1. Find all real numbers $x$ for which

$$
|||| x|-2|-20|-200|=2007
$$
I/1. The value of the expression ||$|x|-2|-20|-200$ can be equal to 2007 or -2007. Since ||$|x|-2|-20|-200 \geq-200$, it must be that ||$|x|-2|-20|-200=2007$ and therefore

$$
|||x|-2|-20|=2207
$$

From this, it follows that ||$x|-2|-20$ is either 2207 or -2207. However, since ||$x|-2|-20 \geq$ -20, it must be that ||$x|-2|-20=2207$ and therefore

$$
|| x|-2|=2227
$$

Since $|x|-2$ cannot be equal to -2227, it must be that $|x|-2=2227$. This gives us $|x|=2229$. The solutions are $x=2229$ and $x=-2229$.

![](https://cdn.mathpix.com/cropped/2024_06_07_f155edc146c61fd862beg-06.jpg?height=611&width=1667&top_left_y=1242&top_left_x=203)

The value of the expression ||$|x|-2|-20|-200$ can be equal to 2007 or $-2007 \ldots \ldots . .1$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_f155edc146c61fd862beg-06.jpg?height=83&width=1633&top_left_y=1335&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_f155edc146c61fd862beg-06.jpg?height=100&width=1633&top_left_y=1418&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_f155edc146c61fd862beg-06.jpg?height=66&width=1633&top_left_y=1509&top_left_x=220)

![](https://cdn.mathpix.com/cropped/2024_06_07_f155edc146c61fd862beg-06.jpg?height=59&width=1633&top_left_y=1581&top_left_x=220)

The solutions are $x=2229$ and $x=-2229$.......................................... 1 . $\mathbf{1}$ point

For solving "from the inside": 2 points for the solutions; 1 point for considering another example besides $x= \pm(2+20+200+2007)$; 2 points for a complete consideration of all 16 possibilities; 2 points for verifying that the obtained $x$ satisfies all conditions.