Navvye/Olympiad-Problems-CSV huggingface dataset:
2000 CentroAmerican
Determine all positive integers $ n$ such that it is possible to tile a $ 15 \times n$ board with pieces shaped like this:



[asy]size(100); draw((0,0)--(3,0)); draw((0,1)--(3,1)); draw((0,2)--(1,2)); draw((2,2)--(3,2)); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,0)--(2,2)); draw((3,0)--(3,2)); draw((5,0)--(6,0)); draw((4,1)--(7,1)); draw((4,2)--(7,2)); draw((5,3)--(6,3)); draw((4,1)--(4,2)); draw((5,0)--(5,3)); draw((6,0)--(6,3)); draw((7,1)--(7,2));[/asy]
2000 CentroAmerican
Let $ ABCDE$ be a convex pentagon. If $ P$, $ Q$, $ R$ and $ S$ are the respective centroids of the triangles $ ABE$, $ BCE$, $ CDE$ and $ DAE$, show that $ PQRS$ is a parallelogram and its area is $ 2/9$ of that of $ ABCD$.
2000 CentroAmerican
Write an integer on each of the 16 small triangles in such a way that every number having at least two neighbors is equal to the difference of two of its neighbors.



Note: Two triangles are said to be neighbors if they have a common side.



[asy]size(100); pair P=(0,0); pair Q=(2, 2*sqrt(3)); pair R=(4,0); draw(P--Q--R--cycle); pair B=midpoint(P--Q); pair A=midpoint(P--B); pair C=midpoint(B--Q); pair E=midpoint(Q--R); pair D=midpoint(Q--E); pair F=midpoint(E--R); pair H=midpoint(R--P); pair G=midpoint(R--H); pair I=midpoint(H--P); draw(A--I); draw(B--H); draw(C--G); draw(I--D); draw(H--E); draw(G--F); draw(C--D); draw(B--E); draw(A--F);[/asy]
2000 CentroAmerican
Let $ ABC$ be an acute-angled triangle. $ C_{1}$ and $ C_{2}$ are two circles of diameters $ AB$ and $ AC$, respectively. $ C_{2}$ and $ AB$ intersect again at $ F$, and $ C_{1}$ and $ AC$ intersect again at $ E$. Also,  $ BE$ meets $ C_{2}$ at $ P$ and $ CF$ meets $ C_{1}$ at $ Q$. Prove that $ AP=AQ$.
2000 CentroAmerican
Let's say we have a nice representation of the positive integer $ n$ if we write it as a sum of powers of 2 in such a way that there are at most two equal powers in the sum (representations differing only in the order of their summands are considered to be the same).



a) Write down the 5 nice representations of 10.



b) Find all positive integers with an even number of nice representations.
1999 CentroAmerican
Suppose that each of the 5 persons knows a piece of information, each piece is different, about a certain event. Each time person $A$ calls person $B$, $A$ gives $B$ all the information that $A$ knows at that moment about the event, while $B$ does not say to $A$ anything that he knew.



(a) What is the minimum number of calls are necessary so that everyone knows about the event?

(b) How many calls are necessary if there were $n$ persons?
1999 CentroAmerican
Find a positive integer $n$ with 1000 digits, all distinct from zero, with the following property: it's possible to group the digits of $n$ into 500 pairs in such a way that if the two digits of each pair are multiplied and then add the 500 products, it results a number $m$ that is a divisor of $n$.
1999 CentroAmerican
The digits of a calculator (with the exception of 0) are shown in the form indicated by the figure below, where there is also a button ``+":

