I found the following problem - there are 5 segments given. Each three of them can be used to form a valid triangle. I need to prove that there is at least one acute triangle among all possible triangles from these segments.
My attempts Suppose all triangles are right or obtuse and let the segments have lengths $$a \leq b \leq c \leq d \leq e$$
Now we have several inequalities which follow from the law of cosines using that the cos function is negative for obtuse angles:$$a^2 + b^2 \geq c^2$$$$a^2 + c^2 \geq d^2$$
There is an equality for each triple of segments. Now, I suppose that the problem can be solved using these inequalities and also using that $a + b > c$ for all triangles, but I cannot find a way to prove the assumption is wrong. Does anyone have ideas how to continue?
For your lengths: $$c^2 \ge a^2+b^2 \\ d^2 \ge b^2+c^2 \ge a^2+2b^2 \\ e^2 \ge c^2+d^2 \ge 2a^2+3b^2 \\ (a+b)^2 \ge e^2 \ge 2a^2+3b^2 \\ 0 \ge (a-b)^2+b^2$$