MathDB

2

Part of 1999 Hungary-Israel Binational

Problems(2)

Hungary-Israel Binational 1999\2

Source: Maximal number of acute triangles

10/30/2008
2n\plus{}1 lines are drawn in the plane, in such a way that every 3 lines define a triangle with no right angles. What is the maximal possible number of acute triangles that can be made in this way?
analytic geometrygraphing linesslopecombinatorics unsolvedcombinatorics
Hungary-Israel Binational 1999\5

Source: Find a point that satisfies a system of inequalities

10/30/2008
The function f(x,y,z)\equal{}\frac{x^2\plus{}y^2\plus{}z^2}{x\plus{}y\plus{}z} is defined for every x,y,z∈R x,y,z \in R whose sum is not 0. Find a point (x0,y0,z0) (x_0,y_0,z_0) such that 0 < x_0^2\plus{}y_0^2\plus{}z_0^2 < \frac{1}{1999} and 1.999<f(x0,y0,z0)<2 1.999 < f(x_0,y_0,z_0) < 2.
functionquadraticsinequalities unsolvedinequalities