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Three of Five Sum to a Multiple of Three

Issue 29 Olympiad Published 19 Oct 2026 By The Chalkboard Desk

Tier Olympiad — Three of Five Sum to a Multiple of Three

The problem

Five integers are written on a blackboard. Prove that three of them can be chosen so that their sum is divisible by . Then show that the number cannot be replaced by .
  • The integers may be negative, zero, or repeated.
  • The three chosen numbers must be distinct positions on the board, not distinct values.

Take a minute

Reveal hint 1
Look at the residues modulo : only three of them exist, and you have five numbers.
Reveal hint 2
Split into two cases: either some residue class holds at least three numbers, or every class holds at most two.

Solution

AnswerThe claim is true for five integers and false for four.

Derivation

  1. Reduce modulo .

  2. Case 1: some class holds three or more.

  3. Case 2: every class holds at most two.

  4. Finish case 2.

  5. Show four is not enough.

  6. State the sharp result.