A 6*6 square is dissected into 9 rectangles by lines parallel to its sides such that all these rectangles have only integer sides. Prove that there are always two congruent rectangles
A 6*6 square has sides 6 units long. 6 can only be factored as 1*2*3 so when you divide each side "by lines parallel to its sides such that all these rectangles have only integer sides" the only possible sides are 2 and 3. There are only 6 possible rectangles: 1*1, 1*2, 1*3, 2*2, 2*3, and 3*3. since there are 9 rectangles and only 6 possible rectangles, some must be the same.
Your argument seems to exclude the vertical division by cuts at 1 and 1 unit from the bottom edge and 1 and 1 units from the right giving pieces of size 1x1, 1x4, 4x1, and 4x4. Now this disection meets the condition to be proven it is not permitted by your construction.
CB