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

## $show=home 1. Find all pairs of natural number$\left(m.n\right)$satisfying$3^{m}-2^{n}=5$. 2. Given a right triangle$ABC$with the right angle$A$and$AB<AC$. Let$AH$be the altitude from the vertex$A$. On$AC$choose$D$such that$AD=AB$. Let$I$be the midpoint of$BD$. Prove that$\widehat{BIH}=\widehat{ACB}$. 3. Can we cover a square-shape area of the size$3,5{\rm m}\times3,5{\rm m}$by rectangle-shape tiles of the size$25{\rm cm}\times100{\rm cm}$without cutting any tile?. 4. Construct a triangle$ABC$given three lines$d_{a},d_{b}$and$d_{c}$which contain perpendicular bisector of$ABC$(assume that they are concurrent at$O$) and given the length of$AH$where$H$is the orthocenter of$ABC$. 5. Solve the equation $$2010-\sqrt{6+\sqrt{6+\sqrt{2016-x}}}=x.$$ 6. Solve the system of equations $$\begin{cases} \frac{12y}{x} & =3+x-2\sqrt{4y-x}\\ \sqrt{y+3}+y & =x^{2}-x-3 \end{cases}.$$ 7. Let$x,y,z$be real numbers such that$x^{2}+y^{2}+z^{2}=8$. Find the maximum and minimum values of the expression $$H=\left|x^{3}-y^{3}\right|+\left|y^{3}-z^{3}\right|+\left|z^{3}-x^{3}\right|.$$ 8. Given an acute triangle$ABC$with the circumcenter$O$. Let$P$be an arbitrary point on the circumcircle of the triangle$ABC$and$P$os different from$B$and$C$. The bisector of the angles$\widehat{CPA}$and$\widehat{APB}$respectively intersects$CA$and$AB$at$E$and$F$. Let$I$,$L$and$K$respectively be the incenters of the incircles of the triangles$PEF$,$PCA$and$PAB$. Prove that$I$,$K$and$L$are colinear. 9. Find positive integers$x,y$such that$x^{3}+y^{3}=x^{2}+12xy+y^{2}$. 10. Given$n$real numbers$a_{1},a_{2},\ldots,a_{n}\left(n\geq3\right)$satisfying $$a_{1}+a_{2}+\ldots+a_{n}\geq n,\quad a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}\geq n^{2}.$$ Prove that$\max\left\{ a_{1},a_{2},\ldots a_{n}\right\} \geq2$. 11. Consider the following sequence of real numbers$\left(a_{n}\right)$: $$\begin{cases} a_{1} & \geq 0 \\ a_{n+1} & =10^{n}a_{n}^{2},\quad n\geq1 \end{cases}.$$ Find all possible values for$a_{i}$so that${\displaystyle \lim_{n\to\infty}a_{n}=0}$. 12. Given an acute triangle$ABC$. Let$E$and$F$respectively be the perpendicular projecions of$B$and$C$on$AC$and$AB$. Let$I$and$J$respectively be the excenters of the excircles relative to the vertices$F$and$E$of the triangles$AFC$and$AEB$. Assume that$BJ$intersects$CI$at$K$. Choose$Q$on the circumcircle of the triangle$BKC$such that circumcircle of the$BKC$such that$\widehat{AQK}=90^{0}$. Prove that$AQ,BI$and$CJ$are concurrent. ##$type=three$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 464
2016 Issue 464
Mathematics & Youth