Navvye/Olympiad-Problems-CSV huggingface dataset:
1995 Austrian-Polish Competition
Consider the cube with the vertices at the points $(\pm 1, \pm  1, \pm  1)$. Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$, where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$, where $s_i = \pm 1$.

(a) If $d = 48$, prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$.

(b) Find a smaller $d$ (the smaller, the better) with the same property.
1995 Austrian-Polish Competition
Prove that for all positive integers $n,m$ and all real numbers $x, y > 0$ the following inequality holds:

$$(n - 1)(m- 1)(x^{n+m} + y^{n+m}) + (n + m - 1)(x^ny^m + x^my^n)\\ \\ \ge

nm(x^{n+m-1}y + xy^{n+m-1}).$$
1994 Austrian-Polish Competition
A function $f: R \to R$ satisfies the conditions:

$f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$.

Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.
1994 Austrian-Polish Competition
The sequences $(a_n)$ and (c_n) are given by $a_0 =\frac12$, $c_0=4$ , and for $n \ge 0$ , $a_{n+1}=\frac{2a_n}{1+a_n^2}$, $c_{n+1}=c_n^2-2c_n+2$

Prove that for all $n\ge 1$, $a_n=\frac{2c_0c_1...c_{n-1}}{c_n}$
1994 Austrian-Polish Competition
A rectangular building consists of $30$ square rooms situated like the cells of a $2 \times 15$ board. In each room there are three doors, each of which leads to another room (not necessarily different). How many ways are there to distribute the doors between the rooms so that it is possible to get from any room to any other one without leaving the building?
1994 Austrian-Polish Competition
The vertices of a regular $n + 1$-gon are denoted by $P_0,P_1,...,P_n$ in some order ($n \ge 2$). Each side of the polygon is assigned a natural number as follows: if the endpoints of the side are $P_i$ and $P_j$, then the assigned number equals $|i - j |$. Let S be the sum of all $n + 1$ assigned numbers.

(a) Given $n$, what is the smallest possible value of $S$?

(b) If $P_0$ is fixed, how many different assignments are there for which $S$ attains the smallest value?
1994 Austrian-Polish Competition
Solve in integers the following equation

$\frac{1}{2}(x + y)(y + z)(z + x) + (x + y + z)^3 = 1 - xyz$.
1994 Austrian-Polish Competition
Let $n > 1$ be an odd positive integer. Assume that positive integers $x_1, x_2,..., x_n \ge 0$ satisfy:

$(x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2$

$(x_3 -x_2)^2 + 2(x_3 +x_2) + 1 = n^2$

...

$(x_1 - x_n)^2 + 2(x_1 + x_n)+ 1 = n^2 $

Show that there exists $j, 1 \le j \le n$, such that $x_j = x_{j+1}$. Here $x_{n+1} = x_1$.
1994 Austrian-Polish Competition
Determine all two-digit positive integers $n =\overline{ab}$ (in the decimal system) with the property that for all integers $x$ the difference $x^a - x^b$ is divisible by $n$.
1994 Austrian-Polish Competition
Given real numbers $a, b$, find all functions $f: R \to R$ satisfying



$f(x,y) = af (x,z) + bf(y,z)$ for all $x,y,z \in R$.
1994 Austrian-Polish Competition
On the plane are given four distinct points $A,B,C,D$ on a line $g$ in this order, at the mutual distances $AB = a, BC = b, CD = c$.

(a) Construct (if possible) a point $P$ outside line $g$ such that $\angle APB =\angle BPC =\angle CPD$.

(b) Prove that such a point $P$ exists if and only if $ (a+b)(b+c) < 4ac$
1993 Austrian-Polish Competition
Solve in positive integers $x,y$ the equation $2^x - 3^y = 7$.
1993 Austrian-Polish Competition
Consider all tetrahedra $ABCD$ in which the sum of the areas of the faces $ABD, ACD, BCD$ does not exceed $1$. Among such tetrahedra, find those with the maximum volume.
1993 Austrian-Polish Competition
Define $f (n) = n + 1$ if $n = p^k > 1$ is a power of a prime number, and $f (n) =p_1^{k_1}+... + p_r^{k_r}$ for natural numbers $n = p_1^{k_1}... p_r^{k_r}$ ($r > 1, k_i > 0$). Given $m > 1$, we construct the sequence $a_0 = m, a_{j+1} = f (a_j)$ for $j \ge  0$ and denote by $g(m)$ the smallest term in this sequence. For each $m > 1$, determine $g(m)$.
1993 Austrian-Polish Competition
The Fibonacci numbers are defined by $ F_0 = 1, F_1 = 1, F_{n+2} = F_{n+1} + F_n$. The positive integers $ A, B$ are such that $ A^{19}$ divides $ B^{93}$ and $ B^{19}$ divides $ A^{93}$. Show that if $ h < k$ are consecutive Fibonacci numbers then $ (AB)^h$ divides $ (A^4 + B^8)^k$
1993 Austrian-Polish Competition
Solve in real numbers the system

