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[Solutions] Baltic Way Mathematical Competition 2007

  1. For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that \[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]
  2. A sequence of integers $a_1,a_2,a_3,\ldots$ is called exact if $$a_n^2-a_m^2=a_{n-m}a_{n+m}$$ for any $n>m$. Prove that there exists an exact sequence with $a_1=1,a_2=0$ and determine $a_{2007}$.
  3. Suppose that $F$, $G$, $H$ are polynomials of degree at most $2n+1$ with real coefficients such that:
    • For all real $x$ we have $F(x)\le G(x)\le H(x)$.
    • There exist distinct real numbers $x_1,x_2,\ldots ,x_n$ such that $F(x_i)=H(x_i)$ for $i=1,2,3,\ldots ,n$.
    • There exists a real number $x_0$ different from $x_1,x_2,\ldots ,x_n$ such that $F(x_0)+H(x_0)=2G(x_0)$.
    Prove that $F(x)+H(x)=2G(x)$ for all real numbers $x$.
  4. Let $a_1,a_2,\ldots ,a_n$ be positive real numbers, and let $$S=a_1+a_2 +\ldots +a_n.$$ Prove that \[(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2 \]
  5. A function $f$ is defined on the set of all real numbers except $0$ and takes all real values except $1$. It is also known that $$f(xy)=f(x)f(-y)-f(x)+f(y)$$ for any $x,y\not= 0$ and that $$f(f(x))=\frac{1}{f(\frac{1}{x})}$$ for any $x\not\in\{ 0,1\}$.  Determine all such functions $f$.
  6. Freddy writes down numbers $1, 2,\ldots ,n$ in some order. Then he makes a list of all pairs $(i, j)$ such that $1\le i<j\le n$ and the $i$-th number is bigger than the $j$-th number in his permutation. After that, Freddy repeats the following action while possible: choose a pair $(i, j)$ from the current list, interchange the $i$-th and the $j$-th number in the current permutation, and delete $(i, j)$ from the list. Prove that Freddy can choose pairs in such an order that, after the process finishes, the numbers in the permutation are in ascending order.
  7. A squiggle is composed of six equilateral triangles with side length $1$ as shown in the figure below. Determine all possible integers $n$ such that an equilateral triangle with side length $n$ can be fully covered with squiggles (rotations and reflections of squiggles are allowed, overlappings are not).
  8. Call a set $A$ of integers non-isolated, if for every $a\in A$ at least one of the numbers $a-1$ and $a+1$ also belongs to $A$. Prove that the number of five-element non-isolated subsets of $\{1, 2,\ldots ,n\}$ is $(n-4)^2$.
  9. A society has to elect a board of governors. Each member of the society has chosen $10$ candidates for the board, but he will be happy if at least one of them will be on the board. For each six members of the society there exists a board consisting of two persons making all of these six members happy.
    Prove that a board consisting of $10$ persons can be elected making every member of the society happy.
  10. We are given an $18\times 18$ table, all of whose cells may be black or white. Initially all the cells are coloured white. We may perform the following operation: choose one column or one row and change the colour of all cells in this column or row. Is it possible by repeating the operation to obtain a table with exactly $16$ black cells?
  11. In triangle $ABC$ let $AD,BE$ and $CF$ be the altitudes. Let the points $P,Q,R$ and $S$ fulfil the following requirements:
    • $P$ is the circumcentre of triangle $ABC$.
    • All the segments $PQ,QR$ and $RS$ are equal to the circumradius of triangle $ABC$.
    • The oriented segment $PQ$ has the same direction as the oriented segment $AD$. Similarly, $QR$ has the same direction as $BE$, and $Rs$ has the same direction as $CF$.
    Prove that $S$ is the incentre of triangle $ABC$.
  12. Let $M$ be a point on the arc $AB$ of the circumcircle of the triangle $ABC$ which does not contain $C$. Suppose that the projections of $M$ onto the lines $AB$ and $BC$ lie on the sides themselves, not on their extensions. Denote these projections by $X$ and $Y$, respectively. Let $K$ and $N$ be the midpoints of $AC$ and $XY$, respectively. Prove that $\angle MNK=90^{\circ}$ .
  13. Let $t_1,t_2,\ldots,t_k$ be different straight lines in space, where $k>1$. Prove that points $P_i$ on $t_i$, $i=1,\ldots,k$, exist such that $P_{i+1}$ is the projection of $P_i$ on $t_{i+1}$ for $i=1,\ldots,k-1$, and $P_1$ is the projection of $P_k$ on $t_1$.
  14. In a convex quadrilateral $ABCD$ we have $ADC = 90^{\circ}$. Let $E$ and $F$ be the projections of $B$ onto the lines $AD$ and $AC$, respectively. Assume that $F$ lies between $A$ and $C$, that $A$ lies between $D$ and $E$, and that the line $EF$ passes through the midpoint of the segment $BD$. Prove that the quadrilateral $ABCD$ is cyclic.
  15. The incircle of the triangle $ABC$ touches the side $AC$ at the point $D$. Another circle passes through $D$ and touches the rays $BC$ and $BA$, the latter at the point $A$. Determine the ratio $\frac{AD}{DC}$.
  16. Let $a$ and $b$ be rational numbers such that $s=a+b=a^2+b^2$. Prove that $s$ can be written as a fraction where the denominator is relatively prime to $6$.
  17. Let $x,y,z$ be positive integers such that $$\frac{x+1}{y}+\frac{y+1}{z}+\frac{z+1}{x}$$ is an integer. Let $d$ be the greatest common divisor of $x,y$ and $z$. Prove that $$d\le \sqrt[3]{xy+yz+zx}.$$
  18. Let $a,b,c,d$ be non-zero integers, such that the only quadruple of integers $(x, y, z, t)$ satisfying the equation \[ax^2+by^2+cz^2+dt^2=0\] is $x=y=z=t=0$. Does it follow that the numbers $a,b,c,d$ have the same sign?
  19. Let $r$ and $k$ be positive integers such that all prime divisors of $r$ are greater than $50$. A positive integer, whose decimal representation (without leading zeroes) has at least $k$ digits, will be called nice if every sequence of $k$ consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of $r$. Prove that if there exist infinitely many nice numbers, then the number $10^k-1$ is nice as well.
  20. Let $a$ and $b$ be positive integers, $b<a$, such that $a^3+b^3+ab$ is divisible by $ab(a-b)$. Prove that $ab$ is a perfect cube.

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MOlympiad: [Solutions] Baltic Way Mathematical Competition 2007
[Solutions] Baltic Way Mathematical Competition 2007
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