Navvye/Olympiad-Problems-CSV huggingface dataset:
2011 Rioplatense Mathematical Olympiad, Level 3
A form  is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, ... , m_n$ satisying $m_1\ge ... \ge m_n$. An angle  in a form  consists of a box $v$ and of all the boxes to the right of $v$ and all the boxes above $v$. The size of a form  or angle  is the number of boxes it contains. Find the maximum number of angles  of size $11$ in a form of size $400$.



source
2011 Rioplatense Mathematical Olympiad, Level 3
Let $d(n)$ be the sum of positive integers divisors of number $n$ and $\phi(n)$ the quantity of integers in the interval $[0,n]$ such that these integers are coprime with $n$. For instance $d(6)=12$ and $\phi(7)=6$.

Determine if the set of the integers $n$ such that, $d(n)\cdot \phi (n)$ is a perfect square, is finite or infinite set.
2010 Rioplatense Mathematical Olympiad, Level 3
Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$.



(a) Is it possible to determine which of the numbers

$a$, $b$, $c$, $d$ is the smallest?

(b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?
2010 Rioplatense Mathematical Olympiad, Level 3
Acute triangle $ABP$, where $AB > BP$, has altitudes $BH$, $PQ$, and $AS$. Let $C$ denote the intersection of lines $QS$ and $AP$, and let $L$ denote the intersection of lines $HS$ and $BC$. If $HS = SL$ and $HL$ is perpendicular to $BC$, find the value of $\frac{SL}{SC}$.
2010 Rioplatense Mathematical Olympiad, Level 3
Find all the functions $f:\mathbb{N}\to\mathbb{R}$ that satisfy

$$ f(x+y)=f(x)+f(y) $$ for all $x,y\in\mathbb{N}$ satisfying $10^6-\frac{1}{10^6} < \frac{x}{y} < 10^6+\frac{1}{10^6}$.



Note: $\mathbb{N}$ denotes the set of positive integers and $\mathbb{R}$ denotes the set of real numbers.
2010 Rioplatense Mathematical Olympiad, Level 3
Let $r_2, r_3,\ldots, r_{1000}$ denote the remainders when a positive odd integer is divided by $2,3,\ldots,1000$, respectively. It is known that the remainders are pairwise distinct and one of them is $0$. Find all values of $k$ for which it is possible that $r_k = 0$.
2010 Rioplatense Mathematical Olympiad, Level 3
Find the minimum and maximum values of $ S=\frac{a}{b}+\frac{c}{d} $ where $a$, $b$, $c$, $d$ are positive integers satisfying $a + c = 20202$ and $b + d = 20200$.
2009 Rioplatense Mathematical Olympiad, Level 3
Find all pairs $(a, b)$ of real numbers with the following property:



Given any real numbers

$c$ and $d$, if both of the equations $x^2+ax+1=c$ and $x^2+bx+1=d$ have real roots, then the equation $x^2+(a+b)x+1=cd$ has real roots.
2009 Rioplatense Mathematical Olympiad, Level 3
Let $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, $I$ be nine points in space such that $ABCDE$, $ABFGH$, and $GFCDI$ are each regular pentagons with side length $1$. Determine the lengths of the sides of triangle $EHI$.
2009 Rioplatense Mathematical Olympiad, Level 3
Call a permutation of the integers $(1,2,\ldots,n)$ $d$-ordered if it does not contains a decreasing subsequence of length $d$. Prove that for every $d=2,3,\ldots,n$,  the number of $d$-ordered permutations of $(1,2,\ldots,n)$ is at most $(d-1)^{2n}$.
2009 Rioplatense Mathematical Olympiad, Level 3
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that

$$f(xy)=\max\{f(x+y),f(x) f(y)\} $$

for all real numbers $x$ and $y$.
2009 Rioplatense Mathematical Olympiad, Level 3
Find all pairs $(a,b)$ of integers with $a>1$ and $b>1$ such that $a$ divides $b+1$ and $b$ divides $a^3-1$.
2009 Rioplatense Mathematical Olympiad, Level 3
Alice and Bob play the following game. It begins with a set of $1000$ $1\times 2$ rectangles. A move consists of choosing two rectangles (a rectangle may consist of one or several $1\times 2$ rectangles combined together) that share a common side length and combining those two rectangles into one rectangle along those sides sharing that common length. The first player who cannot make a move loses. Alice moves first. Describe a winning strategy for Bob.
2008 Rioplatense Mathematical Olympiad, Level 3
In each square of a chessboard with $a$ rows and $b$ columns, a $0$ or $1$ is written satisfying the following conditions.



