MathDB

4

Part of 2023 India Regional Mathematical Olympiad

Problems(2)

RMO 2023 P4

Source:

10/29/2023
Let Ω1,Ω2\Omega_1, \Omega_2 be two intersecting circles with centres O1,O2O_1, O_2 respectively. Let ll be a line that intersects Ω1\Omega_1 at points A,CA, C and Ω2\Omega_2 at points B,DB, D such that A,B,C,DA, B, C, D are collinear in that order. Let the perpendicular bisector of segment ABA B intersect Ω1\Omega_1 at points P,QP, Q; and the perpendicular bisector of segment CDC D intersect Ω2\Omega_2 at points R,SR, S such that P,RP, R are on the same side of ll. Prove that the midpoints of PR,QSP R, Q S and O1O2O_1 O_2 are collinear.
geometry
RMO KV P-4

Source:

10/29/2023
The set XX of NN four-digit numbers formed from the digits 1,2,3,4,5,6,7,81,2,3,4,5,6,7,8 satisfies the following condition:
for any two different digits from 1,2,3,4,,6,7,81,2,3,4,,6,7,8 there exists a number in XX which contains both of them. \\
Determine the smallest possible value of NN.
Digitsnumber theorycombinatorics