2011 Issue 412

  1. Pick $n$ numbers $(n \geq 2)$ from the first hundred natural numbers (from $1$ to $100$) so that the sum of any two distinct numbers is a multiple of $6 .$ What is the largest possible number $n$ so that this can be done?
  2. Given $A=\dfrac{5^{a}}{5^{b+c}}$ and $B=\dfrac{5^{a}+2011}{5^{b+c}+2011}$ where $a, b, c$ are the side lengths of a triangle. Compare $A$ and $B$.
  3. Do there exists three integers $x$, $y$ and $z$ such that $$|x-2005 y|+|y-2007 z|+|z-2009 x|=2011^{x}+2013^{y}+2015^{z} ?$$
  4. Determine the following sum of $2011$ terms $$S=\frac{1}{1^{4}+1^{2}+1}+\ldots+\frac{2011}{2011^{4}+2011^{2}+1}$$
  5. Given a circle $(O),$ a chord $B C$ ($B C$ is not a diameter) and point $A$ moving on the major arc $B C$. Draw a circle $\left(O_{1}\right)$ passing through $B$ and touches $A C$ at $A$, another circle $\left(O_{2}\right)$ passing through $C$ and touches $A B$ at $A .\left(O_{1}\right)$ meets $\left(O_{2}\right)$ at a second point $D$, different from $A$. Prove that line $A D$ always passes through a fixed point.
  6. A quadrilateral $A B C D$ with $A C \perp B D$ is inscribed in a fixed circle $(O ; R)$. Let $p$ be the perimeter of $A B C D$. Prove that $$\frac{A B^{2}}{p-A B}+\frac{B C^{2}}{p-B C}+\frac{C D^{2}}{p-C D}+\frac{D A^{2}}{p-D A} \geq \frac{4 R \sqrt{2}}{3}$$
  7. Solve the system of equations $$\begin{cases}(17-3 x) \sqrt{5-x}+(3 y-14) \sqrt{4-y} &=0 \\ 2 \sqrt{2 x+y+5}+3 \sqrt{3 x+2 y+11} &=x^{2}+6 x+13\end{cases}$$
  8. Prove that the following inequality holds for any triangles $A B C$ $$\cos ^{2} \frac{A-B}{2}+\cos ^{2} \frac{B-C}{2}+\cos ^{2} \frac{C-A}{2} \geq 24 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}.$$
  9. Two circles $\left(C_{1}\right),\left(C_{2}\right)$ are given such that the center $O$ of $\left(C_{2}\right)$ lies on $\left(C_{1}\right) .$ Let $C$, $D$ be their intersection points. Points $A$ and $B$ on $\left(C_{1}\right)$ and $\left(C_{2}\right)$ respectively such that $A C$ touches $\left(C_{2}\right)$ at $C$ and $B C$ touches $\left(C_{1}\right)$ at $C$. The line $A B$ intersects $\left(C_{2}\right)$ at $E$ and $\left(C_{1}\right)$ at $F$. $C E$ meets $\left(C_{1}\right)$ at $G, C F$ meets $G D$ at $H$. Prove that $G O$ intersects $E H$ at the circumcenter of triangle $D E F$.
  10. Let $a_{1}, a_{2} \ldots, a_{n}$ be $n$ positive real numbers such that $$a_{1}^{2}+a_{2}^{2}+\ldots+a_{k}^{2} \leq \frac{k(2 k-1)(2 k+1)}{3}$$ where any $k=\overline{1, n}$. Find the largest possible value of the expression $$P=a_{1}+2 a_{2}+\ldots+n a_{n}.$$
  11. Given a sequence $\left(x_{n}\right)$ such that $$x_{n}=2 n+a \sqrt[3]{8 n^{3}+1}, \forall n=1,2, \ldots$$ where $a$ is any real number.
    a) For what values of $a$ does the sequence has finite limit?
    b) Find $a$ such that $\left(x_{n}\right)$ is eventually increasing.
  12. Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f\left(x^{3}+y^{3}\right)=x^{2} f(x)+y^{2} f(y)$$ where $x, y \in \mathbb{R}$




Mathematics & Youth: 2011 Issue 412
2011 Issue 412
Mathematics & Youth
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