Navvye/Olympiad-Problems-CSV huggingface dataset:
2003 Austrian-Polish Competition
A triangle with sides a, b, c has area S.



The distances of its centroid from the vertices are x, y, z.



Show that:



if (x + y + z)^2 ≤ (a^2 + b^2 + c^2)/2 + 2S√3,

then the triangle is equilateral.
2003 Austrian-Polish Competition
$ABCD$ is a tetrahedron such that we can find a sphere $k(A,B,C)$ through $A, B, C$ which meets the plane $BCD$ in the circle diameter $BC$, meets the plane $ACD$ in the circle diameter $AC$, and meets the plane $ABD$ in the circle diameter $AB$. Show that there exist spheres $k(A,B,D)$, $k(B,C,D)$ and $k(C,A,D)$ with analogous properties.



Austrian-Polish 2003
2003 Austrian-Polish Competition
Put $f(n) = \frac{n^n - 1}{n - 1}$. Show that $n!^{f(n)}$ divides $(n^n)! $.

Find as many positive integers as possible for which $n!^{f(n)+1}$ does not divide $(n^n)!$ .
2003 Austrian-Polish Competition
Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the $ 26$ whose product is a square.





Click to reveal hidden textI think that the upper limit for such subset is 37.
2003 Austrian-Polish Competition
Given reals $x_1 \ge x_2 \ge ... \ge  x_{2003} \ge 0$, show that $$x_1^n  - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n$$for any positive integer $n$.



PS. All the problems of APMC 2003 have beem collected here
2003 Austrian-Polish Competition
What is the smallest number of $5\times 1$ tiles which must be placed on a $31\times 5$ rectangle (each covering exactly $5$ unit squares) so that no further tiles can be placed? How many different ways are there of placing the minimal number (so that further tiles are blocked)? What are the answers for a $52\times 5$ rectangle?
2002 Austrian-Polish Competition
Given a circle $G$ with center $O$ and radius $r$. Let $AB$ be a fixed diameter of $G$. Let $K$ be a fixed point of segment $AO$. Denote by $t$ the line tangent to  at $A$. For any chord $CD$ (other than $AB$) passing through $K$. Let $P$ and $Q$ be the points of intersection of lines $BC$ and $BD$ with $t$.

Prove that the product $AP\cdot AQ$ remains costant as the chord $CD$ varies.
2002 Austrian-Polish Competition
Let $P_{1}P_{2}\dots P_{2n}$ be a convex polygon with an even number of corners. Prove that there exists a diagonal $P_{i}P_{j}$ which is not parallel to any side of the polygon.
2002 Austrian-Polish Competition
Let $ABCD$ be a tetrahedron and let $S$ be its center of gravity. A line through $S$ intersects the surface of $ABCD$ in the points $K$ and $L$. Prove that $$\frac{1}{3}\leq \frac{KS}{LS}\leq 3$$
2002 Austrian-Polish Competition
For each positive integer $n$ find the largest subset $M(n)$ of real numbers possessing the property: $$n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i}\quad \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n)$$ When does the inequality become an equality ?
2002 Austrian-Polish Competition
Let $A$ be the set $\{2,7,11,13\}$. A polynomial $f$ with integer coefficients possesses the following property: for each integer $n$ there exists $p \in A$ such that $p|f(n)$. Prove that there exists $p \in A$ such that $p|f(n)$ for all integers $n$.
2002 Austrian-Polish Competition
The diagonals of a convex quadrilateral $ABCD$ intersect in the point $E$. Let $U$ be the circumcenter of the triangle $ABE$ and $H$ be its orthocenter. Similarly, let $V$ be the circumcenter of the triangle $CDE$ and $K$ be its orthocenter. Prove that $E$ lies on the line $UK$ if and only if it lies on the line $VH$.
2002 Austrian-Polish Competition
Find all real functions $f$ definited on positive integers and satisying:



(a) $f(x+22)=f(x)$,



(b) $f\left(x^{2}y\right)=\left(f(x)\right)^{2}f(y)$



for all positive integers $x$ and $y$.
2002 Austrian-Polish Competition
Determine the number of real solutions of the system $$\left\{ \begin{aligned}\cos x_{1}&= x_{2}\\ &\cdots \\ \cos x_{n-1}&= x_{n}\\ \cos x_{n}&= x_{1}\\ \end{aligned}\right.$$
2002 Austrian-Polish Competition
A set $P$ of $2002$ persons is given. The family of subsets of $P$ containing exactly $1001$ persons has the property that the number of acquaintance pairs in each such subset is the same. (It is assumed that the acquaintance relation is symmetric). Find the best lower estimation of the acquaintance pairs in the set $P$.
2002 Austrian-Polish Competition
For all real number $x$ consider the family $F(x)$ of all sequences $(a_{n})_{n\geq 0}$ satisfying the equation $$a_{n+1}=x-\frac{1}{a_{n}}\quad (n\geq 0).$$ A positive integer $p$ is called a minimal period of the family $F(x)$ if



(a) each sequence $\left(a_{n}\right)\in F(x)$ is periodic with the period $p$,



(b) for each $0<q<p$ there exists $\left(a_{n}\right)\in F(x)$ such that $q$ is not a period of $\left(a_{n}\right)$.



Prove or disprove that for each positive integer $P$ there exists a real number $x=x(P)$ such that the family $F(x)$ has the minimal period $p>P$.
2001 Austrian-Polish Competition
Determine the number of positive integers $a$, so that there exist nonnegative integers $x_0,x_1,\ldots,x_{2001}$ which satisfy the equation

$$ \displaystyle a^{x_0} = \sum_{i=1}^{2001} a^{x_i} $$
2001 Austrian-Polish Competition
Let $n$ be a positive integer greater than $2$. Solve in nonnegative real numbers the following system of equations $$x_{k}+x_{k+1}=x_{k+2}^{2}\quad , \quad k=1,2,\cdots,n$$ where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$.
2001 Austrian-Polish Competition
Let $a,b,c$ be sides of a triangle. Prove that

$$ 2 < \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} - \frac{a^3+b^3+c^3}{abc}\leq 3  $$
2001 Austrian-Polish Competition
Prove that if $a,b,c,d$ are lengths of the successive sides of a quadrangle (not necessarily convex) with the area equal to $S$, then the following inequality holds $$S \leq \frac{1}{2}(ac+bd).$$ For which quadrangles does the inequality become equality?
2001 Austrian-Polish Competition
The fields of the $8\times 8$ chessboard are numbered from $1$ to $64$ in the following manner: For $i=1,2,\cdots,63$ the field numbered by $i+1$ can be reached from the field numbered by $i$ by one move of the knight. Let us choose positive real numbers $x_{1},x_{2},\cdots,x_{64}$. For each white field numbered by $i$ define the number $y_{i}=1+x_{i}^{2}-\sqrt[3]{x_{i-1}^{2}x_{i+1}}$ and for each black field numbered by $j$ define the number $y_{j}=1+x_{j}^{2}-\sqrt[3]{x_{j-1}x_{j+1}^{2}}$ where $x_{0}=x_{64}$ and $x_{1}=x_{65}$. Prove that $$\sum_{i=1}^{64}y_{i}\geq 48$$
2001 Austrian-Polish Competition
Let $k$ be a fixed positive integer. Consider the sequence definited by $$a_{0}=1 \;\; , a_{n+1}=a_{n}+\left\lfloor\root k \of{a_{n}}\right\rfloor \;\; , n=0,1,\cdots$$ where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. For each $k$ find the set $A_{k}$ containing all integer values of the sequence $(\sqrt[k]{a_{n}})_{n\geq 0}$.
2001 Austrian-Polish Competition
Consider the set $A$ containing all positive integers whose decimal expansion contains no $0$, and whose sum $S(N)$ of the digits divides $N$.



(a) Prove that there exist infinitely many elements in $A$ whose decimal expansion contains each digit the same number of times as each other digit.



