MathDB

1

Part of 1980 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1980 Problem 1.1

Source: Bundeswettbewerb Mathematik 1980 Round 1

9/23/2022
Six free cells are given in a row. Players AA and BB alternately write digits from 00 to 99 in empty cells, with AA starting. When all the cells are filled, one considers the obtained six-digit number zz. Player BB wins if zz is divisible by a given natural number nn, and loses otherwise. For which values of nn not exceeding 2020 can BB win independently of his opponent’s moves?
gamenumber theoryDivisibilityDigits
if sum \sqrt[3]{a}+\sqrt[3]{b} is rational then a,b are perfect cubes

Source: 1992 ITAMO p6

1/31/2020
Let aa and bb be integers. Prove that if a3+b3\sqrt[3]{a}+\sqrt[3]{b} is a rational number, then both aa and bb are perfect cubes.
perfect cuberationalnumber theory