Navvye/Olympiad-Problems-CSV huggingface dataset:
2013 IberoAmerican
A set $S$ of positive integers is said to be channeler if for any three distinct numbers $a,b,c \in S$, we have $a\mid bc$, $b\mid ca$, $c\mid ab$.



a) Prove that for any finite set of positive integers $ \{ c_1, c_2, \ldots, c_n \} $ there exist infinitely many positive integers $k$, such that the set $ \{ kc_1, kc_2, \ldots, kc_n \} $ is a channeler set.



b) Prove that for any integer $n \ge 3$ there is a channeler set who has exactly $n$ elements, and such that no integer greater than $1$ divides all of its elements.
2013 IberoAmerican
Let $X$ and $Y$ be the diameter's extremes of a circunference $\Gamma$ and $N$ be the midpoint of one of the arcs $XY$ of $\Gamma$. Let $A$ and $B$ be two points on the segment $XY$. The lines $NA$ and $NB$ cuts $\Gamma$ again in $C$ and $D$, respectively.  The tangents to $\Gamma$ at $C$ and at $D$ meets in $P$. Let $M$ the the intersection point between $XY$ and $NP$. Prove that $M$ is the midpoint of the segment $AB$.
2013 IberoAmerican
Let $A = \{1,...,n\}$ with $n \textgreater 5$. Prove that one can find $B$ a finite set of positive integers such that $A$ is a subset of $B$ and



$\displaystyle\sum_{x \in B} x^2 = \displaystyle\prod_{x \in B} x$
2013 IberoAmerican
Let $\Gamma$ be a circunference and $O$ its center. $AE$ is a diameter of $\Gamma$ and $B$ the midpoint of one of the arcs $AE$ of $\Gamma$. The point $D \ne E$ in on the segment $OE$. The point $C$ is such that the quadrilateral $ABCD$ is a parallelogram, with $AB$ parallel to $CD$ and $BC$ parallel to $AD$. The lines $EB$ and $CD$ meets at point $F$. The line $OF$ cuts the minor arc $EB$ of $\Gamma$ at $I$.



Prove that the line $EI$ is the angle bissector of $\angle BEC$.
2013 IberoAmerican
Let $A$ and $B$ be two sets such that $A \cup B$ is the set of the positive integers, and $A \cap B$ is the empty set. It is known that if two positive integers have a prime larger than $2013$ as their difference, then one of them is in $A$ and the other is in $B$. Find all the possibilities for the sets $A$ and $B$.
2013 IberoAmerican
A beautiful configuration of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.
2012 IberoAmerican
Let $ABCD$ be a rectangle. Construct equilateral triangles $BCX$ and $DCY$, in such a way that both of these triangles share some of their interior points with some interior points of the rectangle. Line $AX$ intersects line $CD$ on $P$, and line $AY$ intersects line $BC$ on $Q$. Prove that triangle $APQ$ is equilateral.
2012 IberoAmerican
A positive integer is called shiny if it can be written as the sum of two not necessarily distinct integers $a$ and $b$ which have the same sum of their digits. For instance, $2012$ is shiny, because $2012 = 2005 + 7$, and both $2005$ and $7$ have the same sum of their digits. Find all positive integers which are not shiny (the dark integers).
2012 IberoAmerican
Let $n$ to be a positive integer. Given a set $\{ a_1, a_2, \ldots, a_n \} $ of integers, where $a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \},$ $\forall i$, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to $0$. If all of these sums have different remainders when divided by $2^n$, we say that $\{ a_1, a_2, \ldots, a_n \} $ is $n$-complete.



For each $n$, find the number of $n$-complete sets.
2012 IberoAmerican
Let $a,b,c,d$ be integers such that the number $a-b+c-d$ is odd and it divides the number $a^2-b^2+c^2-d^2$. Show that, for every positive integer $n$, $a-b+c-d$ divides $a^n-b^n+c^n-d^n$.
2012 IberoAmerican
Let $ABC$ be a triangle, $P$ and $Q$ the intersections of the parallel line to $BC$ that passes through $A$ with the external angle bisectors of angles $B$ and $C$, respectively. The perpendicular to $BP$ at $P$ and the perpendicular to $CQ$ at $Q$ meet at $R$. Let $I$ be the incenter of $ABC$. Show that $AI  = AR$.
2012 IberoAmerican
Show that, for every positive integer $n$, there exist $n$ consecutive positive integers such that none is divisible by the sum of its digits.



