MathDB

3

Part of 2023 Iranian Geometry Olympiad

Problems(3)

tangent (ACE) and (BDM), <B = 3<C

Source: 2023 IGO Intermediate P3

1/18/2024
Let ω\omega be the circumcircle of the triangle ABCABC with ∠B=3∠C\angle B = 3\angle C. The internal angle bisector of ∠A\angle A, intersects ω\omega and BCBC at MM and DD, respectively. Point EE lies on the extension of the line MCMC from MM such that MEME is equal to the radius of ω\omega. Prove that circumcircles of triangles ACEACE and BDMBDM are tangent.
Proposed by Mehran Talaei - Iran
geometrytangent circles
square can be cut into 10 triangles of equal area , with vertex P inside ABCD

Source: 2023 IGO Elementary P3

1/18/2024
Let ABCDABCD be a square with side length 11. How many points PP inside the square (not on its sides) have the property that the square can be cut into 1010 triangles of equal area such that all of them have PP as a vertex?
Proposed by Josef Tkadlec - Czech Republic
geometrycombinatoricscombinatorial geometrysquare
Easy combo geo at IGO

Source: IGO 2023 Advanced P3

1/18/2024
There are several discs whose radii are no more that 11, and whose centers all lie on a segment with length l{l}. Prove that the union of all the discs has a perimeter not exceeding 4l+84l+8.
Proposed by Morteza Saghafian - Iran
geometryperimeter