Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 Cono Sur Olympiad
3. Show that there do not exist positive integers $a$, $b$, $c$ and $d$, pairwise co-prime, such that $ab+cd$, $ac+bd$ and $ad+bc$ are odd divisors of the number

$(a+b-c-d)(a-b+c-d)(a-b-c+d)$.
2012 Cono Sur Olympiad
4.	Find the biggest positive integer $n$, lesser thar $2012$, that has the following property:

If $p$ is a prime divisor of $n$, then $p^2 - 1$ is a divisor of $n$.
2012 Cono Sur Olympiad
5. $A$ and $B$ play alternating turns on a $2012 \times 2013$ board with enough pieces of the following types:



Type $1$: Piece like Type $2$ but with one square at the right of the bottom square.

Type $2$: Piece of $2$ consecutive squares, one over another.

Type $3$: Piece of $1$ square.



At his turn, $A$ must put a piece of the type $1$ on available squares of the board. $B$, at his turn, must put exactly one piece of each type on available squares of the board. The player that cannot do more movements loses. If $A$ starts playing, decide who has a winning strategy.



Note: The pieces can be rotated but cannot overlap; they cannot be out of the board. The pieces of the types $1$, $2$ and $3$ can be put on exactly $3$, $2$ and $1$ squares of the board respectively.
2012 Cono Sur Olympiad
6.	Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.
2011 Cono Sur Olympiad
Find all triplets of positive integers $(x,y,z)$ such that $x^{2}+y^{2}+z^{2}=2011$.
2011 Cono Sur Olympiad
The numbers $1$ through $4^{n}$ are written on a board. In each step, Pedro erases two numbers $a$ and $b$ from the board, and writes instead the number $\frac{ab}{\sqrt{2a^2+2b^2}}$. Pedro repeats this procedure until only one number remains. Prove that this number is less than $\frac{1}{n}$, no matter what numbers Pedro chose in each step.
2011 Cono Sur Olympiad
Let $ABC$ be an equilateral triangle. Let $P$ be a point inside of it such that the square root of the distance of $P$ to one of the sides is equal to the sum of the square roots of the distances of $P$ to the other two sides. Find the geometric place of $P$.
2011 Cono Sur Olympiad
A number $\overline{abcd}$ is called balanced if $a+b=c+d$. Find all balanced numbers with 4 digits that are the sum of two palindrome numbers.
2011 Cono Sur Olympiad
Let $ABC$ be a triangle and $D$ a point in $AC$. If $\angle{CBD} - \angle{ABD} = 60^{\circ}, \hat{BDC} = 30^{\circ}$ and also $AB \cdot BC = BD^{2}$, determine the measure of all the angles of triangle $ABC$.
2011 Cono Sur Olympiad
Let $Q$ be a $(2n+1) \times (2n+1)$ board. Some of its cells are colored black in such a way that every $2 \times 2$ board of $Q$ has at most $2$ black cells. Find the maximum amount of black cells that the board may have.
2010 Cono Sur Olympiad
Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:



The sum of the fractions is equal to $2$.

The sum of the numerators of the fractions is equal to $1000$.



In how many ways can Pedro do this?
2010 Cono Sur Olympiad
On a line, $44$ points are marked and numbered $1, 2, 3,…,44$ from left to right. Various crickets jump around the line. Each starts at point $1$, jumping on the marked points and ending up at point $44$. In addition, each cricket jumps from a marked point to another marked point with a greater number.

When all the crickets have finished jumping, it turns out that for pair $i, j$ with ${1}\leq{i}<{j}\leq{44}$, there was a cricket that jumped directly from point $i$ to point $j$, without visiting any of the points in between the two.

Determine the smallest number of crickets such that this is possible.
2010 Cono Sur Olympiad
Let us define cutting a convex polygon with $n$ sides by choosing a pair of consecutive sides $AB$ and $BC$ and substituting them by three segments $AM, MN$, and $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, the triangle $MBN$ is removed and a convex polygon with $n+1$ sides is obtained.