(Alternative Formulation: Call a number good if it's not divisible by the sum of its digits. Show that for every positive integer $n$ there are $n$ consecutive good numbers.)
2011 IberoAmerican
The number $2$ is written on the board. Ana and Bruno play alternately. Ana begins. Each one, in their turn, replaces the number written by the one obtained by applying exactly one of these operations: multiply the number by $2$, multiply the number by $3$ or add $1$ to the number. The first player to get a number greater than or equal to $2011$ wins. Find which of the two players has a winning strategy and describe it.
2011 IberoAmerican
Find all positive integers $n$ for which exist three nonzero integers $x, y, z$ such that $x+y+z=0$ and:

$$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{n}$$
2011 IberoAmerican
Let $ABC$ be a triangle and $X,Y,Z$ be the tangency points of its inscribed circle with the sides $BC, CA, AB$, respectively. Suppose that $C_1, C_2, C_3$ are circle with chords $YZ, ZX, XY$, respectively, such that $C_1$ and $C_2$ intersect on the line $CZ$ and that $C_1$ and $C_3$ intersect on the line $BY$. Suppose that $C_1$ intersects the chords $XY$ and $ZX$ at $J$ and $M$, respectively; that $C_2$ intersects the chords $YZ$ and $XY$ at $L$ and $I$, respectively; and that $C_3$ intersects the chords $YZ$ and $ZX$ at $K$ and $N$, respectively. Show that $I, J, K, L, M, N$ lie on the same circle.
2011 IberoAmerican
Let $ABC$ be an acute-angled triangle, with $AC \neq BC$ and let $O$ be its circumcenter. Let $P$ and $Q$ be points such that $BOAP$ and $COPQ$ are parallelograms. Show that $Q$ is the orthocenter of $ABC$.
2011 IberoAmerican
Let $x_1,\ldots ,x_n$ be positive real numbers. Show that there exist $a_1,\ldots ,a_n\in\{-1,1\}$ such that:

$$a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_n)^2$$
2011 IberoAmerican
Let $k$ and $n$ be positive integers, with $k \geq 2$. In a straight line there are $kn$ stones of $k$ colours, such that there are $n$ stones of each colour. A step consists of exchanging the position of two adjacent stones. Find the smallest positive integer $m$ such that it is always possible to achieve, with at most $m$ steps, that the $n$ stones are together, if:

a) $n$ is even.

b) $n$ is odd and $k=3$
2010 IberoAmerican
There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?
2010 IberoAmerican
Determine if there are positive integers $a, b$ such that all terms of the sequence defined by

$$ x_{1}= 2010,x_{2}= 2011\\  x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) $$ are integers.
2010 IberoAmerican
The circle $ \Gamma $ is inscribed to the scalene triangle  $ABC$. $ \Gamma $ is tangent to the sides $BC, CA$ and $AB$ at $D, E$ and $F$ respectively. The line $EF$ intersects the line $BC$ at $G$. The circle of diameter $GD$ intersects $ \Gamma $  in $R$ ($ R\neq D $). Let $P$, $Q$ ($ P\neq R , Q\neq R $)  be the intersections of $ \Gamma $  with $BR$ and $CR$, respectively. The lines $BQ$ and $CP$ intersects at $X$. The circumcircle of $CDE$ meets $QR$ at $M$, and the circumcircle of $BDF$ meet $PR$ at $N$. Prove that $PM$, $QN$ and $RX$ are concurrent.



Author: Arnoldo Aguilar, El Salvador
2010 IberoAmerican
The arithmetic, geometric and harmonic mean of two distinct positive integers are different numbers. Find the smallest possible value for the arithmetic mean.
2010 IberoAmerican
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular. Let $O$ be the circumcenter of $ABCD$, $K$ the intersection of the diagonals,  $ L\neq O $ the intersection of the circles circumscribed to $OAC$ and $OBD$, and $G$ the intersection of the diagonals of the quadrilateral whose vertices are the midpoints of the sides of $ABCD$. Prove that $O, K, L$ and $G$ are collinear
2010 IberoAmerican
Around a circular table sit $12$ people, and on the table there are $28$ vases. Two people can see each other, if and only if there is no vase lined with them. Prove that there are at least two people who can be seen.
2009 IberoAmerican
Given a positive integer $ n\geq 2$, consider a set of $ n$ islands arranged in a circle. Between every two neigboring islands two bridges are built as shown in the figure.



