### Decision Problem

Figure 1. Functions for deciding the existence of solutions.

`Developer`` and `Experimental`` contexts are put on `\$ContextPath` so that we do not need to prepend them to the function names.

This shows that is nonnegative for all real x and y.

Let us define two classes of inequalities.

This shows that `i1[2/3]` and `i2[2/3]` have no common solutions.

Here is a graphical representation of the problem. Solutions of `i1` are colored blue and solutions of `i2` are colored red.

This shows that `i1[1/3]` and `i2[1/3]` have common solutions.

Here is a graphical representation of the problem. Solutions of `i1` are colored blue and solutions of `i2` are colored red.

The next example is based on a problem formulated by T. Ono in 1914. He conjectured that for all triangles the following inequality is true.

The variables a, b, and c are lengths of sides of the triangle, and F is its area. A counterexample was found by G. Quijano in 1915. We can generate a counterexample using `InequalityInstance`.

For acute triangles the conjecture is true, and the proof was given by F. Balitrand in 1916. Here is a proof using `ImpliesRealQ`.

Converted by Mathematica      April 24, 2000

[Article Index] [Prev Page][Next Page]