MathDB

13

Part of 2022 Durer Math Competition Finals

Problems(2)

numbers in 6x6 board

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 2 E13

12/12/2022
Write some positive integers in the following table such that ⋅\cdot there is at most one number in each field ⋅\cdot each number is equal to how many numbers there are in edge-adjacent fields, ⋅\cdot edge-adjacent fields cannot have equal numbers. What is the sum of numbers in the resulting table? https://cdn.artofproblemsolving.com/attachments/a/9/63a9c38762a4c895688fff049ed08c96b2c22c.png
combinatorics
computational with tangent circles

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 2 E+13

12/12/2022
Circle k1k_1 has radius 1010, externally touching circle k2k_2 with radius 1818. Circle k3k_3 touches both circles, as well as the line ee determined by their centres. Let k4k_4 be the circle touching k2k_2 and k3k_3 externally (other than k1k_1) whose center lies on line ee. What is the radius of k4k_4?
tangent circlesgeometry