Starting at the island $ X_1$, in how many ways one can one can cross the $ 2n$ bridges so that no bridge is used more than once?
2009 IberoAmerican
Define the succession $ a_{n}$, $ n>0$ as $ n+m$, where $ m$ is the largest integer such that $ 2^{2^{m}}\leq n2^{n}$. Find all numbers that are not in the succession.
2009 IberoAmerican
Let $ C_1$ and $ C_2$ be two congruent circles centered at $ O_1$ and $ O_2$, which intersect at $ A$ and $ B$. Take a point $ P$ on the arc $ AB$ of $ C_2$ which is contained in $ C_1$. $ AP$ meets $ C_1$ at $ C$, $ CB$ meets $ C_2$ at $ D$ and the bisector of $ \angle CAD$ intersects $ C_1$ and $ C_2$ at $ E$ and $ L$, respectively. Let $ F$ be the symmetric point of $ D$ with respect to the midpoint of $ PE$. Prove that there exists a point $ X$ satisfying $ \angle XFL = \angle XDC = 30^\circ$ and $ CX = O_1O_2$.



Author: Arnoldo Aguilar (El Salvador)
2009 IberoAmerican
Given a triangle $ ABC$ of incenter $ I$, let $ P$ be the intersection of the external bisector of angle $ A$ and the circumcircle of $ ABC$, and $ J$ the second intersection of $ PI$ and the circumcircle of $ ABC$. Show that the circumcircles of triangles $ JIB$ and $ JIC$ are respectively tangent to $ IC$ and $ IB$.
2009 IberoAmerican
Consider the sequence $ \{a_n\}_{n\geq1}$ defined as follows: $ a_1 = 1$, $ a_{2k} = 1 + a_k$ and $ a_{2k + 1} = \frac {1}{a_{2k}}$ for every $ k\geq 1$. Prove that every positive rational number appears on the sequence $ \{a_n\}$ exactly once.
2009 IberoAmerican
Six thousand points are marked on a circle, and they are colored using 10 colors in such a way that within every group of 100 consecutive points all the colors are used. Determine the least positive integer $ k$ with the following property: In every coloring satisfying the condition above, it is possible to find a group of $ k$ consecutive points in which all the colors are used.
2008 IberoAmerican
The integers from 1 to $ 2008^2$ are written on each square of a $ 2008 \times 2008$ board. For every row and column the difference between the maximum and minimum numbers is computed. Let $ S$ be the sum of these 4016 numbers. Find the greatest possible value of $ S$.
2008 IberoAmerican
Given a triangle $ ABC$, let $ r$ be the external bisector of $ \angle ABC$. $ P$ and $ Q$ are the feet of the perpendiculars from $ A$ and $ C$ to $ r$. If $ CP \cap BA = M$ and $ AQ \cap BC=N$, show that $ MN$, $ r$ and $ AC$ concur.
2008 IberoAmerican
Let $ P(x) = x^3 + mx + n$ be an integer polynomial satisfying that if $ P(x) - P(y)$ is divisible by 107, then $ x - y$ is divisible by 107 as well, where $ x$ and $ y$ are integers. Prove that 107 divides $ m$.
2008 IberoAmerican
Prove that the equation $$ x^{2008}+ 2008!= 21^{y}$$ doesn't have solutions in integers.
2008 IberoAmerican
Let $ ABC$ a triangle and $ X$, $ Y$ and $ Z$ points at the segments $ BC$, $ AC$ and $ AB$, respectively.Let $ A'$, $ B'$ and $ C'$ the circuncenters of triangles $ AZY$,$ BXZ$,$ CYX$, respectively.Prove that $ 4(A'B'C')\geq(ABC)$ with equality if and only if $ AA'$, $ BB'$ and $ CC'$ are concurrents.



