MathDB

10.9

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(2)

sin x + sin (m x) >= 1.8 - VI Soros Olympiad 1999-00 Round 1 10.9

Source:

5/21/2024
Prove that for any λ>3\lambda > 3 there is a number xx for which sin⁡x+sin⁡(λx)≥1.8.\sin x + \sin (\lambda x) \ge 1.8.
algebratrigonometryinequalities
orthocenter wanted

Source: VI Soros Olympiad 1990-00 R1 10.9 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/28/2024
Given an acute-angled triangle ABCABC, in which PP, MM, NN are the midpoints of the sides ABAB, BCBC, ACAC, respectively. A point HH is taken inside the triangle and perpendiculars HKHK, HSHS, HQHQ are lowered from it to the sides ABAB, BCBC, ACAC, respectively (K∈ABK \in AB, S∈BCS \in BC, Q∈ACQ \in AC). It turned out that MK=MQMK = MQ, NS=NKNS = NK, PS=PQPS=PQ. Prove that HH is the point of intersection of the altitudes of triangle ABCABC.
geometryorthocenter