ... For such spacetimes, (5) cannot hold, so the notion of naked-ness must be defined more carefully.

19 May 2007: 1 log(α1)m such that |N (x) N (x′)| > αn for all but at most γ|B|2. ... Then there must exist for this r a set B such thatg(B) γ1b(B) > αrm2, which implies that |B| > αr/2m and b(B) 6 γ|B|2.

5 Apr 2007: It follows that vi is in particular bounded,i.e. there exists an R such that vi B(R). ... One direction is obvious. For the other, note that if U is an arbitraryopen set, and q = T (p) for p U, then there exists a δ such that p B(

27 Jun 2007: One such assumption is the simplicity of the metric, see, e.g.,[22, 33, 37, 38]. ... Assume that there is a least m 1 such that(2.2) is not true.