MathDB

4

Part of 2023 Czech-Polish-Slovak Junior Match

Problems(2)

coloring 1x1 in a nxn array

Source: 2023 Czech-Polish-Slovak Match Junior, individual p4 CPSJ

5/5/2024
Each field of the n×nn \times n array has been colored either red or blue, with the following conditions met: ∙\bullet if a row and a column contain the same number of red fields, the field at their intersection is red; ∙\bullet if a row and a column contain different numbers of red cells, the field at their intersection is blue. Prove that the total number of blue cells is even.
combinatoricsColoring
2 tangents of different circles intersect on a line

Source: 2023 Czech-Polish-Slovak Match Junior, Team p4 CPSJ

5/5/2024
In triangle ABCABC, the points MM and NN are the midpoints of the sides ABAB and ACAC, respectively. The bisectors of interior angles ∠ABC\angle ABC and ∠BCA\angle BCA intersect the line MNMN at points PP and QQ, respectively. Let pp be the tangent to the circumscribed circle of the triangle AMPAMP passing through point PP, and qq be the tangent to the circumscribed circle of the triangle ANQANQ passing through point QQ. Prove that the lines pp and qq intersect on line BCBC.
geometrycollinear