Found problems: 82
2019 Cono Sur Olympiad, 5
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?
2008 Cono Sur Olympiad, 1
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, 2
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 $UVW$.
2010 Cono Sur Olympiad, 5
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 $ABC$.
2004 Cono Sur Olympiad, 1
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.
2009 Cono Sur Olympiad, 5
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 $K$.
Cono Sur Shortlist - geometry, 2020.G1.4
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 $DE$.
1996 Cono Sur Olympiad, 6
Find all integers $n \leq 3$ such that there is a set $S_n$ formed by $n$ points of the plane that satisfy the following two conditions:
Any three points are not collinear.
No point is found inside the circle whose diameter has ends at any two points of $S_n$.
[b]NOTE: [/b] The points on the circumference are not considered to be inside the circle.
2002 Cono Sur Olympiad, 2
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.
2020 Cono Sur Olympiad, 5
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.
2004 Cono Sur Olympiad, 4
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…
2018 Cono Sur Olympiad, 1
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.
1996 Cono Sur Olympiad, 2
Consider a sequence of real numbers defined by:
$a_{n + 1} = a_n + \frac{1}{a_n}$ for $n = 0, 1, 2, ...$
Prove that, for any positive real number $a_0$, is true that $a_{1996}$ is greater than $63$.
2018 Cono Sur Olympiad, 3
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 $m$.
2020 Cono Sur Olympiad, 1
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.
2023 Cono Sur Olympiad, 2
Grid the plane forming an infinite board. In each cell of this board, there is a lamp, initially turned off. A permitted operation consists of selecting a square of \(3\times 3\), \(4\times 4\), or \(5\times 5\) cells and changing the state of all lamps in that square (those that are off become on, and those that are on become off).
(a) Prove that for any finite set of lamps, it is possible to achieve, through a finite sequence of permitted operations, that those are the only lamps turned on on the board.
(b) Prove that if in a sequence of permitted operations only two out of the three square sizes are used, then it is impossible to achieve that at the end the only lamps turned on on the board are those in a \(2\times 2\) square.
2008 Cono Sur Olympiad, 4
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?
1997 Cono Sur Olympiad, 4
Consider a board with $n$ rows and $4$ columns. In the first line are written $4$ zeros (one in each house). Next, each line is then obtained from the previous line by performing the following operation: one of the houses, (that you can choose), is maintained as in the previous line; the other three are changed:
* if in the previous line there was a $0$, then in the down square $1$ is placed;
* if in the previous line there was a $1$, then in the down square $2$ is placed;
* if in the previous line there was a $2$, then in the down square $0$ is placed;
Build the largest possible board with all its distinct lines and demonstrate that it is impossible to build a larger board.
2010 Cono Sur Olympiad, 4
Pablo and Silvia play on a $2010 \times 2010$ board. To start the game, Pablo writes an integer in every cell. After he is done, Silvia repeats the following operation as many times as she wants: she chooses three cells that form an $L$, like in the figure below, and adds $1$ to each of the numbers in these three cells. Silvia wins if, after doing the operation many times, all of the numbers in the board are multiples of $10$.
Prove that Silvia can always win.
$\begin{array}{|c|c} \cline{1-1} \; & \; \\ \hline \; & \multicolumn{1}{|c|}{\;} \\ \hline \end{array} \qquad \begin{array}{c|c|} \cline{2-2} \; & \; \\ \hline \multicolumn{1}{|c|}{\;} & \; \\ \hline \end{array} \qquad \begin{array}{|c|c} \hline \; & \multicolumn{1}{|c|}{\;} \\ \hline \multicolumn{1}{|c|}{\;} & \; \\ \cline{1-1} \end{array} \qquad \begin{array}{c|c|} \hline \multicolumn{1}{|c|}{\;} & \; \\ \hline \; & \multicolumn{1}{|c|}{\;} \\ \cline{2-2} \end{array}$
2009 Cono Sur Olympiad, 4
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.
[code]
._
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|_| _ _ _
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[/code]
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.
2002 Cono Sur Olympiad, 4
Let $ABCD$ be a convex quadrilateral such that your diagonals $AC$ and $BD$ are perpendiculars. Let $P$ be the intersection of $AC$ and $BD$, let $M$ a midpoint of $AB$. Prove that the quadrilateral $ABCD$ is cyclic, if and only if, the lines $PM$ and $DC$ are perpendiculars.
2010 Cono Sur Olympiad, 6
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, 6
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.
[b]Note:[/b] The rectangles can overlap and they can protrude over the sides of the square.
2017 Cono Sur Olympiad, 1
A positive integer $n$ is called [i]guayaquilean[/i] 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, 3
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]