Let $F$, $D$, and $E$ be points on the sides $[AB]$, $[BC]$, and $[CA]$ of $\triangle ABC$, respectively, such that $\triangle DEF$ is an isosceles right triangle with hypotenuse $[EF]$. The altitude of $\triangle ABC$ passing through $A$ is $10$ cm. If $|BC|=30$ cm, and $EF \parallel BC$, calculate the perimeter of $\triangle DEF$.

The first $9$ positive integers are placed into the squares of a $3\times 3$ chessboard. We are taking the smallest number in a column. Let $a$ be the largest of these three smallest number. Similarly, we are taking the largest number in a row. Let $b$ be the smallest of these three largest number. How many ways can we distribute the numbers into the chessboard such that $a=b=4$?

We call a positive integer good number, if it is divisible by squares of all its prime factors. Show that there are infinitely many pairs of consequtive numbers both are good.