Logic and sets · Symmetric difference · Cardinality counting · Venn regions · Median of sets
Let \(k\) be a positive integer and let \(A\), \(B\), \(C\) be sets such that
\(|A \triangle B| = |B \triangle C| = |C \triangle A| = 2k .\)
Prove that there is exactly one set \(D\) for which
\(|A \triangle D| = |B \triangle D| = |C \triangle D| = k .\)
(For sets \(X\) and \(Y\), \(X \triangle Y = (X \setminus Y) \cup (Y \setminus X)\) denotes the symmetric difference of \(X\) and \(Y\).)
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Serbian National Competition (Drzavno takmicenje) 2018, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source