MathDB

5

Part of 2023 Malaysian IMO Training Camp

Problems(2)

Inversion of a special circle

Source: Malaysian IMO TST 2023 P5

4/30/2023
Let ABCDABCD be a cyclic quadrilateral, with circumcircle ω\omega and circumcenter OO. Let ABAB intersect CDCD at EE, ADAD intersect BCBC at FF, and ACAC intersect BDBD at GG.
The points A1,B1,C1,D1A_1, B_1, C_1, D_1 are chosen on rays GAGA, GBGB, GCGC, GDGD such that:
∙\bullet GA1GA=GB1GB=GC1GC=GD1GD\displaystyle \frac{GA_1}{GA} = \frac{GB_1}{GB} = \frac{GC_1}{GC} = \frac{GD_1}{GD}
∙\bullet The points A1,B1,C1,D1,OA_1, B_1, C_1, D_1, O lie on a circle.
Let A1B1A_1B_1 intersect C1D1C_1D_1 at KK, and A1D1A_1D_1 intersect B1C1B_1C_1 at LL. Prove that the image of the circle (A1B1C1D1)(A_1B_1C_1D_1) under inversion about ω\omega is a line passing through the midpoints of KEKE and LFLF.
Proposed by Anzo Teh Zhao Yang & Ivan Chan Kai Chin
geometry
{x_i-a}+{x_{i+1}-b} at most 1/2024

Source: Malaysian SST 2023 P5

8/27/2023
Find the maximal value of c>0c>0 such that for any n≥1n\ge 1, and for any nn real numbers x1,⋯ ,xnx_1, \cdots, x_n there exists real numbers a,ba ,b such that {xi−a}+{xi+1−b}≤12024\{x_i-a\}+\{x_{i+1}-b\}\le \frac{1}{2024} for at least cncn indices ii. Here, xn+1=x1x_{n+1}=x_1 and {x}\{x\} denotes the fractional part of xx.
Proposed by Wong Jer Ren
algebra