RMM 2010

For a finite non empty set of primes $P$ , let $m(P)$ denote the largest possible number of consecutive positive integers, each of which is divisible by at least one member of $P$ .

(i) Show that $|P|\le m(P)$ , with equality if and only if $\min(P)>|P|$ .

(ii) Show that $m(P)<(|P|+1)(2^{|P|}-1)$ .

(The number $|P|$ is the size of set $P$ )

Dan Schwarz, Romania

For each positive integer $n$ , find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$ , one can find numbers $x_1, x_2, \cdots x_n$ , such that $0 \le x_i \le 1$ satisfying $$|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n$$

Marko Radovanović, Serbia

Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$ , define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$ ). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$ . Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent.

Pavel Kozhevnikov, Russia

Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:

(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);

(ii) $|a_1-b_1|+|a_2-b_2|=2010$ ;

(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$ ;

(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$ .

Massimo Gobbino, Italy

Let $n$ be a given positive integer. Say that a set $K$ of points with integer coordinates in the plane is connected if for every pair of points $R, S\in K$ , there exists a positive integer $\ell$ and a sequence $R=T_0,T_1, T_2,\ldots ,T_{\ell}=S$ of points in $K$ , where each $T_i$ is distance $1$ away from $T_{i+1}$ . For such a set $K$ , we define the set of vectors $$\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}$$ What is the maximum value of $|\Delta(K)|$ over all connected sets $K$ of $2n+1$ points with integer coordinates in the plane?

Grigory Chelnokov, Russia

Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$ , we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$ , $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$ . (Given a set $S$ , we write $f(S)$ for the set $\{f(x)\mid x\in S\})$ . Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$ , $n\geq 0$ . Prove that $f^{\omega}(\mathbb{Q})$ is a finite set.

Dan Schwarz, Romania