Note: $ (XYZ)$ denotes the area of $ XYZ$
2008 IberoAmerican
Biribol is a game played between two teams of 4 people each (teams are not fixed). Find all the possible values of $ n$ for which it is possible to arrange a tournament with $ n$ players in such a way that every couple of people plays a match in opposite teams exactly once.
2007 IberoAmerican
Given an integer $ m$, define the sequence $ \left\{a_{n}\right\}$ as follows:

$$ a_{1}=\frac{m}{2},\ a_{n+1}=a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1$$

Find all values of $ m$ for which $ a_{2007}$ is the first integer appearing in the sequence.



Note: For a real number $ x$, $ \left\lceil x\right\rceil$ is defined as the smallest integer greater or equal to $ x$. For example, $ \left\lceil\pi\right\rceil=4$, $ \left\lceil 2007\right\rceil=2007$.
2007 IberoAmerican
Let $ ABC$ be a triangle with incenter $ I$ and let $ \Gamma$ be a circle centered at $ I$, whose radius is greater than the inradius and does not pass through any vertex. Let $ X_{1}$ be the intersection point of $ \Gamma$ and line $ AB$, closer to $ B$; $ X_{2}$, $ X_{3}$ the points of intersection of $ \Gamma$ and line $ BC$, with $ X_{2}$ closer to $ B$; and let $ X_{4}$ be the point of intersection of $ \Gamma$ with line $ CA$ closer to $ C$. Let $ K$ be the intersection point of lines $ X_{1}X_{2}$ and $ X_{3}X_{4}$. Prove that $ AK$ bisects segment $ X_{2}X_{3}$.
2007 IberoAmerican
Two teams, $ A$ and $ B$, fight for a territory limited by a circumference.



$ A$ has $ n$ blue flags and $ B$ has $ n$ white flags ($ n\geq 2$, fixed). They play alternatively and $ A$ begins the game. Each team, in its turn, places one of his flags in a point of the circumference that has not been used in a previous play. Each flag, once placed, cannot be moved.



Once all $ 2n$ flags have been placed, territory is divided between the two teams. A point of the territory belongs to $ A$ if the closest flag to it is blue, and it belongs to $ B$ if the closest flag to it is white. If the closest blue flag to a point is at the same distance than the closest white flag to that point, the point is neutral (not from $ A$ nor from $ B$). A team wins the game is their points cover a greater area that that covered by the points of the other team. There is a draw if both cover equal areas.

Prove that, for every $ n$, team $ B$ has a winning strategy.
2007 IberoAmerican
In a $ 19\times 19$ board, a piece called dragon moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board.



The draconian distance between two squares is defined as the least number of moves a dragon needs to move from one square to the other.



Let $ C$ be a corner square, and $ V$ the square neighbor of $ C$ that has only a point in common with $ C$. Show that there exists a square $ X$ of the board, such that the draconian distance between $ C$ and $ X$ is greater than the draconian distance between $ C$ and $ V$.
2007 IberoAmerican
Let's say a positive integer $ n$ is atresvido if the set of its divisors (including 1 and $ n$) can be split in in 3 subsets such that the sum of the elements of each is the same. Determine the least number of divisors an atresvido number can have.
2007 IberoAmerican
Let $ \mathcal{F}$ be a family of hexagons $ H$ satisfying the following properties:



i) $ H$ has parallel opposite sides.



ii) Any 3 vertices of $ H$ can be covered with a strip of width 1.



Determine the least $ \ell\in\mathbb{R}$ such that every hexagon belonging to $ \mathcal{F}$ can be covered with a strip of width $ \ell$.



Note: A strip is the area bounded by two parallel lines separated by a distance $ \ell$. The lines belong to the strip, too.
2006 IberoAmerican
In a scalene triangle $ABC$ with $\angle A = 90^\circ,$ the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R.$

The lines $RS$ and $BC$ intersect at $N,$ while the lines $AM$ and $SR$ intersect at $U.$

