MathDB

2

Part of 2018 European Mathematical Cup

Problems(2)

Find all pairs (x; y)

Source: 7th European Mathematical Cup , Junior Category, Q2

12/25/2018
Find all pairs (x;y) (x; y) of positive integers such that xy∣x2+2y−1.xy | x^2 + 2y -1.
number theory
O is the circumcenter of OaObOc

Source: 7th European Mathematical Cup , Senior Category , Q2

12/25/2018
Let ABC be a triangle with∣AB∣<∣AC∣.|AB|< |AC|. Let kk be the circumcircle of △ABC\triangle ABC and let OO be the center of kk. Point MM is the midpoint of the arc BCBC of kk not containing AA. Let DD be the second intersection of the perpendicular line from MM to ABAB with k k and EE be the second intersection of the perpendicular line from MM to ACAC with kk. Points XX and YY are the intersections of CDCD and BEBE with OMOM respectively. Denote by kbk_b and kck_c circumcircles of triangles BDXBDX and CEYCEY respectively. Let GG and HH be the second intersections of kbk_b and kck_c with ABAB and ACAC respectively. Denote by ka the circumcircle of triangle AGH.AGH. Prove that OO is the circumcenter of △OaObOc,\triangle O_aO_bO_c, where Oa,Ob,OcO_a, O_b, O_c are the centers of ka,kb,kck_a, k_b, k_c respectively.
geometry