16
Problems(2)
remove some internal walls in a 2x7 squaretable
Source: 2021 Dürer Math Competition Finals Day2 E16 https://artofproblemsolving.com/community/c2749870_
1/9/2022
Consider a table consisting of squares. Each little square is surrounded by walls (each internal wall belongs to two squares). We would like to remove some internal walls to make it possible to get from any square to any other one without crossing walls. How many ways can we do this while removing the minimal possible number of internal walls?
The figure shows a possible configuration, the remaining walls are marked in red, the removed ones are marked in light pink. Two configurations are considered the same if the same walls are removed.
https://cdn.artofproblemsolving.com/attachments/d/c/1a3d9ab0d0971929e6d484a970d4b1f36f0031.png
combinatorics
convex quadrilateral with angles and sides in 2 arithmetic sequences
Source: 2021 Dürer Math Competition Finals Day2 E+16 https://artofproblemsolving.com/community/c2749870_
1/9/2022
The angles of a convex quadrilateral form an arithmetic sequence in clockwise order, and its side lengths also form an arithmetic sequence (but not necessarily in clockwise order). If the quadrilateral is not a square, and its shortest side has length , then its perimeter is , where and are positive integers. What is the value of ?
geometryanglesarithmetic sequence