Prove that the triangle $UMN$ is isosceles.
2006 IberoAmerican
For n real numbers $a_{1},\, a_{2},\, \ldots\, , a_{n},$ let $d$ denote the difference between the greatest and smallest of them and $S = \sum_{i<j}\left |a_i-a_j \right|.$ Prove that $$(n-1)d\le S\le\frac{n^{2}}{4}d$$ and find when each equality holds.
2006 IberoAmerican
The numbers $1,\, 2,\, \ldots\, , n^{2}$ are written in the squares of an $n \times n$ board in some order. Initially there is a token on the square labelled with $n^{2}.$ In each step, the token can be moved to any adjacent square (by side). At the beginning, the token is moved to the square labelled with the number $1$ along a path with the minimum number of steps. Then it is moved to the square labelled with $2,$ then to square $3,$ etc, always taking the shortest path, until it returns to the initial square. If the total trip takes $N$ steps, find the smallest and greatest possible values of $N.$
2006 IberoAmerican
Find all pairs $(a,\, b)$ of positive integers such that $2a-1$ and $2b+1$ are coprime and $a+b$ divides $4ab+1.$
2006 IberoAmerican
The sides $AD$ and $CD$ of a tangent quadrilateral $ABCD$ touch the incircle $\varphi$ at $P$ and $Q,$ respectively. If $M$ is the midpoint of the chord $XY$ determined by $\varphi$ on the diagonal $BD,$ prove that $\angle AMP = \angle CMQ.$
2006 IberoAmerican
Consider a regular $n$-gon with $n$ odd. Given two adjacent vertices $A_{1}$ and $A_{2},$ define the sequence $(A_{k})$ of vertices of the $n$-gon as follows: For $k\ge 3,\, A_{k}$ is the vertex lying on the perpendicular bisector of $A_{k-2}A_{k-1}.$ Find all $n$ for which each vertex of the $n$-gon occurs in this sequence.
2005 IberoAmerican
Determine all triples of real numbers $(a,b,c)$ such that \begin{eqnarray*} xyz &=& 8 \\ x^2y + y^2z + z^2x &=& 73 \\ x(y-z)^2 + y(z-x)^2 + z(x-y)^2 &=& 98 . \end{eqnarray*}
2005 IberoAmerican
A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$.



Prove that if the flea arrived twice at the point $n$, $n$ positive integer, then it performed at least $\lceil 2\sqrt n\rceil$ jumps.
2005 IberoAmerican
Let $p > 3$ be a prime. Prove that if $$ \sum_{i=1 }^{p-1}{1\over i^p} = {n\over m},  $$ with $\gdc(n,m) = 1$, then $p^3$ divides $n$.
2005 IberoAmerican
Denote by $a \bmod b$ the remainder of the euclidean division of $a$ by $b$. Determine all pairs of positive integers $(a,p)$ such that $p$ is prime and $$ a \bmod p + a\bmod 2p + a\bmod 3p + a\bmod 4p = a + p. $$
2005 IberoAmerican
Let $O$ be the circumcenter of acutangle triangle $ABC$ and let $A_1$ be some point in the smallest arc $BC$ of the circumcircle of $ABC$. Let $A_2$ and $A_3$ points on sides $AB$ and $AC$, respectively, such that $\angle BA_1A_2 = \angle OAC$ and $\angle CA_1A_3 = \angle OAB$.



Prove that the line $A_2A_3$ passes through the orthocenter of $ABC$.
2005 IberoAmerican
Let $n$ be a fixed positive integer. The points $A_1$, $A_2$, $\ldots$, $A_{2n}$ are on a straight line. Color each point blue or red according to the following procedure: draw $n$ pairwise disjoint circumferences, each with diameter $A_iA_j$ for some $i \neq j$ and such that every point $A_k$ belongs to exactly one circumference. Points in the same circumference must be of the same color.



Determine the number of ways of coloring these $2n$ points when we vary the $n$ circumferences and the distribution of the colors.
2004 IberoAmerican
It is given a 1001*1001 board divided in 1*1 squares. We want to amrk m squares in such a way that:

1: if 2 squares are adjacent then one of them is marked.

2: if 6 squares lie consecutively in a row or column then two adjacent squares from them are marked.



