Navvye/Olympiad-Problems-CSV huggingface dataset:
2001 Pan African
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?
2001 Pan African
Let $ABC$ be an equilateral triangle and let $P_0$ be a point outside this triangle, such that $\triangle{AP_0C}$ is an isoscele triangle with a right angle at $P_0$. A grasshopper starts from $P_0$ and turns around the triangle as follows. From $P_0$ the grasshopper jumps to $P_1$, which is the symmetric point of $P_0$ with respect to $A$. From $P_1$, the grasshopper jumps to $P_2$, which is the symmetric point of $P_1$ with respect to $B$. Then the grasshopper jumps to $P_3$ which is the symmetric point of $P_2$ with respect to $C$, and so on. Compare the distance $P_0P_1$ and $P_0P_n$. $n \in N$.
2001 Pan African
Let $n$ be a positive integer, and let $a>0$ be a real number. Consider the equation:

$$ \sum_{i=1}^{n}(x_i^2+(a-x_i)^2)= na^2 $$

How many solutions ($x_1, x_2 \cdots , x_n$) does this equation have, such that:

$$ 0 \leq x_i \leq a, i \in N^+ $$
2001 Pan African
Find the value of the sum:

$$ \sum_{i=1}^{2001} [\sqrt{i}] $$

where $[ {x} ]$ denotes the greatest integer which does not exceed $x$.
2001 Pan African
Let $S_1$ be a semicircle with centre $O$ and diameter $AB$.A circle $C_1$ with centre $P$ is drawn, tangent to $S_1$, and tangent to $AB$ at $O$. A semicircle $S_2$ is drawn, with centre $Q$ on $AB$, tangent to $S_1$ and to $C_1$. A circle $C_2$ with centre $R$ is drawn, internally tangent to $S_1$ and externally tangent to $S_2$ and $C_1$. Prove that $OPRQ$ is a rectangle.
2000 Pan African
Solve for $x \in R$:

$$ \sin^3{x}(1+\cot{x})+\cos^3{x}(1+\tan{x})=\cos{2x} $$
2000 Pan African
Define the polynomials $P_0, P_1, P_2 \cdots$ by:

$$ P_0(x)=x^3+213x^2-67x-2000 $$

$$ P_n(x)=P_{n-1}(x-n), n \in N $$

Find the coefficient of $x$ in $P_{21}(x)$.
2000 Pan African
Let $p$ and $q$ be coprime positive integers such that:

$$ \dfrac{p}{q}=1-\frac12+\frac13-\frac14 \cdots -\dfrac{1}{1334}+\dfrac{1}{1335} $$

Prove $p$ is divisible by 2003.
2000 Pan African
Let $a$, $b$ and $c$ be real numbers such that $a^2+b^2=c^2$, solve the system:

$$ z^2=x^2+y^2 $$

$$ (z+c)^2=(x+a)^2+(y+b)^2 $$

in real numbers $x, y$ and $z$.
2000 Pan African
Let $\gamma$ be circle and let $P$ be a point outside $\gamma$. Let $PA$ and $PB$ be the tangents from $P$ to $\gamma$ (where $A, B \in \gamma$). A line passing through $P$ intersects $\gamma$ at points $Q$ and $R$. Let $S$ be a point on $\gamma$ such that $BS \parallel QR$. Prove that $SA$ bisects $QR$.
2000 Pan African
A company has five directors. The regulations of the company require that any majority (three or more) of the directors should be able to open its strongroom, but any minority (two or less) should not be able to do so. The strongroom is equipped with ten locks, so that it can only be opened when keys to all ten locks are available. Find all positive integers $n$ such that it is possible to give each of the directors a set of keys to $n$ different locks, according to the requirements and regulations of the company.
2019 Rioplatense Mathematical Olympiad, Level 3
Let $ABCDEF$ be a regular hexagon, in the sides $AB$, $CD$, $DE$ and $FA$ we choose four points $P,Q,R$ and $S$ respectively, such that $PQRS$ is a square. Prove that $PQ$ and $BC$ are parallel.
2019 Rioplatense Mathematical Olympiad, Level 3
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that

$f(f(x)^2+f(y^2))=(x-y)f(x-f(y))$
2019 Rioplatense Mathematical Olympiad, Level 3
In the dog dictionary the words are any sequence of letters $A$ and $U$ for example $AA$, $UAU$ and $AUAU$. For each word, your "profundity" will be the quantity of subwords we can obtain by the removal of some letters.

For each positive integer $n$, determine the largest "profundity" of word, in dog dictionary, can have with $n$ letters.

