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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 Nov 2007: p’ and ‘q’ beingliterals—even tho’ A XOR B is not self-dual in general.). ... Show—by considering disjunctive normal forms or otherwise—that whenever A is a self-dual formula there is a formulaB such that A is logically equivalent to B B̂.

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.