Selection tests for the International Mathematical Olympiad 2013

Assume that we have two correspondents, $R$ from Riyadh and $J$ from Jeddah, who are not connected to each other. Let ${\mathcal{J}}_R$ be the set of correspondents from Jeddah connected to $R$ and ${\mathcal{R}}_J$ the set of correspondents from Riyadh connected to $J$. Define the function $$f:{\mathcal{J}}_R⟶{\mathcal{R}}_J$$ in the following way: For each correspondent $J^{\prime }$ $\in {\mathcal{J}}_R$, since the correspondents $J$ and $J^{\prime }$ are different, they share a unique correspondent in ${\mathcal{R}}_J$. Let $f\left(J^{\prime }\right)$ be this unique correspondent. So $f$ is well-defined. Assume that there exist two correspondents $J_1,J_2\in {\mathcal{J}}_R$ such that $f\left(J_1\right)=f\left(J_2\right)=R^{\prime }\in {\mathcal{R}}_J$. This means that the correspondent $R^{\prime }$ from Riyadh shares with the correspondent $R$ two correspondents $J_1,J_2$ from Jeddah. Hence $J_1=J_2$ and the map $f$ is injective. Consider a correspondent $R^{\prime }\in {\mathcal{R}}_J$ and let $J^{\prime }$ be the unique correspondent in ${\mathcal{J}}_R$ that $R^{\prime }$ shares with $R$. Then $R^{\prime }$ is the unique correspondent in ${\mathcal{R}}_J$ that $J^{\prime }$ shares with $J$. Thus $R^{\prime }=f\left(J^{\prime }\right)$ and $f$ is surjective. This proves that $R$ and $J$ have the same number of correspondents from the other city. We deduce from this that if the correspondent Amr from Riyadh is not connected to the correspondent Zayd from Jeddah, the correspondent Amr has exactly eight correspondents from Jeddah. Assume now that the correspondent Amr is connected to the correspondent Zayd from Jeddah. Since there are at least ten correspondents in Riyadh and Zayd is connected only with eight, let $R_1,R_2$ be two correspondents from Riyadh who are not connected to Zayd. Let $J_1$ be the unique correspondent that Amr and $R_1$ share. Let $J_2$ be the unique correspondent that $R_1$ shares with $R_2$. Since $R_1$ is not in contact with Zayd, he has eight correspondents from Jeddah. So there exists at least six correspondents in contact with $R_1$ who are not in contact neither with Amr nor with $R_2$. Let $J$ be one of these correspondents. The correspondent $R_2$ from Riyadh has exactly eight correspondents from Jeddah since he is not connected to the correspondent Zayd. The correspondent $J$ from Jeddah has exactly eight correspondents from Riyadh since he is not connected to the correspondent $R_2$ from Riyadh. The correspondent Amr from Riyadh has exactly eight correspondents from Jeddah since he is not connected to the correspondent $J$ from Jeddah. This proves that in all cases the correspondent Amr from Riyadh has exactly eight correspondents from Jeddah.