Solution option 1. The sum of all ten thought-of numbers will be obtained if we add what the first child said to what the third, fifth, seventh, and ninth children said. Similarly, we add what the second, fourth, sixth, eighth, and tenth children said. Again, we get the sum of all ten thought-of numbers, so the equation $1+3+5+7+9=2+4+6+8+x$ must hold, where $x$ is the number named by the tenth child. From this, $x=5$.

Remark. Where was the condition that the numbers thought of are integers used? Nowhere. The solution works for non-integer numbers as well.
5.4. 10 children stood in a circle. Each thought of an integer and told it to their clockwise neighbor. Then each loudly announced the sum of their number and the number of the counterclockwise neighbor. The first said "10", the next clockwise - "9", the next clockwise - "8", and so on, the ninth said "2". What number did the remaining child say?

Answer, option 1.5.

Answer, option 2. 9.

Answer, option 3. 14.

Answer, option 4. 6.