MathDB

11.10

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(2)

concurrent wanted, mixtilinear incircle from '99

Source: V Soros Olympiad 1998-99 Round 1 11.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/25/2024
Consider a circle tangent to sides ABAB and ACAC (these sides are not equal) of triangle ABCABC and the circumscribed circle around it. Let KK, MM and PP be the touchpoints of this circle with the sides of the triangle and with the circle circumscribed around it, respectively, and let LL be the midpoint of the arc BCBC (not containing AA). Prove that the lines KMKM, PLPL and BCBC intersect at one point.
geometryconcurrency
<ADP wanted , pyramid , 2 spheres

Source: : V Soros Olympiad 1998-99 Round 3 11.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/26/2024
The plane angles at vertex DD of the pyramid ABCDABCD are equal to α\alpha,β\beta and γ\gamma (∠CDB=a\angle CDB = a). An arbitrary point MM is taken on edge CBCB. A ball is inscribed in each of the pyramids ABDMABDM and ACDMACDM. Let us draw through DD a plane distinct from BCDBCD, tangent to both balls and not intersecting the segment connecting the centers of the balls. Let this plane intersect the segment AMAM at point PP. What is ∠ADP\angle ADP equal to?
geometry3D geometrypyramidsphere