4

Part of 2022 Bundeswettbewerb Mathematik

Problems(2)

3-adic divisibility of a divisor sum

Source: Bundeswettbewerb Mathematik 2022, Round 1 - Problem 4

For each positive integer

k

a_k

be the largest divisor of

k

which is not divisible by

3

s_n=a_1+a_2+\dots+a_n

. Show that: (a) The number

s_n

is divisible by

3

iff the number of ones in the ternary expansion of

n

is divisible by

3

. (b) There are infinitely many

n

s_n

is divisible by

3^3

number theorynumber theory proposeddivisorsum of divisors

Enclosing circle of diameter $\sqrt{2}$

Source: Bundeswettbewerb Mathematik 2022, Round 2, Problem 4

Some points in the plane are either colored red or blue. The distance between two points of the opposite color is at most 1.
Prove that there exists a circle with diameter

\sqrt{2}

such that no two points outside of this circle have same color. It is enough to prove this claim for a finite number of colored points.

enclosing circlescolored pointsgeometrycombinatorics