MathDB

4

Part of 2022 Durer Math Competition Finals

Problems(3)

divide divisors into two groups with equal product if their product is a perfect

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 1 E4

11/29/2022
Show that the divisors of a number n≥2n \ge 2 can only be divided into two groups in which the product of the numbers is the same if the product of the divisors of nn is a square number.
number theoryPerfect SquaresDivisors
min no of regular triangles to cover the lines of a diagram

Source: (2021-) 2022 XV Dürer Math Competition Finals Day2 E4

12/1/2022
At least how many regular triangles are needed to cover the lines of the following diagram? https://cdn.artofproblemsolving.com/attachments/e/3/4de2ed2c7cc9421d7d060f0bc537ccaa3838fc.png (Only the perimeter of the triangles is involved in the covering, and the entire perimeter need not be incident on the diagram.)
combinatoricsisosceles
GJ , FB , HK concurrent or parallel , ABCD cyclic and orthodiagonal

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 1 E+4

11/29/2022
ABCDABCD is a cyclic quadrilateral whose diagonals are perpendicular to each other. Let OO denote the centre of its circumcircle and EE the intersection of the diagonals. JJ and KK denote the perpendicular projections of EE on the sides ABAB and BCBC . Let F,GF , G and HH be the midpoint line segments. Show that lines GJGJ , FBFB and HKHK either pass through the same point or are parallel to each other.
geometryconcurrencyconcurrentCyclic