(b) Explain that for each positive integer $k$ there exist an element in $A$ having exactly $k$ digits.
2001 Austrian-Polish Competition
The prism with the regular octagonal base and with all edges of the length equal to $1$ is given. The points $M_{1},M_{2},\cdots,M_{10}$ are the midpoints of all the faces of the prism. For the point $P$ from the inside of the prism denote by $P_{i}$ the intersection point (not equal to $M_{i}$) of the line $M_{i}P$ with the surface of the prism. Assume that the point $P$ is so chosen that all associated with $P$ points $P_{i}$ do not belong to any edge of the prism and on each face lies exactly one point $P_{i}$. Prove that $$\sum_{i=1}^{10}\frac{M_{i}P}{M_{i}P_{i}}=5$$
2001 Austrian-Polish Competition
Let $A$ be a set with $2n$ elements, and let $A_1, A_2...,A_m$ be subsets of $A$e ach one with n elements. Find the greatest possible m, such that it is possible to select these $m$ subsets in such a way that the intersection of any 3 of them has at most one element.
2001 Austrian-Polish Competition
The sequence $a_{1},a_{2},\cdots,a_{2010}$ has the following properties:



(1) each sum of the 20 successive values of the sequence is nonnegative,



(2) $|a_{i}a_{i+1}| \leq 1$ for $i=1,2,\cdots,2009$.



Determine the maximal value of the expression $\sum_{i=1}^{2010}a_{i}$.
2000 Austrian-Polish Competition
Find all polynomials $P(x)$ with real coefficients having the following property: There exists a positive integer n such that the equality $$\sum_{k=1}^{2n+1}(-1)^k \left[\frac{k}{2}\right] P(x + k)=0$$holds for infinitely many real numbers $x$.
2000 Austrian-Polish Competition
In a unit cube, $CG$ is the edge perpendicular to the face $ABCD$. Let $O_1$ be the incircle of square $ABCD$ and $O_2$ be the circumcircle of triangle $BDG$. Determine min$\{XY|X\in O_1,Y\in O_2\}$.
2000 Austrian-Polish Competition
For each integer $n \ge 3$ solve in real numbers the system of equations:

$x_1^3 = x_2 + x_3 + 1$

$...$

$x_{n-1}^3 = x_n+ x_1 + 1$

$x_{n}^3 = x_1+ x_2 + 1$
2000 Austrian-Polish Competition
Find all positive integers $N$ having only prime divisors $2,5$ such that $N+25$ is a perfect square.
2000 Austrian-Polish Competition
For which integers $n \ge 5$ is it possible to color the vertices of a regular$ n$-gon using at most $6$ colors in such a way that any $5$ consecutive vertices have different colors?
2000 Austrian-Polish Competition
Consider the solid $Q$ obtained by attaching unit cubes $Q_1...Q_6$ at the six faces of a unit cube $Q$. Prove or disprove that the space can be filled up with such solids so that no two of them have a common interior point.
2000 Austrian-Polish Competition
Triangle $A_0B_0C_0$ is given in the plane. Consider all triangles $ABC$ such that:

(i) The lines $AB,BC,CA$ pass through $C_0,A_0,B_0$, respectvely,

(ii) The triangles $ABC$ and $A_0B_0C_0$ are similar.

Find the possible positions of the circumcenter of triangle $ABC$.





Austrian-Polish 2000
2000 Austrian-Polish Competition
In the plane are given $27$ points, no three of which are collinear. Four of this points are vertices of a unit square, while the others lie inside the square. Prove that there are three points in this set forming a triangle with area not exceeding $1/48$.
2000 Austrian-Polish Competition
If three nonnegative reals $a$, $b$, $c$ satisfy $a+b+c=1$, prove that



$2 \leq \left(1-a^{2}\right)^{2}+\left(1-b^{2}\right)^{2}+\left(1-c^{2}\right)^{2}\leq \left(1+a\right)\left(1+b\right)\left(1+c\right)$.
2000 Austrian-Polish Competition
The plan of the castle in Baranow Sandomierski can be presented as the graph with $16$ vertices on the picture.

A night guard plans a closed round along the edges of this graph.

(a) How many rounds passing through each vertex exactly once are there? The directions are irrelevant.

