MathDB

1

Part of 2019 Durer Math Competition Finals

Problems(3)

colouring positive rational numbers with blue and red

Source: 2019 Dürer Math Competition Finals E1.1

11/28/2020
a) Is it possible to colour the positive rational numbers with blue and red in a way that there are numbers of both colours and the sum of any two numbers of the same colour is the same colour as them? b) Is it possible to colour the positive rational numbers with blue and red in a way that there are numbers of both colours and the product of any two numbers of the same colour is the same colour as them?
Note: When forming a sum or a product, it is allowed to pick the same number twice.
Coloringcombinatoricsnumber theory
no of integer triangles whose sum of shorter sides is 19

Source: 2019 Dürer Math Competition Finals Day2 E1 https://artofproblemsolving.com/community/c1621835_2019_

1/5/2022
Find the number of non-isosceles triangles (up to congruence) with integral side lengths, in which the sum of the two shorter sides is 1919.
geometrycombinatoricscombinatorial geometry
(\sum a_i )^2 >= sum a_i^3 if a_n non-decreasing, a_0 = 0,a_j-a_i<=j -i for j>1

Source: 2019 Dürer Math Competition Finals E+1.1

11/28/2020
Let ao,a1,a2,..,ana_o,a_1,a_2,..,a_ n be a non-decreasing sequence of n+1n+1 real numbers where a0=0a_0 = 0 and for every j>ij > i we have aj−ai≤j−ia_j - a_i \le j - i. Show that (∑i=0nai)2≥∑i=0nai3\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3
algebrainequalitiesSequenceSum