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

## $show=home 1. Given the following increasing sequence$1,3,5,7,9,\ldots$where all the digits of all the terms are odd. Find the$2016^{{\rm th}}$term of the above sequence. 2. Given an acute triangle$ABC$and the median$AM$. Draw$BH\perp AC$. The line which goes through$A$and is perpendicular to$AM$intersects$BH$at$E$. On the opposite ray of the ray$AE$choose$F$so that$AE=AF$. Prove that$CF\perp AB$. 3. Let$a,b$and$c$be three different nonzero rational numbers such that $$\left(\frac{a}{b-c}\right)^{2}+\left(\frac{b}{c-a}\right)^{2}+\left(\frac{c}{a-b}\right)^{2}\leq2.$$ Prove that $$\sqrt{\left(\frac{b-c}{a}\right)^{2}+\left(\frac{c-a}{b}\right)^{2}+\left(\frac{c-a}{b}\right)^{2}}$$ is a rational number. 4. Given a pentagon$ABCDE$with$\widehat{ACB}=\widehat{ADE}=90^{0}$and$\widehat{CAB}=\widehat{DAE}$. Let$O$be the intersection of$BD$and$CE$. Prove that$AO\perp DC$. 5. Solve the equation $$x^{4}-2x^{3}+\sqrt{2x^{3}+x^{2}+2}-2=0.$$ 6. Given three real numbers$x,y$and$z$such that$x+y+z=3$and$xy+yz+zx=-9$. Find the maximum and minimum values of$xyz$. 7. Let$x,y$and$x$be real numbers satisfying$x^{2}+y^{2}+z^{2}=1$. Find the maximum value of the expression $$P=\left(x^{2}-yz\right)\left(y^{2}-zx\right)\left(z^{2}-xy\right).$$ 8. Let$p$and$R$respectively be the semiperimeter and the circuradius of the circumcircle of a triangle$ABC$. Prove that $$\left(\frac{p}{3R}\right)^{2}+\left(\frac{3R}{p}\right)^{2}\geq\frac{25}{12}.$$ When does the equality happen?. 9. Find all real numbers$k>-2$such that the inequality $$\frac{x}{x+y+kz}+\frac{y}{y+z+kx}+\frac{z}{z+x+ky}\geq\frac{3}{k+2}$$ holds for all nonnegative$x,y$and$z$. 10. Find all natural numbers$x$and$y$such that $$\left(\sqrt{x}-1\right)\left(\sqrt{y}-1\right)=2017.$$ 11. Determine the sequence$\left(x_{n}\right)$as follows $$x_{1}=1;\quad x_{n+1}=\frac{n+1}{n+2}x_{n}+n^{2},\quad n=1,2,\ldots$$ Find${\displaystyle \lim_{n\to\infty}\left(\frac{\sqrt{x_{n}}}{n+1}\right)}$. 12. Assume that$ABC$is a triangle inscribed in a circle$\left(O\right)$and$AD,BE,CF$are its altitudes. The line through$AO$intersects$BC$at$A'$. Let$M$be the midpoint of$BC$and let$S$be the intersection between two tangent lines of$\left(O\right)$at$B$and$C$, The line through$EF$intersects$SD$and$SA'$respectively at$I$and$J$. Show that$BC$is tangent to the circumcircle of the triangle$IJM$at$M$. ##$type=three$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: 2016 Issue 467
2016 Issue 467
Mathematics & Youth