\nopagenumbers
\def\q#1. {{\bf #1. }}
\input amssym.def
\input amssym.tex
\def\geq{\geqslant}
\def\leq{\leqslant}
\centerline{\bf ``TUYMAADA--2003''}
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\centerline{\it Senior league}
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\centerline{\tt First day}
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\q1. 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?
\q2. 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 $\angle APD$.
\q3. 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.
\q4. Find all continuous functions $f(x)$ defined for all $x>0$
such that for every $x$, $y > 0$
$$ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)=
f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) .$$
\medskip
\centerline{\tt Second day}
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\q5. Prove that for every $\alpha_1$, $\alpha_2$, \dots, $\alpha_n$
in the interval $(0,\pi/2)$
$$\left( {1\over \sin \alpha_1} + {1\over \sin \alpha_2} + \ldots
+ {1\over \sin \alpha_n} \right)
\left( {1\over \cos \alpha_1} + {1\over \cos \alpha_2} + \ldots
+ {1\over \cos \alpha_n} \right) \leq $$
$$\leq 2 \left( {1\over \sin 2\alpha_1} + {1\over \sin 2\alpha_2} + \ldots
+ {1\over \sin 2\alpha_n} \right)^2 .$$
\q6. Which number is bigger: the number of positive integers not exceeding
1~000~000 that can be represented by the form $2x^2 - 3y^2$ with
integral $x$ and $y$ or that of positive integers not exceeding
1~000~000 that can be represented by the form $10xy - x^2 - y^2$ with
integral $x$ and $y$?
\q7. In a convex quadrilateral $ABCD$ \ \ $AB\cdot CD=BC\cdot DA$
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 $\angle BCA=\angle DCP$.
\q8. 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 $k$.
\end