MHB Floor Function and Tangent function

[anemone]

Solve the equation $x=\pi\left\lfloor\tan\dfrac{\pi}{x}\right\rfloor$.

 

[kaliprasad]

Solve the equation $x=\pi\left\lfloor\tan\dfrac{\pi}{x}\right\rfloor$.

Spoiler

let us consider the case $x\ge 0$

x has to be multiple of $\pi$
x cannot be zero as RHS is undefined.
x cannot be > 4 as RHS = 0 and LHS > 0

so $x=\pi$ is the possible case

now $\tan \,1$ is between 1 and $\sqrt{3}$ (because $\dfrac{\pi}{4}\lt 1 \lt \dfrac{\pi}{3}$
hence x = $pi$ is the solution

there is no solution for $x\lt 0$

so only solution $\pi$