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2009 Issue 382

  1. Compare $\dfrac{2009}{2008^{2}}$ with the sum (consisting of $2010$ terms) $$\frac{1}{2009}+\frac{2}{2009^{2}}+\frac{3}{2009^{3}}+\ldots+\frac{2009}{2009^{2009}}+\frac{2010}{2009^{2010}}$$
  2. Find a root of the polynomial $P(x)=x^{3}+a x^{2}+b x+c$ given that it has at least one root and $a+2 b+4 c=-\dfrac{1}{2}$.
  3. Let $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$, $a_{6}$, $a_{7}$, $a_{8}$, $a_{9}$ be non negative real numbers whose sum equals $1$. Put $S_{k}=a_{k}+a_{k+1}+a_{k+2}+a_{k+3}$ $(k=1,2, \ldots, 6)$. Determine the smallest possible value of $$M=\max \left\{S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\right\}.$$
  4. Let $m$, $n$, $a$, $b$ and $c$ be real numbers such that the following conditions hold $$\begin{cases}m^{1000}+n^{1000} &=a \\ m^{2000}+n^{2000} &=\dfrac{2 b}{3}\\ m^{5000}+n^{5000} &=\dfrac{c}{36}\end{cases}.$$ Find a formula relating $a$, $b$ and $c$ which does not involve $m$, $n$.
  5. Let $A H$ be the altitude from $A$ of a triangle $A B C$. Choose a point $D$ on the half-plane created by $B C$ which contains $A$ such that $D B=D C=\dfrac{A B}{\sqrt{2}}$. Prove that the lengths of the line segments $B D$, $D H$ and $H A$ are the side lengths of a right triangle.
  6. Determine the maximum possible value of $x^{2}+y^{2}$ where $x$ and $y$ are two integers chosen arbitrarily within the interval $[-2009 ; 2009]$ such that $$\left(x^{2}-2 x y-y^{2}\right)^{2}=4.$$
  7. Consider two polynomials with real coefficients $$P(x)=x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}$$ and $Q(x)=x^{2}+x+2009$. Given that $P(x)$ has $n$ distinct real roots but $P(Q(x))$ does not have any real solution. Prove that $P(2009)>\dfrac{1}{4^{n}}$.
  8. Let $ABCDEF$ be a regular hexagon and let $G$ be the midpoint of $B F$. Choose a point $I$ on $B C$ such that $B I=B G$. Let $H$ be a point on $I G$ such that $\widehat{C D H}=45^{\circ}$ and $K$ is a point on $E F$ such that $\widehat{D K E}=45^{\circ}$. Prove that $D H K$ is an equilateral triangle.

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Mathematics & Youth: 2009 Issue 382
2009 Issue 382
Mathematics & Youth
https://www.molympiad.org/2020/09/2009-issue-382.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2009-issue-382.html
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