$x^3 + y = 3x + 4$

$2y^3 + z = 6y + 6$

$3z^3 + x = 9z + 8$
1993 Austrian-Polish Competition
If $a,b \ge  0$ are real numbers, prove the inequality

$$\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^2\leq\frac{a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b}{4}\leq\frac{a+\sqrt{ab}+b}{3} \leq \sqrt{\left(\frac{a^{2/3}+b^{2/3}}{2}\right)^{3}}$$For each of the inequalities, find the cases of equality.
1993 Austrian-Polish Competition
The sequence $(a_n)$ is defined by $a_0 = 0$ and $a_{n+1} = [\sqrt[3]{a_n +n}]^3$ for $n \ge 0$.

(a) Find $a_n$ in terms of $n$.

(b) Find all $n$ for which $a_n = n$.
1993 Austrian-Polish Competition
Determine all real polynomials $P(z)$ for which there exists a unique real polynomial $Q(x)$ satisfying the conditions

$Q(0)= 0$, $x + Q(y + P(x))= y + Q(x + P(y))$ for all $x,y \in R$.
1993 Austrian-Polish Competition
Point $P$ is taken on the extension of side $AB$ of an equilateral triangle $ABC$ so that $A$ is between $B$ and $P$. Denote by $a$ the side length of triangle $ABC$, by $r_1$ the inradius of triangle $PAC$, and by $r_2$ the exradius of triangle $PBC$ opposite $P$. Find the sum $r_1+r_2$ as a function in $a$.
1992 Austrian-Polish Competition
For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.
1992 Austrian-Polish Competition
Each point on the boundary of a square has to be colored in one color. Consider all right triangles with the vertices on the boundary of the square. Determine the least number of colors for which there is a coloring such that no such triangle has all its vertices of the same color.
1992 Austrian-Polish Competition
For all positive numbers $a, b, c$ prove the inequality $2\sqrt{bc + ca + ab} \le  \sqrt{3} \sqrt[3]{(b + c)(c + a)(a + b)}$.
1992 Austrian-Polish Competition
Let $k$ be a positive integer and $u, v$ be real numbers. Consider $P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c$.

(a) For $k = 2$ prove that if $a, b, c$ are rational then so is $uv$.

(b) Is that also true for $k = 3$?
1992 Austrian-Polish Competition
Given a circle $k$ with center $M$ and radius $r$, let $AB$ be a fixed diameter of $k$ and let $K$ be a fixed point on the segment $AM$. Denote by $t$ the tangent of $k$ at $A$. For any chord $CD$ through $K$ other than $AB$, denote by $P$ and Q the intersection points of $BC$ and $BD$ with $t$, respectively. Prove that $AP\cdot AQ$ does not depend on $CD$.
1992 Austrian-Polish Competition
A function $f: Z \to Z$ has the following properties:

$f (92 + x) = f (92 - x)$

$f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot  92 = 1748$)

$f (1992 + x) = f (1992 - x)$

for all integers $x$. Can all positive divisors of $92$ occur as values of f?
1992 Austrian-Polish Competition
Consider triangles $ABC$ in space.

(a) What condition must the angles $\angle A, \angle B , \angle C$ of  $\triangle ABC$ fulfill in order that there is a point $P$ in space such that $\angle APB, \angle BPC, \angle CPA$ are right angles?

