Navvye/Olympiad-Problems-CSV huggingface dataset:
2002 Cono Sur Olympiad
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.
2002 Cono Sur Olympiad
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.
2002 Cono Sur Olympiad
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 $1 \le k \le \frac{n}{2}$
2002 Cono Sur Olympiad
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.
2001 Cono Sur Olympiad
Each entry in a $2000\times 2000$ array is $0$, $1$, or $-1$. Show that it's possible for all $4000$ row sums and column sums to be distinct.
2001 Cono Sur Olympiad
A sequence $a_1,a_2,\ldots$ of positive integers satisfies the following properties.



$a_1 = 1$

$a_{3n+1} = 2a_n + 1$

$a_{n+1}\ge a_n$

$a_{2001} = 200$



Find the value of $a_{1000}$.



Note. In the original statement of the problem, there was an extra condition:



every positive integer appears at least once in the sequence.



However, with this extra condition, there is no solution, i.e., no such sequence exists. (Try to prove it.) The problem as written above does have a solution.
2001 Cono Sur Olympiad
Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Show that one can choose a vertex from each triangle so that the three chosen points determine a triangle each of whose angles is at most $90^\circ$.
2001 Cono Sur Olympiad
A polygon of area $S$ is contained inside a square of side length $a$. Show that there are two points of the polygon that are a distance of at least $S/a$ apart.
2001 Cono Sur Olympiad
Find all positive integers $m$ for which $2001\cdot S (m) = m$ where $S(m)$ denotes the sum of the digits of $m$.
2001 Cono Sur Olympiad
A function $g$ defined for all positive integers $n$ satisfies



$g(1) = 1$;

for all $n\ge 1$, either $g(n+1)=g(n)+1$ or $g(n+1)=g(n)-1$;

for all $n\ge 1$, $g(3n) = g(n)$; and

$g(k)=2001$ for some positive integer $k$.



Find, with proof, the smallest possible value of $k$.
2000 Cono Sur Olympiad
Call a positive integer descending if, reading left to right, each of its digits (other than its leftmost) is less than or equal to the previous digit. For example, $4221$ and $751$ are descending while $476$ and $455$ are not descending. Determine whether there exists a positive integer $n$ for which $16^n$ is descending.
2000 Cono Sur Olympiad
The numbers $1,2,\ldots,64$ are written in the squares of an $8\times 8$ chessboard, one number to each square. Then $2\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. Find, with proof, the maximum number of tiles that can be placed on the chessboard, and give an example of a distribution of the numbers $1,2,\ldots,64$ into the squares of the chessboard that admits this maximum number of tiles.
2000 Cono Sur Olympiad
Inside a $2\times 2$ square, lines parallel to a side of the square (both horizontal and vertical) are drawn thereby dividing the square into rectangles. The rectangles are alternately colored black and white like a chessboard. Prove that if the total area of the white rectangles is equal to the total area of the black rectangles, then one can cut out the black rectangles and reassemble them into a $1\times 2$ rectangle.
2000 Cono Sur Olympiad
In square $ABCD$ (labeled clockwise), let $P$ be any point on $BC$ and construct square $APRS$ (labeled clockwise). Prove that line $CR$ is tangent to the circumcircle of triangle $ABC$.
2000 Cono Sur Olympiad
Consider the following transformation of the Cartesian plane: choose a lattice point and rotate the plane $90^\circ$ counterclockwise about that lattice point. Is it possible, through a sequence of such transformations, to take the triangle with vertices $(0,0)$, $(1,0)$ and $(0,1)$ to the triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$?
2000 Cono Sur Olympiad
Is there a positive integer divisible by the product of its digits such that this product is greater than $10^{2000}$?
1999 Cono Sur Olympiad
Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible.
1999 Cono Sur Olympiad
Let $ABC$ be a triangle right in $A$. Construct a point $P$ on the hypotenuse $BC$ such that if $Q$ is the foor of the perpendicular drawn from $P$ to side $AC$, then the area of the square of side $PQ$ is equal to the area of the rectangle of sides $PB$ and $PC$. Show construction steps.
1999 Cono Sur Olympiad
There are $1999$ balls in a row, some are red and some are blue (it could be all red or all blue). Under every ball we write a number equal to the sum of the amount of red balls in the right of this ball plus the sum of the amount of the blue balls that are in the left of this ball.

