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[Solutions] International Mathematics Competition for University Students 2011

  1. Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function. A point $x$ is called a shadow point if there exists a point $y\in \mathbb{R}$ with $y>x$ such that $f(y)>f(x).$ Let $a<b$ be real numbers and suppose that
    i) all the points of the open interval $I=(a,b)$ are shadow points;
    ii) $a$ and $b$ are not shadow points.
      Prove that
      a) $f(x)\leq f(b)$ for all $a<x<b;$
      b) $f(a)=f(b).$
    • Does there exist a real $3\times 3$ matrix $A$ such that $\text{tr}(A)=0$ and $A^2+A^t=I?$
      ($\text{tr}(A)$ denotes the trace of $A,\ A^t$ the transpose of $A,$ and $I$ is the identity matrix.)
    • Let $p$ be a prime number. Call a positive integer $n$ interesting if \[x^n-1=(x^p-x+1)f(x)+pg(x)\] for some polynomials $f$ and $g$ with integer coefficients.
      a) Prove that the number $p^p-1$ is interesting.
      b) For which $p$ is $p^p-1$ the minimal interesting number?
    • Let $A_1,A_2,\dots, A_n$ be finite, nonempty sets. Define the function
      \[f(t)=\sum_{k=1}^n \sum_{1\leq i_1<i_2<\dots<i_k\leq n} (-1)^{k-1}t^{|A_{i_1}\cup A_{i_2}\cup \dots\cup A_{i_k}|}.\] Prove that $f$ is nondecreasing on $[0,1].$ ($|A|$ denotes the number of elements in $A.$)
    • Let $n$ be a positive integer and let $V$ be a $(2n-1)$-dimensional vector space over the two-element field. Prove that for arbitrary vectors $v_1,\dots,v_{4n-1} \in V,$ there exists a sequence $1\leq i_1<\dots<i_{2n}\leq 4n-1$ of indices such that $v_{i_1}+\dots+v_{i_{2n}}=0.$
    • Let $(a_n)\subset (\frac{1}{2},1)$. Define the sequence $$x_0=0, \quad x_{n+1}=\frac{a_{n+1}+x_n}{1+a_{n+1}x_n}.$$ Is this sequence convergent? If yes find the limit.
    • An alien race has three genders: male, female and emale. A married triple consists of three persons, one from each gender who all like each other. Any person is allowed to belong to at most one married triple. The feelings are always mutual ( if $x$ likes $y$ then $y$ likes $x$). The race wants to colonize a planet and sends $n$ males, $n$ females and $n$ emales. Every expedition member likes at least $k$ persons of each of the two other genders. The problem is to create as many married triples so that the colony could grow.
      a) Prove that if $n$ is even and $k\geq 1/2$ then there might be no married triple.
      b) Prove that if $k \geq 3n/4$ then there can be formed $n$ married triple (i.e. everybody is in a triple).
    • Calculate $$\sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right).$$
    • Let $f$ be a polynomial with real coefficients of degree $n$. Suppose that $\frac{f(x)-f(y)}{x-y}$ is an integer for all $0 \leq x$.
    • Let $F=A_0A_1...A_n$ be a convex polygon in the plane. Define for all $1 \leq k \leq n-1$ the operation $f_k$ which replaces $F$ with a new polygon $f_k(F)=A_0A_1..A_{k-1}A_k^\prime A_{k+1}...A_n$ where $A_k^\prime$ is the symmetric of $A_k$ with respect to the perpendicular bisector of $A_{k-1}A_{k+1}$. Prove that $$(f_1\circ f_2 \circ f_3 \circ...\circ f_{n-1})^n(F)=F.$$

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    MOlympiad: [Solutions] International Mathematics Competition for University Students 2011
    [Solutions] International Mathematics Competition for University Students 2011
    MOlympiad
    https://www.molympiad.net/2017/06/imc-2011-solutions.html
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