The problem below on Euclidean Geometry may pose a serious challenge to the mathematically gifted students based on common principles and theorems of plane geometry, in particular, congruency of two triangles.

Suppose that we are told that four numbers a , b , c , d lie between -5 and 5. Suppose also that the numbers are constrained so that
5 < a + b < 10 and -10 < c + d < -5 Given this information, what can you deduce about these inequalities?
?? < a + b-c-d < ?? ?? < a-c < ?? ?? < a-c+d-b < ?? ?? < abcd < ?? Did you know ... ?
There are many useful general inequalities in mathematics, such as the AM-GM, Cauchy-Schwarz and Jensen's inequalities. These general inequalities are powerful tools which greatly simplify a wide variety of problems in mathematics, in applications from integration to probability via linear algebra.