Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Argentina Team Selection Test
Each number from the set $\{1,2,3,4,5,6,7,8\}$ is either colored red or blue, following these rules:



a) The number $4$ is colored red, and there is at least one number colored blue.

b) If two numbers $x, y$ have different colors and $x + y \leq 8$, then the number $x + y$ is colored blue.

c) If two numbers $x, y$ have different colors and $x \cdot y \leq 8$, then the number $x \cdot y$ is colored red.



Find all possible ways the numbers from this set can be colored.
2011 Argentina Team Selection Test
A wizard kidnaps $31$ members from party $A$, $28$ members from party $B$, $23$ members from party $C$, and $19$ members from party $D$, keeping them isolated in individual rooms in his castle, where he forces them to work.

Every day, after work, the kidnapped people can walk in the park and talk with each other. However, when three members of three different parties start talking with each other, the wizard reconverts them to the fourth party (there are no conversations with $4$ or more people involved).



a) Find out whether it is possible that, after some time, all of the kidnapped people belong to the same party. If the answer is yes, determine to which party they will belong.

b) Find all quartets of positive integers that add up to $101$ that if they were to be considered the number of members from the four parties, it is possible that, after some time, all of the kidnapped people belong to the same party, under the same rules imposed by the wizard.
2011 Argentina Team Selection Test
Let $ABCD$ be a trapezoid with bases $BC \parallel AD$, where $AD > BC$, and non-parallel legs $AB$ and $CD$. Let $M$ be the intersection of $AC$ and $BD$. Let $\Gamma_1$ be a circumference that passes through $M$ and is tangent to $AD$ at point $A$; let $\Gamma_2$ be a circumference that passes through $M$ and is tangent to $AD$ at point $D$. Let $S$ be the intersection of the lines $AB$ and $CD$, $X$ the intersection of $\Gamma_1$ with the line $AS$, $Y$ the intesection of $\Gamma_2$ with the line $DS$, and $O$ the circumcenter of triangle $ASD$.

Show that $SO \perp XY$.
2011 Argentina Team Selection Test
Determine all positive integers $n$ such that the number $n(n+2)(n+4)$ has at most $15$ positive divisors.
2011 Argentina Team Selection Test
At least $3$ players take part in a tennis tournament. Each participant plays exactly one match against each other participant. After the tournament has ended, we find out that each player has won at least one match. (There are no ties in tennis).

Show that in the tournament, there was at least one trio of players $A,B,C$ such that $A$ beat $B$, $B$ beat $C$, and $C$ beat $A$.
2011 Argentina Team Selection Test
Each square of $1\times 1$, of a $n\times n$ grid is colored using red or blue, in such way that between all the $2\times 2$ subgrids, there are all the possible colorations of a $2\times 2$ grid using red or blue, (colorations that can be obtained by using rotation or symmetry, are said to be different, so there are 16 possibilities). Find:

a) The minimum value of $n$.

b) For that value, find the least possible number of red squares.
2010 Argentina Team Selection Test
In a football tournament there are $8$ teams, each of which plays exacly one match against every other team. If a team $A$ defeats team $B$, then $A$ is awarded $3$ points and $B$ gets $0$ points. If they end up in a tie, they receive $1$ point each.

It turned out that in this tournament, whenever a match ended up in a tie, the two teams involved did not finish with the same final score. Find the maximum number of ties that could have happened in such a tournament.
2010 Argentina Team Selection Test
Let $ABC$ be a triangle with $AB = AC$. The incircle touches $BC$, $AC$ and $AB$ at $D$, $E$ and $F$ respectively. Let $P$ be a point on the arc $\overarc{EF}$ that does not contain $D$. Let $Q$ be the second point of intersection of $BP$ and the incircle of $ABC$. The lines $EP$ and $EQ$ meet the line $BC$ at $M$ and $N$, respectively.

