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Bosnia and Herzegovina JBMO TST 2015 Problem 3

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2015

September 16, 2018
geometrymidpointsperpendicular

Problem Statement

Let ADAD be an altitude of triangle ABCABC, and let MM, NN and PP be midpoints of ABAB, ADAD and BCBC, respectively. Furthermore let KK be a foot of perpendicular from point DD to line ACAC, and let TT be point on extension of line KDKD (over point DD) such that ∣DT∣=∣MN∣+∣DK∣\mid DT \mid = \mid MN \mid + \mid DK \mid. If ∣MP∣=2⋅∣KN∣\mid MP \mid = 2 \cdot \mid KN \mid, prove that ∣AT∣=∣MC∣\mid AT \mid = \mid MC \mid.
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