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.
Find a solution to this equation to 1 dp. Are there any others? Did you know ... ?
Numerical solution of equations forms an important part of real-world mathematics and mathematics applied to science, where equations are often too complex to be solved exactly. Mathematicians have developed many advanced techniques for the numerical solution and exploration of equations.