ISL 1997

Let $R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $R_1 = (1),$ and if $R_{n - 1} = (x_1, \ldots, x_s),$ then

$$R_n = (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).$$

For example, $R_2 = (1, 2),$ $R_3 = (1, 1, 2, 3),$ $R_4 = (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $n > 1,$ then the $k$ th term from the left in $R_n$ is equal to 1 if and only if the $k$ th term from the right in $R_n$ is different from 1.

For each finite set $U$ of nonzero vectors in the plane we define $l(U)$ to be the length of the vector that is the sum of all vectors in $U.$ Given a finite set $V$ of nonzero vectors in the plane, a subset $B$ of $V$ is said to be maximal if $l(B)$ is greater than or equal to $l(A)$ for each nonempty subset $A$ of $V.$

(a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively.

(b) Show that, for any set $V$ consisting of $n \geq 1$ vectors the number of maximal subsets is less than or equal to $2n.$

Let $ABCD$ be a regular tetrahedron and $M,N$ distinct points in the planes $ABC$ and $ADC$ respectively. Show that the segments $MN,BN,MD$ are the sides of a triangle.

(a) Let $n$ be a positive integer. Prove that there exist distinct positive integers $x, y, z$ such that

$$x^{n-1} + y^n = z^{n+1}.$$

(b) Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or to $b.$ Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers $x, y, z$ such that

$$x^a + y^b = z^c.$$

The lengths of the sides of a convex hexagon $ABCDEF$ satisfy $AB = BC$ , $CD = DE$ , $EF = FA$ . Prove that: $$\frac {BC}{BE} + \frac {DE}{DA} + \frac {FA}{FC} \geq \frac {3}{2}.$$

Let $P(x)$ be a polynomial with real coefficients such that $P(x) > 0$ for all $x \geq 0.$ Prove that there exists a positive integer n such that $(1 + x)^n \cdot P(x)$ is a polynomial with nonnegative coefficients.

Let $p$ be a prime number and $f$ an integer polynomial of degree $d$ such that $f(0) = 0,f(1) = 1$ and $f(n)$ is congruent to $0$ or $1$ modulo $p$ for every integer $n$ . Prove that $d\geq p - 1$ .

In an acute-angled triangle $ABC,$ let $AD,BE$ be altitudes and $AP,BQ$ internal bisectors. Denote by $I$ and $O$ the incenter and the circumcentre of the triangle, respectively. Prove that the points $D, E,$ and $I$ are collinear if and only if the points $P, Q,$ and $O$ are collinear.

Let $a_1\geq \cdots \geq a_n \geq a_{n + 1} = 0$ be real numbers. Show that $$\sqrt {\sum_{k = 1}^n a_k} \leq \sum_{k = 1}^n \sqrt k (\sqrt {a_k} - \sqrt {a_{k + 1}}).$$ Proposed by Romania

Does there exist functions $f,g: \mathbb{R}\to\mathbb{R}$ such that $f(g(x)) = x^2$ and $g(f(x)) = x^k$ for all real numbers $x$

a) if $k = 3$ ?

b) if $k = 4$ ?

Let $ABCD$ be a convex quadrilateral. The diagonals $AC$ and $BD$ intersect at $K$ . Show that $ABCD$ is cyclic if and only if $AK \sin A + CK \sin C = BK \sin B + DK \sin D$ .

Let $X,Y,Z$ be the midpoints of the small arcs $BC,CA,AB$ respectively (arcs of the circumcircle of $ABC$ ). $M$ is an arbitrary point on $BC$ , and the parallels through $M$ to the internal bisectors of $\angle B,\angle C$ cut the external bisectors of $\angle C,\angle B$ in $N,P$ respectively. Show that $XM,YN,ZP$ concur.

For every integer $n \geq 2$ determine the minimum value that the sum $\sum^n_{i=0} a_i$ can take for nonnegative numbers $a_0, a_1, \ldots, a_n$ satisfying the condition $a_0 = 1,$ $a_i \leq a_{i+1} + a_{i+2}$ for $i = 0, \ldots, n - 2.$