2011 Issue 409

  1. Without taking common denominator, find the integer $x$ given that $$\left(\frac{2009}{2010}+\frac{2010}{2011}+\frac{2011}{2009}\right) \cdot(x-2011)>3 x-6033$$
  2. In a triangle $A B C$, $M$, $N$, $P$ are midpoints of sides $B C$, $C A$ and $A B$ respectively. Choose the points $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$ on the opposite rays of rays $B A$, $C A$, $C B$, $A B$, $A C$ and $B C$ respectively so that $B A_{1}=C A_{2}$, $C B_{1}=A B_{2}$, $A C_{1}=B C_{2} . A_{0}, B_{0}, C_{0}$ are midpoints of $A_{1} A_{2}, B_{1} B_{2}, C_{1} C_{2}$ respectively. Prove that the lines $A_{0} M, B_{0} N, C_{0} P$ meet at a common point.
  3. Let $b$ be a positive integer with the following properties
    • $b$ equals a sum of three squares.
    • $b$ possess a divisor of the form $a=3 k^{2}+3 k+1$ $(k \in \mathbb{N})$.
    Prove that $b^{n}$ is also a sum of three squares for any positive integer $n$.
  4. Given three non-negative numbers $a, b, c$, prove the inequality $$a+b+c \geq \frac{a-b}{b+2}+\frac{b-c}{c+2}+\frac{c-a}{a+2}.$$ When does equality hold?
  5. Let $A B C$ be a triangle with circumcircle $(O)$, $I$ is the midpoint of side $B C$. $M$ is chosen on $I C$ (differ from both $C$ and $I$). $A M$ meets $(O)$ at $D$. Point $E$ is on $B D$ such that $\widehat{B M E}=\widehat{M A I}$. $E M$ and $D C$ intersect at $F$. Prove that $$\frac{C F}{C D}=\frac{B E}{B D}$$
  6. Solve for $x$ $$\sqrt{\frac{x+2}{2}}-1=\sqrt[3]{3(x-3)^{2}}+\sqrt[3]{9(x-3)}$$
  7. Triangle $A B C$ is inscribed in a fixed circle $(O)$. The medians from $A$, $B$, $C$ meets $(O)$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. Which triangle makes the value of the expression $$p=\frac{A A_{1}^{2}+B B_{1}^{2}+C C_{1}^{2}}{A B^{2}+B C^{2}+C A^{2}}$$ minimum possible?
  8. In a triangle $A B C$, prove that $$\frac{\sin A \cdot \sin B}{\sin ^{2} \frac{C}{2}}+\frac{\sin B \cdot \sin C}{\sin ^{2} \frac{A}{2}}+\frac{\sin C \cdot \sin A}{\sin ^{2} \frac{B}{2}} \geq 9$$
  9. Let $A B C$ be a triangle with points $A'$, $B'$, $C'$ on sides $B C$, $C A$ and $A B$ respectively such that $$\frac{A^{\prime} B}{A^{\prime} C}=\frac{B^{\prime} C}{B^{\prime} A}=\frac{C^{\prime} A}{C^{\prime} B}.$$ $A A^{\prime}$ and $B B^{\prime}$ meet at $D$, $B B$ ' meets $C C$ ' at $E$ and $F$ is the intersection of $C C'$ and $A A '$. Parallel lines to $A A'$, $B B'$, $C C'$ through point $O$ in the interior of $A B C$ meet $B C$, $C A$, $A B$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. Prove that for any point $M$ $$A D\left(M A_{1}-O A_{1}\right)+B E\left(M B_{1}-O B_{1}\right)+C F\left(M C_{1}-O C_{1}\right) \geq 0$$
  10. Given $P=(n+1)^{7}-n^{7}-1$ $(n \in \mathbb{N})$. Prove that there are infinitely many natural numbers $n$ so that $P$ is a perfect square.
  11. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, continuous on $\mathbb{R}$ such that $$f(x y)+f(x+y)=f(x y+x)+f(y),\, \forall x, y \in \mathbb{R}.$$
  12. $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function with the following properties
    • $f(1)=2011$,
    • $f(x+1) f(x)=(f(x))^{2}+f(x)-1, \forall x \in \mathbb{R}$.
    Let $\displaystyle S_{1}=\sum_{i=1}^{n} \frac{1}{f(i)-1}$, $\displaystyle S_{2}=\sum_{i=1}^{n} \frac{1}{f(i)+1}$. Find $\displaystyle\lim_{n \rightarrow+\infty}\left(S_{1}+S_{2}\right)$.




Mathematics & Youth: 2011 Issue 409
2011 Issue 409
Mathematics & Youth
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