Školjka

%V0 In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers $m$ and $n$, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths $m$ and $n$, lie along edges of the squares. Let $S_1$ be the total area of the black part of the triangle and $S_2$ be the total area of the white part. Let $f(m,n) = | S_1 - S_2 |$. a) Calculate $f(m,n)$ for all positive integers $m$ and $n$ which are either both even or both odd. b) Prove that $f(m,n) \leq \frac 12 \max \{m,n \}$ for all $m$ and $n$. c) Show that there is no constant $C\in\mathbb{R}$ such that $f(m,n) < C$ for all $m$ and $n$.

%V0 An $n \times n$ matrix whose entries come from the set $S = \{1, 2, \ldots , 2n - 1\}$ is called a silver matrix if, for each $i = 1, 2, \ldots , n$, the $i$-th row and the $i$-th column together contain all elements of $S$. Show that: (a) there is no silver matrix for $n = 1997$; (b) silver matrices exist for infinitely many values of $n$.

%V0 It is known that $\angle BAC$ is the smallest angle in the triangle $ABC$. The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$. The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ at $V$ and $W$, respectively. The lines $BV$ and $CW$ meet at $T$. Show that $AU = TB + TC$. Alternative formulation: Four different points $A,B,C,D$ are chosen on a circle $\Gamma$ such that the triangle $BCD$ is not right-angled. Prove that: (a) The perpendicular bisectors of $AB$ and $AC$ meet the line $AD$ at certain points $W$ and $V,$ respectively, and that the lines $CV$ and $BW$ meet at a certain point $T.$ (b) The length of one of the line segments $AD, BT,$ and $CT$ is the sum of the lengths of the other two.

%V0 Find all pairs $(a,b)$ of positive integers that satisfy the equation: $a^{b^2} = b^a$.

%V0 Let $x_1$, $x_2$, $\ldots$, $x_n$ be real numbers satisfying the conditions: $ |x_1 + x_2 + \dots + x_n| = 1 $ and $|x_i| \leq \frac{n+1}{2}$, for $i = 1, 2, \dots, n$ Show that there exists a permutation $y_1$, $y_2$, $\ldots$, $y_n$ of $x_1$, $x_2$, $\ldots$, $x_n$ such that $$| y_1 + 2 y_2 + \cdots + n y_n | \leq \frac {n + 1}{2}.$$

%V0 For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4) = 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1. Prove that, for any integer $n \geq 3$ we have $2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.