Tuymaada Mathematical Olympiad 2003

A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares. What maximum number of...

  1. A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares. What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices?
  2. In a quadrilateral $ABCD$ sides $AB$ and $CD$ are equal, $\angle A=150^\circ$, $\angle B=44^\circ$, $\angle C=72^\circ$. Perpendicular bisector of the segment $AD$ meets the side $BC$ at point $P$. Find $\angle APD$.
  3. Alphabet $A$ contains $n$ letters. $S$ is a set of words of finite length composed of letters of $A$. It is known that every infinite sequence of letters of $A$ begins with one and only one word of $S$. Prove that the set $S$ is finite.
  4. Find all continuous functions $f(x)$ defined for all $x>0$ such that for every $x$, $y > 0$ \[ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) . \]
  5. Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$ \[\begin{align}&\left(\frac{1}{\sin\alpha_{1}}+\frac{1}{\sin\alpha_{2}}+\ldots+\frac{1}{\sin\alpha_{n}}\right)\left(\frac{1}{\cos\alpha_{1}}+\frac{1}{\cos\alpha_{2}}+\ldots+\frac{1}{\cos\alpha_{n}}\right)\\\leq&\left(\frac{1}{\sin2\alpha_{1}}+\frac{1}{\sin2\alpha_{2}}+\ldots+\frac{1}{\sin2\alpha_{n}}\right)^{2}.\end{align}\]
  6. Which number is bigger : the number of positive integers not exceeding 1000000 that can be represented by the form $2x^{2}-3y^{2}$ with integral $x$ and $y$ or that of positive integers not exceeding 1000000 that can be represented by the form $10xy-x^{2}-y^{2}$ with integral $x$ and $y?$
  7. In a convex quadrilateral $ABCD$ we have $AB\cdot CD=BC\cdot DA$ and $2\angle A+\angle C=180^\circ$. Point $P$ lies on the circumcircle of triangle $ABD$ and is the midpoint of the arc $BD$ not containing $A$. It is known that the point $P$ lies inside the quadrilateral $ABCD$. Prove that $\angle BCA=\angle DCP$
  8. Given are polynomial $f(x)$ with non-negative integral coefficients and positive integer $a.$ The sequence $\{a_{n}\}$ is defined by $a_{1}=a,$ $a_{n+1}=f(a_{n}).$ It is known that the set of primes dividing at least one of the terms of this sequence is finite. Prove that $f(x)=cx^{k}$ for some non-negative integral $c$ and $k.$

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: Tuymaada Mathematical Olympiad 2003
Tuymaada Mathematical Olympiad 2003
MATHEMATICAL OLYMPIAD PROBLEMS
http://www.molympiad.ml/2017/10/tuymaada-mathematical-olympiad-2003.html
http://www.molympiad.ml/
http://www.molympiad.ml/
http://www.molympiad.ml/2017/10/tuymaada-mathematical-olympiad-2003.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