## IMC 2016 – Day 2 – Problem 8

**Problem 8.** Let be a positive integer and denote by the ring of integers modulo . Suppose that there exists a function satisfying the following three properties:

- (i) ,
- (ii) ,
- (iii) for all .

Prove that modulo .

## IMC 2016 – Day 1 – Problem 1

**Problem 1.** Let be continuous on and differentiable on . Suppose that has infinitely many zeros, but there is no with .

- (a) Prove that .
- (b) Give an example of such a function.

## There isn’t such a function

Prove that there is no continuous function such that and .

## Number of even images is very small

I found on Math Stack Exchange a generalisation of a problem I proposed in the Romanian Mathematical Olympiad:

Let be a continuous function such that and . Moreover assume is finite for all . Prove that is countable.

My problem was to prove that there doesn’t exist a function with the property that the equation has an even number of solutions for all .

## Function on Subsets

Let be a function defined on the subsets of a finite set . Prove that if and for all subsets , then assumes at most distinct values.

*Miklos Schweitzer, 2001*

## Reccurent function sequence SEEMOUS 2010

Suppose is a continuous function, and define the sequence in the following way:

.

a) Prove that the series converges for any .

b) Find an explicit formula in terms of for the above series.

*Seemous 2010, Problem 1*

## Cute Problem with functions

Let and A function. Define and . Assume that with and coprimes ( ), such that . Find all functions in the following cases:

i) .

ii) such that

iii) Any set such that there are no three collinear points in .