ISL 2000

Let $a, b, c$ be positive integers satisfying the conditions $b > 2a$ and $c > 2b.$ Show that there exists a real number $\lambda$ with the property that all the three numbers $\lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $\left(\frac {1}{3}, \frac {2}{3} \right].$

Find all pairs of functions $f : \mathbb R \to \mathbb R$ , $g : \mathbb R \to \mathbb R$ such that $$f \left( x + g(y) \right) = xf(y) - y f(x) + g(x) \text{for all } x, y\in\mathbb{R}.$$

The function $F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every $n \geq 0,$

(i) $F(4n) = F(2n) + F(n),$ (ii) $F(4n + 2) = F(4n) + 1,$ (iii) $F(2n + 1) = F(2n) + 1.$

Prove that for each positive integer $m,$ the number of integers $n$ with $0 \leq n < 2^m$ and $F(4n) = F(3n)$ is $F(2^{m + 1}).$

A nonempty set $A$ of real numbers is called a $B_3$ -set if the conditions $a_1, a_2, a_3, a_4, a_5, a_6 \in A$ and $a_1 + a_2 + a_3 = a_4 + a_5 + a_6$ imply that the sequences $(a_1, a_2, a_3)$ and $(a_4, a_5, a_6)$ are identical up to a permutation. Let $A = \{a_0 = 0 < a_1 < a_2 < \cdots \}$ , $B = \{b_0 = 0 < b_1 < b_2 < \cdots \}$ be infinite sequences of real numbers with $D(A) = D(B),$ where, for a set $X$ of real numbers, $D(X)$ denotes the difference set $\{|x-y|\mid x, y \in X \}.$ Prove that if $A$ is a $B_3$ -set, then $A = B.$

For a polynomial $P$ of degree 2000 with distinct real coefficients let $M(P)$ be the set of all polynomials that can be produced from $P$ by permutation of its coefficients. A polynomial $P$ will be called $n$ -independent if $P(n) = 0$ and we can get from any $Q \in M(P)$ a polynomial $Q_1$ such that $Q_1(n) = 0$ by interchanging at most one pair of coefficients of $Q.$ Find all integers $n$ for which $n$ -independent polynomials exist.

A staircase-brick with 3 steps of width 2 is made of 12 unit cubes. Determine all integers $n$ for which it is possible to build a cube of side $n$ using such bricks.

Attachments:

Let $n \geq 4$ be a fixed positive integer. Given a set $S = \{P_1, P_2, \ldots, P_n\}$ of $n$ points in the plane such that no three are collinear and no four concyclic, let $a_t,$ $1 \leq t \leq n,$ be the number of circles $P_iP_jP_k$ that contain $P_t$ in their interior, and let $$m(S)=a_1+a_2+\cdots + a_n.$$ Prove that there exists a positive integer $f(n),$ depending only on $n,$ such that the points of $S$ are the vertices of a convex polygon if and only if $m(S) = f(n).$

In the plane we have $n$ rectangles with parallel sides. The sides of distinct rectangles lie on distinct lines. The boundaries of the rectangles cut the plane into connected regions. A region is nice if it has at least one of the vertices of the $n$ rectangles on the boundary. Prove that the sum of the numbers of the vertices of all nice regions is less than $40n$ . (There can be nonconvex regions as well as regions with more than one boundary curve.)

Let $A_1A_2 \ldots A_n$ be a convex polygon, $n \geq 4.$ Prove that $A_1A_2 \ldots A_n$ is cyclic if and only if to each vertex $A_j$ one can assign a pair $(b_j, c_j)$ of real numbers, $j = 1, 2, \ldots, n,$ so that $A_iA_j = b_jc_i - b_ic_j$ for all $i, j$ with $1 \leq i < j \leq n.$

The tangents at $B$ and $A$ to the circumcircle of an acute angled triangle $ABC$ meet the tangent at $C$ at $T$ and $U$ respectively. $AT$ meets $BC$ at $P$ , and $Q$ is the midpoint of $AP$ ; $BU$ meets $CA$ at $R$ , and $S$ is the midpoint of $BR$ . Prove that $\angle ABQ=\angle BAS$ . Determine, in terms of ratios of side lengths, the triangles for which this angle is a maximum.

Ten gangsters are standing on a flat surface, and the distances between them are all distinct. At twelve o'clock, when the church bells start chiming, each of them fatally shoots the one among the other nine gangsters who is the nearest. At least how many gangsters will be killed?

Determine all positive integers $n\geq 2$ that satisfy the following condition: for all $a$ and $b$ relatively prime to $n$ we have $$a \equiv b \pmod n \text{if and only if} ab\equiv 1 \pmod n.$$

Prove that there exist infinitely many positive integers $n$ such that $p = nr,$ where $p$ and $r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.