bash - Multiple conditions with arithmetic, comparison and regular operators in if statement

Here is a variation on Inian's answer taking advantage of arithmetic operations not requiring $ to expand variables, them accepting logical operators, and the fact that the first test for equality of all numbers allows the following test to be simplified.

Please note that checking if the values read actually are integer would be a good idea to avoid unexpected behavior.

#!/bin/bash
read X
read Y
read Z
if
  (( X+Y>Z || X+Z>Y || Y+Z>X ))
then
  if
    (( X==Y && Y==Z ))
  then
      echo "All numbers are equal"
  elif
    (( X==Y || X==Z || Z==Y ))
  then
    echo "Two of the numbers are equal"
  else
    echo "All three numbers are different"
  fi
fi

The $(( )) for of arithmetic expression expands to the result of the evaluation of the expression found inside. The (( )) for acts as a command that returns 0 if the expression is a test that results in a "true" value OR if it evaluates to a non-zero number, and a non-zero value otherwise. This second form is very useful for tests.

As an aside, I like using the properties of (( )) to handle on/off options in scripts. For instance, ((state_variable)) will evaluate to "false" if the variable is null or 0, and "true" otherwise, which maps nicely to how such a variable is intuitively expected to behave.