(b) Let $d$ be the longest of the edges $PA,PB,PC$ and let $h$ be the longest altitude of $\triangle ABC$. Show that $\frac{1}{3}\sqrt6 h \le d \le h$.
1992 Austrian-Polish Competition
Let $n\ge 3$ be a given integer. Nonzero real numbers $a_1,..., a_n$ satisfy:

$\frac{-a_1-a_2+a_3+...a_n}{a_1}=\frac{a_1-a_2-a_3+a_4+...a_n}{a_2}=...=\frac{a_1+...+a_{n-2}-a_{n-1}-a_n}{a_{n-1}}=\frac{-a_1+a_2+...+a_{n-1}-a_n}{a_{n}}$

What values can be taken by the product

$\frac{a_2+a_3+...a_n}{a_1}\cdot \frac{a_1+a_3+a_4+...a_n}{a_2}\cdot ...\cdot \frac{+a_1+a_2+...+a_{n-1}}{a_{n}}$ ?
1992 Austrian-Polish Competition
Given an integer $n > 1$, consider words composed of $n$ letters $A$ and $n$ letters $B$. A word $X_1...X_{2n}$ is said to belong to set $R(n)$ (respectively, $S(n)$) if no initial segment (respectively, exactly one initial segment) $X_1...X_k$ with $1 \le k < 2n$ consists of equally many letters $A$ and $B$. If $r(n)$ and $s(n)$ denote the cardinalities of $R(n)$ and $S(n)$ respectively, compute $s(n)/r(n)$.
1991 Austrian-Polish Competition
Show that there are infinitely many integers $m \ge 2$ such that $m \choose 2$ $= 3$ $n \choose 4$ holds for some integer $n \ge 4$. Give the general form of all such $m$.
1991 Austrian-Polish Competition
Find all solutions $(x,y,z)$ to the system

$(x^2 - 6x + 13)y = 20$

$(y^2 - 6y + 13)z = 20$

$(z^2 - 6z + 13)x = 20$.
1991 Austrian-Polish Competition
Given two distinct points $A_1,A_2$ in the plane, determine all possible positions of a point $A_3$ with the following property: There exists an array of (not necessarily distinct) points $P_1,P_2,...,P_n$ for some $n \ge 3$ such that the segments $P_1P_2,P_2P_3,...,P_nP_1$ have equal lengths and their midpoints are $A_1, A_2, A_3, A_1, A_2, A_3, ...$ in this order.
1991 Austrian-Polish Competition
Let $P(x)$ be a real polynomial with $P(x) \ge 0$ for $0 \le x \le  1$. Show that there exist polynomials $P_i (x) (i = 0, 1,2)$ with $P_i (x) \ge 0$ for all real x such that $P (x) = P_0 (x) + xP_1 (x)( 1- x)P_2 (x)$.
1991 Austrian-Polish Competition
If $x,y, z$ are arbitrary positive numbers with $xyz = 1$, prove the inequality

$$x^2+y^2+z^2 + xy+yz + zx \ge 2(\sqrt{x} +\sqrt{y}+ \sqrt{z})$$.
1991 Austrian-Polish Competition
Suppose that there is a point $P$ inside a convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal areas. Prove that one of the diagonals bisects the area of $ABCD$.
1991 Austrian-Polish Competition
For a given positive integer $n$ determine the maximum value of the function $f (x) = \frac{x + x^2 +...+ x^{2n-1}}{(1 + x^n)^2}$ over all $x \ge  0$ and find all positive $x$ for which the maximum is attained.
1991 Austrian-Polish Competition
Consider the system of congruences $xy \equiv - 1$ (mod $z$), $yz \equiv 1$ (mod $x$), $zx \equiv 1$ (mod $y$).

Find the number of triples $(x,y, z) $ of distinct positive integers satisfying this system such that one of the numbers $x,y, z$ equals $19$.
1991 Austrian-Polish Competition
For a positive integer $n$ denote $A = \{1,2,..., n\}$. Suppose that $g : A\to A$ is a fixed function with $g(k) \ne  k$ and $g(g(k)) = k$ for $k \in  A$. How many functions $f: A \to  A$ are there such that $f(k)\ne  g(k)$ and $f(f(f(k))= g(k)$ for $k \in A$?
1990 Austrian-Polish Competition
The distinct points $X_1, X_2, X_3, X_4, X_5, X_6$ all lie on the same side of the line $AB$. The six triangles $ABX_i$ are all similar. Show that the $X_i$ lie on a circle.
1990 Austrian-Polish Competition
Find all solutions in positive integers to $a^A = b^B = c^C = 1990^{1990}abc$, where $A = b^c, B = c^a, C = a^b$.
1990 Austrian-Polish Competition
Show that there are two real solutions to:

$x + y^2 + z^4 = 0$

$y + z^2 + x^4 = 0$

$z + x^2 + y^5 = 0$.
1990 Austrian-Polish Competition
Find all solutions in positive integers to:

