MathDB

13

Part of 2016 Hanoi Open Mathematics Competitions

Problems(2)

H midpoint of DE iff F midpoint of BC (2016 HOMC Junior Q13)

Source:

7/20/2019
Let HH be orthocenter of the triangle ABCABC. Let d1,d2d_1, d_2 be lines perpendicular to each-another at HH. The line d1d_1 intersects AB,ACAB, AC at D,ED, E and the line d_2 intersects BCB C at FF. Prove that HH is the midpoint of segment DEDE if and only if FF is the midpoint of segment BCBC.
midpointsorthocenterequal segmentsgeometry
|2a + b| >=4, |ax^2 + bx + c| <=1 for x \in [-1, 1] (HOMC 2016 S Q13)

Source:

9/8/2019
Find all triples (a,b,c)(a,b,c) of real numbers such that ∣2a+b∣≥4|2a + b| \ge 4 and ∣ax2+bx+c∣≤1|ax^2 + bx + c| \le 1 ∀x∈[−1,1] \forall x \in [-1, 1].
algebraSysteminequalities