Problem - 3835
Let the lengths of five line segments be $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$, respectively, where $a_1 \ge a_2\ge a_3\ge a_4\ge a_5$. If any three of these five line segments can form a triangle, then prove at least one of such triangle is acute.
Assume that there is no acute triangle, then $$a_1^2\ge a_2^2 + a_3^2,\quad a_2^2 \ge a_3^2 + a_4^2,\quad a_3^2\ge a_4^2 + a_5^2$$
Adding these three relations leads to $$a_1^2 \ge a_3^2 + 2a_4^2 + a_5^2\implies a_1^2 >a_4^2+2a_4a_5+a_5^2=(a_4+a_5)^2\implies a_1 > a_4+a_5 $$
This contradicts the fact that $a_1$, $a_4$, and $a_5$ can form a triangle.