Rmo 1993 Exclusive

If $a, b, c$ are positive real numbers such that $a+b+c=1$, prove that: $$ \fracab+c + \fracbc+a + \fracca+b \ge \frac32 $$

A classic proof appeared in the 1993 set requiring students to demonstrate that the tens digit of any power of 3 is even . This remains a popular exercise for teaching modular arithmetic and pattern recognition. rmo 1993

The RMO 1993 has had a significant impact on the model railway community: If $a, b, c$ are positive real numbers

Second radical: ( x+8-6\sqrtx-1 = (t^2+1)+8 - 6t = t^2 - 6t + 9 = (t-3)^2 ). rmo 1993

This is a classic geometry problem testing cyclic quadrilateral properties.

( x-1 \ge 0 \Rightarrow x \ge 1 ).