58th Ukrainian National Mathematical Olympiad

Without loss of generality, we can say that after each draw they both got 0 peso. Then Alyokhin in $k^{th}$ game could get $a_k\cdot 3^{k-1}$, where $a_k\in \{-1;0;1\}$. Let us show that his gain after the $k^{th}$ game could be from $A_k=3^0+3^1+\cdots +3^{k-1}$ to $(-A_k)$. Moreover, each possible result is yielded by the unique variant of the results of the games. Indeed, for $k=1,2$ everything is obvious from the simple search. Suppose after $k$ games he could obtain by one variant from $(-A_k)$ to $A_k$. Let us add to each of the results the result of the $(k+1{)}^{th}$ game. If it was a draw, then no result would change. If Alyokhin wins, he will get all the results from $3^k-A_k$ to $3^k+A_k$.

$$3^k-A_k=3^k-3^{k-1}-3^{k-2}-\cdots -3-1=A_k+1=3^{k-1}+3^{k-2}+\cdots +3+1+1\iff$$ $$3^k=2\cdot 3^{k-1}+\cdots +2\cdot 3^2+2\cdot 3+2\cdot 1+1=2\cdot 3^{k-1}+\cdots +2\cdot 3^2+2\cdot 3+3=$$ $$3^k=2\cdot 3^{k-1}+\cdots +2\cdot 3^3+3\cdot 3^2=2\cdot 3^{k-1}+\cdots +3\cdot 3^3=2\cdot 3^{k-1}+3\cdot 3^{k-2}=3^k,$$ Then the result after $(k+1{)}^{th}$ game will also will every value from $(-A_{k+1})$ to $A_{k+1}$, and each result follows from a unique combination of results of the games.

Now we can create a table of the results, which Alyokhin can achieve: 1 game: $[-1;1]$; 3 game: $[-13;13]$; 5 game: $[-161;161]$; 7 game: $[-1133;1133]$; 2 game: $[-4;4]$; 4 game: $[-40;40]$; 6 game: $[-404;404]$; 8 game: $[-3320;3320]$.

Therefore, after 8 games, 8 other games resulted in a draw. From the obtained ranges it is easy to find the corresponding set of the results: $$2018=2187-169=2187-243+74=2187-243+81-7=2187-243+81-9+3-1$$ $$\Rightarrow 2018=3^7-3^5+3^4-3^2+3^1-3^0.$$ Hence, they both won 3 games each, and two games finished in a draw.