MathDB

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Part of 2021 Bundeswettbewerb Mathematik

Problems(2)

Dividing a cube into three cuboids with integral side lengths

Source: Bundeswettbewerb Mathematik 2021, Round 1 - Problem 1

4/12/2021
A cube with side length 1010 is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?
3D geometryinequalities
Each natural number has a multiple with same sum of digits as its square

Source: Bundeswettbewerb Mathematik (German Federal Math Competiton) Round 2 2021

9/5/2021
Let Q(n)Q(n) denote the sum of the digits of nn in its decimal representation. Prove that for every positive integer kk, there exists a multiple nn of kk such that Q(n)=Q(n2)Q(n)=Q(n^2).
number theory