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Two players $A$ and $B$ play in the following manner: $A$ turns on the calculator and presses a digit, and then presses the button ``+". $A$ passes the calculator to $B$, which presses a digit in the same row or column with the one pressed by $A$ that is not the same as the last one pressed by $A$; and then presses + and returns the calculator to $A$, repeating the operation in this manner successively. The first player that reaches or exceeds the sum of 31 loses the game. Which of the two players have a winning strategy and what is it?
1999 CentroAmerican
In the trapezoid $ABCD$ with bases $AB$ and $CD$, let $M$ be the midpoint of side $DA$. If $BC=a$, $MC=b$ and $\angle MCB=150^\circ$, what is the area of trapezoid $ABCD$ as a function of $a$ and $b$?
1999 CentroAmerican
Let $a$ be an odd positive integer greater than 17 such that $3a-2$ is a perfect square. Show that there exist distinct positive integers $b$ and $c$ such that $a+b,a+c,b+c$ and $a+b+c$ are four perfect squares.
1999 CentroAmerican
Denote $S$ as the subset of $\{1,2,3,\dots,1000\}$ with the property that none of the sums of two different elements in $S$ is in $S$. Find the maximum number of elements in $S$.
2021 Cono Sur Olympiad
We say that a positive integer is guarani if the sum of the number with its reverse is a number that only has odd digits. For example, $249$ and $30$ are guarani, since $249 + 942 = 1191$ and $30 + 03 = 33$.

a) How many $2021$-digit numbers are guarani?

b) How many $2023$-digit numbers are guarani?
2021 Cono Sur Olympiad
Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle  of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.
2021 Cono Sur Olympiad
In a tennis club, each member has exactly $k > 0$ friends, and a tournament is organized in rounds such that each pair of friends faces each other in matches exactly once. Rounds are played in simultaneous matches, choosing pairs until they cannot choose any more (that is, among the unchosen people, there is not a pair of friends which has its match pending). Determine the maximum number of rounds the tournament can have, depending on $k$.
2021 Cono Sur Olympiad
In a heap there are $2021$ stones. Two players $A$ and $B$ play removing stones of the pile, alternately starting with $A$. A valid move for $A$ consists of remove $1, 2$ or $7$ stones. A valid move for B is to remove $1, 3, 4$ or $6$ stones. The player who leaves the pile empty after making a valid move wins. Determine if some of the players have a winning strategy. If such a strategy exists, explain it.
2021 Cono Sur Olympiad
Given an integer $n \geq 3$, determine if there are $n$ integers $b_1, b_2, \dots , b_n$, distinct two-by-two (that is, $b_i \neq b_j$ for all $i \neq j$) and a polynomial $P(x)$ with coefficients integers, such that $P(b_1) = b_2, P(b_2) = b_3, \dots , P(b_{n-1}) = b_n$ and $P(b_n) = b_1$.
2021 Cono Sur Olympiad
Let $ABC$ be a scalene triangle with circle $\Gamma$. Let $P,Q,R,S$ distinct points on the $BC$ side, in that order, such that $\angle BAP = \angle CAS$ and $\angle BAQ = \angle CAR$. Let $U, V, W, Z$ be the intersections, distinct from $A$, of the $AP, AQ, AR$ and $AS$ with $\Gamma$, respectively. Let $X = UQ \cap SW$, $Y = PV \cap ZR$, $T = UR \cap VS$ and $K = PW \cap ZQ$. Suppose that the points $M$ and $N$ are well determined, such that $M = KX \cap TY$ and $N = TX \cap KY$. Show that $M, N, A$ are collinear.
2020 Cono Sur Olympiad
Ari and Beri play a game using a deck of $2020$ cards with exactly one card with each number from $1$ to $2020$. Ari gets a card with a number $a$ and removes it from the deck. Beri sees the card, chooses another card from the deck with a number $b$ and removes it from the deck. Then Beri writes on the board exactly one of the trinomials $x^2-ax+b$ or $x^2-bx+a$ from his choice. This process continues until no cards are left on the deck. If at the end of the game every trinomial written on the board has integer solutions, Beri wins. Otherwise, Ari wins. Prove that Beri can always win, no matter how Ari plays.
2020 Cono Sur Olympiad
Given $2021$ distinct positive integers non divisible by $2^{1010}$, show that it's always possible to choose $3$ of them $a$, $b$ and $c$, such that $|b^2-4ac|$ is not a perfect square.
2020 Cono Sur Olympiad
Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$. Points $F$ and $E$ are chosen in $AB$ and $BC$, respectively, such that $AC=CF$ and $EB=EF$. The line $AM$ intersects $\omega$ in $D\neq A$. The line $DE$ intersects the line $FM$ in $G$. Prove that $G$ lies on $\omega$.
2020 Cono Sur Olympiad
Let $ABC$ be an acute scalene triangle. $D$ and $E$ are variable points in the half-lines $AB$ and $AC$ (with origin at $A$) such that the symmetric of $A$ over $DE$ lies on $BC$. Let $P$ be the intersection of the circles with diameter $AD$ and $AE$. Find the locus of $P$ when varying the line segment $DE$.
2020 Cono Sur Olympiad
There is a pile with $15$ coins on a table. At each step, Pedro choses one of the piles in the table with $a>1$ coins and divides it in two piles with $b\geq1$ and $c\geq1$ coins and writes in the board the product $abc$. He continues until there are $15$ piles with $1$ coin each. Determine all possible values that the final sum of the numbers in the board can have.
2020 Cono Sur Olympiad
A $4$ x $4$ square board is called $brasuca$ if it follows all the conditions:



