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[Solutions] International Mathematics Competition for University Students 2009

  1. Suppose that $f,g:\mathbb{R}\to \mathbb{R}$ satisfying \[ f(r)\le g(r)\quad \forall r\in \mathbb{Q} \] Does this imply $f(x)\le g(x)$, $\forall x\in \mathbb{R}$ if
    (a) $f$ and $g$ are non-decreasing ?
    (b) $f$ and $g$ are continuous?
  2. Let $A,B,C$ be real square matrices of the same order, and suppose $A$ is invertible. Prove that \[ (A-B)C=BA^{-1}\implies C(A-B)=A^{-1}B \]
  3. In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from $1$ to $n$ and let $a_i$ be the number of friends of the $i^{\text{th}}$ resident. Suppose that \[ \sum_{i=1}^{n}a_i^2=n^2-n \] Let $k$ be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of $k.$
  4. Let $p(z)=a_0+a_1z+a_2z^2+\cdots+a_nz^n$ be a complex polynomial. Suppose that $1=c_0\ge c_1\ge \cdots \ge c_n\ge 0$ is a sequence of real numbers which form a convex sequence. (That is $2c_k\le c_{k-1}+c_{k+1}$ for every $k=1,2,\cdots ,n-1$) and consider the polynomial \[ q(z)=c_0a_0+c_1a_1z+c_2a_2z^2+\cdots +c_na_nz^n .\] Prove that \[ \max_{|z|\le 1}q(z)\le \max_{|z|\le 1}p(z) \]
  5. Let $n$ be a positive integer. An $n$-simplex in $\mathbb{R}^n$ is given by $n+1$ points $P_0, P_1,\cdots , P_n$, called its vertices, which do not all belong to the same hyperplane. For every $n$-simplex $\mathcal{S}$ we denote by $v(\mathcal{S})$ the volume of $\mathcal{S}$, and we write $C(\mathcal{S})$ for the center of the unique sphere containing all the vertices of $\mathcal{S}$.
    Suppose that $P$ is a point inside an $n$-simplex $\mathcal{S}$. Let $\mathcal{S}_i$ be the $n$-simplex obtained from $\mathcal{S}$ by replacing its $i^{\text{th}}$ vertex by $P$. Prove that \[\sum_{j=0}^{n}v(\mathcal{S}_j)C(\mathcal{S}_j)=v(\mathcal{S})C(\mathcal{S}) \]
  6. Let $\ell$ be a line and $P$ be a point in $\mathbb{R}^3$. Let $S$ be the set of points $X$ such that the distance from $X$ to $\ell$ is greater than or equal to two times the distance from $X$ to $P$. If the distance from $P$ to $\ell$ is $d>0$, find $\text{Volume}(S)$.
  7. Suppose $f:\mathbb{R}\to \mathbb{R}$ is a two times differentiable function satisfying $f(0)=1,f^{\prime}(0)=0$ and for all $x\in [0,\infty)$, it satisfies \[ f^{\prime \prime}(x)-5f^{\prime}(x)+6f(x)\ge 0 \] Prove that, for all $x\in [0,\infty)$, \[ f(x)\ge 3e^{2x}-2e^{3x} \]
  8. Let $A,B\in \mathcal{M}_n(\mathbb{C})$ be two $n \times n$ matrices such that \[ A^2B+BA^2=2ABA. \] Prove there exists $k\in \mathbb{N}$ such that \[ (AB-BA)^k=\mathbf{0}_n.\] Here $\mathbf{0}_n$ is the null matrix of order $n$.
  9. Let $p$ be a prime number and $\mathbf{W}\subseteq \mathbb{F}_p[x]$ be the smallest set satisfying the following :
    (a) $x+1\in \mathbf{W}$ and $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$
    (b) For $\gamma_1,\gamma_2$ in $\mathbf{W}$, we also have $\gamma(x)\in \mathbf{W}$, where $\gamma(x)$ is the remainder $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$.
    How many polynomials are in $\mathbf{W}?$
  10. Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S\subseteq \mathbb{M}$, denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$.
    Say that a vector subspace $T\subseteq \mathbb{M}$ is a covering matrix space if \[ \bigcup_{A\in T, A\ne \mathbf{0}} \ker A =\mathbb{R}^p \] Such a $T$ is minimal if it doesn't contain a proper vector subspace $S\subset T$ such that $S$ is also a covering matrix space.
    (a) Let $T$ be a minimal covering matrix space and let $n=\dim (T)$. Prove that \[ \delta(T)\le \dbinom{n}{2}\] (b) Prove that for every integer $n$ we can find $m$ and $p$, and a minimal covering matrix space $T$ as above such that $\dim T=n$ and $\delta(T)=\dbinom{n}{2}$

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MOlympiad: [Solutions] International Mathematics Competition for University Students 2009
[Solutions] International Mathematics Competition for University Students 2009
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https://www.molympiad.net/2017/06/imc-2009-solutions.html
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