IR-MO 2013

Planet Tarator is a planet in the Yoghurty way galaxy. This planet has a shape of convex $1392$ -hedron. On earth we don't have any other information about sides of planet tarator. We have discovered that each side of the planet is a country, and has it's own currency. Each two neighbour countries have their own constant exchange rate, regardless of other exchange rates. Anybody who travels on land and crosses the border must change all his money to the currency of the destination country, and there's no other way to change the money. Incredibly, a person's money may change after crossing some borders and getting back to the point he started, but it's guaranteed that crossing a border and then coming back doesn't change the money. On a research project a group of tourists were chosen and given same amount of money to travel around the Tarator planet and come back to the point they started. They always travel on land and their path is a nonplanar polygon which doesn't intersect itself. What is the maximum number of tourists that may have a pairwise different final amount of money?Note 1: Tourists spend no money during travel!Note 2: The only constant of the problem is 1392, the number of the sides. The exchange rates and the way the sides are arranged are unknown. Answer must be a constant number, regardless of the variables.Note 3: The maximum must be among all possible polyhedras.

Time allowed for this problem was 90 minutes.

An equation $P(x)=Q(y)$ is called Interesting if $P$ and $Q$ are polynomials with degree at least one and integer coefficients and the equations has an infinite number of answers in $\mathbb{N}$ . An interesting equation $P(x)=Q(y)$ yields in interesting equation $F(x)=G(y)$ if there exists polynomial $R(x) \in \mathbb{Q} [x]$ such that $F(x) \equiv R(P(x))$ and $G(x) \equiv R(Q(x))$ . (a) Suppose that $S$ is an infinite subset of $\mathbb{N} \times \mathbb{N}$ . $S$ is an answer of interesting equation $P(x)=Q(y)$ if each element of $S$ is an answer of this equation. Prove that for each $S$ there's an interesting equation $P_0(x)=Q_0(y)$ such that if there exists any interesting equation that $S$ is an answer of it, $P_0(x)=Q_0(y)$ yields in that equation. (b) Define the degree of an interesting equation $P(x)=Q(y)$ by $max\{deg(P),deg(Q)\}$ . An interesting equation is called primary if there's no other interesting equation with lower degree that yields in it. Prove that if $P(x)=Q(y)$ is a primary interesting equation and $P$ and $Q$ are monic then $(deg(P),deg(Q))=1$ .

Time allowed for this question was 2 hours.

Let $A_1A_2A_3A_4A_5$ be a convex 5-gon in which the coordinates of all of it's vertices are rational. For each $1\leq i \leq 5$ define $B_i$ the intersection of lines $A_{i+1}A_{i+2}$ and $A_{i+3}A_{i+4}$ . ( $A_i=A_{i+5}$ ) Prove that at most 3 lines from the lines $A_iB_i$ ( $1\leq i \leq 5$ ) are concurrent.

Time allowed for this problem was 75 minutes.