(b) How many non-selfintersecting rounds (taking directions into account) containing each edge of the graph exactly once are there?
1999 Austrian-Polish Competition
Find the number of $6$-tuples $(A_1,A_2,...,A_6)$ of subsets of $M = \{1,..., n\}$ (not necessarily different) such that each element of $M$ belongs to zero, three, or six of the subsets $A_1,...,A_6$.
1999 Austrian-Polish Competition
Find the best possible $k,k'$ such that $$k<\frac{v}{v+w}+\frac{w}{w+x}+\frac{x}{x+y}+\frac{y}{y+z}+\frac{z}{z+v}<k'$$

for all positive reals $v,w,x,y,z$.
1999 Austrian-Polish Competition
Given an integer $n \ge 2$, find all sustems of $n$ functions$ f_1,..., f_n : R \to R$ such that for all $x,y \in R$

$f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0$

$f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0$

$...$

$f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0$
1999 Austrian-Polish Competition
Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$, $l $ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$, $m$ meets $CA$ at $C)1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$. Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.
1999 Austrian-Polish Competition
A sequence of integers $(a_n)$ satisfies $a_{n+1} = a_n^3 + 1999$ for $n = 1,2,....$

Prove that there exists at most one $n$ for which $a_n$ is a perfect square.
1999 Austrian-Polish Competition
Solve in the nonnegative real numbers the system of equations



$x_n^2 + x_nx{n-1} + x_{n-1}^4 = 1$ for $n = 1,2,..., 1999$,

$x_0 = x_{1999}$.
1999 Austrian-Polish Competition
Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$.
1999 Austrian-Polish Competition
Let $P,Q,R$ be points on the same side of a line $g$ in the plane. Let $M$ and $N$ be the feet of the perpendiculars from $P$ and $Q$ to $g$ respectively. Point $S$ lies between the lines $PM$ and $QN$ and satisfies and satisfies $PM = PS$ and $QN = QS$. The perpendicular bisectors of $SM$ and $SN$ meet in a point $R$. If the line $RS$ intersects the circumcircle of triangle $PQR$ again at $T$, prove that $S$ is the midpoint of $RT$.



Austrian-Polish 1999
1999 Austrian-Polish Competition
A point in the cartesian plane with integer coordinates is called a lattice point. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold:

(i) The endpoints of each selected segment are lattice points;

(ii) Each selected segment is parallel to a coordinate axis or to one of the lines $y = \pm x$,

(iii) Each selected segment contains exactly five lattice points, all of which are selected,

(iv) Every two selected segments have at most one common point.

A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and segments is a position. Prove or disprove that there exists an initial position such that the game can have infinitely many moves.
1998 Austrian-Polish Competition
Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$. Prove the inequality  $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$
1998 Austrian-Polish Competition
For n points $$ P_1;P_2;...;P_n $$ in that order on a straight line. We colored each point by 1 in 5 white, red, green, blue, and purple. A coloring is called acceptable if two consecutive points $$ P_i;P_{i+1} (i=1;2;...n-1) $$ is the same color or 2 points with at least one of 2 points are colored white. How many ways acceptable color?
1998 Austrian-Polish Competition
Find all pairs of real numbers $(x, y)$ satisfying the following system of

equations

$2-x^{3}=y, 2-y^{3}=x$.
1998 Austrian-Polish Competition
For positive integers $m, n$, denote $$S_m(n)=\sum_{1\le k \le n} \left[ \sqrt[k^2]{k^m}\right]$$Prove that $S_m(n) \le n + m  (\sqrt[4]{2^m}-1)$
1998 Austrian-Polish Competition
Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).
1998 Austrian-Polish Competition
Different points $A,B,C,D,E,F$ lie on circle $k$ in this order. The tangents to $k$ in the points $A$ and $D$ and the lines $BF$ and $CE$ have a common point $P$. Prove that the lines $AD,BC$ and $EF$ also have a common point or are parallel.