Let $P_6$ be a regular hexagon with area $1$. $P_6$ is cut and the polygon $P_7$ is obtained. Then $P_7$ is cut in one of seven ways and polygon $P_8$ is obtained, and so on. Prove that, regardless of how the cuts are made, the area of $P_n$ is always greater than $2/3$.
2010 Cono Sur Olympiad
Pablo and Silvia play a game on a $2010\times{2010}$ board. First Pablo fills in each cell with an integer. Then Silvia repeats the following operation as many times as she likes: she picks three cells that form an L, like in the figure, and adds 1 to each number in the three cells. Silvia wins if she makes all of the number multiples of $10$.

Demonstrate that Silvia can always pick a succession of operations such that she wins the game.

._       _   _ _   _ _

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|_|_| |_|_| |_|     |_|
2010 Cono Sur Olympiad
The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D, E$, and $F$ respectively. Let $\omega_a, \omega_b$ and $\omega_c$ be the circumcircles of triangles $EAF, DBF$, and $DCE$, respectively. The lines $DE$ and $DF$ cut $\omega_a$ at $E_a\neq{E}$ and $F_a\neq{F}$, respectively. Let $r_A$ be the line $E_{a}F_a$. Let $r_B$ and $r_C$ be defined analogously. Show that the lines $r_A$, $r_B$, and $r_C$ determine a triangle with its vertices on the sides of triangle $ABC$.
2010 Cono Sur Olympiad
Determine if there exists an infinite sequence $a_0, a_1, a_2, a_3,...$ of nonegative integers that satisfies the following conditions:

(i)	All nonegative integers appear in the sequence exactly once.

(ii)	The succession

$b_n=a_{n}+n,$, $n\geq0$,

is formed by all prime numbers and each one appears exactly once.
2009 Cono Sur Olympiad
The four circles in the figure determine 10 bounded regions. $10$ numbers summing to $100$ are written in these regions, one in each region. The sum of the numbers contained in each circle is equal to $S$ (the same quantity for each of the four circles). Determine the greatest and smallest possible values of $S$.
2009 Cono Sur Olympiad
A hook consists of three segments of longitude $1$ forming two right angles as demonstrated in the figure.



|_|



We have a square of side length $n$ divided into $n^2$ squares of side length $1$ by lines parallel to its sides. Hooks are placed on this square in such a way that each segment of the hook covers one side of a little square. Two segements of a hook cannot overlap.

Determine all possible values of n for which it is possible to cover the sides of the $n^2$ small squares.
2009 Cono Sur Olympiad
Let $A$, $B$, and $C$ be three points such that $B$ is the midpoint of segment $AC$ and let $P$ be a point such that $<PBC=60$. Equilateral triangle $PCQ$ is constructed such that $B$ and $Q$ are on different half=planes with respect to $PC$, and the equilateral triangle $APR$ is constructed in such a way that $B$ and $R$ are in the same half-plane with respect to $AP$. Let $X$ be the point of intersection of the lines $BQ$ and $PC$, and let $Y$ be the point of intersection of the lines $BR$ and $AP$. Prove that $XY$ and $AC$ are parallel.
2009 Cono Sur Olympiad
Andrea and Bruno play a game on a table with $11$ rows and $9$ columns. First Andrea divides the table in $33$ zones. Each zone is formed by $3$ contiguous cells aligned vertically or horizontally, as shown in the figure.

._

|_|

|_|  _ _ _

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Then, Bruno writes one of the numbers $0, 1, 2, 3, 4, 5$ in each cell in such a way that the sum of the numbers in each zone is equal to $5$. Bruno wins if the sum of the numbers written in each of the $9$ columns of the table is a prime number. Otherwise, Andrea wins. Show that Bruno always has a winning strategy.
2009 Cono Sur Olympiad
Given a succession $C$ of $1001$ positive real numbers (not necessarily distinct), and given a set $K$ of distinct positive integers, the permitted operation is: select a number $k\in{K}$, then select $k$ numbers in $C$, calculate the arithmetic mean of those $k$ numbers, and replace each of those $k$ selected numbers with the mean.



