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

## $show=home 1. Let$S(n)$be denote the sum of the digits of$n$. Find a positive integer$n$such that$S(n)=n^{2}-2009 n+11$. 2. Inside a square$A B C D$thoose two points$P$,$Q$such that$B P$and$D Q$are parallel and$B P^{2}+D O^{2}=P Q^{2}$. Find the measure of the angle$P A Q$. 3. Compare$2008$with the sum$S$of$2009$terms $$S=\frac{2008+2007}{2009+2008}+\frac{2008^{2}+2007^{2}}{2009^{2}+2008^{2}}+\ldots +\frac{2008^{2009}+2007^{2009}}{2009^{2009}+2008^{2009}}.$$ 4. Let$a$,$b$,$c$be three positive numbers such that$a+b+c=1$. Find the least value of the expression $$P=\frac{9}{1-2(a b+b c+c a)}+\frac{2}{a b c}.$$ 5. Let$B$,$C$be two fixed points on the circle$(\omega)$such that$B C$does not pass through the center of$(\omega) .$On the major arc$B C$, choose a point$A$differs from$B$and$C$. Another point$M$moves on the line segment$B C$. The lines passing through$M$and parallel to$A B$,$A C$intersect$A C$and$A B$at$F$and$E$, respectively. When$M$moves on the line segment$B C$and for each point$A$, let$x$be the least possible length of$EF$. Find the positions of$A$and$M$such that$x$is greatest possible. 6. Find the greatest and the least value of the expression $$A=\frac{y^{2}}{25}+\frac{t^{2}}{144}$$ where$x$,$y$,$z$,$t$satisfy the system of equations $$\begin{cases}x^{2}+y^{2}+2 x+4 y-20 &=0 \\ t^{2}+z^{2}-2 t-143 &= 0\\ x t+y z-x+t+2 z-61 &\geq 0\end{cases}$$ 7. Consider the sequence$\left(u_{n}\right)$defined as follow $$u_{0}=9,\, u_{1}=161,\quad u_{n}=18 u_{n-1}-u_{n-2},\,\forall n=2,3, \ldots$$ Prove that for any$n$,$\dfrac{u_{n}^{2}-1}{5}$is always a perfect square. 8. Let$P$be an arbitrary point inside a given triangle$A B C$. Let$A'$,$B'$,$C'$be the orthogonal projection of$P$on$B C$,$C A$,$A B$respectively. Let$I$be the incenter and$r$be the inradius of the triangle$A B C$. Find the least value of the expression $$P A^{\prime}+P B^{\prime}+P C^{\prime}+\frac{P I^{2}}{2 r}$$ ##$type=three$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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2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
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Mathematics & Youth: 2009 Issue 385
2009 Issue 385
Mathematics & Youth