Prove that the four points $P, F, B, M$ lie on a circle and $\frac{EM}{EN} = \frac{BF}{BP}$.
2010 Argentina Team Selection Test
Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that

$$f(x+xy+f(y)) = \left(f(x)+\frac{1}{2}\right) \left(f(y)+\frac{1}{2}\right)$$holds for all real numbers $x,y$.
2010 Argentina Team Selection Test
Two players, $A$ and $B$, play a game on a board which is a rhombus of side $n$ and angles of $60^{\circ}$ and $120^{\circ}$, divided into $2n^2$ equilateral triangles, as shown in the diagram for $n=4$.

$A$ uses a red token and $B$ uses a blue token, which are initially placed in cells containing opposite corners of the board (the $60^{\circ}$ ones). In turns, players move their token to a neighboring cell (sharing a side with the previous one). To win the game, a player must either place his token on the cell containing the other player's token, or get to the opposite corner to the one where he started.

If $A$ starts the game, determine which player has a winning strategy.
2010 Argentina Team Selection Test
Let $p$ and $q$ be prime numbers. The sequence $(x_n)$ is defined by $x_1 = 1$, $x_2 = p$ and $x_{n+1} = px_n - qx_{n-1}$ for all $n \geq 2$.

Given that there is some $k$ such that $x_{3k} = -3$, find $p$ and $q$.
2010 Argentina Team Selection Test
Suppose $a_1, a_2, ..., a_r$ are integers with $a_i \geq 2$ for all $i$ such that $a_1 + a_2 + ... + a_r = 2010$.

Prove that the set $\{1,2,3,...,2010\}$ can be partitioned in $r$ subsets $A_1, A_2, ..., A_r$ each with $a_1, a_2, ..., a_r$ elements respectively, such that the sum of the numbers on each subset is divisible by $2011$.

Decide whether this property still holds if we replace $2010$ by $2011$ and $2011$ by $2012$ (that is, if the set to be partitioned is $\{1,2,3,...,2011\}$).
2009 Argentina Team Selection Test
On a $ 50 \times 50$ board, the centers of several unit squares are colored black. Find the maximum number of centers that can be colored black in such a way that no three black points form a right-angled triangle.
2009 Argentina Team Selection Test
Let $ a_1, a_2, ..., a_{300}$ be nonnegative real numbers, with $ \sum_{i=1}^{300} a_i = 1$.



Find the maximum possible value of $ \sum_{i \neq j, i|j} a_ia_j$.
2009 Argentina Team Selection Test
Let $ ABC$ be a triangle, $ B_1$ the midpoint of side $ AB$ and $ C_1$ the midpoint of side $ AC$. Let $ P$ be the point of intersection ($ \neq A$) of the circumcircles of triangles $ ABC_1$ and $ AB_1C$. Let $ Q$ be the point of intersection ($ \neq A$) of the line $ AP$ and the circumcircle of triangle $ AB_1C_1$.



Prove that $ \frac{AP}{AQ} = \frac{3}{2}$.
2009 Argentina Team Selection Test
Find all positive integers $ n$ such that $ 20^n - 13^n - 7^n$ is divisible by $ 309$.
2009 Argentina Team Selection Test
There are several contestants at a math olympiad. We say that two contestants $ A$ and $ B$ are indirect friends if there are contestants $ C_1, C_2, ..., C_n$ such that $ A$ and $ C_1$ are friends, $ C_1$ and $ C_2$ are friends, $ C_2$ and $ C_3$ are friends, ..., $ C_n$ and $ B$ are friends. In particular, if $ A$ and $ B$ are friends themselves, they are indirect friends as well.

Some of the contestants were friends before the olympiad. During the olympiad, some contestants make new friends, so that after the olympiad every contestant has at least one friend among the other contestants. We say that a contestant is special if, after the olympiad, he has exactly twice as indirect friends as he had before the olympiad.

Prove that the number of special contestants is less or equal than $ \frac{2}{3}$ of the total number of contestants.
2009 Argentina Team Selection Test
Let $ n \geq 3$ be an odd integer. We denote by $ [-n,n]$ the set of all integers greater or equal than $ -n$ and less or equal than $ n$.