If $K$ is a set such that for each $C$ we can reach, by a sequence of permitted operations, a state where all the numbers are equal, determine the smallest possible value of the maximum element of $K$.
2009 Cono Sur Olympiad
Sebastian has a certain number of rectangles with areas that sum up to 3 and with side lengths all less than or equal to $1$. Demonstrate that with each of these rectangles it is possible to cover a square with side $1$ in such a way that the sides of the rectangles are parallel to the sides of the square.



Note: The rectangles can overlap and they can protrude over the sides of the square.
2008 Cono Sur Olympiad
We define $I(n)$ as the result when the digits of $n$ are reversed. For example, $I(123)=321$, $I(2008)=8002$. Find all integers $n$, $1\leq{n}\leq10000$ for which $I(n)=\lceil{\frac{n}{2}}\rceil$.

Note: $\lceil{x}\rceil$ denotes the smallest integer greater than or equal to $x$. For example, $\lceil{2.1}\rceil=3$, $\lceil{3.9}\rceil=4$, $\lceil{7}\rceil=7$.
2008 Cono Sur Olympiad
Let $P$ be a point in the interior of triangle $ABC$. Let $X$, $Y$, and $Z$ be points on sides $BC$, $AC$, and $AB$ respectively, such that

$<PXC=<PYA=<PZB$.

Let $U$, $V$, and $W$ be points on sides $BC$, $AC$, and $AB$, respectively, or on their extensions if necessary, with $X$ in between $B$ and $U$, $Y$ in between $C$ and $V$, and $Z$ in between $A$ and $W$, such that $PU=2PX$, $PV=2PY$, and $PW=2PZ$. If the area of triangle $XYZ$ is $1$, find the area of triangle $UVW$.
2008 Cono Sur Olympiad
Two friends $A$ and $B$ must solve the following puzzle. Each of them receives a number from the set $\{1,2,…,250\}$, but they don’t see the number that the other received. The objective of each friend is to discover the other friend’s number. The procedure is as follows: each friend, by turns, announces various not necessarily distinct positive integers: first $A$ says a number, then $B$ says one, $A$ says a number again, etc., in such a way that the sum of all the numbers said is $20$. Demonstrate that there exists a strategy that $A$ and $B$ have previously agreed on such that they can reach the objective, no matter which number each one received at the beginning of the puzzle.
2008 Cono Sur Olympiad
What is the largest number of cells that can be colored in a $7\times7$ table in such a way that any $2\times2$ subtable has at most 2 colored cells?
2008 Cono Sur Olympiad
Let $ABC$ be an isosceles triangle with base $AB$. A semicircle $\Gamma$ is constructed with its center on the segment AB and which is tangent to the two legs, $AC$ and $BC$. Consider a line tangent to $\Gamma$ which cuts the segments $AC$ and $BC$ at $D$ and $E$, respectively. The line perpendicular to $AC$ at $D$ and the line perpendicular to $BC$ at $E$ intersect each other at $P$. Let $Q$ be the foot of the perpendicular from $P$ to $AB$. Show that

$\frac{PQ}{CP}=\frac{1}{2}\frac{AB}{AC}$.
2008 Cono Sur Olympiad
A palindrome is a number that is the same when its digits are reversed. Find all numbers that have at least one multiple that is a palindrome.
2007 Cono Sur Olympiad
Find all pairs $(x,y)$ of nonnegative integers that satisfy $$x^3y+x+y=xy+2xy^2.$$
2007 Cono Sur Olympiad
Given are $100$ positive integers whose sum equals their product. Determine the minimum number of $1$s that may occur among the $100$ numbers.
2007 Cono Sur Olympiad
Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ where $D$, $E$, $F$ lie on $BC$, $AC$, $AB$, respectively. Let $M$ be the midpoint of $BC$. The circumcircle of triangle $AEF$ cuts the line $AM$ at $A$ and $X$. The line $AM$ cuts the line $CF$ at $Y$. Let $Z$ be the point of intersection of $AD$ and $BX$. Show that the lines $YZ$ and $BC$ are parallel.
2007 Cono Sur Olympiad
Some cells of a $2007\times 2007$ table are colored. The table is charrua if none of the rows and none of the columns are completely colored.



