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2009 Issue 388

  1. Prove that a number of the form $(\overline{33 \ldots 3})^{2}$ with $k$ $(k>0)$ digits $3$ can always be written as the difference between of the number whose digits are 1 and the one whose digits are $2$.
  2. Find all real numbers $a$ such that $|3 a-2| \leq 1$ and $A=\dfrac{3 a-1}{4 a^{4}+a^{2}}$ is an integer.
  3. Find the greatest value of the expression $$P=3 \sqrt{x}+8 \sqrt{y}$$ where $x, y$ are two non-negative numbers satisfying $17 x^{2}-72 x y+90 y^{2}-9=0$
  4. Solve the equation $$\sqrt{x-2}+\sqrt{4-x}+\sqrt{2 x-5}=2 x^{2}-5 x$$
  5. From a point $A$ outside a given circle witlı center $O$, construct two tangent lines $A B$ and $A C$ with $B, C$ are the tangency points. On $O B$, choose $N$ such that $B N=2 O N$. The perpendicular bisector of the line segment $C N$ meets $O A$ at $M .$ Determine the ratio $\dfrac{A M}{A O}$
  6. Find the least value of the expression $$A=\frac{1}{a^{4}(b+1)(c+1)}+\frac{1}{b^{4}(c+1)(a+1)}+\frac{1}{c^{4}(a+1)(b+1)}$$ where $a, b, c$ are positive real numbers satisfying $a b c=1$.
  7. Let $a, b$ be two real numbers in the interval $(0 ; 1) .$ A sequence $\left(u_{n}\right)$ $(n=0,1,2, \ldots)$ is given by $$u_{0}=a, u_{1}=b,\quad u_{n+2}=\frac{1}{2010} \cdot u_{n+1}^{4}+\frac{2009}{2010} \sqrt[4]{u_{n}},\,\forall n \in \mathbb{N}.$$ Prove that $\left(u_{n}\right)$ has a finite limit, and find that limit.
  8. Let $l_a$, $l_b$, $l_c$ be respectively the lengths of the interior angle-bisectors from the three vertices $A$, $B$, $C$ of a triangle $A B C$. Let $R$ be its circumradius. Prove that $$\frac{l_{a}+l_{b}+l_{c}}{R} \leq 2\left(\cos \frac{A}{2} \cos \frac{B}{2}+\cos \frac{B}{2} \cos \frac{C}{2}+\cos \frac{C}{2} \cos \frac{A}{2}\right).$$ When does the equality occur?
  9. Let $A B$ be a fixed chord on a given circle $(O) .$ A point $C$ moves on the circle and let $M$ be a point on the zigzag $A C B$ (consisting of two line segments $A C$ and $C B)$ such that it divides this zigzag into two parts of equal length. Find the locus of $M$ when $C$. moves around the circle $(O)$.
  10. Let $m, n, d$ be positive integers such that $m<n$ and $\gcd(d, m)=1$, $\gcd(d, n)=1$. (Here, $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b$.) Prove that in an arithmetic progression of $n$ terms with common difference $d$, there always exist two distinct terms whose product is a multiple of $m n$.
  11. A function $f: \mathbb{R} \rightarrow \mathbb{R}$ is given with the following properties
    • $f(0)=1$
    • $f(x) \leq 1,\, \forall x \in \mathbb{R}$
    • $\displaystyle f\left(x+\frac{11}{24}\right)+f(x)=f\left(x+\frac{1}{8}\right)+f\left(x+\frac{1}{3}\right)$
    Put $\displaystyle F(x)=\sum_{n=0}^{2009} f(x+n)$. Find $F(2009)$
  12. Given two positive degrees polynomials with real coefficients $$P(x)=x^{n}+a_{1} x^{n-1}+\ldots+a_{n-1} x+a_{n}$$ and $$Q(x)=x^{m}+b_{1} x^{m-1}+\ldots+b_{m-1} x+b_{m}$$ where $Q(x)$ has exactly $m$ real roots and $P(x)$ is divisible by $Q(x)$. Prove that if there exists a $k \in\{1,2, \ldots, m\}$ such that $\left|b_{k}\right|>C_{m}^{k} .2009^{k},$ then there also exists an $i \in\{1,2, \ldots, n\}$ such that $|a,|>2008$

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Mathematics & Youth: 2009 Issue 388
2009 Issue 388
Mathematics & Youth
https://www.molympiad.org/2020/09/2009-issue-388.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2009-issue-388.html
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