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 $|a,|>2008$




Mathematics & Youth: 2009 Issue 388
2009 Issue 388
Mathematics & Youth
Loaded All Posts Not found any posts VIEW ALL Readmore Reply Cancel reply Delete By Home PAGES POSTS View All RECOMMENDED FOR YOU LABEL ARCHIVE SEARCH ALL POSTS Not found any post match with your request Back Home Sunday Monday Tuesday Wednesday Thursday Friday Saturday Sun Mon Tue Wed Thu Fri Sat January February March April May June July August September October November December Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec just now 1 minute ago $$1$$ minutes ago 1 hour ago $$1$$ hours ago Yesterday $$1$$ days ago $$1$$ weeks ago more than 5 weeks ago Followers Follow THIS CONTENT IS PREMIUM Please share to unlock Copy All Code Select All Code All codes were copied to your clipboard Can not copy the codes / texts, please press [CTRL]+[C] (or CMD+C with Mac) to copy