In the sequence of numbers that we get with this balls we have exactly three numbers that appears an odd number of times, which numbers could these three be?
1999 Cono Sur Olympiad
Let $A$ be a six-digit number, three of which are colored and equal to $1, 2$, and $4$.

Prove that it is always possible to obtain a number that is a multiple of $7$, by performing only one of the following operations: either delete the three colored figures, or write all the numbers of $A$ in some order.
1999 Cono Sur Olympiad
Give a square of side $1$. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to $3/4$.
1999 Cono Sur Olympiad
An ant walks across the floor of a circular path of radius $r$ and moves in a straight line, but sometimes stops. Each time it stops, before resuming the march, it rotates  $60^o$ alternating the direction (if the last time it turned  $60^o$ to its right, the next one does it $60^o$ to its left, and vice versa). Find the maximum possible length of the path the ant goes through. Prove that the length found is, in fact, as long as possible.



Figure: turn $60^o$ to the right .
1998 Cono Sur Olympiad
We have $98$ cards, in each one we will write one of the numbers: $1, 2, 3, 4,...., 97, 98$.

We can order the $98$ cards, in a sequence such that two consecutive numbers $X$ and $Y$ and the number $X - Y$ is greater than $48$, determine how and how many ways we can make this sequence!!
1998 Cono Sur Olympiad
Let $H$ be the orthocenter of the triangle $ABC$, $M$ is the midpoint of the segment $BC$. Let $X$ be the point of the intersection of the line $HM$ with arc $BC$(without $A$) of the circumcircle of $ABC$, let $Y$ be the point of intersection of the line $BH$ with the circle, show that $XY = BC$.
1998 Cono Sur Olympiad
Prove that, least $30$% of the natural numbers $n$ between $1$ and $1000000$ the first digit of $2^n$ is $1$.
1998 Cono Sur Olympiad
Find all functions $R-->R$ such that:

$f(x^2) - f(y^2) + 2x + 1 = f(x + y)f(x - y)$
1998 Cono Sur Olympiad
In Terra Brasilis there are $n$ houses where $n$ goblins live, each in a house. There are one-way routes such that:

- each route joins two houses,

- in each house exactly one route begins,

- in each house exactly one route ends.

If a route goes from house $A$ to house $B$, then we will say that house $B$ is next to house $A$. This relationship is not symmetric, that is: in this situation, not necessarily house $A$ is next to house $B$.

Every day, from day $1$, each goblin leaves the house where he is and arrives at the next house. A legend of Terra Brasilis says that when all the goblins return to the original position, the world will end.

a) Show that the world will end.

b) If $n = 98$, show that it is possible for elves to build and guide the routes so that the world does not end before $300,000$ years.
1998 Cono Sur Olympiad
The mayor of a city wishes to establish a transport system with at least one bus line, in which:

- each line passes exactly three stops,

- every two different lines have exactly one stop in common,

- for each two different bus stops there is exactly one line that passes through both.

Determine the number of bus stops in the city.
1997 Cono Sur Olympiad
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.
1997 Cono Sur Olympiad
Show that, exist infinite triples $(a, b, c)$ where $a, b, c$ are natural numbers, such that:

$2a^2 + 3b^2 - 5c^2 = 1997$
1997 Cono Sur Olympiad
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.
1997 Cono Sur Olympiad
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 $9(n - 1)$.
1997 Cono Sur Olympiad
Let $ABC$ be a acute-angle triangle and $X$ be point in the plane of this triangle.

