Balkan Mathematical Oympiad Team Selection Test 2014

Prove that for $n\ge 2$ the following inequality holds $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left...

  1. Prove that for $n\ge 2$ the following inequality holds $$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$
  2. Solve the following equation in $\mathbb{R}$ $$\left(x-\frac{1}{x}\right)^\frac{1}{2}+\left(1-\frac{1}{x}\right)^\frac{1}{2}=x.$$
  3. From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$.In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$, which intersects $PA$ and $PB$ in $C$ and $D$. Prove that $M$ is the midpoint $CD$.
  4. Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x)f(y)=f(x+y)+xy$$ for all $x,y\in \mathbb{R}$.
  5. Find all non-negative integers $k,n$ which satisfy $$2^{2k+1} + 9\cdot 2^k+5=n^2.$$

COMMENTS

Name

Balkan,1,Bosonia,1,Brazil,1,Bulgary,1,Canada,2,Caucasus,2,CentroAmerican,1,CGMO,1,China,5,Cono Sur,2,France,1,Germany,2,Greece,2,IberoAmerican,1,IMO,1,India,2,Indonedia,1,International,42,Italy,1,Itan,1,Japan,1,JBMO,2,Kazakhstan,1,Korea,2,Macedonia,1,Mediterrane,7,Mediterranean,3,MEMO,1,Mexico,1,Miklós Schweitzer,1,Moldova,1,National,30,Olympic Revenge,1,Paenza,1,Paraguayan,1,Rusia,1,TST,9,Turkey,1,Tuymaada,19,Undergraduate,1,Zhautykov,1,
ltr
item
MATHEMATICAL OLYMPIAD PROBLEMS: Balkan Mathematical Oympiad Team Selection Test 2014
Balkan Mathematical Oympiad Team Selection Test 2014
MATHEMATICAL OLYMPIAD PROBLEMS
http://www.molympiad.ml/2017/10/balkan-mathematical-oympiad-team-selection-test-2014.html
http://www.molympiad.ml/
http://www.molympiad.ml/
http://www.molympiad.ml/2017/10/balkan-mathematical-oympiad-team-selection-test-2014.html
true
3289146460604631361
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