# TÀI LIỆU TOÁN - WWW.MOLYMPIAD.NET - ĐỀ THI TOÁN

## $show=home 1. Compare$\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. ##$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide ## Anniversary_$type=three$c=12$title=oot$h=1$m=hide\$rm=hide

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Mathematics & Youth: 2009 Issue 382
2009 Issue 382
Mathematics & Youth