If a row and a column intersect in a square with a $0$, then that row and column have the same number of $0$s.

If a row and a column intersect in a square with a $1$, then that row and column have the same number of $1$s.



Find all pairs $(a,b)$ for which this is possible.
2008 Rioplatense Mathematical Olympiad, Level 3
On a line, there are $n$ closed intervals (none of which is a single point) whose union we denote by $S$. It's known that for every real number $d$, $0<d\le 1$, there are two points in $S$ that are a distance $d$ from each other.



(a) Show that the sum of the lengths of the

$n$ closed intervals is larger than $\frac{1}{n}$.

(b) Prove that, for each positive integer $n$, the $\frac{1}{n}$ in the statement of part (a) cannot be replaced with a larger number.
2008 Rioplatense Mathematical Olympiad, Level 3
Find all integers $k\ge 2$ such that for all integers $n\ge 2$, $n$ does not divide the greatest odd divisor of $k^n+1$.
2008 Rioplatense Mathematical Olympiad, Level 3
Can the positive integers be partitioned into $12$ subsets such that for each positive integer $k$, the numbers $k, 2k,\ldots,12k$ belong to different subsets?
2008 Rioplatense Mathematical Olympiad, Level 3
In triangle $ABC$, where $AB<AC$, let $X$, $Y$, $Z$ denote the points where the incircle is tangent to $BC$, $CA$, $AB$, respectively. On the circumcircle of $ABC$, let $U$ denote the midpoint of the arc $BC$ that contains the point $A$. The line $UX$ meets the circumcircle again at the point $K$. Let $T$ denote the point of intersection of $AK$ and $YZ$. Prove that $XT$ is perpendicular to $YZ$.
2008 Rioplatense Mathematical Olympiad, Level 3
Consider a collection of stones whose total weight is $65$ pounds and each of whose stones is at most $w$ pounds. Find the largest number $w$ for which any such collection of stones can be divided into two groups whose total weights differ by at most one pound.



Note: The weights of the stones are not necessarily integers.
2007 Rioplatense Mathematical Olympiad, Level 3
Determine the values of $n \in  N$ such that a square of side $n$ can be split into a square of side  $1$ and five rectangles whose side measures are $10$ distinct natural numbers   and all greater than $1$.
2007 Rioplatense Mathematical Olympiad, Level 3
Let $ABC$ be a triangle with  incenter $I$ . The circle of center $I$ which passes through $B$ intersects $AC$ at points $E$ and $F$, with $E$ and $F$ between $A $ and $C$ and different from each other. The circle circumscribed to triangle $IEF$ intersects segments $EB$ and  $FB$ at $Q$ and $R$, respectively. Line  $QR$ intersects the sides $A B$ and $B C$ at $P$ and  $S$, respectively.

If $a , b$ and $c$ are the measures of the sides $B C, CA$ and $A B$, respectively, calculate the measurements of $B P$ and $B S$.
2007 Rioplatense Mathematical Olympiad, Level 3
Let $p > 3$ be a prime number and $ x$ an integer, denote by $r ( x )\in \{ 0 , 1 , ... , p - 1 \}$ to the rest of $x$ modulo $p$ . Let $x_1, x_2, ... , x_k$ ( $2 < k < p$) different integers  modulo $p$ and not divisible by $p$.

We say that a number $a \in \{ 1 , 2 ,..., p -1 \}$ is good  if $r ( a x_1) < r ( a x_2) <...< r ( a x_k)$.

Show that there are at most  $\frac{2 p}{k + 1}-{ 1}$ good  numbers.
2007 Rioplatense Mathematical Olympiad, Level 3
Find all functions   $ f:Z\to Z$ with the following property: if  $x+y+z=0$, then $f(x)+f(y)+f(z)=xyz.$
2007 Rioplatense Mathematical Olympiad, Level 3
Divide each side of a triangle into $50$ equal parts, and each point of the division is joined to the opposite vertex by a segment. Calculate the number of intersection points determined by these segments.



Clarification : the vertices of the original triangle are not considered points of intersection or division.
2007 Rioplatense Mathematical Olympiad, Level 3
Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$-small if there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $ n - 1$ $n$-small  sets .



Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .
2006 Rioplatense Mathematical Olympiad, Level 3
(a) For each integer $k\ge 3$, find a positive integer $n$ that can be represented as the sum of exactly $k$ mutually distinct positive divisors of $n$.

(b) Suppose that $n$ can be expressed as the sum of exactly $k$ mutually distinct positive divisors of $n$ for some $k\ge 3$. Let $p$ be the smallest prime divisor of $n$. Show that $$\frac1p+\frac1{p+1}+\cdots+\frac{1}{p+k-1}\ge1.$$
2006 Rioplatense Mathematical Olympiad, Level 3
Let $ABCD$ be a convex quadrilateral with $AB = AD$ and $CB = CD$. The bisector of $\angle BDC$ intersects $BC$ at $L$, and $AL$ intersects $BD$ at $M$, and it is known that $BL = BM$. Determine the value of $2\angle BAD + 3\angle BCD$.
2006 Rioplatense Mathematical Olympiad, Level 3
The numbers $1, 2,\ldots, 2006$ are written around the circumference of a circle. A move consists of exchanging two adjacent numbers. After a sequence of such moves, each number ends up $13$ positions to the right of its initial position. lf the numbers $1, 2,\ldots, 2006$  are partitioned into $1003$ distinct pairs, then show that in at least one of the moves, the two numbers of one of the pairs were exchanged.
2006 Rioplatense Mathematical Olympiad, Level 3
The acute triangle $ABC$ with $AB\neq AC$ has circumcircle $\Gamma$, circumcenter $O$, and orthocenter $H$. The midpoint of $BC$ is $M$, and the extension of the median $AM$ intersects $\Gamma$ at $N$. The circle of diameter $AM$ intersects $\Gamma$ again at $A$ and $P$. Show that the lines $AP$, $BC$, and $OH$ are concurrent if and only if $AH = HN$.
2006 Rioplatense Mathematical Olympiad, Level 3
A given finite number of lines in the plane, no two of which are parallel and no three of which are concurrent, divide the plane into finite and infinite regions. In each finite region we write $1$ or $-1$. In one operation, we can choose any triangle made of three of the lines (which may be cut by other lines in the collection) and multiply by $-1$ each of the numbers in the triangle. Determine if it is always possible to obtain $1$ in all the finite regions by successively applying this operation, regardless of the initial distribution of  $1$s and $-1$s.
2006 Rioplatense Mathematical Olympiad, Level 3
An infinite sequence $x_1,x_2,\ldots$ of positive integers satisfies $$ x_{n+2}=\gcd(x_{n+1},x_n)+2006 $$ for each positive integer $n$. Does there exist such a sequence which contains exactly $10^{2006}$ distinct numbers?
2005 Rioplatense Mathematical Olympiad, Level 3
Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$.
2005 Rioplatense Mathematical Olympiad, Level 3
In trapezoid $ABCD$, the sum of the lengths of the bases $AB$ and $CD$ is equal to the length of the diagonal $BD$. Let $M$ denote the midpoint of $BC$, and let $E$ denote the reflection of $C$ about the line $DM$. Prove that $\angle AEB=\angle ACD$.
2005 Rioplatense Mathematical Olympiad, Level 3
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$. (Note: $n$ is written in the usual base ten notation.)
2005 Rioplatense Mathematical Olympiad, Level 3
Let $P$ be a point inside triangle $ABC$ and let $R$ denote the circumradius of triangle $ABC$. Prove that

$$ \frac{PA}{AB\cdot AC}+\frac{PB}{BC\cdot BA}+\frac{PC}{CA\cdot CB}\ge\frac{1}{R}.$$
2005 Rioplatense Mathematical Olympiad, Level 3
Consider all finite sequences of positive real numbers each of whose terms is at most $3$ and the sum of whose terms is more than $100$. For each such sequence, let $S$ denote the sum of the subsequence whose sum is the closest to $100$, and define the defect of this sequence to be the value $|S-100|$. Find the maximum possible value of the defect.
2005 Rioplatense Mathematical Olympiad, Level 3
Let $k$ be a positive integer. Show that for all $n>k$ there exist convex figures $F_{1},\ldots, F_{n}$ and $F$ such that there doesn't exist a subset of $k$ elements from $F_{1},..., F_{n}$ and $F$ is covered for this elements, but $F$ is covered for every subset of $k+1$ elements from $F_{1}, F_{2},....., F_{n}$.
2004 Rioplatense Mathematical Olympiad, Level 3
Find all polynomials $P(x)$ with real coefficients such that $$xP\bigg(\frac{y}{x}\bigg)+yP\bigg(\frac{x}{y}\bigg)=x+y$$ for all nonzero real numbers $x$ and $y$.
2004 Rioplatense Mathematical Olympiad, Level 3
Find the smallest integer $n$ such that each subset of $\{1,2,\ldots, 2004\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$.
2004 Rioplatense Mathematical Olympiad, Level 3
In a convex hexagon $ABCDEF$, triangles $ACE$ and $BDF$ have the same circumradius $R$. If triangle $ACE$ has inradius $r$, prove that