Let $M,N,P$ be the orthogonal projections of $X$ in the lines that contains the altitudes of this triangle

Determine the positions of the point $X$ such that the triangle $MNP$ is congruent to $ABC$
1996 Cono Sur Olympiad
In the following figure, the largest square is divided into two squares and three rectangles, as shown:



The area of each smaller square is equal to $a$ and the area of each small rectangle is equal to $b$. If $a+b=24$ and the root square of $a$ is a natural number, find all possible values for the area of the largest square.
1996 Cono Sur Olympiad
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 $63$.
1996 Cono Sur Olympiad
A shop sells bottles with this capacity: $1L, 2L, 3L,..., 1996L$, the prices of bottles satifies this $2$ conditions:

$1$. Two bottles have the same price, if and only if, your capacities satifies

$m - n = 1000$

$2$. The price of bottle $m$($1001>m>0$) is $1996 - m$ dollars.

Find all pair(s) $m$ and $n$ such that:

a) $m + n = 1000$

b) the cost is smallest possible!!!

c) with the pair, the shop can measure $k$ liters, with $0<k<1996$(for all $k$ integer)

Note: The operations to measure are:

i) To fill or empty any one of two bottles

ii)Pass water of a bottle for other bottle

We can measure $k$ liters when the capacity of one bottle plus the capacity of other bottle is equal to $k$
1996 Cono Sur Olympiad
The sequence $0, 1, 1, 1, 1, 1,....,1$ where have $1$ number zero and $1995$ numbers one.

If we choose two or more numbers in this sequence(but not the all $1996$ numbers) and substitute one number by arithmetic mean of the numbers selected, we obtain a new sequence with $1996$ numbers!!!

Show that, we can repeat this operation until we have all $1996$ numbers are equal

Note: It's not necessary to choose the same quantity of numbers in each operation!!!
1996 Cono Sur Olympiad
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
1996 Cono Sur Olympiad
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 $S_n$.



NOTE:  The points on the circumference are not considered to be inside the circle.
1995 Cono Sur Olympiad
Find a number with $3$ digits, knowing that the sum of its digits is $9$, their product is $24$ and also the number read from right to left is $\frac{27}{38}$ of the original.
1995 Cono Sur Olympiad
There are ten points marked on a circumference, numbered from $1$ to $10$ and join all points with segments. I color the segments, with red someones and others with blue. Without changing the colors of the segments, renumber all the points from the $1$ to the $10$. Will be possible to color the segments and to renumber the points so that those numbers that were jointed with red are jointed now with blue and the numbers that were jointed with blue they are jointed now with red?
1995 Cono Sur Olympiad
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.
1995 Cono Sur Olympiad
We write the digits of $1995$ in the following way:



$199511999955111999999555......$



1. Determine how many digits we have to write such that the sum of the written digits is $2880$.

2.Which digit is in position number $1995$?
1995 Cono Sur Olympiad
The semicircle with centre $O$ and the diameter $AC$ is divided in two arcs $AB$ and $BC$ with ratio $1: 3$. $M$ is the midpoint of the radium $OC$. Let $T$ be the point of arc $BC$ such that the area of the cuadrylateral $OBTM$ is maximum. Find such area in fuction of the radium.
1995 Cono Sur Olympiad
Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function).



1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$.



2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$
1994 Cono Sur Olympiad
The positive integrer number $n$ has $1994$ digits. $14$ of its digits are $0$'s and the number of times that the other digits: $1, 2, 3, 4, 5, 6, 7, 8, 9$ appear are in proportion $1: 2: 3: 4: 5: 6: 7: 8: 9$, respectively. Prove that $n$ is not a perfect square.
1994 Cono Sur Olympiad
Consider a circle $C$ with diameter $AB=1$. A point $P_0$ is chosen on $C$, $P_0 \ne A$, and starting in $P_0$ a sequence of points $P_1, P_2, \dots, P_n, \dots$ is constructed on $C$, in the following way: $Q_n$ is the symmetrical point of $A$ with respect of $P_n$ and the straight line that joins $B$ and $Q_n$ cuts $C$ at $B$ and $P_{n+1}$ (not necessary different). Prove that it is possible to choose $P_0$ such that:



i $\angle {P_0AB} < 1$.



