MathDB

1

Part of 2016 Danube Mathematical Olympiad

Problems(2)

Can The Sum Be 2016?

Source: Mathematical Danube Competition 2016, Juniors P1

4/21/2022
Let S=x1x2+x3x4+⋯+x2015x2016,S=x_1x_2+x_3x_4+\cdots+x_{2015}x_{2016}, where x1,x2,…,x2016∈{3−2,3+2}.x_1,x_2,\ldots,x_{2016}\in\{\sqrt{3}-\sqrt{2},\sqrt{3}+\sqrt{2}\}. Can SS be equal to 2016?2016?
Cristian Lazăr
romaniaalgebra
Problem 1, Geometric Inequality

Source: Danube Contest 2016, Romania

10/29/2017
1.Let ABCABC be a triangle, DD the foot of the altitude from AA and MM the midpoint of the side BCBC. Let SS be a point on the closed segment DMDM and let P,QP, Q the projections of SS on the lines ABAB and ACAC respectively. Prove that the length of the segment PQPQ does not exceed one quarter the perimeter of the triangle ABCABC.
geometric inequalityinequalities