2

Part of 2021 Bundeswettbewerb Mathematik

Problems(2)

Writing 3/n as sum of two reciprocals in exactly 2021 ways

Source: Bundeswettbewerb Mathematik 2021, Round 1 - Problem 2

\frac{3}{10}

can be written as a sum of two reciprocals in exactly two ways:

\frac{3}{10}=\frac{1}{5}+\frac{1}{10}=\frac{1}{4}+\frac{1}{20}

a) In how many ways can

\frac{3}{2021}

be written as a sum of two reciprocals? b) Is there a positive integer

n

not divisible by

3

with the property that

\frac{3}{n}

can be written as a sum of two reciprocals in exactly

2021

number theorynumber theory proposedreciprocal sum

Looking for a 4-cycle in a graph

Source: Bundeswettbewerb Mathematik 2021, Round 2 - Problem 2

A school has 2021 students, each of which knows at least 45 of the other students (where "knowing" is mutual).
Show that there are four students who can be seated at a round table such that each of them knows both of her neighbours.

combinatoricscombinatorics proposedgraph theorygraph cycles