2008 Issue 372

  1. Let $$A=\frac{1}{1^{2}}+\frac{1}{2^{3}}+\frac{1}{3^{4}}+\ldots+\frac{1}{2007^{2008}}.$$ Prove that $A$ is not an integer.
  2. Let $P(x)=x^{3}-7 x^{2}+14 x-8 .$ Prove that for every natural number $n,$ there exists a triple of distinct intergers $a_{1}$, $a_{2}$, $a_{3}$ such that the following two conditions are satisfied
    • $\left|a_{i}-a_{j}\right|<5, \forall i, j \in\{1,2,3\}$
    •  $P\left(a_{i}\right) \neq 0$ and $5^{n} \mid P\left(a_{i}\right)$ for all $i \in\{1,2,3\}$
  3. Find all pairs of intergers $x$, $y$ such that $$\sqrt[n]{x+\sqrt[n]{x+\ldots+\sqrt[n]{x}}}=y$$ ($m$ times) where $m$, $n$ are positive intergers which are greater than $2$.
  4. Given $x>2$. Prove the inequality $$\frac{x}{2}+\frac{8 x^{3}}{(x-2)(x+2)^{2}}>9.$$
  5. A pentagon $A B C D E$ is inscribed in a circle with center at $O$ and radius $R$ such that $A B=C D=E A=R$. Let $M$, $N$ be respectively the midpoints of $B C$ and $D E$ Prove that $A M N$ is an equilateral triangle.
  6. Do there exist two distinct positive integers $a$, $b$ such that $b^{n}+n$ is $a$ multiple of $a^{n}+n$ for every postive integer $n ?$.
  7. Let $S$ be the set of all pairs of real numbers $(\alpha, \beta)$ such that the equation $$x^{3}-6 x^{2}+\alpha x-\beta=0$$ has three real roots (not necessarily distinct) and they are all greater than $1$. Find the maximum value of $T=8 \alpha-3 \beta$ for $(\alpha, \beta) \in S$.
  8. Let $A B C$ be an acute triangle and denote by $Q$ the center of its Euler's circle. The circumcircle of $A B C,$ which has radius $R,$ meets $A Q, B Q,$ and $C Q$ respectively at $M$, $N$ and $P$ Prove the inequality $$\frac{1}{Q M}+\frac{1}{Q N}+\frac{1}{Q P} \geq \frac{3}{R}.$$
  9. Find the interger part of $$A=\sqrt[n]{1-\frac{x}{2008}}+\sqrt[m]{1+\frac{x}{2008}}$$ where $x$ is a real number in $[-2008 ; 2008]$, and $m$, $n$ are natural numbers, $m \geq n \geq 2$.
  10. Let $S$ be denote the set of all $n$-tuples $(n>1)$ of real numbers $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ such that $$3\sum_{i=1}^{n} a_i^{2}=502.$$ a) Prove that $\displaystyle \min _{1 \leq \leq j \leq n}\left|a_{j}-a_{i}\right| \leq \sqrt{\frac{2008}{n\left(n^{2}-1\right)}}$.
    b) Give an example of an $n$ -tuple $\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ such that the above condition holds and for which there is an equality in a).
  11. A function $f(x),$ whose domain is the interval $[1 ;+\infty),$ has the following two properties $$f(1)=\frac{1}{2008},\quad f(x)+2007(f(x+1))^{2}=f(x+1),\,\forall x \in[1 ;+\infty).$$ Find the limit $$\lim_{n \rightarrow+\infty}\left(\frac{f(2)}{f(1)}+\frac{f(3)}{f(2)}+\ldots+\frac{f(n+1)}{f(n)}\right).$$
  12. The angle-bisectors $A A_{1}$, $B B_{1}$, $C C_{1}$ of a triangle $A B C$ with perimeter $p$ meet $B_{1} C_{1}$, $C_{1} A_{1},$ and $A_{1} B_{1}$ respectively at $A_{2}$, $B_{2},$ and $C_{2}$. The line through $A_{2}$ and parallel to $B C$ meets $A B$, $A C$ at $A_{3}$, $A_{4}$. Construct the points $B_{3}$, $B_{4}$ and $C_{3}$, $C_{4}$ in a similar way. Prove the inequality $$A B_{4}+B C_{4}+C A_{4}+B A_{3}+C B_{3}+A C_{3} \leq p.$$ When does equality holds?




Mathematics & Youth: 2008 Issue 372
2008 Issue 372
Mathematics & Youth
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