MathDB

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Part of 2003 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Ineq x^2 + y^2 + z^2 = 1 implies x^2yz + xy^2z + xyz^2 ≤ 1/3

Source: XII Rioplatense Mathematical Olympiad (2003), Level 3

8/8/2011
Let xx, yy, and zz be positive real numbers satisfying x2+y2+z2=1x^2+y^2+z^2=1. Prove that x2yz+xy2z+xyz2≤13.x^2yz+xy^2z+xyz^2\le\frac{1}{3}.
inequalitiesinequalities unsolved
Ruler-&-compass construction of triangle with minimum area

Source: XII Rioplatense Mathematical Olympaid (2003), Level 3

8/9/2011
Inside right angle XAYXAY, where AA is the vertex, is a semicircle Γ\Gamma whose center lies on AXAX and that is tangent to AYAY at the point AA. Describe a ruler-and-compass construction for the tangent to Γ\Gamma such that the triangle enclosed by the tangent and angle XAYXAY has minimum area.
geometryinequalitiesinradiusgeometry unsolved