MathDB

3

Part of 2022 Durer Math Competition Finals

Problems(3)

n numbered students , n pieces of paper

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 1 E3

11/29/2022
nn students, numbered from 11 to nn are sitting next to each other in a class. In the beginning the 11st student has nn pieces of paper in one pile. The goal of the students is to distribute the nn pieces in a way that everyone gets exactly one. The teacher claps once in a minute and for each clap the students can choose one of the following moves (or do nothing): ∙\bullet They divide one of their piles of paper into two smaller piles. ∙\bullet They give one of their piles of paper to the student with the next number. At least how many times does the teacher need to clap in order to make it possible for the students to distribute all the pieces of paper amongst themselves?
combinatorics
3 palaces, each rotating on a duck leg

Source: (2021-) 2022 XV Dürer Math Competition Finals Day2 E3

12/1/2022
Three palaces, each rotating on a duck leg, make a full round in 3030, 5050, and 7070 days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards?
algebranumber theory
sum (x - yz)^2/(x + yz)^2 <1 if x,y,z>0 with x+y+z = 1, x> yz, y > zx, z > xy

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 1 E+3

11/29/2022
Let x,y,zx, y, z denote positive real numbers for which x+y+z=1x+y+z = 1 and x>yzx > yz, y>zxy > zx, z>xyz > xy. Prove that (x−yzx+yz)2+(y−zxy+zx)2+(z−xyz+xy)2<1.\left(\frac{x - yz}{x + yz}\right)^2+ \left(\frac{y - zx}{y + zx}\right)^2+\left(\frac{z - xy}{z + xy}\right)^2< 1.
algebrainequalities