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Those are questions for Elementary School Math Olympiads in my country but both of us were having a hard time figuring them out.
For question no. 9, if only that number 9 is located right above number 13 it would be easy to solve. However, we don't know whether the placement of 9 above is intentional or not. Can someone help us solve either question?

2.

3. Originally Posted by Monoxdifly
So the vertices of the quadrilateral, in particular, $C$, don't lie on the grid nodes? Then I doubt there is a nice answer.

Originally Posted by Monoxdifly
So I can invent any rule for arranging these numbers? For example, I can place 15 to the right of 14 or I can place it in the beginning of the next line. And I can skip 0, 1, 2, etc. places after 15 or in any other place, just like 1 place is skipped after 8 for no particular reason. Then I doubt there is a nice answer.

4. For the first problem, #8, it looks like you are not expected calculate any precise values but to count rectangles, including estimating areas of partial rectangles.

For #9, I would suspect that the space between 8 and 9 shouldn't be there and write
1 2
3 4 5
6 7 8 9
10 11 12 12 14
15 16 17 18 19 20
21 22 23 24 25 26 27
28 29 30 31 32 33 34 35

So the number under 25 is 32.