# Bulgarian National Mathematics Olympiad 1998 Solutions

1. Let $n$ be a natural number. Find the least natural number $k$ for which there exist $k$ sequences of $0$ and $1$ of length $2n+2$ with the following property: any sequence of $0$ and $1$ of length $2n+2$ coincides with some of these $k$ sequences in at least $n+2$ positions.
2. The polynomials $P_n(x,y), n=1,2,...$ are defined by $P_1(x,y)=1$, $P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y).$ Prove that $P_{n}(x,y)=P_{n}(y,x)$ for all $x,y \in \mathbb{R}$ and $n$.
3. On the sides of a non-obtuse triangle $ABC$ a square, a regular $n$-gon and a regular $m$-gon ($m,n > 5$) are constructed externally, so that their centers are vertices of a regular triangle. Prove that $m = n = 6$ and find the angles of $\triangle ABC$.
4. Let $a_1,a_2,\cdots ,a_n$ be real numbers, not all zero. Prove that the equation $\sqrt{1+a_1x}+\sqrt{1+a_2x}+\cdots +\sqrt{1+a_nx}=n$ has at most one real nonzero root.
5. Let m and n be natural numbers such that $3m|(m+3)^n+1$. Prove that $$\frac{(m+3)^n+1}{3m}$$ is odd
6. The sides and diagonals of a regular $n$-gon $R$ are colored in $k$ colors so that
• For each color $a$ and any two vertices $A$,$B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$.
• The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors.
Prove that $k\leq 2$.
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