$$ \text{Area}(ABCDEF)\le\frac{R}{r}\cdot\text{Area}(ACE).$$
2004 Rioplatense Mathematical Olympiad, Level 3
How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$?
2004 Rioplatense Mathematical Olympiad, Level 3
A collection of cardboard circles, each with a diameter of at most $1$, lie on a $5\times 8$ table without overlapping or overhanging the edge of the table. A cardboard circle of diameter $2$ is added to the collection. Prove that this new collection of cardboard circles can be placed on a $7\times 7$ table without overlapping or overhanging the edge.
2004 Rioplatense Mathematical Olympiad, Level 3
Consider a partition of $\{1,2,\ldots,900\}$ into $30$ subsets $S_1,S_2,\ldots,S_{30}$ each with $30$ elements. In each $S_k$, we paint the fifth largest number blue. Is it possible that, for $k=1,2,\ldots,30$, the sum of the elements of $S_k$ exceeds the sum of the blue numbers?
2003 Rioplatense Mathematical Olympiad, Level 3
Let $x$, $y$, and $z$ be positive real numbers satisfying $x^2+y^2+z^2=1$. Prove that $$x^2yz+xy^2z+xyz^2\le\frac{1}{3}.$$
2003 Rioplatense Mathematical Olympiad, Level 3
Triangle $ABC$ is inscribed in the circle $\Gamma$. Let $\Gamma_a$ denote the circle internally tangent to $\Gamma$ and also tangent to sides $AB$ and $AC$. Let $A'$ denote the point of tangency of $\Gamma$ and $\Gamma_a$. Define $B'$ and $C'$ similarly. Prove that $AA'$, $BB'$ and $CC'$ are concurrent.
2003 Rioplatense Mathematical Olympiad, Level 3
An $8\times 8$ chessboard is to be tiled (i.e., completely covered without overlapping) with pieces of the following shapes:

[asy]

unitsize(.6cm);

draw(unitsquare,linewidth(1));

draw(shift(1,0)*unitsquare,linewidth(1));

draw(shift(2,0)*unitsquare,linewidth(1));

label("\footnotesize $1\times 3$ rectangle",(1.5,0),S);

draw(shift(8,1)*unitsquare,linewidth(1));

draw(shift(9,1)*unitsquare,linewidth(1));

draw(shift(10,1)*unitsquare,linewidth(1));

draw(shift(9,0)*unitsquare,linewidth(1));

label("\footnotesize T-shaped tetromino",(9.5,0),S);

[/asy] The $1\times 3$ rectangle covers exactly three squares of the chessboard, and the T-shaped tetromino covers exactly four squares of the chessboard.



(a) What is the maximum number of pieces that can be used?

(b) How many ways are there to tile the chessboard using this maximum number of pieces?
2003 Rioplatense Mathematical Olympiad, Level 3
Inside right angle $XAY$, where $A$ is the vertex, is a semicircle $\Gamma$ whose center lies on $AX$ and that is tangent to $AY$ at the point $A$. Describe a ruler-and-compass construction for the tangent to $\Gamma$ such that the triangle enclosed by the tangent and angle $XAY$ has minimum area.
2003 Rioplatense Mathematical Olympiad, Level 3
Let $n$ and $k$ be positive integers. Consider $n$ infinite arithmetic progressions of nonnegative integers with the property that among any $k$ consecutive nonnegative integers, at least one of $k$ integers belongs to one of the $n$ arithmetic progressions. Let $d_1,d_2,\ldots,d_n$ denote the differences of the arithmetic progressions, and let $d=\min\{d_1,d_2,\ldots,d_n\}$. In terms of $n$ and $k$, what is the maximum possible value of $d$?
2003 Rioplatense Mathematical Olympiad, Level 3
Without overlapping, hexagonal tiles are placed inside an isosceles right triangle of area $1$ whose hypotenuse is horizontal. The tiles are similar to the figure below, but are not necessarily all the same size.[asy]

