OPM 2012

A five-digit positive integer $abcde_{10}$ ( $a\neq 0$ ) is said to be a range if its digits satisfy the inequalities $a<b>c<d>e$ . For example, $37452$ is a range. How many ranges are there?

In triangle $[ABC]$ , the bissector of the angle $\angle{BAC}$ intersects the side $[BC]$ at $D$ . Suppose that $\overline{AD}=\overline{CD}$ . Find the lengths $\overline{BC}$ , $\overline{AC}$ and $\overline{AB}$ that minimize the perimeter of $[ABC]$ , given that all the sides of the triangles $[ABC]$ and $[ADC]$ have integer lengths.

Find the number of positive integers $n$ such that $1\leq n\leq 1000$ and $n$ is divisible by $\lfloor \sqrt[3]{n} \rfloor$ .

Let $[ABC]$ be a triangle. Points $D$ , $E$ , $F$ and $G$ are such $E$ and $F$ are on the lines $AC$ and $BC$ , respectively, and $[ACFG]$ and $[BCED]$ are rhombus. Lines $AC$ and $BG$ meet at $H$ ; lines $BC$ and $AD$ meet at $I$ ; lines $AI$ and $BH$ meet at $J$ . Prove that $[JICH]$ and $[ABJ]$ have equal area.