$x_14 + 14x_1x_2 + 1 = y_1^4$

$x_24 + 14x_2x_3 + 1 = y_2^4$

$...$

$x_n4 + 14x_nx_1 + 1 = y_n^4$.
1990 Austrian-Polish Competition
Let $n>1$ be an integer and let $f_1$, $f_2$, ..., $f_{n!}$ be the $n!$ permutations of $1$, $2$, ..., $n$. (Each $f_i$ is a bijective function from $\{1,2,...,n\}$ to itself.) For each permutation $f_i$, let us define $S(f_i)=\sum^n_{k=1} |f_i(k)-k|$. Find $\frac{1}{n!} \sum^{n!}_{i=1} S(f_i)$.
1990 Austrian-Polish Competition
$p(x)$ is a polynomial with integer coefficients. The sequence of integers $a_1, a_2, ... , a_n$ (where $n > 2$) satisfies $a_2 = p(a_1), a_3 = p(a_2), ... , a_n = p(a_{n-1}), a_1 = p(a_n)$. Show that $a_1 = a_3$.
1990 Austrian-Polish Competition
$D_n$ is a set of domino pieces. For each pair of non-negative integers $(a, b)$ with $a \le b \le n$, there is one domino, denoted $[a, b]$ or $[b, a]$ in $D_n$. A ring  is a sequence of dominoes $[a_1, b_1], [a_2, b_2], ... , [a_k, b_k]$ such that $b_1 = a_2, b_2 = a_3, ... , b_{k-1} = a_k$ and $b_k = a_1$. Show that if $n$ is even there is a ring which uses all the pieces. Show that for n odd, at least $(n+1)/2$ pieces are not used in any ring. For $n$ odd, how many different sets of $(n+1)/2$ are there, such that the pieces not in the set can form a ring?
1990 Austrian-Polish Competition
We are given a supply of $a \times b$ tiles with $a$ and $b$ distinct positive integers. The tiles are to be used to tile a $28 \times  48$ rectangle. Find $a, b$ such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find $a, b$ such that the tile has the largest possible area and there is more than one possible tiling.
1990 Austrian-Polish Competition
$a_1, a_2, ... , a_n$ is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if $x_i$ all belong to the same set, then $a_1x_1 + a_2x_2 + ... + a_nx_n$ is non-zero.
1989 Austrian-Polish Competition
Show that $(\sum_{i=1}^{n}x_iy_iz_i)^2 \le (\sum_{i=1}^{n}x_i^3) (\sum_{i=1}^{n}y_i^3) (\sum_{i=1}^{n}z_i^3)$ for any positive reals $x_i, y_i, z_i$.
1989 Austrian-Polish Competition
Each point of the plane is colored by one of the two colors. Show that there exists an equilateral triangle with monochromatic vertices.
1989 Austrian-Polish Competition
Find all natural numbers $N$ (in decimal system) with the following properties:

(i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes,

(ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.
1989 Austrian-Polish Competition
Let $P$ be a convex polygon in the plane. Show that there exists a circle containing the entire polygon $P$ and having at least three adjacent vertices of $P$ on its boundary.
1989 Austrian-Polish Competition
Let $A$ be a vertex of a cube $\omega$ circumscribed about a sphere $k$ of radius $1$. We consider lines $g$ through $A$ containing at least one point of $k$. Let $P$ be the intersection point of $g$ and $k$ closer to $A$, and $Q$ be the second intersection point of $g$ and $\omega$. Determine the maximum value of $AP\cdot AQ$ and characterize the lines $g$ yielding the maximum.
1989 Austrian-Polish Competition
A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.
1989 Austrian-Polish Competition
Functions $f_0, f_1,f_2,...$ are recursively defined by

$f_0(x) = x$ and $f_{2k+1} (x) = 3^{f_{2k}(x)}$ and $f_{2k+2} = 2^{f_{2k+1}(x)}$, $k = 0,1,2,...$ for all $x \in  R$.

