Problem
ALG-B2-M12-P010 Set 3. Triangle substitution
#10
★★★★★ Level 5 of 5
Let \(a,b,c\) be the sides of a triangle. Prove \[\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}\ge3.\]
Hint. Put \(a=y+z\), \(b=z+x\), \(c=x+y\).
Then the denominators are \(2x,2y,2z\), and the sum is \(\frac12\sum\frac{y+z}{x}\). It contains three pairs \(\frac{x}{y}+\frac{y}{x}\), each at least \(2\). Hence the sum is at least \(3\).
Mock set 3: the problem is intended for independent method selection without an explicit cue in the statement.