Found problems: 82
2010 Cono Sur Olympiad, 1
Pedro must choose two irreducible fractions, each with a positive numerator and denominator such that:
[list]
[*]The sum of the fractions is equal to $2$.
[*]The sum of the numerators of the fractions is equal to $1000$.
[/list]
In how many ways can Pedro do this?
Cono Sur Shortlist - geometry, 2009.G5.3
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.
1996 Cono Sur Olympiad, 5
We want to cover totally a square(side is equal to $k$ integer and $k>1$) with this rectangles:
$1$ rectangle ($1\times 1$), $2$ rectangles ($2\times 1$), $4$ rectangles ($3\times 1$),...., $2^n$ rectangles ($n + 1 \times 1$), such that the rectangles can't overlap and don't exceed the limits of square.
Find all $k$, such that this is possible and for each $k$ found you have to draw a solution
Cono Sur Shortlist - geometry, 2003.G5.4
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 $ABC$.
1997 Cono Sur Olympiad, 2
Let $C$ be a circunference, $O$ is your circumcenter, $AB$ is your diameter and $R$ is any point in $C$ ($R$ is different of $A$ and $B$)
Let $P$ be the foot of perpendicular by $O$ to $AR$, in the line $OP$ we match a point $Q$, where $QP$ is $\frac{OP}{2}$ and the point $Q$ isn't in the segment $OP$.
In $Q$, we will do a parallel line to $AB$ that cut the line $AR$ in $T$.
Denote $H$ the point of intersections of the line $AQ$ and $OT$.
Show that $H$, $B$ and $R$ are collinears.
Cono Sur Shortlist - geometry, 2018.G1.1
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.
2017 Cono Sur Olympiad, 6
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 $a_1,a_2,a_3,\ldots$
2009 Cono Sur Olympiad, 1
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, 3
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.
2010 Cono Sur Olympiad, 3
Let us define [i]cutting[/i] 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 [i]cut[/i] 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 $2/3$.
2018 Cono Sur Olympiad, 2
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.
2002 Cono Sur Olympiad, 5
Consider the set $A = \{1, 2, ..., n\}$. For each integer $k$, let $r_k$ be the largest quantity of different elements of $A$ that we can choose so that the difference between two numbers chosen is always different from $k$. Determine the highest value possible of $r_k$, where $1 \le k \le \frac{n}{2}$
Cono Sur Shortlist - geometry, 2018.G2.5
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, 4
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 $m$.
2011 Cono Sur Olympiad, 3
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 $P$.
2002 Cono Sur Olympiad, 3
Arnaldo and Bernardo play a Super Naval Battle. Each has a board $n \times n$. Arnaldo puts boats on his board (at least one but not known how many). Each boat occupies the $n$ houses of a line or a column and the boats they can not overlap or have a common side. Bernardo marks $m$ houses (representing shots) on your board. After Bernardo marked the houses, Arnaldo says which of them correspond to positions occupied by ships. Bernardo wins, and then discovers the positions of all Arnaldo's boats. Determine the lowest value of $m$ for which Bernardo can guarantee his victory.
2006 Cono Sur Olympiad, 6
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.
2017 Cono Sur Olympiad, 2
Let $A(XYZ)$ be the area of the triangle $XYZ$. A non-regular convex polygon $P_1 P_2 \ldots P_n$ is called [i]guayaco[/i] 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.
2004 Cono Sur Olympiad, 5
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 $1$.
2003 Cono Sur Olympiad, 1
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.
2020 Cono Sur Olympiad, 3
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 $\omega$.
2008 Cono Sur Olympiad, 5
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
$\frac{PQ}{CP}=\frac{1}{2}\frac{AB}{AC}$.
2017 Cono Sur Olympiad, 4
Let $ABC$ an acute triangle with circumcenter $O$. Points $X$ and $Y$ are chosen such that:
[list]
[*]$\angle XAB = \angle YCB = 90^\circ$[/*]
[*]$\angle ABC = \angle BXA = \angle BYC$[/*]
[*]$X$ and $C$ are in different half-planes with respect to $AB$[/*]
[*]$Y$ and $A$ are in different half-planes with respect to $BC$[/*]
[/list]
Prove that $O$ is the midpoint of $XY$.
2003 Cono Sur Olympiad, 3
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 $BR\leq{RS}$.
2019 Cono Sur Olympiad, 3
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.