ii In the sequence that starts with $P_0$ there are $2$ points, $P_k$ and $P_j$, such that $\triangle {AP_kP_j}$ is equilateral.
1994 Cono Sur Olympiad
Let $p$ be a positive real number given. Find the minimun vale of $x^3+y^3$, knowing that $x$ and $y$ are positive real numbers such that $xy(x+y)=p$.
1994 Cono Sur Olympiad
Pedro and Cecilia play the following game: Pedro chooses a positive integer number $a$ and Cecilia wins if she finds a positive integrer number $b$, prime with $a$, such that, in the factorization of $a^3+b^3$ will appear three different prime numbers. Prove that Cecilia can always win.
1994 Cono Sur Olympiad
Solve the following equation in integers with gcd (x, y) = 1



$x^2 + y^2 = 2 z^2$
1994 Cono Sur Olympiad
Consider a $\triangle {ABC}$, with $AC \perp BC$. Consider a point $D$ on $AB$ such that $CD=k$, and the radius of the inscribe circles on $\triangle {ADC}$ and $\triangle {CDB}$ are equals. Prove that the area of $\triangle {ABC}$ is equal to $k^2$.
1993 Cono Sur Olympiad
On a table there is a pile with $ T$ tokens which incrementally shall be converted into piles with three tokens each. Each step is constituted of selecting one pile removing one of its tokens. And then the remaining pile is separated into two piles. Is there a sequence of steps that can accomplish this process?



a.) $ T = 1000$ (Cono Sur)

b.) $ T = 2001$ (BWM)
1993 Cono Sur Olympiad
Consider a circle with centre $O$, and $3$ points on it, $A,B$ and $C$, such that $\angle {AOB}< \angle {BOC}$. Let $D$ be the midpoint on the arc $AC$ that contains the point $B$. Consider a point $K$ on $BC$ such that $DK \perp BC$. Prove that $AB+BK=KC$.
1993 Cono Sur Olympiad
Find the number of elements that a set $B$ can have, contained in $(1, 2, ... , n)$, according to the following property: For any elements $a$ and $b$ on $B$ ($a \ne b$), $(a-b) \not| (a+b)$.
1993 Cono Sur Olympiad
On a chess board ($8*8$) there are written the numbers $1$ to $64$: on the first line, from left to right, there are the numbers $1, 2, 3, ... , 8$; on the second line, from left to right, there are the numbers $9, 10, 11, ... , 16$;etc. The $\"+\"$ and $\"-\"$ signs are put to each number such that, in each line and in each column, there are $4$ $\"+\"$ signs and $4$ $\"-\"$ signs. Then, the $64$ numbers are added. Find all the possible values of this sum.
1993 Cono Sur Olympiad
Prove that there exists a succession $a_1, a_2, ... , a_k, ...$, where each $a_i$ is a digit ($a_i \in (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)$ ) and $a_0=6$, such that, for each positive integrer $n$, the number $x_n=a_0+10a_1+100a_2+...+10^{n-1}a_{n-1}$ verify that $x_n^2-x_n$ is divisible by $10^n$.
1993 Cono Sur Olympiad
Prove that, given a positive integrer $n$, there exists a positive integrer $k_n$ with the following property: Given any $k_n$ points in the space, $4$ by $4$ non-coplanar, and associated integrer numbers between $1$ and $n$ to each sharp edge that meets $2$ of this points, there's necessairly a triangle determined by $3$ of them, whose sharp edges have associated the same number.
1992 Cono Sur Olympiad
Find a positive integrer number $n$ such that, if yor put a number $2$ on the left and a number $1$ on the right, the new number is equal to $33n$.
1992 Cono Sur Olympiad
Let $P$ be a point outside the circle $C$. Find two points $Q$ and $R$ on the circle, such that $P,Q$ and $R$ are collinear and $Q$ is the midpopint of the segmenet $PR$. (Discuss the number of solutions).
1992 Cono Sur Olympiad
Consider the set $S$ of $100$ numbers: $1; \frac{1}{2}; \frac{1}{3}; ... ; \frac{1}{100}$.

Any two numbers, $a$ and $b$, are eliminated in $S$, and the number $a+b+ab$ is added. Now, there are $99$ numbers on $S$.

