MathDB

1

Part of 1986 Vietnam National Olympiad

Problems(2)

Inequality

Source: Vietnam NMO 1986 Problem 1

2/6/2009
Let 12≤a1,a2,…,an≤5 \frac{1}{2}\le a_1, a_2, \ldots, a_n \le 5 be given real numbers and let x1,x2,…,xn x_1, x_2, \ldots, x_n be real numbers satisfying 4x_i^2\minus{} 4a_ix_i \plus{} \left(a_i \minus{} 1\right)^2 \le 0. Prove that \sqrt{\sum_{i\equal{}1}^n\frac{x_i^2}{n}}\le\sum_{i\equal{}1}^n\frac{x_i}{n}\plus{}1
inequalitiesfunctioninequalities unsolved
Find the locus of points

Source: Vietnam NMO 1986 Problem 4

2/6/2009
Let ABCD ABCD be a square of side 2a 2a. An equilateral triangle AMB AMB is constructed in the plane through AB AB perpendicular to the plane of the square. A point S S moves on AB AB such that SB\equal{}x. Let P P be the projection of M M on SC SC and E E, O O be the midpoints of AB AB and CM CM respectively. (a) Find the locus of P P as S S moves on AB AB. (b) Find the maximum and minimum lengths of SO SO.
geometry unsolvedgeometry