Find the greater one of the numbers $f_{10}(1)$ and $f_9(2)$.
1989 Austrian-Polish Competition
$ABC$ is an acute-angled triangle and $P$ a point inside or on the boundary. The feet of the perpendiculars from $P$ to $BC, CA, AB$ are $A', B', C'$ respectively. Show that if $ABC$ is equilateral, then $\frac{AC'+BA'+CB'}{PA'+PB'+PC'}$ is the same for all positions of $P$, but that for any other triangle it is not.
1989 Austrian-Polish Competition
Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$) of squares of adjacent positive integers.
1988 Austrian-Polish Competition
Let $P(x)$ be a polynomial with integer coefficients. Show that if $Q(x) = P(x) +12$ has at least six distinct integer roots, then $P(x)$ has no integer roots.
1988 Austrian-Polish Competition
If $a_1 \le  a_2 \le  .. \le  a_n$ are natural numbers ($n \ge 2$), show that the inequality $$\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1}  >0$$holds for all $n$-tuples $(x_1,...,x_n) \ne (0,..., 0)$ of real numbers if and only if $a_2 \ge 2$.
1988 Austrian-Polish Competition
In a ABCD cyclic quadrilateral 4 points K, L ,M, N are taken on AB , BC , CD and DA , respectively such that KLMN is a parallelogram. Lines AD, BC and KM have a common point. And also lines AB, DC and NL have a common point. Prove that KLMN is rhombus.
1988 Austrian-Polish Competition
Determine all strictly increasing functions $f: R \to R$ satisfying $f (f(x) + y) = f(x + y) + f (0)$ for all $x,y \in R$.
1988 Austrian-Polish Competition
Two sequences $(a_k)_{k\ge 0}$ and $(b_k)_{k\ge 0}$ of integers are given by $b_k = a_k + 9$ and $a_{k+1} = 8b_k + 8$ for $k\ge  0$. Suppose that the number $1988$ occurs in one of these sequences. Show that the sequence $(a_k)$ does not contain any nonzero perfect square.
1988 Austrian-Polish Competition
Three rays $h_1,h_2,h_3$ emanating from a point $O$ are given, not all in the same plane. Show that if for any three points $A_1,A_2,A_3$ on $h_1,h_2,h_3$ respectively, distinct from $O$, the triangle $A_1A_2A_3$ is acute-angled, then the rays $h_1,h_2,h_3$ are pairwise orthogonal.
1988 Austrian-Polish Competition
Each side of a regular octagon is colored blue or yellow. In each step, the sides are simultaneously recolored as follows: if the two neighbors of a side have different colors, the side will be recolored blue, otherwise it will be recolored yellow. Show that after a finite number of moves all sides will be colored yellow. What is the least value of the number $N$ of moves that always lead to all sides being yellow?
1988 Austrian-Polish Competition
We are given $1988$ unit cubes. Using some or all of these cubes, we form three quadratic boards $A, B,C$ of dimensions $a \times  a \times  1$, $b \times b \times  1$, and $c \times  c \times 1$ respectively, where $a \le  b \le c$. Now we place board $B$ on board $C$ so that each cube of $B$ is precisely above a cube of $C$ and $B$ does not overlap $C$. Similarly, we place $A$ on $B$. This gives us a three-floor tower. What choice of $a, b$ and $c$ gives the maximum number of such three-floor towers?
1988 Austrian-Polish Competition
For a rectangle $R$ with integral side lengths, denote by $D(a, b)$ the number of ways of covering $R$ by congruent rectangles with integral side lengths formed by a family of cuts parallel to one side of $R$. Determine the perimeter $P$ of the rectangle $R$ for which $\frac{D(a,b)}{a+b}$ is maximal.
1987 Austrian-Polish Competition
Three pairwise orthogonal chords of a sphere $S$ are drawn through a given point $P$ inside $S$. Prove that the sum of the squares of their lengths does not depend on their directions.
1987 Austrian-Polish Competition
Let $n$ be the square of an integer whose each prime divisor has an even number of decimal digits. Consider $P(x) = x^n - 1987x$. Show that if $x,y$ are rational numbers with $P(x) = P(y)$, then $x = y$.
1987 Austrian-Polish Competition
A function $f: R \to  R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$. Given $a \in R$, define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge  0$. Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$  exists and determine its value.
1987 Austrian-Polish Competition
Does the set $\{1,2,3,...,3000\}$ contain a subset $ A$ consisting of 2000 numbers that $x\in A$ implies $2x \notin A$ ?!!  :?:
1987 Austrian-Polish Competition
The Euclidian three-dimensional space has been partitioned into three nonempty sets $A_1,A_2,A_3$. Show that one of these sets contains, for each $d > 0$, a pair of points at mutual distance $d$.
1987 Austrian-Polish Competition
Let $C$ be a unit circle and $n \ge 1$ be a fixed integer. For any set $A$ of $n$ points $P_1,..., P_n$ on $C$ define $D(A) = \underset{d}{max}\,  \underset{i}{min}\delta (P_i, d)$, where $d$ goes over all diameters of $C$ and $\delta (P, \ell)$ denotes the distance from point $P$ to line $\ell$. Let $F_n$ be the family of all such sets $A$. Determine $D_n =  \underset{A\in F_n}{min} D(A)$ and describe all sets $A$ with $D(A) = D_n$.
1987 Austrian-Polish Competition
For any natural number $n= \overline{a_k...a_1a_0}$ $(a_k \ne 0)$ in decimal system write $p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k$, $s(n)=a_0+ a_1+ ... + a_k$, $n^*= \overline{a_0a_1...a_k}$. Consider $P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}$ and let $Q$ be the set of numbers in $P$ with all digits greater than $1$.

