Prove an inequality involving powers of two Let $a,b,c$ be mutually distinct positive integers.
Prove 
$$2^{a+b} + 2^{b+c} + 2^{a+c} < 2^{a+b+c} + 1$$
Any hints on how to start proving it?

Note that the left-hand side is a sum of distinct powers of $2$, so (assuming WLOG that $a < b < c$)
$$2^{a+b}+2^{b+c}+2^{a+c}<\sum_{k=0}^{b+c}2^k=2^{b+c+1}-1<2^{a+b+c}+1.$$
