parmenides51 wrote: Prove theat for an arbitrary triangle holds the inequality $$a \cos A+ b \cos B + c \cos C \le p ,$$where $a, b, c$ are the sides of the triangle, $A, B, C$ are the angles, $p$ is the semiperimeter. https://artofproblemsolving.com/community/c6h2044665p14481559 https://artofproblemsolving.com/community/c6h1548966p14481598 Let $\triangle{\text{ABC}}$ be a triangle with sides $a,b$ and $c$ and $x$ be any non-negative real number.Then show that: $$a^x\cos A+b^x\cos B+c^x\cos C\leq\frac{a^x+b^x+c^x}{2}.$$For all acute angle triangle $ABC$ prove that$$ acos B+ bcos C+ ccosA <\frac{3}{5}(a+b+c),$$In $\triangle ABC$ , Prove that\[acos\frac{A}{2}+bcos\frac{B}{2}+ccos\frac{C}{2}\le\frac{\sqrt{3}}{2}(a+b+c).\]