2011 Issue 408

  1. How many integers $n$ are there such that $-1964 \leq n \leq 2011$ and the fraction $\dfrac{n^{2}+2}{n+9}$ is reducible?
  2. Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence satisfying the conditions $$a_{2}=3,\, a_{50}=300,\quad a_{n}+a_{n+1}=a_{n+2},\quad \forall n \geq 1 .$$ Find the sum of the first $48$ terms $S=a_{1}+a_{2}+\ldots+a_{48}$.
  3. Let $p$ be a prime number. Let $x, y$ be nonzero natural numbers such that $\dfrac{x^{2}+p y^{2}}{x y}$ is also a natural number. Prove that $$\frac{x^{2}+p y^{2}}{x y}=p+1$$
  4. Solve the system of equations $$\begin{cases}x-2 \sqrt{y+1} &=3 \\ x^{3}-4 x^{2} \sqrt{y+1}-9 x-8 y &=-52-4 x y \end{cases}$$
  5. Let $A B$ be a fixed line segment. Point $M$ is such that $M A B$ is an acute triangle. Let $H$ be the orthocenter of $M A B$, $I$ is the midpoint of $A B$, $D$ is the projection of $H$ onto $MI$. Prove that the product $M I$. $D I$ does not depend on the position of $M$.
  6. Let $a, b, c$ be real numbers such that $\sin a+\sin b+\sin c \geq \dfrac{3}{2}$. Prove the inequality $$\sin \left(a-\frac{\pi}{6}\right)+\sin \left(b-\frac{\pi}{6}\right)+\sin \left(c-\frac{\pi}{6}\right) \geq 0$$
  7. Denote by $[x]$ the largest integer not exceeding $x$. Solve the equation $$x^{2}-(1+[x]) x+2011=0.$$
  8. Let $E$ be the center of the nine-point circle (the Euler's circle) of triangle $A B C$ with edge-lengths $B C=a$, $A C=b$, $A B=c$; $E_{1}$, $E_{2}$, $E_{3}$ are respectively the projections of $E$ onto $B C$, $C A$, $A B$ and let $R$ be the circumradius of triangle $A B C$. Prove that $$\frac{S_{E_{1} E_{2} E_{3}}}{S_{A B C}}=\frac{a^{2}+b^{2}+c^{2}}{16 R^{2}}-\frac{5}{16}$$
  9. Find the minimum value of the expression $$\tan B+\tan C-\tan A \tan A+\tan C-\tan B \quad \tan A+\tan B-\tan C$$ where $A$, $B$, $C$ are three angles of an acute triangle $A B C$ and $C \geq A$
  10. a) Prove that for each positive integer $n,$ the equation $$x+x^{2}+x^{3}+\ldots+2011 x^{2 n+1}=2009$$ has a unique real root.
    b) Let $x_{n}$ be denote the real solution in part a). Prove that $0<x_{n}<\dfrac{2010}{2011}$.
  11. Let $\left(u_{n}\right)$ be a sequence given by $$u_{1}=2011,\quad u_{n+1}=\frac{\pi}{8}\left(\cos u_{n}+\frac{\cos 2 u_{n}}{2}+\frac{\cos 3 u_{n}}{3}\right),\, \forall n \geq 1.$$ Prove that the sequence $\left(u_{n}\right)$ has a finite limit.
  12. Let $A_{1} B_{1} C_{1} D_{1}$ and $A_{2} B_{2} C_{2} D_{2}$ be two squares in opposite direction (that is, if the vertices $A_{1}$, $B_{1}$, $C_{1}$, $D_{1}$ are in clockwise order, then $A_{2}$, $B_{2}$, $C_{2}$, $D_{2}$ are ordered counterclockwise) with centers $O_{1}$, $O_{2}$ suppose that $D_{2}$, $D_{1}$ are respectively in $A_{1} B_{1}$, $A_{2} B_{2}$. Prove that the lines $B_{1} B_{2}$, $C_{1} C_{2}$ and $O_{1} O_{2}$ are concurrent.




Mathematics & Youth: 2011 Issue 408
2011 Issue 408
Mathematics & Youth
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