MathDB

4

Part of 2022 Rioplatense Mathematical Olympiad

Problems(3)

Rays, incircle, angles...

Source: Rioplatense L-3 2022 #4

12/13/2022
Let ABCABC be a triangle with incenter II. Let DD be the point of intersection between the incircle and the side BCBC, the points PP and QQ are in the rays IBIB and ICIC, respectively, such that ∠IAP=∠CAD\angle IAP=\angle CAD and ∠IAQ=∠BAD\angle IAQ=\angle BAD. Prove that AP=AQAP=AQ.
geometryincenter
Angle = Angle (twice)

Source: Rioplatense L-2 2022 #4

12/13/2022
Let ABCDABCD be a parallelogram and MM is the intersection of ACAC and BDBD. The point NN is inside of the △AMB\triangle AMB such that ∠AND=∠BNC\angle AND=\angle BNC. Prove that ∠MNC=∠NDA\angle MNC=\angle NDA and ∠MND=∠NCB\angle MND=\angle NCB.
geometryparallelogram
Rioplatense L-1 2022 #4

Source:

12/13/2022
Let LL be the number formed by 20222022 digits equal to 11, that is, L=1111…111L=1111\dots 111. Compute the sum of the digits of the number 9L2+2L9L^2+2L.
number theory