Olympiad problems

2006 IMO, 5

Tags: algebra , polynomial , roots , IMO , IMO 2006 , IMO Shortlist , Dan Schwarz

Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.

2005 IMO, 6

Tags: combinatorics , IMO , IMO 2005 , graph theory , IMO Shortlist , Dan Schwarz , Evan admits orz

In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac 25$ of the contestants. Moreover, no contestant solved all the $6$ problems. Show that there are at least $2$ contestants who solved exactly $5$ problems each. [i]Radu Gologan and Dan Schwartz[/i]

2005 IMO Shortlist, 6

Tags: combinatorics , IMO , IMO 2005 , graph theory , IMO Shortlist , Dan Schwarz , Evan admits orz

In a mathematical competition, in which $6$ problems were posed to the participants, every two of these problems were solved by more than $\frac 25$ of the contestants. Moreover, no contestant solved all the $6$ problems. Show that there are at least $2$ contestants who solved exactly $5$ problems each. [i]Radu Gologan and Dan Schwartz[/i]

2006 IMO Shortlist, 4

Tags: algebra , polynomial , roots , IMO , IMO 2006 , IMO Shortlist , Dan Schwarz

Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.