5,6 |
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|  | Classical combinatorics (miscellaneous). |  |
|  | Examples and counterexamples. Constructions |  |
|  | Estimation + example |  |

Each of thirty sixth-graders has one pen, one pencil, and one ruler. After their participation in the Olympiad, it turned out that 26 students lost their pen, 23 - their ruler, and 21 - their pencil. Find the smallest possible number of sixth-graders who lost all three items.
From the condition, it follows that four sixth-graders have a pen, seven have a ruler, and nine have a pencil. Thus, at least one item can be owned by no more than $4+7+9=20$ people. Therefore, no fewer than $30-20=10$ people have lost all three items.

All three items will be lost by exactly 10 people if each of the other twenty loses exactly two items.

## Answer

10 sixth-graders.