2012 Federal Competition For Advanced Students, Part 1

Part of Austrian MO National Competition

Subcontests

(4)

1

Function such that (m,n)|f(m)+f(n)

Determine all functions

f: \mathbb{Z}\to\mathbb{Z}

satisfying the following property: For each pair of integers

m

n

(not necessarily distinct),

\mathrm{gcd}(m, n)

f(m) + f(n)

n\in\mathbb{Z}

\mathrm{gcd}(m, n)=\mathrm{gcd}(|m|, |n|)

\mathrm{gcd}(n, 0)=n

1

Number of colorings using three colors

Consider a stripe of

n

fieds, numbered from left to right with the integers

1

n

in ascending order. Each of the fields is colored with one of the colors

1

2

3

. Even-numbered fields can be colored with any color. Odd-numbered fields are only allowed to be colored with the odd colors

1

3

. How many such colorings are there such that any two neighboring fields have different colors?

1

Diophantine equation: n!+An = n^k

Determine all solutions

(n, k)

of the equation n!+An = n^k with

n, k \in\mathbb{N}

A = 7

A = 2012

1

Gamma is the second-largest angle

ABC

be a scalene (i.e. non-isosceles) triangle. Let

U

be the center of the circumcircle of this triangle and

I

the center of the incircle. Assume that the second point of intersection different from

C

of the angle bisector of

\gamma = \angle ACB

with the circumcircle of

ABC

lies on the perpendicular bisector of

UI

\gamma

is the second-largest angle in the triangle

ABC