Player $ A$ chooses an arbitrary positive integer $ k$, then player $ B$ picks a subset of $ k$ (distinct) elements from $ [-n,n]$. Let this subset be $ S$.

If all numbers in $ [-n,n]$ can be written as the sum of exactly $ n$ distinct elements of $ S$, then player $ A$ wins the game. If not, $ B$ wins.

Find the least value of $ k$ such that player $ A$ can always win the game.
2008 Argentina Team Selection Test
-Find all the functions from $ \mathbb R^+ \to \mathbb R^+$ that satisfy:

$ x^2(f(x)+f(y))=(x+y)(f(yf(x)))$
2008 Argentina Team Selection Test
Triangle $ ABC$ is inscript in a circumference $ \Gamma$. A chord $ MN=1$ of $ \Gamma$ intersects the sides $ AB$ and $ AC$ at $ X$ and $ Y$ respectively, with $ M$, $ X$, $ Y$, $ N$ in that order in $ MN$. Let $ UV$ be the diameter of $ \Gamma$ perpendicular to $ MN$ with $ U$ and $ A$ in the same semiplane respect to $ MN$. Lines $ AV$, $ BU$ and $ CU$ cut $ MN$ in the ratios $ \frac{3}{2}$, $ \frac{4}{5}$ and $ \frac{7}{6}$ respectively (starting counting from $ M$). Find $ XY$
2008 Argentina Team Selection Test
Let $ d(n)$ be the number of positive divisors of the natural number $ n$.   Find all $ n$ such that  $ \frac {n} {d(n)}=p$   where $ p$ is a prime number



Daniel
2008 Argentina Team Selection Test
Let $ ABC$ be a triangle, $ D$, $ E$ and $ F$ the points of tangency of the incircle with sides $ BC$, $ CA$, $ AB$ respectively. Let $ P$ be the second point of intersection of $ CF$ and the incircle. If $ ABPE$ is a cyclic quadrilateral prove that $ DP$ is parellel to $ AB$
2008 Argentina Team Selection Test
Show that in acute triangle ABC we have:

$\frac{a^5+b^5+c^5}{a^4+b^4+c^4} \ge \sqrt{3}R$
2007 Argentina Team Selection Test
Let $ X$, $ Y$, $ Z$ be distinct positive integers having exactly two digits in such a way that:



$ X=10a +b$

$ Y=10b +c$

$ Z=10c +a$



($ a,b,c$ are digits)



Find all posible values of $ gcd(X,Y,Z)$
2007 Argentina Team Selection Test
Let $ ABCD$ be a trapezium of parallel sides $ AD$ and $ BC$ and non-parallel sides $ AB$ and $ CD$

Let $ I$ be the incenter of $ ABC$. It is known that exists a point $ Q \in AD$ with $ Q \neq A$ and $ Q \neq D$ such that if $ P$ is a point of the intersection of the bisectors of $ \widehat{ CQD}$ and $ \widehat{CAD}$ then $ PI \parallel AD$ Prove that $ PI=BQ$
2007 Argentina Team Selection Test
Find all real values of $ x>1$ which satisfy:



$ \frac{x^2}{x-1} + \sqrt{x-1}  +\frac{\sqrt{x-1}}{x^2} = \frac{x-1}{x^2} + \frac{1}{\sqrt{x-1}} + \frac{x^2}{\sqrt{x-1}}$
2007 Argentina Team Selection Test
Let $ d_1,d_2 \ldots, d_r$ be the positive divisors of $ n$

$ 1=d_1<d_2< \ldots <d_r=n$

If $ (d_7)^2 + (d_{15})^2= (d_{16})^2$ find all posible values of $ d_{17}$
2007 Argentina Team Selection Test
For natural $ n$ we define $ s(n)$ as the sum of digits of $ n$ (in base ten)

Does there exist a positive real constant $ c$ such that for all natural $ n$ we have



