KURSCHAK 1995

Given in the plane is a lattice and a grid rectangle with sides parallel to the coordinate axes. We divide the rectangle into grid triangles with area $\frac12$ . Prove that the number of right angled triangles is at least twice as much as the shorter side of the rectangle.

(A grid polygon is a polygon such that both coordinates of each vertex is an integer.)

Consider a polynomial in $n$ variables with real coefficients. We know that if every variable is $\pm1$ , the value of the polynomial is positive, or negative if the number of $-1$ 's is even, or odd, respectively. Prove that the degree of this polynomial is at least $n$ .

Points $A$ , $B$ , $C$ , $D$ are such that no three of them are collinear. Let $E=AB\cap CD$ and $F=BC\cap DA$ . Let $k_1$ , $k_2$ and $k_3$ denote the circles with diameter $\overline{AC}$ , $\overline{BD}$ and $\overline{EF}$ , respectively. Prove that either $k_1,k_2,k_3$ pass through one point, or no two of them intersect.