10.3. Several different real numbers are written on the board. It is known that the sum of any three of them is rational, while the sum of any two of them is irrational. What is the largest number of numbers that can be written on the board? Justify your answer.
Solution: The set of three numbers $\sqrt{2}, \sqrt{3}, -\sqrt{2}-\sqrt{3}$, as is easily verified, satisfies the condition of the problem. We will show that no more than three numbers can be written on the board.

Assume the contrary, and let $a_{i} (i=\overline{1,4})$ be some four numbers from this set. Then the numbers $a_{1}+a_{2}+a_{3}$ and $a_{1}+a_{2}+a_{4}$ are rational, so the number $\left(a_{1}+a_{2}+a_{3}\right)-\left(a_{1}+a_{2}+a_{4}\right)=a_{3}-a_{4}$ is also rational. Similarly, the rationality of the number $a_{3}-a_{2}$ can be proven. Then the number $\left(a_{3}-a_{4}\right)+\left(a_{3}-a_{2}\right)-\left(a_{2}+a_{3}+a_{4}\right)=a_{3}$ is rational. The same can be shown for the rationality of all numbers $a_{i}$. But then the sum of any two of them is rational, which contradicts the condition of the problem.

Answer: Three numbers.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and justified answer | 7 points |
| There is a correct example of three numbers, but there are inaccuracies in the proof (which is basically correct) of its optimality | 5 points |
| Correct proof that 4 numbers cannot exist (in the absence of an example of 3 numbers) | 3 points |
| In a correct (in general) proof that 4 numbers cannot exist, there are inaccuracies and no example of 3 numbers | 2 points |
| An example of three numbers is given, but the proof of its optimality is missing | 1 point |
| Correct answer without justification (or with incorrect justification) | 0 points |
| Proof that there are fewer than 5 (or even more) numbers | not evaluated |