International Mathematics Competition for University Students 2008 Solutions

  1. Find all continuous functions $f: \mathbb{R}\to \mathbb{R}$ such that \[ f(x)-f(y)\in \mathbb{Q}\quad \text{ for all }\quad x-y\in\mathbb{Q} \]
  2. Denote by $\mathbb{V}$ the real vector space of all real polynomials in one variable, and let $\gamma :\mathbb{V}\to \mathbb{R}$ be a linear map. Suppose that for all $f,g\in \mathbb{V}$ with $\gamma(fg)=0$ we have $\gamma(f)=0$ or $\gamma(g)=0$. Prove that there exist $c,x_0\in \mathbb{R}$ such that \[ \gamma(f)=cf(x_0)\quad \forall f\in \mathbb{V}\]
  3. Let $p$ be a polynomial with integer coefficients and let $a_1<a_2<\cdots <a_k$ be integers. Given that $p(a_i)\ne 0\forall\; i=1,2,\cdots, k$.
    (a) Prove $\exists\; a\in \mathbb{Z}$ such that \[ p(a_i)\mid p(a)\;\;\forall i=1,2,\dots ,k \] (b) Does there exist $a\in \mathbb{Z}$ such that \[ \prod_{i=1}^{k}p(a_i)\mid p(a) \]
  4. We say a triple of real numbers $ (a_1,a_2,a_3)$ is better than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers special when they are nonnegative and their sum is $ 1$.
    For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| = n$, such that any special triple is bettered by at least one element of $ S$?
  5. Does there exist a finite group $ G$ with a normal subgroup $ H$ such that $ |\text{Aut } H| > |\text{Aut } G|$? Disprove or provide an example. Here the notation $ |\text{Aut } X|$ for some group $ X$ denotes the number of isomorphisms from $ X$ to itself.
  6. For a permutation $ \sigma\in S_n$ with $ (1,2,\dots,n)\mapsto(i_1,i_2,\dots,i_n)$, define \[ D(\sigma) = \sum_{k = 1}^n |i_k - k| .\] Let \[ Q(n,d) = \left|\left\{\sigma\in S_n : D(\sigma) = d\right\}\right|. \] Show that when $ d \geq 2n$, $ Q(n,d)$ is an even number.
  7. Let $ n, k$ be positive integers and suppose that the polynomial $ x^{2k}-x^k+1$ divides $ x^{2n}+x^n+1$. Prove that $ x^{2k}+x^k+1$ divides $ x^{2n}+x^n+1$.
  8. Two different ellipses are given. One focus of the first ellipse coincides with one focus of the second ellipse. Prove that the ellipses have at most two points in common.
  9. Let $ n$ be a positive integer. Prove that $ 2^{n-1}$ divides \[ \sum_{0\leq k < n/2} \binom{n}{2k+1}5^k.\]
  10. Let $ \mathbb{Z}[x]$ be the ring of polynomials with integer coefficients, and let $ f(x), g(x) \in\mathbb{Z}[x]$ be nonconstant polynomials such that $ g(x)$ divides $ f(x)$ in $ \mathbb{Z}[x]$. Prove that if the polynomial $ f(x)-2008$ has at least 81 distinct integer roots, then the degree of $ g(x)$ is greater than 5.
  11. Let $ n$ be a positive integer, and consider the matrix $ A = (a_{ij})_{1\leq i,j\leq n}$ where $ a_{ij} = 1$ if $ i+j$ is prime and $ a_{ij} = 0$ otherwise. Prove that $ |\det A| = k^2$ for some integer $ k$.
  12. Let $ \mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$, and suppose that $ S$ is a set of points (not necessarily countable) in $ \mathcal{H}$ such that the distance between any two distinct points in $ S$ is equal to $ d$. Show that there is a point $ y\in\mathcal{H}$ such that \[ \left\{\frac{\sqrt{2}}{d}(x-y): \ x\in S\right\}\] is an orthonormal system of vectors in $ \mathcal{H}$.

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