$show=home

2008 Issue 369

  1. For each integer $n$ greater than $6,$ denote by $A_{n}$ the collection of integers which are less than $n$ and not less than $\dfrac{n}{2} .$ Find $n$ such that there are no perfect square in $A_{n}$.
  2. Let $A B C$ be an acute triangle with $\widehat{B A C}=60^{\circ} . E$ and $F$ are two points on $A C$ and $A B$ respectively such that $\widehat{E B C}=\widehat{F C B}=30^{\circ}$. Prove that $$B F=F E=E C \geq \dfrac{B C}{2}.$$
  3. Find four distinct integers $a, b, c, d$ in the set $\{10 ; 21 ; 37 ; 51\}$ such that $$a b+b c-a d=637.$$
  4. Solve for $x$ $$(x+3) \sqrt{(4-x)(12+x)}=28-x.$$
  5. Let $A B C$ be a triangle with $\widehat{B A C} \neq 45^{\circ}$ and $\widehat{A I O}=90^{\circ}$ where $(O)$ and $(I)$ are its circumcircle and incircle, respectively. Choose a point $D$ on the ray $B C$ such that $B D=A B+A C$. The tangent line through $D$ touches $(O)$ at $E$. The tangent line through $B$ of $(O)$ meets $D E$ at $F$; $C F$ meets $(O)$ at another point denoted by $K$. Let $G$ be the centroid of the triangle $A B C$. Prove that $I G$ is parallel to $E K$.
  6. Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive integers such that the following two properties hold
    • $a_{i}<2008$ for all $i=1,2, \ldots, n$
    • The greatest common divisor of any pair of numbers is greater than $2008$. 
    Prove that $\displaystyle\sum_{i=1}^{n} \frac{1}{a_{i}}<2$.
  7. Prove the following inequality $$(3 a+2 b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \leq \frac{45}{2}$$ where $a, b,$ and $c$ are in the interval $[1 ; 2]$ When does equality occur?
  8. Given a triangle $A B C$ with circumcircle $(O)$ and three sides $B C=a$, $C A=b$, $A B^{\prime}=c .$ Denote by $A_{1}$, $B_{1},$ and $C_{1}$ the midpoints of $B C$, $C A,$ and $A B$ respectively; and $A_{2}$, $B_{2}$, $C_{2}$ are the midpoint of the arcs $\widehat{B C}$ (which does not contain $A$), $\widehat{C A}$ (which does not contains $B$), and $\widehat{A B}$ (which does not contain $C$). Draw the circles $\left(Q_{1}\right),\left(O_{2}\right),$ and $\left(O_{3}\right)$ whose diameters are $A_{1} A_{2}$, $B_{1} B_{2},$ and $C_{1} C_{2}$ respectively. Prove the inequality $$\mathcal{P}_{A/\left(O_{1}\right)}+\mathcal{P}_{B /\left(O_{2}\right)}+\mathcal{P}_{C /\left(O_{3}\right)} \geq \frac{(a+b+c)^{2}}{3}.$$ When does equality occur? 
  9. Find all positive integers $x, y, z, n$ such that $$x !+y !+z !=5 . n !$$ where $k !=1 \times 2 \times \ldots \times k$.
  10. Let $a$ and $b$ be two real numbers in the open interval $(0 ; 4) .$ A sequence $\left(a_{n}\right),$ $(n=0,1, \ldots)$ is constructed by the following recursive formula $$a_{0}=a,\, a_{1}=b,\quad a_{n+2}=\frac{2\left(a_{n+1}+a_{n}\right)}{\sqrt{a_{n+1}}+\sqrt{a_{n}}}.$$ Prove that the sequence $\left(a_{n}\right)$ (n=0,1, \ldots)$ converges and find its limit.
  11. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $$f(f(x))+f(x)=\left(26^{3^{2008}}+\left(26^{32008}\right)^{2}\right) x.$$
  12. Let $S$ denote the surface area of a given tetrahedron. Prove that the sum of areas of the six angle-bisectors of this tetrahedron does not exceed $\dfrac{\sqrt{6}}{2} S$.

$type=three$c=3$source=random$title=oot$p=1$h=1$m=hide$rm=hide

Anniversary_$type=three$c=12$title=oot$h=1$m=hide$rm=hide

Name

2006,1,2007,12,2008,12,2009,12,2010,12,2011,12,2012,12,2013,12,2014,12,2015,12,2016,12,2017,12,2018,11,2019,12,2020,12,2021,6,Anniversary,4,
ltr
item
Mathematics & Youth: 2008 Issue 369
2008 Issue 369
Mathematics & Youth
https://www.molympiad.org/2020/09/2008-issue-369.html
https://www.molympiad.org/
https://www.molympiad.org/
https://www.molympiad.org/2020/09/2008-issue-369.html
true
8958236740350800740
UTF-8
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