After doing this operation $99$ times, there's only $1$ number on $S$. What values can this number take?
1992 Cono Sur Olympiad
Prove that there aren't any positive integrer numbers $x,y,z$ such that $x^2+y^2=3z^2$.
1992 Cono Sur Olympiad
In a $\triangle {ABC}$, consider a point $E$ in $BC$ such that $AE \perp BC$. Prove that $AE=\frac{bc}{2r}$, where $r$ is the radio of the circle circumscripte, $b=AC$ and $c=AB$.
1992 Cono Sur Olympiad
Consider a $m*n$ board. On each box there's a non-negative integrer number assigned. An operation consists on choosing any two boxes with $1$ side in common, and add to this $2$ numbers the same integrer number (it can be negative), so that both results are non-negatives.

What conditions must be satisfied initially on the assignment of the boxes, in order to have, after some operations, the number $0$ on every box?.
1991 Cono Sur Olympiad
Let $A, B$ and $C$ be three non-collinear points and $E$ ($\ne B$) an arbitrary point not in the straight line $AC$. Construct the parallelograms $ABCD$ and $AECF$. Prove that $BE \parallel DF$.
1991 Cono Sur Olympiad
Two people, $A$ and $B$, play the following game: $A$ start choosing a positive integrer number and then, each player in it's turn, say a number due to the following rule:



If the last number said was odd, the player add $7$ to this number;

If the last number said was even, the player divide it by $2$.



The winner is the player that repeats the first number said. Find all numbers that $A$ can choose in order to win. Justify your answer.
1991 Cono Sur Olympiad
It is known that the number of real solutions of the following system if finite. Prove that this system has an even number of solutions:



$(y^2+6)(x-1)=y(x^2+1)$



$(x^2+6)(y-1)=x(y^2+1)$
1991 Cono Sur Olympiad
A game consists in $9$ coins (blacks or whites) arrenged in the following position (see picture 1). If you choose $1$ coin on the border of the square, this coin and it's neighbours change their color. If you choose the coin at the centre, it doesn't change it's color, but the other $8$ coins do. Here is an example of $9$ white coins, and the changes of their colors, choosing the coin said: (see picture 2).

Is it possible, starting with $9$ white coins, to have $9$ black coins?.
1991 Cono Sur Olympiad
Given a square $ABCD$ with side $1$, and a square inside $ABCD$ with side $x$, find (in terms of $x$) the radio $r$ of the circle tangent to two sides of $ABCD$ and touches the square with side $x$. (See picture).
1991 Cono Sur Olympiad
Given a positive integrer number $n$ ($n\ne 0$), let $f(n)$ be the average of all the positive divisors of $n$. For example, $f(3)=\frac{1+3}{2}=2$, and $f(12)=\frac{1+2+3+4+6+12}{6}=\frac{14}{3}$.



a Prove that $\frac{n+1}{2} \ge f(n)\ge \sqrt{n}$.



b Find all $n$ such that $f(n)=\frac{91}{9}$.
1989 Cono Sur Olympiad
Two isosceles triangles with sidelengths $x,x,a$ and $x,x,b$ ($a \neq b$) have equal areas. Find $x$.
1989 Cono Sur Olympiad
Find the sum$$1+11+111+\cdots+\underbrace{111\ldots111}_{n\text{ digits}}.$$
1989 Cono Sur Olympiad
A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$. Let $f$ be a function such that:



$f(n)=0$, if n is perfect

$f(n)=0$, if the last digit of n is 4

$f(a.b)=f(a)+f(b)$



Find $f(1998)$
1989 Cono Sur Olympiad
Let $n$ be square with 4 digits, such that all its digits are less than 6. If we add 1 to each digit the resulting number is another square. Find $n$
1989 Cono Sur Olympiad
Let $ABCD$ be a square with diagonals $AC$ and $BD$, and $P$ a point in one of the sides of the square. Show that the sum of the distances from P to the diagonals is constant.
1989 Cono Sur Olympiad
Show that reducing the dimensions of a cuboid we can't get another cuboid with half the volume and half the surface.