MathDB

10.4

Part of VMEO III 2006

Problems(2)

ax^2 + bx + c = 0 has real root in [\beta, \beta + \alpha]

Source: 2006 VMEO III Juniors 10.4 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463155_vmeo_iii

9/11/2021
Find the least real number α\alpha such that there is a real number β\beta so that for all triples of real numbers (a,b,c)(a, b,c) satisfying 2006a+10b+c=02006a + 10b + c = 0, the equation ax2+bx+c=0ax^2 + bx + c = 0 always has real root in the interval [β,β+α][\beta, \beta + \alpha].
algebratrinomial
VMEO III problem 4, October 2006

Source:

7/30/2007
Given a convex polygon G G, show that there are three vertices of G G which form a triangle so that it's perimeter is not less than 70% of the polygon's perimeter.
geometryperimetertrigonometryinequalitiescombinatorics proposedcombinatorics