$ \frac{s(n)}{s(n^2)} \le c$ ?
2006 Argentina Team Selection Test
Let $ A$ be a set of natural numbers in which if $ a$ , $ b$ belong to $ A$ ($ a>b$) then either $ a+b$ or $ a-b$ belong to $ A$  ( both cases may be posible at the same time). Decide wheter there is or not a set $ A$ consisting on exactly $ 100$ elements which has four elements $ x$, $ y$ , $ z$ , $ w$ ( not necesarilly distinct) that satisfy  $ x-y=512$ and $ z-w=460$



Daniel
2006 Argentina Team Selection Test
In a circumference with center $ O$ we draw two equal chord $ AB=CD$ and if $ AB \cap CD =L$ then $ AL>BL$ and $ DL>CL$

We consider $ M \in AL$ and $ N \in DL$ such that $ \widehat {ALC} =2 \widehat {MON}$

Prove that the chord determined by extending $ MN$ has the same as length as both $ AB$ and $ CD$
2006 Argentina Team Selection Test
Find all integers solutions for

$ xy+yz+zx-xyz=2$
2006 Argentina Team Selection Test
Let $p$ be a prime with $p>5$, and let $S=\{p-n^2 \vert n \in \mathbb{N}, {n}^{2}<p \}$. Prove that $S$ contains two elements $a$ and $b$ such that $a \vert b$ and $1<a<b$.
2006 Argentina Team Selection Test
Let $ n$ be a natural number, and we consider the sequence $ a_1, a_2 \ldots , a_{2n}$ where  $ a_i \in (-1,0,1)$

If we make the sum of consecutive members of the sequence, starting from one with an odd index and finishing in one with and even index, the result is $ \le 2$ and $ \ge -2$

How many sequence are there satisfying this conditions?
2005 Argentina Team Selection Test
Find all pairs of integers $(m,n)$ such that an $m\times n$ board can be totally covered with $1\times 3$ and $2 \times 5$ pieces.
2005 Argentina Team Selection Test
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$ we have $$f(xf(x)+f(y)) = f(x)^2 + y$$
2005 Argentina Team Selection Test
Given the triangle $ABC$ we consider the points $X,Y,Z$ such that the triangles $ABZ,BCX,CAZ$ are equilateral, and they don't have intersection with $ABC$. Let $B'$ be the midpoint of $BC$, $N'$ the midpoint of $CY$, and $M,N$ the midpoints of $AZ,CX$, respectively. Prove that $B'N' \bot MN$.
2005 Argentina Team Selection Test
We have $ 150$ numbers $ x_1,x_2, \cdots , x_{150}$ each of which is either $ \sqrt 2 +1$ or $ \sqrt 2 -1$



We calculate the following sum:



$ S=x_1x_2 +x_3x_4+ x_5x_6+ \cdots   + x_{149}x_{150}$



Can we choose the $ 150$ numbers such that $ S=121$?

And what about $ S=111$?
2005 Argentina Team Selection Test
Let $n,p$ be integers such that $n>1$ and $p$ is a prime. If $n\mid p-1$ and $p\mid n^3-1$, show that $4p-3$ is a perfect square.
2005 Argentina Team Selection Test
We say that a group of $k$ boys is $n-acceptable$ if removing any boy from the group one can always find, in the other $k-1$ group,  a group of $n$ boys such that everyone knows each other. For each $n$, find the biggest $k$ such that in any group of $k$ boys that is $n-acceptable$ we must always have a group of $n+1$ boys such that everyone knows each other.
2006 Austrian-Polish Competition
Let $M(n)=\{n,n+1,n+2,n+3,n+4,n+5\}$ be a set of 6 consecutive integers. Let's take all values of the form $$\frac{a}{b}+\frac{c}{d}+\frac{e}{f}$$ with the set $\{a,b,c,d,e,f\}=M(n)$.

