MathDB

3

Part of 2007 Hungary-Israel Binational

Problems(2)

Maximizing the Area of a Quadrilateral

Source: Hungary-Israel Mathematical Competition 2007 Problem 3

11/19/2007
Let AB AB be the diameter of a given circle with radius 1 1 unit, and let P P be a given point on AB AB. A line through P P meets the circle at points C C and D D, so a convex quadrilateral ABCD ABCD is formed. Find the maximum possible area of the quadrilateral.
geometrytrigonometryfunctioncalculusderivativeinequalitiesgeometry unsolved
Prove the degree of f(x) is at least n.

Source: Hungary-Israel Mathematical Competition 2007 Problem 6

11/19/2007
Let t≥3 t \ge 3 be a given real number and assume that the polynomial f(x) f(x) satisfies |f(k)\minus{}t^k|<1, for k\equal{}0,1,2,\ldots ,n. Prove that the degree of f(x)f(x) is at least nn.
algebrapolynomialalgebra unsolved