[Solutions] Junior Balkan Mathematical Olympiad 2018

  1. Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$. 
  2. Find max number $n$ of numbers of three digits such that 
    • Each has digit sum $9$,
    • No one contains digit $0$,
    • Each $2$ have different unit digits,
    • Each $2$ have different decimal digits,
    • Each $2$ have different hundreds digits.
  3. Let $k>1$ be a positive integer and $n>2018$ an odd positive integer. The non-zero rational numbers $x_1,x_2,\ldots,x_n$ are not all equal and $$x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1}$$Find the minimum value of $k$, such that the above relations hold. 
  4. Let $\triangle ABC$ and $A'$, $B'$, $C'$ the symmetrics of vertex over opposite sides. The intersection of the circumcircles of $\triangle ABB'$ and $\triangle ACC'$ is $A_1$. $B_1$ and $C_1$ are defined similarly. Prove that lines $AA_1$, $BB_1$ and $CC_1$ are concurent.

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