Let $$\frac{x}{u}+\frac{y}{v}+\frac{z}{w}=\frac{xvw+yuw+zuv}{uvw}$$ be the greatest of all these values.



a) show: for all odd $n$ hold: $\gcd (xvw+yuw+zuv, uvw)=1$ iff $\gcd (x,u)=\gcd (y,v)=\gcd (z,w)=1$.

b) for which positive integers $n$ hold $\gcd (xvw+yuw+zuv, uvw)=1$?
2006 Austrian-Polish Competition
Find all polynomials $P(x)$ with real coefficients satisfying the equation $$(x+1)^{3}P(x-1)-(x-1)^{3}P(x+1)=4(x^{2}-1) P(x)$$ for all real numbers $x$.
2006 Austrian-Polish Competition
$ABCD$ is a tetrahedron.

Let $K$ be the center of the incircle of $CBD$.

Let $M$ be the center of the incircle of $ABD$.

Let $L$ be the gravycenter of $DAC$.

Let $N$ be the gravycenter of $BAC$.



Suppose $AK$, $BL$, $CM$, $DN$ have one common point.

Is $ABCD$ necessarily regular?
2006 Austrian-Polish Competition
A positive integer $d$ is called nice iff for all positive integers $x,y$ hold: $d$ divides $(x+y)^{5}-x^{5}-y^{5}$ iff $d$ divides $(x+y)^{7}-x^{7}-y^{7}$ .

a) Is 29 nice?

b) Is 2006 nice?

c) Prove that infinitely many nice numbers exist.
2006 Austrian-Polish Competition
Prove that for all positive integers $n$ and all positive reals $a,b,c$ the following inequality holds: $$\frac{a^{n+1}}{a^{n}+a^{n-1}b+\ldots+b^{n}}+\frac{b^{n+1}}{b^{n}+b^{n-1}c+\ldots+c^{n}}+\frac{c^{n+1}}{c^{n}+c^{n-1}a+\ldots+a^{n}}\\ \ge \frac{a+b+c}{n+1}$$
2006 Austrian-Polish Competition
Let $D$ be an interior point of the triangle $ABC$.

$CD$ and $AB$ intersect at $D_{c}$,

$BD$ and $AC$ intersect at $D_{b}$,

$AD$ and $BC$ intersect at $D_{a}$.

Prove that there exists a triangle $KLM$ with orthocenter $H$ and the feet of altitudes $H_{k}\in LM, H_{l}\in KM, H_{m}\in KL$, so that

$(AD_{c}D) = (KH_{m}H)$

$(BD_{c}D) = (LH_{m}H)$

$(BD_{a}D) = (LH_{k}H)$

$(CD_{a}D) = (MH_{k}H)$

$(CD_{b}D) = (MH_{l}H)$

$(AD_{b}D) = (KH_{l}H)$



where $(PQR)$ denotes the area of the triangle $PQR$
2006 Austrian-Polish Competition
Find all nonnegative integers $m,n$ so that $$\sum_{k=1}^{2^{m}}\lfloor \frac{kn}{2^{m}}\rfloor\in \{28,29,30\}$$
2006 Austrian-Polish Competition
Let $A\subset \{x|0\le x<1\}$ with the following properties:

1. $A$ has at least 4 members.

2. For all pairwise different $a,b,c,d\in A$, $ab+cd\in A$ holds.

Prove: $A$ has infinetly many members.
2006 Austrian-Polish Competition
We have an 8x8 chessboard with 64 squares. Then we have 3x1 dominoes which cover exactly 3 squares. Such dominoes can only be moved parallel to the borders of the chessboard and also only if the passing squares are free. If no dominoes can be moved, then the position is called stable.

a. Find the smalles number of covered squares neccessary for a stable position.

b. Prove: There exist a stable position with only one square uncovered.

c. Find all Squares which are uncoverd in at least one position of b).
2006 Austrian-Polish Competition
Let $ABCDS$ be a (not neccessarily straight) pyramid with a rectangular base $ABCD$ and acute triangular faces $ABS,BCS,CDS,DAS$. We consider all cuboids which are inscribed inside the pyramid with its base being in the plane $ABCD$ and its upper vertexes are in the triangular faces (one in each).