unitsize(.85cm);

draw((0,0)--(1,0)--(1,1)--(2,2)--(-1,2)--(0,1)--(0,0),linewidth(1));

draw((0,2)--(0,1)--(1,1)--(1,2),dashed);

label("\footnotesize $a$",(0.5,0),S);

label("\footnotesize $a$",(0,0.5),W);

label("\footnotesize $a$",(1,0.5),E);

label("\footnotesize $a$",(0,1.5),E);

label("\footnotesize $a$",(1,1.5),W);

label("\footnotesize $a$",(-0.5,2),N);

label("\footnotesize $a$",(0.5,2),N);

label("\footnotesize $a$",(1.5,2),N);

[/asy] The longest side of each tile is parallel to the hypotenuse of the triangle, and the horizontal side of length $a$ of each tile lies between this longest side of the tile and the hypotenuse of the triangle. Furthermore, if the longest side of a tile is farther from the hypotenuse than the longest side of another tile, then the size of the first tile is larger or equal to the size of the second tile. Find the smallest value of $\lambda$ such that every such configuration of tiles has a total area less than $\lambda$.
2002 Rioplatense Mathematical Olympiad, Level 3
Determine all pairs $(a, b)$ of positive integers for which    $\frac{a^2b+b}{ab^2+9}$ is an integer number.
2002 Rioplatense Mathematical Olympiad, Level 3
Let $\lambda$ be a real number such that the inequality $0 <\sqrt {2002} - \frac {a} {b} <\frac {\lambda} {ab}$ holds for an infinite number of pairs $ (a, b)$ of positive integers. Prove that $\lambda \geq 5  $.
2002 Rioplatense Mathematical Olympiad, Level 3
Let $ABC$ be a triangle with $\angle C=60^o$. The point $P$ is the symmetric of $A$ with respect to the point of tangency of the circle inscribed with the side $BC$ . Show that if the perpendicular bisector of the $CP$ segment intersects the line containing the angle - bisector of $\angle B$ at the point $Q$, then the triangle $CPQ$ is equilateral.
2002 Rioplatense Mathematical Olympiad, Level 3
Let $a, b$ and $c$ be positive real numbers. Show that $\frac{a+b}{c^2}+ \frac{c+a}{b^2}+ \frac{b+c}{a^2}\ge \frac{9}{a+b+c}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$
2002 Rioplatense Mathematical Olympiad, Level 3
$ABC$ is any triangle. Tangent at $C$ to circumcircle ($O$) of $ABC$ meets $AB$ at $M$. Line perpendicular to $OM$ at $M$ intersects $BC$ at $P$ and $AC$ at $Q$. P.T. $MP=MQ$.
2002 Rioplatense Mathematical Olympiad, Level 3
Daniel chooses a positive integer $n$ and tells Ana. With this information, Ana chooses a positive integer $k$ and tells Daniel. Daniel draws $n$ circles on a piece of paper and chooses $k$ different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the $k$ points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of $k$ that allows Ana to achieve her goal regardless of how Daniel chose the $n$ circumferences and the $k$ points.
2001 Rioplatense Mathematical Olympiad, Level 3
Find all integer numbers $a, b, m$ and $n$, such that the following two equalities are verified:

$a^2+b^2=5mn$ and  $m^2+n^2=5ab$
2001 Rioplatense Mathematical Olympiad, Level 3
Let $ABC$ be an acute triangle and $A_1, B_1$ and $C_1$, points on the sides $BC, CA$ and $AB$, respectively, such that $CB_1 = A_1B_1$ and $BC_1 = A_1C_1$. Let $D$ be the symmetric of $A_1$ with respect to $B_1C_1, O$ and $O_1$ are the circumcenters of triangles $ABC$ and $A_1B_1C_1$, respectively. If $A \ne D, O \ne O_1$ and $AD$ is perpendicular to $OO_1$, prove that $AB = AC$.
2001 Rioplatense Mathematical Olympiad, Level 3
For every integer $n > 1$, the sequence $\left( {{S}_{n}} \right)$ is defined by ${{S}_{n}}=\left\lfloor {{2}^{n}}\underbrace{\sqrt{2+\sqrt{2+...+\sqrt{2}}}}_{n\ radicals} \right\rfloor $

where  $\left\lfloor x \right\rfloor$  denotes the floor function of $x$. Prove that ${{S}_{2001}}=2\,{{S}_{2000}}+1$.

