MathDB

6

Part of 2008 IMC

Problems(2)

IMC 2008 Day 1 P6 - Permutation Distances

Source: Problem 6

7/30/2008
For a permutation σ∈Sn \sigma\in S_n with (1,2,…,n)↦(i1,i2,…,in) (1,2,\dots,n)\mapsto(i_1,i_2,\dots,i_n), define D(\sigma) \equal{} \sum_{k \equal{} 1}^n |i_k \minus{} k| Let Q(n,d) \equal{} \left|\left\{\sigma\in S_n : D(\sigma) \equal{} d\right\}\right| Show that when d≥2n d \geq 2n, Q(n,d) Q(n,d) is an even number.
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IMC 2008 Day 2 P6 - Hilbert space

Source: Problem 6

7/28/2008
Let H \mathcal{H} be an infinite-dimensional Hilbert space, let d>0 d>0, and suppose that S S is a set of points (not necessarily countable) in H \mathcal{H} such that the distance between any two distinct points in S S is equal to d d. Show that there is a point y∈H y\in\mathcal{H} such that \left\{\frac{\sqrt{2}}{d}(x\minus{}y): \ x\in S\right\} is an orthonormal system of vectors in H \mathcal{H}.
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