MathDB

3

Part of 2018 European Mathematical Cup

Problems(2)

The line MN bisects the segment \overline{AH}

Source: 7th European Mathematical Cup , Junior Category, Q3

12/25/2018
Let ABCABC be an acute triangle with ∣AB∣<∣AC∣ |AB | < |AC |and orthocenter HH. The circle with center A and radius∣AC∣ |AC | intersects the circumcircle of △ABC\triangle ABC at point DD and the circle with center AA and radius∣AB∣ |AB | intersects the segment AD‾\overline{AD} at point K.K. The line through KK parallel to CDCD intersects BCBC at the point L. L. If MM is the midpoint of BC‾\overline{BC} and N is the foot of the perpendicular from HH to AL,AL, prove that the line MN MN bisects the segment AH‾\overline{AH}.
geometry
There exist a bounded set of positive real numbers S

Source: 7th European Mathematical Cup , Senior Category , Q3

12/25/2018
For which real numbers k>1k > 1 does there exist a bounded set of positive real numbers SS with at least 33 elements such that k(a−b)∈Sk(a - b)\in S for all a,b∈Sa,b\in S with a>b?a > b? Remark: A set of positive real numbers SS is bounded if there exists a positive real number MM such that x<Mx < M for all x∈S.x \in S.
algebraemcmath olympiad