Austrian-Polish 1998
1998 Austrian-Polish Competition
Consider all pairs $(a, b)$ of natural numbers such that the product $a^ab^b$ written in decimal system ends with exactly $98$ zeros. Find the pair $(a, b)$ for which the product $ab$ is the smallest.
1998 Austrian-Polish Competition
In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called admissible, where $n > 2$ is a given natural number. The value  of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$.)
1998 Austrian-Polish Competition
Given a triangle $ABC$, points $K,L,M$ are the midpoints of the sides $BC,CA,AB$, and points $X,Y,Z$ are the midpoints of the arcs $BC,CA,AB$ of the circumcircle not containing $A,B,C$ respectively. If $R$ denotes the circumradius and $r$ the inradius of the triangle, show that $r+KX+LY+MZ=2R$.
1997 Austrian-Polish Competition
Let $P$ be the intersection of lines $l_1$ and $l_2$. Let $S_1$ and $S_2$ be two

circles externally tangent at $P$ and both tangent to $l_1$, and let $T_1$

and $T_2$ be two circles externally tangent at $P$ and both tangent to $l_2$.

Let $A$ be the second intersection of $S_1$ and $T_1, B$ that of $S_1$ and $T_2,

C$ that of $S_2$ and $T_1$, and $D$ that of $S_2$ and $T_2$. Show that the points $A,B,C,D$ are concyclic if and only if $l_1$ and $l_2$ are perpendicular.
1997 Austrian-Polish Competition
Each square of an $n \times m$ board is assigned a pair of coordinates $(x,y)$ with $1 \le x  \le m$ and $1  \le y  \le n$. Let $p$ and $q$ be positive integers. A pawn can be moved from the square $(x,y)$ to $(x',y')$ if and only if $|x - x'| = p$ and $|y- y'| = q$. There is a pawn on each square. We want to move each pawn at the same time so that no two pawns are moved onto the same square. In how many ways can this be done?
1997 Austrian-Polish Competition
Numbers $\frac{49}{1}, \frac{49}{2}, ... , \frac{49}{97}$ are writen on a blackboard. Each time, we can replace two numbers (like $a, b$) with $2ab-a-b+1$. After $96$ times doing that prenominate action, one number will be left on the board. Find all the possible values fot that number.
1997 Austrian-Polish Competition
In a trapezoid $ABCD$ with $AB // CD$, the diagonals $AC$ and $BD$ intersect at point $E$. Let $F$ and $G$ be the orthocenters of the triangles $EBC$ and $EAD$. Prove that the midpoint of $GF$ lies on the perpendicular from $E$ to $AB$.



Austrian-Polish 1997

2013 Rioplatense L1 P2
1997 Austrian-Polish Competition
Let $p_1,p_2,p_3,p_4$ be four distinct primes. Prove that there is no polynomial $Q(x) = ax^3 + bx^2 + cx + d$ with integer coefficients such that $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.
1997 Austrian-Polish Competition
Show that there is no integer-valued function on the integers such that $f(m+f(n))=f(m)-n$ for all $m,n$.
1997 Austrian-Polish Competition
(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$, .

(b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$

(c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$.
1997 Austrian-Polish Competition
Let $X$ be a set with $n$ elements. Find the largest number of subsets of $X$, each with $3$ elements, so that no two of them are disjoint.
1997 Austrian-Polish Competition
Given a parallelepiped $P$, let $V_P$ be its volume, $S_P$ the area of its surface and $L_P$ the sum of the lengths of its edges. For a real number $t \ge 0$, let $P_t$ be the solid consisting of all points $X$ whose distance from some point of $P$ is at most $t$. Prove that the volume of the solid $P_t$ is given by the formula $V(P_t) =V_P + S_Pt + \frac{\pi}{4}  L_P  t^2 + \frac{4\pi}{3} t^3$.
1996 Austrian-Polish Competition
Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties:

(i) $n$ has exactly $k$ digits (in decimal representation),

(ii) all the digits of $n$ are odd,

(iii) $n$ is divisible by $5$,

(iv) the number $m = n/5$ has $k$ odd digits
1996 Austrian-Polish Competition
A convex hexagon $ ABCDEF$ satisfies the following conditions:

1) $ AB\parallel DE$, $ BC\parallel EF$, and $ CD\parallel FA$.

2) The distances between these pairs of parallel lines are the same.

3) $ \angle FAB = \angle CDE = 90^\circ$

Prove that the diagonals $ BE$ and $ CF$ of the hexagon intersect with angle $ 45$ degrees.



