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9.8

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(2)

locus of circumcenters - VI Soros Olympiad 1999-00 Round 1 9.8

Source:

5/21/2024
Given a line ℓ\ell and a ray pp on a plane with its origin on this line. Two fixed circles (not necessarily equal) are constructed, inscribed in the two formed angles. On ray pp, point AA is taken so that the tangents from AA to the given circles, different from pp, intersect line ℓ\ell at points BB and CC, and at the same time triangle ABCABC contains the given circles. Find the locus of the centers of the circles inscribed in triangle ABCABC (as AA moves).
geometrycircumcircle
sum \sqrt{1+k^2} sin(a_k-a_{1000}) (VI Soros Olympiad 1990-00 R1 9.8)

Source:

5/27/2024
Let ana_n denote an angle from the interval for each (0,π2)\left( 0, \frac{\pi}{2}\right) , the tangent of which is equal to nn . Prove that 1+12sin⁡(a1−a1000)+1+22sin⁡(a2−a1000)+...+1+20002sin⁡(a2000−a1000)=sin⁡a1000\sqrt{1+1^2} \sin(a_1-a_{1000}) + \sqrt{1+2^2} \sin(a_2-a_{1000})+...+\sqrt{1+2000^2} \sin(a_{2000}-a_{1000}) = \sin a_{1000}
algebratrigonometrySum