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

## $show=home 1. Write the numbers$1,2,3, \ldots, 2007$in an arbitrary order and let$A$be the resulting number. Can$A+2008^{2007}+2009$be a perfect square?. 2. Consider the following two polynomials $$f(x)=(x-2)^{2008}+(2 x-3)^{2007}+2006 x$$ and $$g(y)=y^{2009}-2007 y^{2008}+2005 y^{2007}.$$ Let$s$be denote the sum of all the coefficients of$f(x)$(after expansion). Find$s$, and the value of$g(s)$. 3. Find all positive integer solutions of the following system of two equations $$\begin{cases} x+y+z &= 15 \\ x^{3}+y^{3}+z^{3} &= 495\end{cases}.$$ 4. Let$a, b, c$be non-negative real numbers such that$a^{2}+b^{2}+c^{2}=1 .$Find the maximum value of the expression $$(a+b+c)^{3}+a(2 b c-1)+b(2 a c-1)+c(2 a b-1).$$ 5. Given a triangle$A B C$where$\widehat{A B C}$is not a right angle. Let$A H$and$A M$denote, the altitude and the median throught vertex$A$. Choose a point$E$on the ray$A B$and$F$on the ray$A C$such that$M E=M F=M A .$Let$K$be reflection point of$H$over$M .$Prove that the four points$E$,$M$,$K$and$F$lie on a single circle. 6. Solve for$x$$$\sqrt{x+\sqrt{x^{2}-1}}=\frac{9 \sqrt{2}}{4}(x-1) \sqrt{x-1}.$$ 7. Prove that in any acute triangle$A B C$, the following inequality holds $$\frac{\tan A}{\tan B}+\frac{\tan B}{\tan C}+\frac{\tan C}{\tan A} \geq \frac{\sin 2 A}{\sin 2 B}+\frac{\sin 2 B}{\sin 2 C}+\frac{\sin 2 C}{\sin 2 A}$$ 8. The incircle of a triangle$A B C$meets$B C$,$C A$and$A B$respectively at$A_{1}$,$B_{1}$,$C_{1}$. Let$p$,$S$,$R$be respectively, half of the perimeter, the area and the circumradius of$A B C$. Let$p_{1}$be half of the perimeter of$A_{1} B_{1} C_{1}$. Prove the inequality $$p_{1}^{2} \leq \frac{p S}{2 R}.$$ When does equality occur? 9. Let$A(A \subset \mathbb{N})$be a non-empty set satisfying the condition: If$a \in A$then$4 a$and$[\sqrt{a}]$are also in$A([x]$is the integer part of$x$). Prove that$A=\mathbb{N}$. 10. Let$a$be a natural number which is greater than$3$and consider the sequence$\left(u_{n}\right)(n=1,2, \ldots)$defined inductively by$u_{1}=a$and $$u_{n+1}=u_{n}-\left[\frac{u_{n}}{2}\right]+1,\,\forall n=1,2, \ldots.$$ Prove that there exists$k \in \mathbb{N}^{*}$such that$u_{n}=u_{k}$for all$n \geq k$. 11. Find all polynomials with real coefficients$P(x)$,$Q(x)$and$R(x)$such that $$\sqrt{P(x)}-\sqrt{Q(x)}=R(x),\,\forall x.$$ 12. Let$A B C D$be a tetrahedron with the centroid$G$and the circumradius$R$. Prove that $$G A+G B+G C+G D+4 R \geq \frac{2}{\sqrt{6}}(A B+A C+A D+B C+C D+D B).$$ ##$type=three$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: 2008 Issue 373
2008 Issue 373
Mathematics & Youth