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2013 pairwise different numbers game

Source: 2014 Saudi Arabia Pre-TST 4.3

September 13, 2020
combinatorics

Problem Statement

Fatima and Asma are playing the following game. First, Fatima chooses 20132013 pairwise different numbers, called a1,a2,...,a2013a_1, a_2, ..., a_{2013}. Then, Asma tries to know the value of each number a1,a2,...,a2013a_1, a_2, ..., a_{2013}.. At each time, Asma chooses 1≤i<j≤20131 \le i < j \le 2013 and asks Fatima ''What is the set {ai,aj}\{a_i,a_j\}?'' (For example, if Asma asks what is the set {ai,aj}\{a_i,a_j\}, and a1=17a_1 = 17 and a2=13a_2 = 13, Fatima will answer {13.17}\{13. 17\}). Find the least number of questions Asma needs to ask, to know the value of all the numbers a1,a2,...,a2013a_1, a_2, ..., a_{2013}.
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