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Miklós Schweitzer 2000, Problem 4

Source: Miklós Schweitzer 2000

July 30, 2016
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Problem Statement

Let a1<a2<a3a_1<a_2<a_3 be positive integers. Prove that there are integers x1,x2,x3x_1,x_2,x_3 such that ∑i=13∣xi∣>0\sum_{i=1}^3 |x_i | >0, ∑i=13aixi=0\sum_{i=1}^3 a_ix_i= 0 and max⁡1≤i≤3∣xi∣<23a3+1\max_{1\le i\le 3} | x_i|<\frac{2}{\sqrt{3}}\sqrt{a_3}+1.
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