TITLE: V.I. Arnold says Russian students can't solve this problem, but American students can -- why?
QUESTION [294 upvotes]: In a book of word problems by V.I Arnold, the following appears:


The hypotenuse of a right-angled  triangle  (in  a  standard  American  examination) is 10 inches, the altitude dropped onto it is 6 inches. Find the area of the triangle.

American school students had been coping successfully with this problem for over a decade. But then Russian school students arrived from Moscow, and none of them was able to solve it as had their American peers (giving 30 square inches as the answer). Why?

Here's the book. I assume the answer is some joke at the expense of the Americans, but I don't get it. Possibly a joke about inches? Anyone?

REPLY [4 votes]: Let $\Delta ABC$ be our triangle, $\measuredangle ACB=90^{\circ}$ and $CD$ be an altitude of the triangle.
Thus, by AM-GM $$6=CD=\sqrt{AD\cdot BD}\leq\frac{AD+BD}{2}=\frac{10}{2}=5,$$ which is a contradiction, which says this triangle does not exist.