FI-MO 2009

In a plane, the point $(x,y)$ has temperature $x^2+y^2-6x+4y$ . Determine the coldest point of the plane and its temperature.

A polynomial $P$ has integer coefficients and $P(3)=4$ and $P(4)=3$ . For how many $x$ we might have $P(x)=x$ ?

The circles $\mathcal{Y}_0$ and $\mathcal{Y}_1$ lies outside each other. Let $O_0$ be the center of $\mathcal{Y}_0$ and $O_1$ be the center of $\mathcal{Y}_1$ . From $O_0$ , draw the rays which are tangents to $\mathcal{Y}_1$ and similarty from $O_1$ , draw the rays which are tangents to $\mathcal{Y}_0$ . Let the intersection points of rays and circle $\mathcal{Y}_i$ be $A_i$ and $B_i$ . Show that the line segments $A_0B_0$ and $A_1B_1$ have equal lengths.

We say that the set of step lengths $D\subset \mathbb{Z}_+=\{1,2,\ldots\}$ is excellent if it has the following property: If we split the set of integers into two subsets $A$ and $\mathbb{Z}\setminus{A}$ , at least other set contains element $a-d,a,a+d$ (i.e. $\{a-d,a,a+d\} \subset A$ or $\{a-d,a,a+d\}\in \mathbb{Z}\setminus A$ from some integer $a\in \mathbb{Z},d\in D$ .) For example the set of one element $\{1\}$ is not excellent as the set of integer can be split into even and odd numbers, and neither of these contains three consecutive integer. Show that the set $\{1,2,3,4\}$ is excellent but it has no proper subset which is excellent.

As in the picture below, the rectangle on the left hand side has been divided into four parts by line segments which are parallel to a side of the rectangle. The areas of the small rectangles are $A,B,C$ and $D$ . Similarly, the small rectangles on the right hand side have areas $A^\prime,B^\prime,C^\prime$ and $D^\prime$ . It is known that $A\leq A^\prime$ , $B\leq B^\prime$ , $C\leq C^\prime$ but $D\leq B^\prime$ .

$A$ $B$ $C$ $D$ $A'$ $B'$ $C'$ $D'$

Prove that the big rectangle on the left hand side has area smaller or equal to the area of the big rectangle on the right hand side, i.e. $A+B+C+D\leq A^\prime+B^\prime+C^\prime+D^\prime$ .