(a) What is the maximum number

$k$ of colored cells that a charrua table can have?

(b) For such $k$, calculate the number of distinct charrua tables that exist.
2007 Cono Sur Olympiad
Let $ABCDE$ be a convex pentagon that satisfies all of the following:



There is a circle $\Gamma$ tangent to each of the sides.

The lengths of the sides are all positive integers.

At least one of the sides of the pentagon has length $1$.

The side $AB$ has length $2$.



Let $P$ be the point of tangency of $\Gamma$ with $AB$.



(a) Determine the lengths of the segments

$AP$ and $BP$.

(b) Give an example of a pentagon satisfying the given conditions.
2007 Cono Sur Olympiad
Show that for each positive integer $n$, there is a positive integer $k$ such that the decimal representation of each of the numbers $k, 2k,\ldots, nk$ contains all of the digits $0, 1, 2,\ldots, 9$.
2006 Cono Sur Olympiad
Let $ABCD$ be a convex quadrilateral, let $E$ and $F$ be the midpoints of the sides $AD$ and $BC$, respectively. The segment $CE$ meets $DF$ in $O$. Show that if the lines $AO$ and $BO$ divide the side $CD$ in 3 equal parts, then $ABCD$ is a parallelogram.
2006 Cono Sur Olympiad
Two players, A and B, play the following game: they retire coins of a pile which contains initially 2006 coins. The players play removing alternatingly, in each move, from 1 to 7 coins, each player keeps the coins that retires. If a player wishes he can pass(he doesn't retire any coin), but to do that he must pay 7 coins from the ones he retired from the pile in past moves. These 7 coins are taken to a separated box and don't interfere in the game any more. The winner is the one who retires  the last coin, and A starts the game. Determine which player can win for sure, it doesn't matter how the other one plays. Show the winning strategy and explain why it works.
2006 Cono Sur Olympiad
Let $n$ be a natural number. The finite sequence $\alpha$ of positive integer terms, there are $n$ different numbers ($\alpha$ can have repeated terms). Moreover, if from one from its terms any we subtract 1, we obtain a sequence which has, between its terms, at least $n$ different positive numbers. What's the minimum value of the sum of all the terms of $\alpha$?
2006 Cono Sur Olympiad
Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same

list of numbers that Daniel wrote from top to down. Find the greatest length of the Daniel's list can have.
2006 Cono Sur Olympiad
Find all positive integer number $n$ such that $[\sqrt{n}]-2$ divides $n-4$ and $[\sqrt{n}]+2$ divides $n+4$. Note: $[r]$ denotes the integer part of $r$.
2006 Cono Sur Olympiad
We divide the plane in squares shaped of side 1, tracing straight lines parallel bars to the coordinate axles. Each square is painted of black white or. To each as, we recolor all simultaneously squares, in accordance with the following rule: each square $Q$ adopts the color that more appears in the

configuration of five squares indicated in the figure. The recoloration process is repeated indefinitely.



Determine if exists an initial coloration with black a finite amount of squares such that always has at least one black square, not mattering how many seconds if had passed since the beginning of the process.
2005 Cono Sur Olympiad
Let $a_n$ be the last digit of the sum of the digits of $20052005...2005$, where the $2005$ block occurs $n$ times. Find $a_1 +a_2 + \dots +a_{2005}$.
2005 Cono Sur Olympiad
Let $ABC$ be an acute-angled triangle and let $AN$, $BM$ and $CP$ the altitudes with respect to the sides $BC$, $CA$ and $AB$, respectively. Let $R$, $S$ be the pojections of $N$ on the sides $AB$, $CA$, respectively, and let $Q$, $W$ be the projections of $N$ on the altitudes $BM$ and $CP$, respectively.



