MathDB

x = f(x) + f(f(x)) - f(f(f(x))) if x \in H={-2019,...,2020}

Source: 2020 Dürer Math Competition Finals E1.5 E+1.2

11/29/2020

Let

H = \{-2019,-2018, ...,-1, 0, 1, 2, ..., 2020\}

. Describe all functions

f : H \to H

for which a)

x = f(x) - f(f(x))

holds for every

x \in H

. b)

x = f(x) + f(f(x)) - f(f(f(x)))

holds for every

x \in H

. c)

x = f(x) + 2f(f(x)) - 3f(f(f(x)))

holds for every

x \in H

.
PS. (a) + (b) for category E 1.5, (b) + (c) for category E+ 1.2

algebrafunctional equationfunctional

4 consecutive sides of equiangular hexagon have lengths 7, 6, 3, 5 in this order

Source: 2020 Dürer Math Competition Finals Day2 E5 https://artofproblemsolving.com/community/c1622639_2020_

1/7/2022

The hexagon

ABCDEF

has all angles equal . We know that four consecutive sides of the hexagon have lengths

7, 6, 3

and

5

in this order. What is the sum of the lengths of the two remaining sides?

equal angleshexagongeometry

no of orientations of a connected 3 -regular graph on 2n vertices

Source: 2020 Dürer Math Competition Finals E+ 1. 5

11/30/2020

Prove that the number of orientations of a connected

3

-regular graph on

2n

vertices where the number of vertices with indegree

0

and outdegree

0

are equal, is exactly

2^{n+1}

{2n} \choose {n}

.

combinatoricsgraph theorygraph

sum of all numbers on the paper having exactly 2 proper divisors v2

Source: 2020 Dürer Math Competition Finals Day2 E+5 https://artofproblemsolving.com/community/c1622639_2020_

1/7/2022

On a piece of paper, we write down all positive integers

n

such that all proper divisors of

n

are less than

30

. We know that the sum of all numbers on the paper having exactly one proper divisor is

2397

. What is the sum of all numbers on the paper having exactly two proper divisors?
We say that

k

is a proper divisor of the positive integer

n

if

k | n

and

1 < k < n

.

number theorydivisor

5