51st Kürschák 1950 Problem 1 Several people visited a library yesterday. Each one visited the library just once (in the course of yesterday). Amongst any three of them, there were two who met in the library. Prove that there were two moments T and T' yesterday such that everyone who visited the library yesterday was in the library at T or T' (or both).
Problem 1095
Official solution
Let the people be P 1 , ... P n and suppose P i is present during the time interval [a i ,b i ]. Let a M = max a i , b m = min b i . Now we claim that given any of the intervals [a i ,b i ] we have a M ∈ [a i ,b i ] or b m ∈ [a i ,b i ]. Suppose not. So a M ∉ [a i ,b i ]. We know that a i ≤ a M , so we must have b i < a M . Thus the intervals [a i ,b i ] and [a M ,b M ] are disjoint with [a i ,b i ] lying entirely to the left of [a M ,b M ]. Similarly, b m ∉ [a i ,b i ], but b m ≤ b i , so b m < b i and [a m , b m ] and [a i ,b i ] are disjoint with [a m , b m ] lying entirely to the left of [a i ,b i ]. Thus the three intervals [a m , b m ], [a i ,b i ] and [a M ,b M ] are disjoint. Contradiction. So we can take T, T' to be a M and b m . 51st Kürschák 1950 © John Scholes [email protected] 29 Oct 2003 Last corrected/updated 29 Oct 03