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However, since a sensible solution can only have whole people or animals, most of the problems have only one solution consisting of positive integers. In each case, Alcuin gives a solution and proves that it is correct, without describing how the solution was found. Problem 26: There is a field that is 150 feet long.
However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
The two circles in the Two points, one line problem where the line through P and Q is not parallel to the given line l, can be constructed with compass and straightedge by: Draw the line m through the given points P and Q. The point G is where the lines l and m intersect; Draw circle C that has PQ as diameter. Draw one of the tangents from G to ...
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. [1]The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. [2]
Vertical line of equation x = a Horizontal line of equation y = b. Each solution (x, y) of a linear equation + + = may be viewed as the Cartesian coordinates of a point in the Euclidean plane. With this interpretation, all solutions of the equation form a line, provided that a and b are not both zero. Conversely, every line is the set of all ...
This application was the motivation for Paul Erdős to find his solution for the no-three-in-line problem. [13] It remained the best area lower bound known for the Heilbronn triangle problem from 1951 until 1982, when it was improved by a logarithmic factor using a construction that was not based on the no-three-in-line problem. [14]