MathDB

5

Part of 2022 Durer Math Competition Finals

Problems(4)

areas of rectangles by n x k lines on a rectangle

Source: (2021-) 2022 XV 15th Dürer r Math Competition Finals Day 1 E5 E+2

11/29/2022
Annie drew a rectangle and partitioned it into nn rows and kk columns with horizontal and vertical lines. Annie knows the area of the resulting n⋅kn \cdot k little rectangles while Benny does not. Annie reveals the area of some of these small rectangles to Benny. Given nn and kk at least how many of the small rectangle’s areas did Annie have to reveal, if from the given information Benny can determine the areas of all the n⋅kn \cdot k little rectangles? For example in the case n=3n = 3 and k=4k = 4 revealing the areas of the 1010 small rectangles if enough information to find the areas of the remaining two little rectangles. https://cdn.artofproblemsolving.com/attachments/b/1/c4b6e0ab6ba50068ced09d2a6fe51e24dd096a.png
geometryrectanglecombinatoricsareascombinatorial geometry
smallest n fow which the radius of the n-th circle is an integer >1

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 2 E5

12/12/2022
Benedek draws circles with the same center in the following way. The first circle he draws has radius 11. Next, he draws a second circle such that the ring between the first and second circles has twice the area of the first circle. Next, he draws a third circle such that the ring between the second and third circles is three times the area of the first circle, and so on (see the diagram). What is the smallest nn fow which the radius of the nn-th circle is an integer greater than 11? https://cdn.artofproblemsolving.com/attachments/e/2/afa6d5ead6f2252aa821028370a3768912e674.png
geometryareascircles
ratio of angles in cylic quad

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 2 E+5

12/12/2022
On a circle kk, we marked four points (A,B,C,D)(A, B, C, D) and drew pairwise their connecting segments. We denoted angles as seen on the diagram. We know that α1:α2=2:5\alpha_1 : \alpha_2 = 2 : 5, β1:β2=7:11\beta_1 : \beta_2 = 7 : 11, and γ1:γ2=10:3\gamma_1 : \gamma_2 = 10 : 3. If δ1:δ2=p:q\delta_1 : \delta_2 = p : q, where pp and qq are coprime positive integers, then what is pp? https://cdn.artofproblemsolving.com/attachments/c/e/b532dd164a7cf99cea7b3b7d98f81796622da5.png
ratiogeometryangles
after k minutes, everyone will have a number that is divisible by n

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 1 E+5

11/29/2022
nn people sitting at a round table. In the beginning, everyone writes down a positive number nn on piece of paper in front of them. From now on, in every minute, they write down the number that they get if they subtract the number of their right-hand neighbour from their own number. They write down the new number and erase the original. Give those number nn that there exists an integer kk in a way that regardless of the starting numbers, after kk minutes, everyone will have a number that is divisible by nn.
combinatoricsnumber theory