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.




Mathematics & Youth: 2009 Issue 382
2009 Issue 382
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy