China Girls Mathematical Olympiad 2014

$\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$. The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$...

  1. $\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$. The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$, and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$. If $DE \parallel O_1A$, prove that $DC \perp CO_2$.
  2. Let $x_1,x_2,\ldots,x_n $ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor $ be a permutation of $1,2,\ldots,n$. Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor $ is the largest integer not greater than $x$).
  3. There are $ n$ students; each student knows exactly $d $ girl students and $d $ boy students ("knowing" is a symmetric relation). Find all pairs $ (n,d) $ of integers.
  4. For an integer $m\geq 4,$ let $T_{m}$ denote the number of sequences $a_{1},\dots,a_{m}$ such that the following conditions hold
    i) For all $i=1,2,\dots,m$ we have $a_{i}\in \{1,2,3,4\}$,
    ii) $a_{1} = a_{m} = 1$ and $a_{2}\neq 1$,
    iii) For all $i=3,4\cdots, m, a_{i}\neq a_{i-1}, a_{i}\neq a_{i-2}.$
    Prove that there exists a geometric sequence of positive integers $\{g_{n}\}$ such that for $n\geq 4$ we have that \[ g_{n} - 2\sqrt{g_{n}} < T_{n} < g_{n} + 2\sqrt{g_{n}}.\]
  5. Let $a$ be a positive integer, but not a perfect square; $r$ is a real root of the equation $x^3-2ax+1=0$. Prove that $ r+\sqrt{a}$ is an irrational number.
  6. In acute triangle $ABC$, $AB > AC$. $D$ and $E$ are the midpoints of $AB$, $AC$ respectively. The circumcircle of $ADE$ intersects the circumcircle of $BCE$ again at $P$. The circumcircle of $ADE$ intersects the circumcircle $BCD$ again at $Q$. Prove that $AP = AQ$.
  7. Given a finite nonempty set $X$ with real values, let $f(X) = \frac{1}{|X|} \displaystyle\sum\limits_{a\in X} a$, where $\left\lvert X \right\rvert$ denotes the cardinality of $X$. For ordered pairs of sets $(A,B)$ such that $A\cup B = \{1, 2, \dots , 100\}$ and $A\cap B = \emptyset$ where $1\leq |A| \leq 98$, select some $p\in B$, and let $A_{p} = A\cup \{p\}$ and $B_{p} = B - \{p\}.$ Over all such $(A,B)$ and $p\in B$ determine the maximum possible value of $$(f(A_{p})-f(A))(f(B_{p})-f(B)).$$
  8. Let $n$ be a positive integer, and set $S$ be the set of all integers in $\{1,2,\dots,n\}$ which are relatively prime to $n$. Set $S_1 = S \cap \left(0, \frac n3 \right]$, $S_2 = S \cap \left( \frac n3, \frac {2n}3 \right]$, $S_3 = S \cap \left( \frac{2n}{3}, n \right]$. If the cardinality of $S$ is a multiple of $3$, prove that $S_1$, $S_2$, $S_3$ have the same cardinality.



Balkan,1,Bosonia,1,Brazil,1,Bulgary,1,Canada,1,CentroAmerican,1,CGMO,1,China,5,Cono Sur,2,France,1,Germany,2,Greece,2,IberoAmerican,1,IMO,1,India,2,Indonedia,1,International,40,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,29,Olympic Revenge,1,Paenza,1,Paraguayan,1,Rusia,1,TST,9,Turkey,1,Tuymaada,19,Undergraduate,1,Zhautykov,1,
MATHEMATICAL OLYMPIAD PROBLEMS: China Girls Mathematical Olympiad 2014
China Girls Mathematical Olympiad 2014
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