• each box contains one of the numbers $0, 1, 2, 3, 4$ or $5$;

• the sum of the numbers in each line is $5$;

• the sum of the numbers in each column is $5$;

• the sum of the numbers on each diagonal of four squares is $5$;

• the number written in the upper left box of the board is less than or equal to the other numbers

the board;

• when dividing the board into four $2$ × $2$ squares, in each of them the sum of the four

numbers is $5$.



How many $"brasucas"$ boards are there?
2019 Cono Sur Olympiad
Martin has two boxes $A$ and $B$. In the box $A$ there are $100$ red balls numbered from $1$ to $100$, each one with one of these numbers. In the box $B$ there are $100$ blue balls numbered from $101$ to $200$, each one with one of these numbers. Martin chooses two positive integers $a$ and $b$, both less than or equal to $100$, and then he takes out $a$ balls from box $A$ and $b$ balls from box $B$, without replacement. Martin's goal is to have two red balls and one blue ball among all balls taken such that the sum of the numbers of two red balls equals the number of the blue ball.



What is the least possible value of $a+b$ so that Martin achieves his goal for sure? For such a minimum value of $a+b$, give an example of $a$ and $b$ satisfying the goal and explain why every $a$ and $b$ with smaller sum cannot accomplish the aim.
2019 Cono Sur Olympiad
We say that a positive integer $M$ with $2n$ digits is hypersquared if the following three conditions are met:



$M$ is a perfect square.

The number formed by the first $n$ digits of $M$ is a perfect square.

The number formed by the last $n$ digits of $M$ is a perfect square and has exactly $n$ digits (its first digit is not zero).



Find a hypersquared number with $2000$ digits.
2019 Cono Sur Olympiad
Let $n\geq 3$ an integer. Determine whether there exist permutations $(a_1,a_2, \ldots, a_n)$ of the numbers $(1,2,\ldots, n)$ and $(b_1, b_2, \ldots, b_n)$ of the numbers $(n+1,n+2,\ldots, 2n)$ so that $(a_1b_1, a_2b_2, \ldots a_nb_n)$ is a strictly increasing arithmetic progression.
2019 Cono Sur Olympiad
Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.
2019 Cono Sur Olympiad
Let $n\geq 3$ a positive integer. In each cell of a $n\times n$ chessboard one must write $1$ or $2$ in such a way the sum of all written numbers in each $2\times 3$ and $3\times 2$ sub-chessboard is even. How many different ways can the chessboard be completed?
2019 Cono Sur Olympiad
Let $ABC$ be an acute-angled triangle with $AB< AC$, and let $H$ be its orthocenter. The circumference with diameter $AH$ meets the circumscribed circumference of $ABC$ at $P\neq A$. The tangent to the circumscribed circumference of $ABC$ through $P$ intersects line $BC$ at $Q$. Show that $QP=QH$.
2018 Cono Sur Olympiad
Let $ABCD$ be a convex quadrilateral, where $R$ and $S$ are points in $DC$ and $AB$, respectively, such that $AD=RC$ and $BC=SA$. Let $P$, $Q$ and $M$ be the midpoints of $RD$, $BS$ and $CA$, respectively. If $\angle MPC + \angle MQA = 90$, prove that $ABCD$ is cyclic.
2018 Cono Sur Olympiad
Prove that every positive integer can be formed by the sums of powers of 3, 4 and 7, where do not appear two powers of the same number and with the same exponent.