Find the minimun number of squares we most mark.
2004 IberoAmerican
In the plane are given a circle with center $ O$ and radius $ r$ and a point $ A$ outside the circle. For any point $ M$ on the circle, let $ N$ be the diametrically opposite point. Find the locus of the circumcenter of triangle $ AMN$ when $ M$ describes the circle.
2004 IberoAmerican
Let $ n$ and $ k$ be positive integers such as either $ n$ is odd or both $ n$ and $ k$ are even. Prove that exists integers $ a$ and $ b$ such as $ GCD(a,n) = GCD(b,n) = 1$ and $ k = a + b$
2004 IberoAmerican
Determine all pairs $ (a,b)$ of positive integers, each integer having two decimal digits, such that $ 100a+b$ and $ 201a+b$ are both perfect squares.
2004 IberoAmerican
Given a scalene triangle $ ABC$. Let $ A'$, $ B'$, $ C'$ be the points where the internal bisectors of the angles $ CAB$, $ ABC$, $ BCA$ meet the sides $ BC$, $ CA$, $ AB$, respectively. Let the line $ BC$ meet the perpendicular bisector of $ AA'$ at $ A''$. Let the line $ CA$ meet the perpendicular bisector of $ BB'$ at $ B'$. Let the line $ AB$ meet the perpendicular bisector of $ CC'$ at $ C''$. Prove that $ A''$, $ B''$ and $ C''$ are collinear.
2004 IberoAmerican
Given a set $ \mathcal{H}$ of points in the plane, $ P$ is called an "intersection point of $ \mathcal{H}$" if distinct points $ A,B,C,D$ exist in $ \mathcal{H}$ such that lines $ AB$ and $ CD$ are distinct and intersect in $ P$.

Given a finite set $ \mathcal{A}_{0}$ of points in the plane, a sequence of sets is defined as follows: for any $ j\geq0$, $ \mathcal{A}_{j+1}$ is the union of $ \mathcal{A}_{j}$ and the intersection points of $ \mathcal{A}_{j}$.

Prove that, if the union of all the sets in the sequence is finite, then $ \mathcal{A}_{i}=\mathcal{A}_{1}$ for any $ i\geq1$.
2003 IberoAmerican
$(a)$There are two sequences of numbers, with $2003$ consecutive integers each, and a table of $2$ rows and $2003$ columns

$\begin{array}{|c|c|c|c|c|c|} \hline\ \ &\ &\ &\cdots\cdots&\ &\ \\ \hline \ &\ &\ &\cdots\cdots&\ &\ \\ \hline \end{array}$

Is it always possible to arrange the numbers in the first sequence in the first row and the second sequence in the second row, such that the sequence obtained of the $2003$ column-wise sums form a new sequence of $2003$ consecutive integers?

$(b)$ What if $2003$ is replaced with $2004$?
2003 IberoAmerican
Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.
2003 IberoAmerican
Pablo copied from the blackboard the problem:



Consider all the sequences of

$2004$ real numbers $(x_0,x_1,x_2,\dots, x_{2003})$ such that: $x_0=1, 0\le x_1\le 2x_0,0\le x_2\le 2x_1\ldots ,0\le x_{2003}\le 2x_{2002}$. From all these sequences, determine the sequence which minimizes $S=\cdots$



As Pablo was copying the expression, it was erased from the board. The only thing that he could remember was that $S$ was of the form $S=\pm x_1\pm x_2\pm\cdots\pm x_{2002}+x_{2003}$. Show that, even when Pablo does not have the complete statement, he can determine the solution of the problem.
2003 IberoAmerican
Let $M=\{1,2,\dots,49\}$ be the set of the first $49$ positive integers. Determine the maximum integer $k$ such that the set $M$ has a subset of $k$ elements such that there is no $6$ consecutive integers in such subset. For this value of $k$, find the number of subsets of $M$ with $k$ elements with the given property.
2003 IberoAmerican
In a square $ABCD$, let $P$ and $Q$ be points on the sides $BC$ and $CD$ respectively, different from its endpoints, such that $BP=CQ$. Consider points $X$ and $Y$ such that $X\neq Y$, in the segments $AP$ and $AQ$ respectively. Show that, for every $X$ and $Y$ chosen, there exists a triangle whose sides have lengths $BX$, $XY$ and $DY$.
2003 IberoAmerican
The sequences $(a_n),(b_n)$ are defined by $a_0=1,b_0=4$ and for $n\ge 0$

$$a_{n+1}=a_n^{2001}+b_n,\ \ b_{n+1}=b_n^{2001}+a_n$$

Show that $2003$ is not divisor of any of the terms in these two sequences.