MathDB

11.10

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(2)

sum sin (4n-3)a_n/ (1-cos (4n-3)a_n) =kn

Source: IV Soros Olympiad 1997-98 Corr. 11.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/1/2024
Let an=π2na_n = \frac{\pi}{2n}, where nn is a natural number. Prove that for any k=1k = 1,22,......, nn holds the equality sin⁡kan1−cos⁡an+sin⁡5kan1−cos⁡5an+sin⁡9kan1−cos⁡9an+...+sin⁡(4n−3)an1−cos⁡(4n−3)an=kn\frac{\sin ka_n}{1-\cos a_n}+\frac{\sin 5ka_n}{1-\cos 5a_n}+\frac{\sin 9ka_n}{1-\cos 9a_n}+...+\frac{\sin (4n-3)a_n}{1-\cos (4n-3)a_n}=kn
algebratrigonometrySum
perimeter of ABC = k (BC)

Source: IV Soros Olympiad 1997-98 R3 11.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/2/2024
The perimeter of triangle ABCABC is kk times larger than side BCBC, AB≠ACAB \ne AC. In what ratio does the median to side BCBC divide the diameter of the circle inscribed in this triangle, perpendicular to this side?
geometryperimeterratio