.
2001 Rioplatense Mathematical Olympiad, Level 3
Find all functions $f: R \to R$ such that, for any $x, y \in R$:

$f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}$
2001 Rioplatense Mathematical Olympiad, Level 3
Let $ABC$ be a acute-angled triangle with centroid $G$, the angle bisector of $\angle ABC$ intersects $AC$ in $D$. Let $P$ and $Q$ be points in $BD$ where $\angle PBA = \angle PAB$ and $\angle QBC = \angle QCB$. Let $M$ be the midpoint of $QP$, let $N$ be a point in the line $GM$ such that $GN = 2GM$(where $G$ is the segment $MN$), prove that:

$\angle ANC + \angle ABC = 180$
2001 Rioplatense Mathematical Olympiad, Level 3
For $m = 1, 2, 3, ...$ denote $S(m)$ the sum of the digits of $m$, and let  $f(m)=m+S(m)$.

Show that for each positive integer $n$, there exists a number that appears exactly $n$ times in the sequence $f(1),f(2),...,f(m),...$
2000 Rioplatense Mathematical Olympiad, Level 3
Let $a$ and $b$ be positive integers such that the number $b^2 + (b + l)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.
2000 Rioplatense Mathematical Olympiad, Level 3
In a triangle $ABC$, points $D, E$ and $F$ are considered on the sides $BC, CA$ and $AB$ respectively, such that the areas of the triangles $AFE, BFD$ and $CDE$ are equal. Prove that $$\frac{(DEF) }{ (ABC)} \ge \frac{1}{4}$$

Note: $(XYZ)$  is the area of triangle $XYZ$.
2000 Rioplatense Mathematical Olympiad, Level 3
Let $n > 1$ a integer, for each numbers $X_1, X_2, ...X_n$ such that $X_1^2 + X_2^2 + X_3^2 + ... + X_n^2 = 1$, denote:

$m = min { | X_i - X_j | : 0 < i < j < n + 1}$



Find the maximum value of $m$
2000 Rioplatense Mathematical Olympiad, Level 3
Let $a, b$ and $c$ be positive integers such that $a^2 + b^2 + 1 = c^2$ . Prove that $[a/2] + [c / 2]$ is even.



Note: $[x]$ is the integer part of $x$.
2000 Rioplatense Mathematical Olympiad, Level 3
Let $ABC$ be a triangle with $AB < AC$, let $L$ be midpoint of arc $BC$(the point $A$ is not in this arc) of the circumcircle $w$($ABC$). Let $E$ be a point in $AC$ where $AE = \frac{AB + AC}{2}$, the line $EL$ intersects $w$ in $P$.

If $M$ and $N$ are the midpoints of $AB$ and $BC$, respectively, prove that $AL, BP$ and $MN$ are concurrents
2000 Rioplatense Mathematical Olympiad, Level 3
Let $g(x) = ax^2 + bx + c$ a quadratic function with real coefficients such that the equation $g(g(x)) = x$ has four distinct real roots. Prove that there isn't a function $f$: $R--R$ such that $f(f(x)) = g(x)$ for all $x$ real
2021 Romanian Masters In Mathematics
Let $T_1, T_2, T_3, T_4$ be pairwise distinct collinear points such that $T_2$ lies between $T_1$ and $T_3$, and $T_3$ lies between $T_2$ and $T_4$. Let $\omega_1$ be a circle through $T_1$ and $T_4$; let $\omega_2$ be the circle through $T_2$ and internally tangent to $\omega_1$ at $T_1$; let $\omega_3$ be the circle through $T_3$ and externally tangent to $\omega_2$ at $T_2$; and let $\omega_4$ be the circle through $T_4$ and externally tangent to $\omega_3$ at $T_3$. A line crosses $\omega_1$ at $P$ and $W$, $\omega_2$ at $Q$ and $R$, $\omega_3$ at $S$ and $T$, and $\omega_4$ at $U$ and $V$, the order of these points along the line being $P,Q,R,S,T,U,V,W$. Prove that $PQ + TU = RS + VW$



Geza Kos, Hungary