Problem

Source: IV International Festival of Young Mathematicians Sozopol 2013, Theme for 10-12 grade

Tags: algebra, equation



Let $a,b,c,$ and $d$ be real numbers and $k\geq l\geq m$ and $p\geq q\geq r$. Prove that $f(x)=a(x+1)^k (x+2)^p+b(x+1)^l (x+2)^q+c(x+1)^m (x+2)^r-d=0$ has no more than 14 positive roots.