MathDB

3

Part of 2019 Czech-Polish-Slovak Junior Match

Problems(2)

fill the n x n table with numbers 1, 2 and -3 so that each sum is 0

Source: Czech-Polish-Slovak Match Junior 2019, individual p3 CPSJ

1/20/2020
Determine all positive integers nn such that it is possible to fill the n×nn \times n table with numbers 1,21, 2 and −3-3 so that the sum of the numbers in each row and each column is 00.
combinatoricsnumbers in a table
cyclic ABCD wanted, perpendicular diagonals given

Source: Czech-Polish-Slovak Match Junior 2019, team p3 CPSJ

1/20/2020
Let ABCDABCD be a convex quadrilateral with perpendicular diagonals, such that ∠BAC=∠ADB\angle BAC = \angle ADB, ∠CBD=∠DCA\angle CBD = \angle DCA, AB=15AB = 15, CD=8CD = 8. Show that ABCDABCD is cyclic and find the distance between its circumcenter and the intersection point of its diagonals.
geometrycircumcirclecyclic quadrilateral