Find the locus of the midpoints of these cuboids.
2005 Austrian-Polish Competition
For a convex $n$-gon $P_n$, we say that a convex quadrangle $Q$ is a diagonal-quadrangle of $P_n$, if its vertices are vertices of $P_n$ and its sides are diagonals of $P_n$. Let $d_n$ be the number of diagonal-quadrangles of a convex $n$-gon. Determine $d_n$ for all $n\geq 8$.
2005 Austrian-Polish Competition
Determine all polynomials $P$ with integer coefficients satisfying

$$P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}$$
2005 Austrian-Polish Competition
Let $a_0, a_1, a_2, ... , a_n$ be real numbers, which fulfill the following two conditions:

a) $0 = a_0 \leq a_1 \leq a_2 \leq ... \leq a_n$.

b) For all $0 \leq i < j \leq n$ holds: $a_j - a_i \leq j-i$.



Prove that

$$\left( \displaystyle \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.$$
2005 Austrian-Polish Competition
Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that

$$\frac{a^p - a}{p}=b^2.$$
2005 Austrian-Polish Competition
Given is a convex quadrilateral $ABCD$ with $AB=CD$. Draw the triangles $ABE$ and $CDF$ outside $ABCD$ so that $\angle{ABE} = \angle{DCF}$ and $\angle{BAE}=\angle{FDC}$. Prove that the midpoints of $\overline{AD}$, $\overline{BC}$ and $\overline{EF}$ are collinear.
2005 Austrian-Polish Competition
Determine all monotone functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$, so that for all $x, y \in \mathbb{Z}$

$$f(x^{2005} + y^{2005}) = (f(x))^{2005} + (f(y))^{2005}$$
2005 Austrian-Polish Competition
For each natural number $n\geq 2$, solve the following system of equations in the integers $x_1, x_2, ..., x_n$:

$$(n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n$$

where

$$S=x_1^2+x_2^2+\dots+x_n^2.$$
2005 Austrian-Polish Competition
Given the sets $R_{mn} = \{ (x,y) \mid x=0,1,\dots,m; y=0,1,\dots,n \}$, consider functions $f:R_{mn}\to \{-1,0,1\}$ with the following property: for each quadruple of points $A_1,A_2,A_3,A_4\in R_{mn}$ which form a square with side length $0<s<3$, we have

$$f(A_1)+f(A_2)+f(A_3)+f(A_4)=0.$$

For each pair $(m,n)$ of positive integers, determine $F(m,n)$, the number of such functions $f$ on $R_{mn}$.
2005 Austrian-Polish Competition
Consider the equation $x^3 + y^3 + z^3 = 2$.

a) Prove that it has infinitely many integer solutions $x,y,z$.

b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.
2005 Austrian-Polish Competition
Determine all pairs $(k,n)$ of non-negative integers such that the following inequality holds $\forall x,y>0$:

$$1+ \frac{y^n}{x^k} \geq \frac{(1+y)^n}{(1+x)^k}.$$
2004 Austrian-Polish Competition
Let $S(n)$ be the sum of digits for any positive integer n (in decimal notation).

Let $N=\displaystyle\sum_{k=10^{2003}}^{10{^{2004}-1}} S(k)$. Determine $S(N)$.
2004 Austrian-Polish Competition
In a triangle $ABC$ let $D$ be the intersection of the angle bisector of $\gamma$, angle at $C$, with the side $AB.$ And let $F$ be the area of the triangle $ABC.$ Prove the following inequality:





$$2 \cdot \ F \cdot \left( \frac{1}{AD} -\frac{1}{BD} \right) \leq AB.$$
2004 Austrian-Polish Competition
Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative:



$

\begin{matrix}



a  - \sqrt{1-b^2} + \sqrt{1-c^2} = d \\
b -  \sqrt{1-c^2} + \sqrt{1-d^2} = a \\
c -  \sqrt{1-d^2} + \sqrt{1-a^2} = b \\
d -  \sqrt{1-a^2} + \sqrt{1-b^2} = c \\


\end{matrix}

$
2004 Austrian-Polish Competition
Determine all $n \in \mathbb{N}$ for which $n^{10} + n^5 + 1$ is prime.
2004 Austrian-Polish Competition
Determine all $n$ for which the system with of equations can be solved in $\mathbb{R}$:



$$\sum^{n}_{k=1} x_k = 27$$



and



$$\prod^{n}_{k=1} x_k = \left( \frac{3}{2} \right)^{24}.$$
2004 Austrian-Polish Competition
For $n=2^m$ (m is a positive integer) consider the set $M(n) = \{ 1,2,...,n\}$ of natural numbers.

