Olympiad problems

2002 Cono Sur Olympiad, 1

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?

2016 Cono Sur Olympiad, 6

Tags: number theory , cono sur

We say that three different integers are [i]friendly[/i] 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.

2010 Cono Sur Olympiad, 2

Tags: combinatorics , cono sur

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.

2003 Cono Sur Olympiad, 5

Tags: combinatorics , cono sur

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.

1995 Cono Sur Olympiad, 3

Tags: geometry , rectangle , quadratics , algebra , cono sur

Let $ABCD$ be a rectangle with: $AB=a$, $BC=b$. Inside the rectangle we have to exteriorly tangents circles such that one is tangent to the sides $AB$ and $AD$,the other is tangent to the sides $CB$ and $CD$. 1. Find the distance between the centers of the circles(using $a$ and $b$). 2. When the radiums of both circles change the tangency point between both of them changes, and describes a locus. Find that locus.

2016 Cono Sur Olympiad, 1

Tags: number theory , Perfect Squares , cono sur

Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an [i]special[/i] 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. [b]Note:[/b] If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.

2008 Cono Sur Olympiad, 6

Tags: number theory , cono sur

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.