Found problems: 82
1997 Cono Sur Olympiad, 5
Let $n$ be a natural number $n>3$.
Show that in the multiples of $9$ less than $10^n$, exist more numbers with the sum of your digits equal to $9(n - 2)$ than numbers with the sum of your digits equal to $9(n - 1)$.
2016 Cono Sur Olympiad, 4
Let $S(n)$ be the sum of the digits of the positive integer $n$. Find all $n$ such that $S(n)(S(n)-1)=n-1$.
2019 Cono Sur Olympiad, 4
Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.
2003 Cono Sur Olympiad, 6
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 $n^n$.
2016 Cono Sur Olympiad, 3
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.
2004 Cono Sur Olympiad, 3
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 $C_n\geq\frac{4}{9}(10^{n}-1)$.
Cono Sur Shortlist - geometry, 2003.G7.3
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 $BR\leq{RS}$.
2002 Cono Sur Olympiad, 6
Let $n$ a positive integer, $n > 1$. The number $n$ is wonderful if the number is divisible by sum of the your prime factors.
For example; $90$ is wondeful, because $90 = 2 \times 3^2\times 5$ and $2 + 3 + 5 = 10, 10$ divides $90$.
Show that, exist a number "wonderful" with at least $10^{2002}$ distinct prime numbers.
2003 Cono Sur Olympiad, 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 $ABC$.
2003 Cono Sur Olympiad, 2
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 $2^{2003}.$
1997 Cono Sur Olympiad, 3
Show that, exist infinite triples $(a, b, c)$ where $a, b, c$ are natural numbers, such that:
$2a^2 + 3b^2 - 5c^2 = 1997$
2008 Cono Sur Olympiad, 3
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.
Cono Sur Shortlist - geometry, 2020.G3.3
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 $\omega$.
2018 Cono Sur Olympiad, 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 $IH=BI+IC$.
2004 Cono Sur Olympiad, 6
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 $p$?
2017 Cono Sur Olympiad, 5
Let $a$, $b$ and $c$ positive integers. Three sequences are defined as follows:
[list]
[*] $a_1=a$, $b_1=b$, $c_1=c$[/*]
[*] $a_{n+1}=\lfloor{\sqrt{a_nb_n}}\rfloor$, $\:b_{n+1}=\lfloor{\sqrt{b_nc_n}}\rfloor$, $\:c_{n+1}=\lfloor{\sqrt{c_na_n}}\rfloor$ for $n \ge 1$[/*]
[/list]
[list = a]
[*]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 $b+c=2a-1$.[/*]
[/list]
2016 Cono Sur Olympiad, 2
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 $s_1 + s_2 + \cdots + s_{200}$.
2009 Cono Sur Olympiad, 2
A [i]hook[/i] consists of three segments of longitude $1$ forming two right angles as demonstrated in the figure.
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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.
2016 Cono Sur Olympiad, 5
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.
2004 Cono Sur Olympiad, 2
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.
2023 Cono Sur Olympiad, 3
In a half-plane, bounded by a line \(r\), equilateral triangles \(S_1, S_2, \ldots, S_n\) are placed, each with one side parallel to \(r\), and their opposite vertex is the point of the triangle farthest from \(r\).
For each triangle \(S_i\), let \(T_i\) be its medial triangle. Let \(S\) be the region covered by triangles \(S_1, S_2, \ldots, S_n\), and let \(T\) be the region covered by triangles \(T_1, T_2, \ldots, T_n\).
Prove that \[\text{area}(S) \leq 4 \cdot \text{area}(T).\]
2023 Cono Sur Olympiad, 1
A list of \(n\) positive integers \(a_1, a_2,a_3,\ldots,a_n\) is said to be [i]good[/i] if it checks simultaneously:
\(\bullet a_1<a_2<a_3<\cdots<a_n,\)
\(\bullet a_1+a_2^2+a_3^3+\cdots+a_n^n\le 2023.\)
For each \(n\ge 1\), determine how many [i]good[/i] lists of \(n\) numbers exist.
2018 Cono Sur Olympiad, 6
A sequence $a_1, a_2,\dots, a_n$ of positive integers is [i]alagoana[/i], 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) [i]alagoana[/i].
Cono Sur Shortlist - geometry, 2009.G1.6
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.
2020 Cono Sur Olympiad, 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 $DE$.