Day 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? Proposed by A. Golovanov
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 $\angle APD.$ Proposed by F. Bakharev
Click for solution We have that $ \angle{D} = 94$, and denote $ \angle{PDA} = \angle{PAD} = x$, therefore $ \angle{APD} = 180 - 2x$. Therefore we have that: 1) $ \angle{CDP} = 94 - x$, so $ \angle{CPD} = x + 14$, 2) $ \angle{BAP} = 150 - x$, so $ \angle{APB} = x - 14$. Now from the sin law we have: 1) $ \frac {AP}{\sin{44}} = \frac {AB}{\sin{(x - 14)}}$. 2) $ \frac {DP}{\sin{72}} = \frac {CD}{\sin{(x + 14)}}$. By dividing them we obtain: $ \sin{72}\cdot \sin{(x - 14)} = \sin{44}\cdot \sin{(x + 14)}$ $ \iff \cos{(86 - x)} = \cos{(x - 30)}\iff 2x = 116 \iff x = 58$. Thus, $ \angle{APD} = 180 - 116 = 64$., and the problem is solved.
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. Proposed by F. Bakharev
Find all continuous functions $f(x)$ defined for all $x>0$ such that for every $x$, $y > 0$ \[ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) . \] Proposed by F. Petrov
Day 2
Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$ \[\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq\] \[\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.\] Proposed by A. Khrabrov
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 $y?$ Proposed by A. Golovanov
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 $\angle BCA=\angle DCP$ Proposed by S. Berlov
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 $k.$ Proposed by F. Petrov For those of you who liked this problem.Check this thread out.