Find the area of the right triangle with hypotenuse 10 and height (dropped from the right angle) 6.
This problem was on a math test for quite some time, until it turned out that the best students (and only the best students) could not solve it.
No, the area is not 30. Yes, it is a trick question. The problem was eventually removed from the test because such triangle does not exist. Maximum possible height of a right triangle with hypotenuse 10 is 5. To see why, try drawing such triangle, or find one of its legs.
Source: problem 6 in Арнольд В.И., Задачи для детей от 5 до 15 лет