MathDB

3

Part of 2023 Malaysian IMO Training Camp

Problems(2)

Three circles meet at a point

Source: Own. Malaysian IMO TST 2023 P3

4/29/2023
Let ABCABC be an acute triangle with AB≠ACAB\neq AC. Let D,E,FD, E, F be the midpoints of the sides BCBC, CACA, and ABAB respectively, and M,NM, N be the midpoints of minor arc BCBC not containing AA and major arc BACBAC respectively. Suppose W,X,Y,ZW, X, Y, Z are the incenter, DD-excenter, EE-excenter, and FF-excenter of triangle DEFDEF respectively.
Prove that the circumcircles of the triangles ABCABC, WNXWNX, YMZYMZ meet at a common point.
Proposed by Ivan Chan Kai Chin
geometry
Strange recurrence

Source: Own. Malaysian SST 2023 P3

8/30/2023
A sequence of reals a1,a2,⋯a_1, a_2, \cdots satisfies for all m>1m>1, am+1am−1=am2−a12a_{m+1}a_{m-1}=a_m^2-a_1^2 Prove that for all m>n>1m>n>1, the sequence satisfies the equation am+nam−n=am2−an2a_{m+n}a_{m-n}=a_m^2-a_n^2
Proposed by Ivan Chan Kai Chin
algebra