MathDB

3

Part of 2021 Saint Petersburg Mathematical Olympiad

Problems(3)

3D geo with n circles having a common point

Source: St Petersburg 2021 11.3

12/23/2021
In the pyramid SA1A2⋯AnSA_1A_2 \cdots A_n, all sides are equal. Let point XiX_i be the midpoint of arc AiAi+1A_iA_{i+1} in the circumcircle of △SAiAi+1\triangle SA_iA_{i+1} for 1≤i≤n1 \le i \le n with indices taken mod nn. Prove that the circumcircles of X1A2X2,X2A3X3,⋯ ,XnA1X1X_1A_2X_2, X_2A_3X_3, \cdots, X_nA_1X_1 have a common point.
geometry3D geometrypyramidcircumcircle
Strange pentagon geo

Source: St Petersburg 2021 10.3

12/23/2021
Given a convex pentagon ABCDEABCDE, points A1,B1,C1,D1,E1A_1, B_1, C_1, D_1, E_1 are such that AA1⊥BE,BB1⊥AC,CC1⊥BD,DD1⊥CE,EE1⊥DA.AA_1 \perp BE, BB_1 \perp AC, CC_1 \perp BD, DD_1 \perp CE, EE_1 \perp DA. In addition, AE1=AB1,BC1=BA1,CB1=CD1AE_1 = AB_1, BC_1 = BA_1, CB_1 = CD_1 and DC1=DE1DC_1 = DE_1. Prove that ED1=EA1ED_1 = EA_1
geometrypentagon
Inequality in a cyclic quad

Source: St Petersburg 2021 9.3

12/23/2021
Given is cyclic quadrilateral ABCDABCD with∠A=3A = 3∠BB. On the ABAB side is chosen point C1C_1, and on side BCBC - point A1A_1 so that AA1=AC=CC1AA_1 = AC = CC_1. Prove that 3A1C1>BD3A_1C_1>BD.
geometry