Prove that there exists an order $a_1, a_2, ..., a_n$ of the elements of M(n), so that for all $1\leq i < j < k \leq n$ holds: $a_j - a_i \neq a_k - a_j$.
2004 Austrian-Polish Competition
Determine all functions $f:\mathbb{Z}^+\to \mathbb{Z}$ which satisfy the following condition for all pairs $(x,y)$ of relatively prime positive integers:

$$f(x+y) = f(x+1) + f(y+1).$$
2004 Austrian-Polish Competition
a.) Prove that for $n = 4$ or $n \geq 6$ each triangle $ABC$ can be decomposed in $n$ similar (not necessarily congruent) triangles.



b.) Show: An equilateral triangle can neither be composed in 3 nor 5 triangles.



c.) Is there a triangle $ABC$ which can be decomposed in 3 and 5 triangles, analogously to a.). Either give an example or prove that there is not such a triangle.
2004 Austrian-Polish Competition
Given are the sequences

$$ (..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...);  (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)$$

of positive real numbers. For each integer $n$ the following inequalities hold:

$$a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})$$

$$b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})$$

$$c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})$$

Determine $a_{2005}$, $b_{2005}$, $c_{2005}$, if $a_0 = 26, b_0 = 6, c_0 = 2004$.
2004 Austrian-Polish Competition
For each polynomial $Q(x)$ let $M(Q)$ be the set of non-negative integers $x$ with $0 < Q(x) < 2004.$ We consider polynomials $P_n(x)$ of the form



$$P_n(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_{n-1} \cdot x + 1$$



with coefficients $a_i \in \{ \pm1\}$ for $i = 1, 2, \ldots, n-1.$



For each $n = 3^k, k > 0$ determine:



a.) $m_n$ which represents the maximum of elements in $M(P_n)$ for all such polynomials $P_n(x)$



b.) all polynomials $P_n(x)$ for which $|M(P_n)| = m_n.$
2003 Austrian-Polish Competition
Find all real polynomials $p(x) $ such that $p(x-1)p(x+1)= p(x^2-1)$.
2003 Austrian-Polish Competition
The sequence $a_0, a_1, a_2, ..$ is defined by $a_0 = a, a_{n+1} = a_n + L(a_n)$, where $L(m)$ is the last digit of $m$ (eg $L(14) = 4$). Suppose that the sequence is strictly increasing. Show that infinitely many terms must be divisible by $d = 3$. For what other d is this true?
2003 Austrian-Polish Competition
$ABC$ is a triangle. Take $a = BC$ etc as usual.

Take points $T_1, T_2$ on the side $AB$ so that $AT_1 = T_1T_2 = T_2B$. Similarly, take points $T_3, T_4$ on the side BC so that $BT_3 = T_3T_4 = T_4C$, and points $T_5, T_6$ on the side $CA$ so that $CT_5 = T_5T_6 = T_6A$.

Show that if $a' = BT_5, b' = CT_1, c'=AT_3$, then there is a triangle $A'B'C'$ with sides $a', b', c'$ ($a' = B'C$' etc).

In the same way we take points $T_i'$ on the sides of $A'B'C' $ and put $a''  = B'T_6', b''  = C'T_2', c'' = A'T_4'$.

Show that there is a triangle $A'' B'' C'' $ with sides $a'' b'' , c''$  and that it is similar to $ABC$.

Find $a'' /a$.



Austrian-Polish 2003
2003 Austrian-Polish Competition
A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.