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

## $show=home 1. Let$m$be a posivite interger number. Determine the decimal digits$x$and$y(x \neq 0)$such that the number$A=\overline{x y 5}+100 m(m+5)$is a perfect square. 2. Which number is greater?$A=4023^{2}$or$B$where $$\frac{B}{12}=\frac{\left(3^{4}+3^{2}+1\right)\left(5^{4}+5^{2}+1\right) \ldots\left(2011^{4}+2011^{2}+1\right)}{\left(2^{4}+2^{2}+1\right)\left(4^{4}+4^{2}+1\right) \ldots\left(2010^{4}+2010^{2}+1\right)}$$ 3. Let$A B C$be an acute triangle. Find the integer part of $$M=\frac{\sin A}{\sin B+\sin C}+\frac{\sin B}{\sin C+\sin A}+\frac{\sin C}{\sin A+\sin B}$$ 4. Solve the system of equations $$\begin{cases} 2 x^{2}+3 x y &=3 y-13 \\ 3 y^{2}+2 x y &=2 x+11 \end{cases}$$ 5. The inscribed circle of a triangle$A B C(A B \neq A C)$meets edges$A B$,$B C$,$C A$at$E$,$D$and$F$respectively.$D P$is the altitude from$D$onto$E F$. The perpendicular bisector of$B C$cuts line$E F$at$Q$. Prove that the quadrilateral$B P Q C$is cyclic. 6. Find the minimum value of the expression $$P=\sqrt{2 x^{2}+2 y^{2}-2 x+2 y+1}+\sqrt{2 x^{2}+2 y^{2}+2 x-2 y+1}+\sqrt{2 x^{2}+2 y^{2}+4 x+4 y+4}$$ 7. Solve the equation$f(g(x))+g(2+f(x))=23$where $$f(x)=\frac{x^{2}}{2}-4 x+9 ;\quad g(x)=\left\{\begin{array}{l}14 & \text { iff } x \geq 3 \\ 2^{x}+\dfrac{12}{5-x} & \text { iff } x<3\end{array}\right.$$ 8. Let$A B C$be a triangle;$h_{a}$,$h_{b}$,$h_{c}$are the lengths of the three altitudes;$R$and$r$are its circumradius and inradius, respectively. Prove the inequality $$h_{a}+h_{b}+h_{c}-9 r \leq 2(R-2 r)$$ 9. In convex quadrilateral$A B C D,$the circumcircles of triangles$A B C$,$A C D$cut the rays$O D$,$O B$at$E$,$F$respectively. Prove that quadrangle$A B C D$is circumscribed if and only if the quadrangle$A E C F$is circumscribed. 10. A sequence$\left(u_{n}\right)$is given by $$u_{1}=a,\quad u_{n+1}=\frac{k+u_{n}}{1-u_{n}},\,k>0,\,\forall n \in \mathbb{N}^{*}.$$ If$u_{13}=u_{1}$, find all possible values of$k$. 11. Let$\left(x_{n}\right)$be a sequence given by $$x_{1}=0,\quad x_{n+1}=\left(\frac{1}{27}\right)^{x_{n}},\,\forall n \in \mathbb{N}^{*} .$$ Prove that the sequence converges and find its limit. 12. Let$X=\{0,1,2,3, \ldots, n\}$where$n$is a positive number. If$x=\left(x_{1}, x_{2}, x_{3}\right)$,$y=\left(y_{1}, y_{2}, y_{3}\right)$in$X^{3},$denote$x R y$if$x_{k} \leq y_{k}$for$k=1,2,3 .$Find the smallest positive integer$p$such that any subset$A$of$X^{3}$containing$p$elements has the following property: If$x$,$y$are any two elements in$A,$then either$x R y$or$y R x$. ##$type=three$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: 2011 Issue 406
2011 Issue 406
Mathematics & Youth