Example: $2= 7^0 + 7^0$ and $22=3^2 + 3^2+4^1$ are not valid representations, but $2=3^0+7^0$ and $22=3^2+3^0+4^1+4^0+7^1$ are valid representations.
2018 Cono Sur Olympiad
Define the product $P_n=1! \cdot 2!\cdot 3!\cdots (n-1)!\cdot n!$

a) Find all positive integers $m$, such that $\frac {P_{2020}}{m!}$ is a perfect square.

b) Prove that there are infinite many value(s) of $n$, such that $\frac {P_{n}}{m!}$ is a perfect square, for at least two positive integers $m$.
2018 Cono Sur Olympiad
For each interger $n\geq 4$, we consider the $m$ subsets $A_1, A_2,\dots, A_m$ of $\{1, 2, 3,\dots, n\}$, such that

$A_1$ has exactly one element, $A_2$ has exactly two elements,...., $A_m$ has exactly $m$ elements and none of these subsets is contained in any other set. Find the maximum value of $m$.
2018 Cono Sur Olympiad
Let $ABC$ be an acute-angled triangle with $\angle BAC = 60^{\circ}$ and with incenter $I$ and circumcenter $O$. Let $H$ be the point diametrically opposite(antipode) to $O$ in the circumcircle of $\triangle BOC$. Prove that $IH=BI+IC$.
2018 Cono Sur Olympiad
A sequence $a_1, a_2,\dots, a_n$ of positive integers is alagoana, if for every $n$ positive integer, one have these two conditions

I- $a_{n!} = a_1\cdot a_2\cdot a_3\cdots a_n$

II- The number $a_n$ is the $n$-power of a positive integer.

Find all the sequence(s) alagoana.
2017 Cono Sur Olympiad
A positive integer $n$ is called guayaquilean if the sum of the digits of $n$ is equal to the sum of the digits of $n^2$. Find all the possible values that the sum of the digits of a guayaquilean number can take.
2017 Cono Sur Olympiad
Let $A(XYZ)$ be the area of the triangle $XYZ$. A non-regular convex polygon $P_1 P_2 \ldots P_n$ is called guayaco if exists a point $O$ in its interior such that $$A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).$$Show that, for every integer $n \ge 3$, a guayaco polygon of $n$ sides exists.
2017 Cono Sur Olympiad
Let $n$ be a positive integer. In how many ways can a $4 \times 4n$ grid be tiled with the following tetromino?

[asy]

size(4cm);

draw((1,0)--(3,0)--(3,1)--(0,1)--(0,0)--(1,0)--(1,2)--(2,2)--(2,0));

[/asy]
2017 Cono Sur Olympiad
Let $ABC$ an acute triangle with circumcenter $O$. Points $X$ and $Y$ are chosen such that:



$\angle XAB = \angle YCB = 90^\circ$

$\angle ABC = \angle BXA = \angle BYC$

$X$ and $C$ are in different half-planes with respect to $AB$

$Y$ and $A$ are in different half-planes with respect to $BC$



Prove that $O$ is the midpoint of $XY$.
2017 Cono Sur Olympiad
Let $a$, $b$ and $c$ positive integers. Three sequences are defined as follows:



$a_1=a$, $b_1=b$, $c_1=c$

$a_{n+1}=\lfloor{\sqrt{a_nb_n}}\rfloor$, $\:b_{n+1}=\lfloor{\sqrt{b_nc_n}}\rfloor$, $\:c_{n+1}=\lfloor{\sqrt{c_na_n}}\rfloor$ for $n \ge 1$





Prove that for any $a$, $b$, $c$, there exists a positive integer $N$ such that $a_N=b_N=c_N$.

Find the smallest $N$ such that $a_N=b_N=c_N$ for some choice of $a$, $b$, $c$ such that $a \ge 2$ y $b+c=2a-1$.
2017 Cono Sur Olympiad
The infinite sequence $a_1,a_2,a_3,\ldots$ of positive integers is defined as follows: $a_1=1$, and for each $n \ge 2$, $a_n$ is the smallest positive integer, distinct from $a_1,a_2, \ldots , a_{n-1}$ such that:

$$\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}}$$is an integer. Prove that all positive integers appear on the sequence $a_1,a_2,a_3,\ldots$
2016 Cono Sur Olympiad
Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an special number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers.

