2

Part of 1992 IberoAmerican

Problems(2)

Source: Iberoamerican Olympiad 1992, Problem 2

Given the positive real numbers

a_{1}<a_{2}<\cdots<a_{n}

, consider the function

f(x)=\frac{a_{1}}{x+a_{1}}+\frac{a_{2}}{x+a_{2}}+\cdots+\frac{a_{n}}{x+a_{n}}

Determine the sum of the lengths of the disjoint intervals formed by all the values of

x

f(x)>1

functionalgebrapolynomialalgebra proposed

Construction of a trapezoid

Source: Iberoamerican Olympiad 1992, Problem 5

\Gamma

and the positive numbers

h

m

, construct with straight edge and compass a trapezoid inscribed in

\Gamma

, such that it has altitude

h

and the sum of its parallel sides is

m

geometrytrapezoidtrigonometryrhombuscircumcircleperpendicular bisectorgeometry proposed