First round – City competition

MATHEMATICAL COMPETITIONS IN CROATIA IN 2015 If $a_1$ and $z_1$ are connected by bus, let us put $A^{\prime }=\{z_1,g,a_1,\dots ,a_n\}$ and $Z^{\prime }=\{z_2,\dots ,z_m\}$. Then $(A^{\prime },Z^{\prime })$ is a good pair and the number of elements of $A^{\prime }\cup Z^{\prime }$ is greater than the number of elements of $A\cup Z$, which contradicts the assumption. If $a_1$ and $z_1$ are connected by train, let us put $A^{\prime \prime }=\{a_2,\dots ,a_n\}$ and $Z^{\prime \prime }=\{a_1,g,z_1,z_2,\dots ,z_m\}$. Then $(A^{\prime \prime },Z^{\prime \prime })$ is a good pair and the number of elements of $A^{\prime \prime }\cup Z^{\prime \prime }$ is greater than the number of elements of $A\cup Z$, which contradicts the assumption. Since all cases lead to contradiction, we conclude that the assumption was wrong and that every city is either in the set $A$ or in the set $Z$. 3.4. Let $F^{\prime }$ be the intersection of the circumcircle of the triangle $ABD$ and the line $BG$ (different from $B$).