(a) Show that $P$ is infinite.

(b) Show that $Q$ is finite.

(c) Write down all the elements of $Q$.
1987 Austrian-Polish Competition
A circle of perimeter $1$ has been dissected into four equal arcs $B_1, B_2, B_3, B_4$. A closed smooth non-selfintersecting curve $C$ has been composed of translates of these arcs (each $B_j$ possibly occurring several times). Prove that the length of $C$ is an integer.
1987 Austrian-Polish Competition
Let $M$ be the set of all points $(x,y)$ in the cartesian plane, with integer coordinates satisfying $1 \le x \le 12$ and $1 \le y \le 13$.

(a) Prove that every $49$-element subset of $M$ contains four vertices of a rectangle with sides parallel to the coordinate axes.

(b) Give an example of a $48$-element subset of $M$ without this property.
1986 Austrian-Polish Competition
A non-right triangle $A_1A_2A_3$ is given. Circles $C_1$ and $C_2$ are tangent at $A_3, C_2$ and $C_3$ are tangent at $A_1$, and $C_3$ and $C_1$ are tangent at $A_2$. Points $O_1,O_2,O_3$ are the centers of $C_1, C_2, C_3$, respectively. Supposing that the triangles $A_1A_2A_3$ and $O_1O_2O_3$ are similar, determine their angles.
1986 Austrian-Polish Competition
The monic polynomial $P(x) = x^n + a_{n-1}x^{n-1} +...+ a_0$ of degree $n > 1$ has $n$ distinct negative roots. Prove that $a_1P(1) > 2n^2a_o$
1986 Austrian-Polish Competition
Each point in space is colored either blue or red. Show that there exists a unit square having exactly $0, 1$ or $4$ blue vertices.
1986 Austrian-Polish Competition
Find all triples (m,n,N) of positive integers numbers m,n and N  such that

$m^N-n^N=2^{100}$ with N>1
1986 Austrian-Polish Competition
Find all real solutions of the system of equations

$x^2 + y^2 + u^2 + v^2 = 4$

$xu + yv + xv + yu = 0$

$xyu + yuv + uvx + vxy = - 2$

$xyuv = -1$
1986 Austrian-Polish Competition
Let $M$ be the set of all tetrahedra whose inscribed and circumscribed spheres are concentric. If the radii of these spheres are denoted by $r$ and $R$ respectively, find the possible values of $R/r$ over all tetrahedra from $M$ .
1986 Austrian-Polish Competition
Let $k$ and $n$ be integers with $0 < k < n^2/4$ such that k has no prime divisor greater than $n$. Prove that $k$ divides $n!$.
1986 Austrian-Polish Competition
Pairwise distinct real numbers are arranged into an $m \times n$ rectangular array. In each row the entries are arranged increasingly from left to right. Each column is then rearranged in decreasing order from top to bottom. Prove that in the reorganized array, the rows remain arranged increasingly.
1986 Austrian-Polish Competition
Find all continuous monotonic functions $f : R \to R$ that satisfy $f (1) = 1$ and $f(f (x)) = f (x)^2$ for all $x \in R$.
1985 Austrian-Polish Competition
Suppose that $n\ge 8$ persons $P_1,P_2,\dots,P_n$ meet at a party. Assume that $P_k$ knows $k+3$ persons for $k=1,2,\dots,n-6$. Further assume that each of $P_{n-5},P_{n-4},P_{n-3}$ knows $n-2$ persons, and each of $P_{n-2},P_{n-1},P_n$ knows $n-1$ persons. Find all integers $n\ge 8$ for which this is possible.



(It is understood that "to know" is a symmetric nonreflexive relation: if $P_i$ knows $P_j$ then $P_j$ knows $P_i$; to say that $P_i$ knows $p$ persons means: knows $p$ persons other than herself/himself.)
1985 Austrian-Polish Competition
In a convex quadrilateral of area $1$, the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$. Prove this.