(a) Show that $R$, $Q$, $W$, $S$ are collinear.

(b) Show that $MP=RS-QW$.
2005 Cono Sur Olympiad
The monetary unit of a certain country is called Reo, and all the coins circulating are integers values of Reos. In a group of three people, each one has 60 Reos in coins (but we don't know what kind of coins each one has). Each of the three people can pay each other any integer value between 1 and 15 Reos, including, perhaps with change. Show that the three persons together can pay exactly (without change) any integer value between 45 and 135 Reos, inclusive.
2005 Cono Sur Olympiad
Let $ABC$ be a isosceles triangle, with $AB=AC$.  A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.
2005 Cono Sur Olympiad
We say that a number of 20 digits is special if its impossible to represent it as an product of a number of 10 digits by a number of 11 digits. Find the maximum quantity of consecutive numbers that are specials.
2005 Cono Sur Olympiad
On the cartesian plane we draw circunferences of radii 1/20 centred in each lattice point. Show that any circunference of radii 100 in the cartesian plane intersect at least one of the small circunferences.
2004 Cono Sur Olympiad
Maxi chose $3$ digits, and by writing down all possible permutations of these digits, he obtained $6$ distinct $3$-digit numbers. If exactly one of those numbers is a perfect square and exactly three of them are prime, find Maxi’s $3$ digits.

Give all of the possibilities for the $3$ digits.
2004 Cono Sur Olympiad
Given a circle $C$ and a point $P$ on its exterior, two tangents to the circle are drawn through $P$, with $A$ and $B$ being the points of tangency. We take a point $Q$ on the minor arc $AB$ of $C$. Let $M$ be the intersection of $AQ$ with the line perpendicular to $AQ$ that goes through $P$, and let $N$ be the intersection of $BQ$ with the line perpendicular to $BQ$ that goes through $P$.

Show that, by varying $Q$ on the minor arc $AB$, all of the lines $MN$ pass through the same point.
2004 Cono Sur Olympiad
Let $n$ be a positive integer. We call $C_n$ the number of positive integers $x$ less than $10^n$ such that the sum of the digits of $2x$ is less than the sum of the digits of $x$.

Show that $C_n\geq\frac{4}{9}(10^{n}-1)$.
2004 Cono Sur Olympiad
Arnaldo selects a nonnegative integer $a$ and Bernaldo selects a nonnegative integer $b$. Both of them secretly tell their number to Cernaldo, who writes the numbers $5$, $8$, and $15$ on the board, one of them being the sum $a+b$.

Cernaldo rings a bell and Arnaldo and Bernaldo, individually, write on different slips of paper whether they know or not which of the numbers on the board is the sum $a+b$ and they turn them in to Cernaldo.

If both of the papers say NO, Cernaldo rings the bell again and the process is repeated.

It is known that both Arnaldo and Bernaldo are honest and intelligent.

What is the maximum number of times that the bell can be rung until one of them knows the sum?



Personal note: They really phoned it in with the names there…
2004 Cono Sur Olympiad
Using cardboard equilateral triangles of side length $1$, an equilateral triangle of side length $2^{2004}$ is formed. An equilateral triangle of side $1$ whose center coincides with the center of the large triangle is removed.

Determine if it is possible to completely cover the remaining surface, without overlaps or holes, using only pieces in the shape of an isosceles trapezoid, each of which is created by joining three equilateral triangles of side $1$.
2004 Cono Sur Olympiad
Let $m$, $n$ be positive integers. On an $m\times{n}$ checkerboard, divided into $1\times1$ squares, we consider all paths that go from upper right vertex to the lower left vertex, travelling exclusively on the grid lines by going down or to the left. We define the area of a path as the number of squares on the checkerboard that are below this path. Let $p$ be a prime such that $r_{p}(m)+r_{p}(n)\geq{p}$, where $r_{p}(m)$ denotes the remainder when $m$ is divided by $p$ and $r_{p}(n)$ denotes the remainder when $n$ is divided by $p$.

