BG-MO 1975

Find all pairs of natural numbers $(m,n)$ bigger than $1$ for which $2^m+3^n$ is the square of whole number.

I. Tonov

Let $F$ be a polygon the boundary of which is a broken line with vertices in the knots (units) of a given in advance regular square network. If $k$ is the count of knots of the network situated over the boundary of $F$ , and $\ell$ is the count of the knots of the network lying inside $F$ , prove that if the surface of every square from the network is $1$ , then the surface $S$ of $F$ is calculated with the formulae: $$S=\frac k2+\ell-1$$ V. Chukanov

Let $f(x)=a_0x^3+a_1x^2+a_2x+a_3$ be a polynomial with real coefficients ( $a_0\ne0$ ) such that $|f(x)|\le1$ for every $x\in[-1,1]$ . Prove that

(a) there exist a constant $c$ (one and the same for all polynomials with the given property), for which (b) $|a_0|\le4$ .

V. Petkov

In the plane are given a circle $k$ with radii $R$ and the points $A_1,A_2,\ldots,A_n$ , lying on $k$ or outside $k$ . Prove that there exist infinitely many points $X$ from the given circumference for which $$\sum_{i=1}^n A_iX^2\ge2nR^2.$$ Does there exist a pair of points on different sides of some diameter, $X$ and $Y$ from $k$ , such that $$\sum_{i=1}^n A_iX^2\ge2nR^2\text{ and }\sum_{i=1}^n A_iY^2\ge2nR^2?$$ H. Lesov

Let the subbishop (a bishop is the figure moving only by a diagonal) be a figure moving only by diagonal but only in the next cells (squares) of the chessboard. Find the maximal count of subbishops over a chessboard $n\times n$ , no two of which are not attacking.

V. Chukanov

Some of the faces of a convex polyhedron $M$ are painted in blue, others are painted in white and there are no two walls with a common edge. Prove that if the sum of surfaces of the blue walls is bigger than half surface of $M$ then it may be inscribed a sphere in the polyhedron given $(M)$ .

(H. Lesov)