VN-TST 2007

Given two sets $A, B$ of positive real numbers such that: $|A| = |B| =n$ ; $A \neq B$ and $S(A)=S(B)$ , where $|X|$ is the number of elements and $S(X)$ is the sum of all elements in set $X$ . Prove that we can fill in each unit square of a $n\times n$ square with positive numbers and some zeros such that:

a) the set of the sum of all numbers in each row equals $A$ ;

b) the set of the sum of all numbers in each column equals $A$ .

c) there are at least $(n-1)^{2}+k$ zero numbers in the $n\times n$ array with $k=|A \cap B|$ .

Let $ABC$ be an acute triangle with incricle $(I)$ . $(K_{A})$ is the cricle such that $A\in (K_{A})$ and $AK_{A}\perp BC$ and it in-tangent for $(I)$ at $A_{1}$ , similary we have $B_{1},C_{1}$ . a) Prove that $AA_{1},BB_{1},CC_{1}$ are concurrent, called point-concurrent is $P$ . b) Assume circles $(J_{A}),(J_{B}),(J_{C})$ are symmetry for excircles $(I_{A}),(I_{B}),(I_{C})$ across midpoints of $BC,CA,AB$ ,resp. Prove that $P_{P/(J_{A})}=P_{P/(J_{B})}=P_{P/(J_{C})}$ .

Note. If $(O;R)$ is a circle and $M$ is a point then $P_{M/(O)}=OM^{2}-R^{2}$ .

Find all continuous functions $f: \mathbb{R}\to\mathbb{R}$ such that for all real $x$ we have $$f(x)=f\left(x^{2}+\frac{x}{3}+\frac{1}{9}\right).$$

Let $A\subset \{1,2,\ldots,4014\}$ , $|A|=2007$ , such that $a$ does not divide $b$ for all distinct elements $a,b\in A$ . For a set $X$ as above let us denote with $m_{X}$ the smallest element in $X$ . Find $\min m_{A}$ (for all $A$ with the above properties).

Let $A_{1}A_{2}\ldots A_{9}$ be a regular $9-$ gon. Let $\{A_{1},A_{2},\ldots,A_{9}\}=S_{1}\cup S_{2}\cup S_{3}$ such that $|S_{1}|=|S_{2}|=|S_{3}|=3$ . Prove that there exists $A,B\in S_{1}$ , $C,D\in S_{2}$ , $E,F\in S_{3}$ such that $AB=CD=EF$ and $A \neq B$ , $C\neq D$ , $E\neq F$ .