Note: If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.
2016 Cono Sur Olympiad
For every $k= 1,2, \ldots$ let $s_k$  be the number of pairs $(x,y)$ satisfying the equation $kx + (k+1)y = 1001 - k$ with $x$, $y$ non-negative integers. Find $s_1 + s_2 + \cdots + s_{200}$.
2016 Cono Sur Olympiad
There are $ 2016 $ positions marked around a circle, with a token on one of them. A legitimate move is to move the token either 1 position or 4 positions from its location, clockwise. The restriction is that the token can not occupy the same position more than once. Players $ A $ and $ B $ take turns making moves. Player $ A $ has the first move. The first player who cannot make a legitimate move loses. Determine which of the two players has a winning strategy.
2016 Cono Sur Olympiad
Let $S(n)$ be the sum of the digits of the positive integer $n$. Find all $n$ such that $S(n)(S(n)-1)=n-1$.
2016 Cono Sur Olympiad
Let $ABC$ be a triangle inscribed on a circle with center $O$. Let $D$ and $E$ be points on the sides $AB$ and $BC$,respectively, such that $AD = DE = EC$. Let $X$ be the intersection of the angle bisectors of $\angle ADE$ and $\angle DEC$.

If $X \neq O$, show that, the lines $OX$ and $DE$ are perpendicular.
2016 Cono Sur Olympiad
We say that three different integers are friendly if one of them divides the product of the other two. Let $n$ be a positive integer.



a) Show that, between $n^2$ and $n^2+n$, exclusive, does not exist any triplet of friendly numbers.



b) Determine if for each $n$ exists a triplet of friendly numbers between $n^2$ and $n^2+n+3\sqrt{n}$ , exclusive.
2015 Cono Sur Olympiad
Show that, for any integer $n$, the number $n^3 - 9n + 27$ is not divisible by $81$.
2015 Cono Sur Olympiad
$3n$ lines are drawn on the plane ($n > 1$), such that no two of them are parallel and no three of them are concurrent. Prove that, if $2n$ of the lines are coloured red and the other $n$ lines blue, there are at least two regions of the plane such that all of their borders are red.



Note: for each region, all of its borders are contained in the original set of lines, and no line passes through the region.
2015 Cono Sur Olympiad
Given a acute triangle $PA_1B_1$ is inscribed in the circle $\Gamma$ with radius $1$. for all  integers $n \ge 1$ are defined:

$C_n$ the foot of the perpendicular from $P$ to $A_nB_n$

$O_n$ is the center of $\odot (PA_nB_n)$

$A_{n+1}$ is the foot of the perpendicular from $C_n$ to $PA_n$

$B_{n+1} \equiv PB_n \cap O_nA_{n+1}$



If $PC_1 =\sqrt{2}$, find the length of $PO_{2015}$



SourceCono Sur Olympiad - 2015 - Day 1 - Problem 3
2015 Cono Sur Olympiad
Let $ABCD$ be a convex quadrilateral such that $\angle{BAD} = 90^{\circ}$ and its diagonals $AC$ and $BD$ are perpendicular. Let $M$ be the midpoint of side $CD$, and $E$ be the intersection of $BM$ and $AC$. Let $F$ be a point on side $AD$ such that $BM$ and $EF$ are perpendicular. If $CE = AF\sqrt{2}$ and $FD = CE\sqrt{2}$, show that $ABCD$ is a square.
2015 Cono Sur Olympiad
Determine if there exists an infinite sequence of not necessarily distinct positive integers $a_1, a_2, a_3,\ldots$ such that for any positive integers $m$ and $n$ where $1 \leq m < n$, the number $a_{m+1} + a_{m+2} + \ldots + a_{n}$ is not divisible by $a_1 + a_2 + \ldots + a_m$.
2015 Cono Sur Olympiad
Let $S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}$. Two subsets $A$ and $B$ of $S$ are said to be friends if the following conditions are true:



They do not share any elements.

They both have the same number of elements.

The product of all elements from $A$ equals the product of all elements from $B$.



Prove that there are two subsets of $S$ that are friends such that each one of them contains at least $738$ elements.
2014 Cono Sur Olympiad
Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$.



Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$
2014 Cono Sur Olympiad
A pair of positive integers $(a,b)$ is called charrua if there is a positive integer $c$ such that $a+b+c$ and $a\times b\times c$ are both square numbers; if there is no such number $c$, then the pair is called non-charrua.



a) Prove that there are infinite non-charrua pairs.

b) Prove that there are infinite positive integers $n$ such that $(2,n)$ is charrua.
2014 Cono Sur Olympiad
Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$.



Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other.



Note: the pieces can be rotated and flipped over.
2014 Cono Sur Olympiad
Show that the number $n^{2} - 2^{2014}\times 2014n + 4^{2013} (2014^{2}-1)$ is not prime, where $n$ is a positive integer.
2014 Cono Sur Olympiad
Let $ABCD$ be an inscribed quadrilateral in a circumference with center $O$ such that it lies inside $ABCD$ and $\angle{BAC} = \angle{ODA}$. Let $E$ be the intersection of $AC$ with $BD$. Lines $r$ and $s$ are drawn through $E$ such that $r$ is perpendicular to $BC$, and $s$ is perpendicular to $AD$. Let $P$ be the intersection of $r$ with $AD$, and $M$ the intersection of $s$ with $BC$. Let $N$ be the midpoint of $EO$.



Prove that $M$, $N$, and $P$ lie on a line.
2014 Cono Sur Olympiad
Let $F$ be a family of subsets of $S = \left \{ 1,2,...,n \right \}$ ($n \geq 2$). A valid play is to choose two disjoint sets $A$ and $B$ from $F$ and add $A \cup B$ to $F$ (without removing $A$ and $B$).



Initially, $F$ has all the subsets that contain only one element of $S$. The goal is to have all subsets of $n - 1$ elements of $S$ in $F$ using valid plays.



Determine the lowest number of plays required in order to achieve the goal.
2013 Cono Sur Olympiad
Four distinct points are marked in a line. For each point, the sum of the distances from said point to the other three is calculated; getting in total 4 numbers.



Decide whether these 4 numbers can be, in some order:

a) $29,29,35,37$

b) $28,29,35,37$

c) $28,34,34,37$
2013 Cono Sur Olympiad
In a triangle $ABC$, let $M$ be the midpoint of $BC$ and $I$ the incenter of $ABC$. If $IM$ = $IA$, find the least possible measure of $\angle{AIM}$.
2013 Cono Sur Olympiad
Nocycleland is a country with $500$ cities and $2013$ two-way roads, each one of them connecting two cities. A city $A$ neighbors $B$ if there is one road that connects them, and a city $A$ quasi-neighbors $B$ if there is a city $C$ such that $A$ neighbors $C$ and $C$ neighbors $B$.

It is known that in Nocycleland, there are no pair of cities connected directly with more than one road, and there are no four cities $A$, $B$, $C$ and $D$ such that $A$ neighbors $B$, $B$ neighbors $C$, $C$  neighbors $D$, and $D$ neighbors $A$.

Show that there is at least one city that quasi-neighbors at least $57$ other cities.
2013 Cono Sur Olympiad
Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is given one of $k$ available colors, with the only condition that if the union of two different subsets $A$ and $B$ is $M$, then $A$ and $B$ are given different colors. What is the least possible value of $k$?
2013 Cono Sur Olympiad
Let $d(k)$ be the number of positive divisors of integer $k$. A number $n$ is called balanced if $d(n-1) \leq d(n) \leq d(n+1)$ or $d(n-1) \geq d(n) \geq d(n+1)$.

Show that there are infinitely many balanced numbers.
2013 Cono Sur Olympiad
Let $ABCD$ be a convex quadrilateral. Let $n \geq 2$ be a whole number. Prove that there are $n$ triangles with the same area that satisfy all of the following properties:



a) Their interiors are disjoint, that is, the triangles do not overlap.

b) Each triangle lies either in $ABCD$ or inside of it.

c) The sum of the areas of all of these triangles is at least $\frac{4n}{4n+1}$ the area of $ABCD$.
2012 Cono Sur Olympiad
1.	Around a circumference are written $2012$ number, each of with is equal to $1$ or $-1$. If there are not $10$ consecutive numbers that sum $0$, find all possible values of the sum of the $2012$ numbers.
2012 Cono Sur Olympiad
2.	In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.