How many paths have an area that is a multiple of $p$?
2003 Cono Sur Olympiad
In a soccer tournament between four teams, $A$, $B$, $C$, and $D$, each team plays each of the others exactly once.

a) Decide if, at the end of the tournament, it is possible for the quantities of goals scored and goals allowed for each team to be as follows:

$\begin{tabular}{ c|c|c|c|c } 

 {} & A & B & C & D \\
\hline 

 Goals scored & 1 & 3 & 6 & 7 \\
\hline 

 Goals allowed & 4 & 4 & 4 & 5 \\ 

\end{tabular}$

If the answer is yes, give an example for the results of the six games; in the contrary, justify your answer.

b) Decide if, at the end of the tournament, it is possible for the quantities of goals scored and goals allowed for each team to be as follows:

$\begin{tabular}{ c|c|c|c|c } 

 {} & A & B & C & D \\
\hline 

 Goals scored & 1 & 3 & 6 & 13 \\
\hline 

 Goals allowed & 4 & 4 & 4 & 11 \\ 

\end{tabular}$

If the answer is yes, give an example for the results of the six games; in the contrary, justify your answer.
2003 Cono Sur Olympiad
Define the sequence $\{a_n\}$ in the following manner:

$a_1=1$

$a_2=3$

$a_{n+2}=2a_{n+1}a_{n}+1$ ; for all $n\geq1$

Prove that the largest power of $2$ that divides $a_{4006}-a_{4005}$ is $2^{2003}.$
2003 Cono Sur Olympiad
Let $ABC$ be an acute triangle such that $\angle{B}=60$. The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$, respectively $(M\neq{A},$ $N\neq{C})$. The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$, respectively. Prove that $BR\leq{RS}$.
2003 Cono Sur Olympiad
In an acute triangle $ABC$, the points $H$, $G$, and $M$ are located on $BC$ in such a way that $AH$, $AG$, and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$, $AB=10$, and $AC=14$. Find the area of triangle $ABC$.
2003 Cono Sur Olympiad
Let $n=3k+1$, where $k$ is a positive integer. A triangular arrangement of side $n$ is formed using circles with the same radius, as is shown in the figure for $n=7$.

Determine, for each $k$, the largest number of circles that can be colored red in such a way that there are no two mutually tangent circles that are both colored red.
2003 Cono Sur Olympiad
Show that there exists a sequence of positive integers $x_1, x_2,…x_n,…$ that satisfies the following two conditions:

(i)	Every positive integer appears exactly once,

(ii)	For every $n=1,2,…$ the partial sum $x_1+x_2+…+x_n$ is divisible by $n^n$.
2002 Cono Sur Olympiad
Students in the class of Peter practice the addition and multiplication of integer numbers.The teacher writes the numbers from $1$ to $9$ on nine cards, one for each number, and  places them in an ballot box. Pedro draws three cards, and must calculate the sum and the product of the three corresponding numbers. Ana and Julián do the same, emptying the ballot box. Pedro informs the teacher that he has picked three consecutive numbers whose product is $5$

times the sum. Ana informs that she has no prime number, but two consecutive and that the product of these three numbers is $4$ times the sum of them. What numbers did Julian remove?
2002 Cono Sur Olympiad
Given a triangle $ABC$, with right $\angle A$, we know: the point $T$ of tangency of the circumference inscribed in $ABC$ with the hypotenuse $BC$, the point $D$ of intersection of the angle bisector of $\angle B$ with side AC and the point E of intersection of the angle bisector of $\angle C$ with side $AB$ . Describe a construction with ruler and compass for points $A$, $B$, and $C$. Justify.