2010 Issue 392

  1. Does there exist a pair of integers $x$, $y$ such that $$x^{3}-y^{3}=10 \times 10 \times 2010$$
  2. Let $n$ be a natural number, greater than $1$. Prove that $$\frac{1+n}{1+n^{n+1}}>\left(\frac{1+n^{n}}{1+n^{n+1}}\right)^{n}$$
  3. Let $a, b, c, x, y, z$ be positive integers satisfying the conditions $$\begin{cases}\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z} &=2010 \\ a x^{3}=b y^{3}=c z^{3}\end{cases}.$$ Prove that $$x+y+z \geq \frac{3}{670}.$$
  4. Let $A B C$ be a triangle whose sides satisfying the relation $$B C^{2}+A B \cdot A C-A B^{2}=0.$$ Determine the sum $\hat{A}+\dfrac{2}{3} \hat{B}$.
  5. Two orthogonal diameters $A E$ and $B F$ of a circle center $O$ radius $R$ are given. $A$ point $C$ is chosen on the minor arc $E F$. The chord $A C$ intersects $B F$ at $P$ and the chord $B C$ meets $A E$ at $Q$. Find the area of the quadrilateral $A P Q B$ in term of $R$.
  6. Solve the system of equations $$\begin{cases}\sqrt{y^{2}-8 x+9}-\sqrt[3]{x y+12-6 x} &\leq 1 \\ \sqrt{2(x-y)^{2}+10 x-6 y+12}-\sqrt{y} &=\sqrt{x+2}\end{cases}.$$
  7. A quadrilateral $A B C D$ is inscribed in a circle centered $I$ and circumscribed another circle with center at $O$. The diagonals $A C$ and $B D$ intersect at $E$. Prove that $E$, $I$ and $O$ are collinear.
  8. Let $\left(x_{n}\right)$ be a sequence given by $$x_{1}=5, \quad x_{n+1}=x_{n}^{2}-2,\,\forall n \geq 1 .$$ Find
    a) $\displaystyle\lim _{n \rightarrow+\infty} \frac{x_{n+1}}{x_{1} x_{2} \ldots x_{n}}$.
    b) $\displaystyle\lim _{n \rightarrow+\infty}\left(\frac{1}{x_{1}}+\frac{1}{x_{1} x_{2}}+\ldots+\frac{1}{x_{1} x_{2} \ldots x_{n}}\right)$.
  9. Prove that the equation $P(x)=2^{x}$ where $P(x)$ is a polynomial of degree $n,$ has less than $n+1$ roots.
  10. Find all continuous functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the condition $$f(2010 x-f(y))=f(2009 x)-f(y)+x,\, \forall x, y \in \mathbb{R}$$
  11. Let $\left(a_{n}\right)$ be a sequence with $$a_{1}=a_{2}=1,\quad a_{n+2}=a_{n+1}+a_{n},\,\forall n \geq 1.$$ Find all pair of positive integers $a$, $b$, $a<b$ so that $a_{n}-2 n a^{n}$ is a multible of $b$ for any $n \geq 1$.
  12. $I$ and $R$ are the incenter and the circumradius of a gven triangle $A B C$. $I A$, $I B$, $I C$ intersect the circumcircle at $A_{1}$, $B_{1}$, $C_{1}$ respectively. Prove the inequality $$2 R+\frac{L A+I B+I C}{3} \leq L A_{1}+I B_{1}+I C_{1} \leq \frac{5}{2} R+\frac{I A+I B+I C}{6}$$




Mathematics & Youth: 2010 Issue 392
2010 Issue 392
Mathematics & Youth
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