## Task 3 - 110833

Determine all real numbers $x$ for which an isosceles triangle $\triangle A B C$ with side lengths $a=-5 x+12 ; b=3 x+20 ; c=4 x+16$ exists!

(Consider which conditions $a, b$ and $c$ must satisfy!)
A triangle $\triangle A B C$ with three real numbers $a, b, c$ as side lengths exists if and only if the inequalities
(1) $\quad a>0$
(4) $a>0$
(5) $b>0$
(6) $c<a+b$

are all satisfied. It is isosceles if and only if $a=b$ or $a=c$ or $b=c$. The condition $a=b$ is equivalent to $-5 x+12=3 x+20$, which simplifies to $x=-1$.

This results in $a=b=17$ and $c=12$, and (1) to (6) are all satisfied.

The condition $a=c$ is equivalent to $-5 x+12=4 x+16$, which simplifies to $x=-\frac{4}{9}$. This results in $a=c=\frac{128}{8}$ and $b=\frac{56}{3}$, and (1) to (6) are all satisfied.

The condition $b=c$ is equivalent to $3 x+20=4 x+16$, which simplifies to $x=4$. This results in $b=c=32$ and $a=-8$, meaning (1) is not satisfied.

Thus, there is exactly one isosceles triangle for $x=-1$ and $x=-\frac{4}{9}$.