$ \bullet$ Thank you dear Babis Stergiou for your translation.  :P
1996 Austrian-Polish Competition
The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0,P_{1}(x)=x$ and $$P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2$$ For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$.
1996 Austrian-Polish Competition
Real numbers $x,y,z, t$ satisfy $x + y + z +t = 0$ and $x^2+ y^2+ z^2+t^2 = 1$.



Prove that $- 1 \le xy + yz + zt + tx \le 0$.
1996 Austrian-Polish Competition
A sphere $S$ divides every edge of a convex polyhedron $P$ into three equal parts. Show that there exists a sphere tangent to all the edges of $P$.
1996 Austrian-Polish Competition
Given natural numbers $n > k > 1$, find all real solutions $x_1,..., x_n$ of the system

$$x_i^3(x_i^2 + x_{i+1}^2+...  +x_{i+k-1}^2) = x_{i-1}^2$$for 1 $\le i \le n$. Here $x_{n+i} = x_i$ for all$ i$.
1996 Austrian-Polish Competition
Prove there are no such integers $ k, m $ which satisfy $ k \ge 0, m \ge 0 $ and $ k!+48=48(k+1)^m $.
1996 Austrian-Polish Competition
Show that there is no polynomial $P(x)$ of degree $998$ with real coefficients which satisfies $P(x^2 + 1) = P(x)^2 - 1$ for all $x$.
1996 Austrian-Polish Competition
For any triple $(a, b, c)$ of positive integers, not all equal, We are given sufficiently many rectangular blocks of size $a  \times b \times c$. We use these blocks to fill up a cubic box of edge $10$.

(a) Assume we have used at least $100$ blocks. Show that there are two blocks, one of which is a translate of the other.

(b) Find a number smaller than $100$ (the smaller, the better) for which the above statement still holds.
1995 Austrian-Polish Competition
Determine all real solutions $(a_1,...,a_n)$ of the following system of equations:

$a_3 = a_2 + a_1$

$a_4 = a_3 + a_2$

$...$

$a_n = a_{n-1} + a_{n-2}$

$a_1= a_n +a_{n-1}$

$a_2 = a_1 + a_n$.
1995 Austrian-Polish Competition
Let $X= \{A_1, A_2, A_3, A_4\}$ be a set of four distinct points in the plane. Show that there exists a subset $Y$ of $X$ with the property that there is no (closed) disk $K$ such that $K\cap X = Y$.
1995 Austrian-Polish Competition
Let $P(x) = x^4 + x^3 + x^2 + x + 1$. Show that there exist two non-constant polynomials $Q(y)$ and $R(y)$ with integer coefficients such that for all $Q(y) \cdot R(y)= P(5y^2)$ for all $y$ .
1995 Austrian-Polish Competition
Determine all polynomials $P(x)$ with real coefficients such that

$P(x)^2 + P\left(\frac{1}{x}\right)^2=  P(x^2)P\left(\frac{1}{x^2}\right)$ for all $x$.
1995 Austrian-Polish Competition
$ABC$ is an equilateral triangle. $A_{1}, B_{1}, C_{1}$ are the midpoints of $BC, CA, AB$ respectively. $p$ is an arbitrary line through $A_{1}$. $q$ and $r$ are lines parallel to $p$ through $B_{1}$ and $C_{1}$ respectively. $p$ meets the line $B_{1}C_{1}$ at $A_{2}$. Similarly, $q$ meets $C_{1}A_{1}$ at $B_{2}$, and $r$ meets $A_{1}B_{1}$ at $C_{2}$. Show that the lines $AA_{2}, BB_{2}, CC_{2}$ meet at some point $X$, and that $X$ lies on the circumcircle of $ABC$.
1995 Austrian-Polish Competition
The Alpine Club organizes four mountain trips for its $n$ members. Let $E_1, E_2, E_3, E_4$ be the teams participating in these trips. In how many ways can these teams be formed so as to satisfy

$E_1 \cap E_2 \ne\varnothing$, $E_2 \cap E_3 \ne\varnothing$ , $E_3 \cap E_4 \ne\varnothing$ ?
1995 Austrian-Polish Competition
Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal:

(i) $|x|$ is a square of an integer;

(ii) $y$ is a squarefree number.