MathDB

5

Part of 2016 Czech-Polish-Slovak Junior Match

Problems(2)

fill the fields of the table 10 x10 with no 1-100, minimum

Source: Czech-Polish-Slovak Junior Match 2016, individual p5 CPSJ

1/20/2020
Determine the smallest integer jj such that it is possible to fill the fields of the table 10×1010\times 10 with numbers from 11 to 100100 so that every 1010 consecutive numbers lie in some of the j×jj\times j squares of the table.
Czech Republic
combinatorics
circumcenter of ABN is midpoint of incenter's segment of ABC and ABM

Source: Czech-Polish-Slovak Junior Match 2016, team p5 CPSJ

1/20/2020
Let ABCABC be a triangle with AB:AC:BC=5:5:6AB : AC : BC =5:5:6. Denote by MM the midpoint of BCBC and by NN the point on the segment BCBC such that BN=5⋅CNBN = 5 \cdot CN. Prove that the circumcenter of triangle ABNABN is the midpoint of the segment connecting the incenters of triangles ABCABC and ABMABM.
Slovakia
geometryincenterCircumcentermidpoint