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2011 Issue 406

  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 $y R x$.

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Mathematics & Youth: 2011 Issue 406
2011 Issue 406
Mathematics & Youth
https://www.molympiad.org/2020/09/2011-issue-406.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2011-issue-406.html
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