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Prove that there are 100 natural number

Source: Iranian 3rd round Number Theory exam P6

September 22, 2014
modular arithmeticarithmetic seriesnumber theory proposednumber theory

Problem Statement

Prove that there are 100 natural number a1<a2<...<a99<a100a_1 < a_2 < ... < a_{99} < a_{100} ( ai<106 a_i < 10^6) such that A , A+A , 2A , A+2A , 2A + 2A are five sets apart ?
A={a1,a2,...,a99,a100}A = \{a_1 , a_2 ,... , a_{99} ,a_{100}\}
2A={2ai∣1≤i≤100}2A = \{2a_i \vert 1\leq i\leq 100\}
A+A={ai+aj∣1≤i<j≤100}A+A = \{a_i + a_j \vert 1\leq i<j\leq 100\}
A+2A={ai+2aj∣1≤i,j≤100}A + 2A = \{a_i + 2a_j \vert 1\leq i,j\leq 100\}
2A+2A={2ai+2aj∣1≤i<j≤100}2A + 2A = \{2a_i + 2a_j \vert 1\leq i<j\leq 100\}
(20 ponits )
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