SMO 2005 Senior Round 2

The digits of a 3-digit number are interchanged so that none of the digits retain their original positions. The difference of the two numbers is a 2-digit number and is a perfect square.

**Source clarification.** The printed statement ends here and omits an explicit request. Its accompanying solution identifies the possible differences. Under that intended interpretation, determine all possible positive differences and justify your answer. This clarification is editorial; the question remains marked for review.

Consider the nonconvex quadrilateral $ABCD$ with $\angle C>180^\circ$. Let the side $CD$ extended meet $AB$ at $F$ and the side $BC$ extended meet $AD$ at $E$. A line intersects the interiors of the sides $AB,AD,BC,CD$ at points $K,L,J,I$, respectively. Prove that if $DI=CF$ and $BJ=CE$, then $KJ=IL$.

Let $S$ be a subset of $\{1,2,3,\ldots,24\}$ with $|S|=10$. Show that $S$ has two $2$-element subsets $\{x,y\}$ and $\{u,v\}$ such that $x+y=u+v$.

Determine whether there exists a positive integer $n$ such that $n!$ begins with $2005$. Justify your answer.