Note: The word $AAUUA$ has "profundity" $14$ because your subwords are $A, U, AU, AA, UU, UA, AUU, UUA, AAU, AUA, AAA, AAUU, AAUA, AUUA$.
2019 Rioplatense Mathematical Olympiad, Level 3
Prove that there are infinite triples $(a,b,c)$ of positive integers $a,b,c>1$, $gcd(a,b)=gcd(b,c)=gcd(c,a)=1$ such that $a+b+c$ divides $a^b+b^c+c^a$.
2019 Rioplatense Mathematical Olympiad, Level 3
Let $ABC$ be a triangle with $AB<AC$ and circuncircle $\omega$. Let $M$ and $N$ be the midpoints of $AC$ and $AB$ respectively and $G$ is the centroid of $ABC$. Let $P$ be the foot of perpendicular of $A$ to the line $BC$, and the point $Q$ is the intersection of $GP$ and  $\omega$($Q,P,G$ are collinears in this order). The line $QM$ cuts $\omega$ in $M_1$ and the line $QN$ cuts $\omega$ in $N_1$. If $K$ is the intersection of $BM_1$ and $CN_1$ prove that $P$, $G$ and $K$ are collinears.
2019 Rioplatense Mathematical Olympiad, Level 3
Let $\alpha>1$ be a real number such that the sequence $a_n=\alpha\lfloor \alpha^n\rfloor- \lfloor \alpha^{n+1}\rfloor$, with $n\geq 1$, is periodic, that is, there is a positive integer $p$ such that $a_{n+p}=a_n$ for all $n$. Prove that $\alpha$ is an integer.
2018 Rioplatense Mathematical Olympiad, Level 3
Determine if there are $2018$ different positive integers such that the sum of their squares is a perfect cube and the sum of their cubes is a perfect square.
2018 Rioplatense Mathematical Olympiad, Level 3
Let $P$ be a point outside a circumference $\Gamma$, and let $PA$ be one of the tangents from $P$ to $\Gamma$. Line  $l$ passes through $P$ and intersects $\Gamma$ at $B$ and $C$, with $B$ between $P$ and $C$. Let $D$ be the symmetric of $B$ with respect to $P$. Let $\omega_1$ and $\omega_2$ be the circles circumscribed to the triangles $DAC$ and $PAB$ respectively. $\omega_1$ and $\omega _2$ intersect at $E \neq A$. Line $EB$ cuts back to $\omega _1 $ in $F$. Prove that $CF = AB$.
2018 Rioplatense Mathematical Olympiad, Level 3
Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23}  + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$
2018 Rioplatense Mathematical Olympiad, Level 3
Let $ABC$ be an acute triangle with $AC> AB$. be $\Gamma$ the circumcircle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smallest arc  $BC$ of this circumference. Let $E$ and $F$ points of the segments $AB$ and $AC$ respectively such that $AE = AF$. Let $P \neq A$ be the second point of intersection of the circumcircle circumscribed to $AEF$ with $\Gamma$. Let $G$ and $H$ be the intersections of lines $PE$ and $PF$ with $\Gamma$ other than $P$, respectively. Let $J$ and $K$ be the points of intersection of lines $DG$ and $DH$ with lines $AB$ and $AC$ respectively. Show that the $JK$ line passes through the midpoint of $BC$
2018 Rioplatense Mathematical Olympiad, Level 3
Let $n$ be a positive integer. Find all $n$- rows $( a_1 , a_2 ,..., a_n )$ of different positive integers such that $$ \frac{(a_1 + d ) (a_2 + d ) \cdot\cdot\cdot ( a_n + d )}{a_1a_2\cdot \cdot \cdot a_n }$$is integer for every integer $d\ge 0$
2018 Rioplatense Mathematical Olympiad, Level 3
A company has $n$ employees. It is known that each of the employees works at least one of the $7$ days of the week, with the exception of an employee who does not work any of the $7$ days. Furthermore, for any two of these $n$ employees, there are at least $3$ days of the week in which one of the two works that day and the other does not (it is not necessarily the same employee who works those days). Determine the highest possible value of $n$.
2017 Rioplatense Mathematical Olympiad, Level 3
Problem 1

Let $a$ be a fixed positive integer. Find the largest integer $b$ such that

$(x+a)(x+b)=x+a+b$, for some integer $x$.



Problem 2

One have $n$ distinct circles(with the same radius) such that for any $k+1$ circles there are (at least) two circles that intersects in two points. Show that for each line $l$ one can make $k$ lines, each one parallel with $l$, such that each circle has (at least) one point of intersection with some of these lines.
2017 Rioplatense Mathematical Olympiad, Level 3
Problem 3

Show that there are infinitely many pairs of positive integers $(m,n)$, with $m<n$, such that

$m$ divides $n^{2016}+n^{2015}+\dots+n^2+n+1$ and $n$ divides $m^{2016}+m^{2015} +\dots+m^2+m+1$.



Problem 4

Is there a number $n$ such that one can write $n$ as the sum of $2017$ perfect squares and (with at least) $2017$ distinct ways?
2017 Rioplatense Mathematical Olympiad, Level 3
Problem 5

Let $ABC$ be a triangle and $I$ is your incenter, let $P$ be a point in $AC$ such that $PI$ is perpendicular to $AC$, and let $D$ be the reflection of $B$ to circumcenter of the circumcircle of $\triangle ABC$. The line $DI$ intersects again the circumcircle of $\triangle ABC$ in the point $Q$. Prove that $QP$ is the angle bisector of the angle $\angle AQC$.



Problem 6

For each fixed positiver integer $n$, $n\geq 4$ and $P$ an integer, let $(P)_n \in [1, n]$ be the smallest positive residue of $P$ modulo $n$. Two sequences $a_1, a_2, \dots, a_k$ and $b_1, b_2, \dots, b_k$ with the terms in $[1, n]$ are defined as equivalents, if there is $t$ positive integer, gcd$(t,n)=1$, such that the sequence $(ta_1)_n, \dots, (ta_k)_n$ is a permutation of $b_1, b_2, \dots, b_k$.

Let $\alpha$ a sequence of size $n$ and your terms are in $[1, n]$, such that each term appears $h$ times in the sequence $\alpha$ and $2h\geq n$.

Show that $\alpha$ is equivalent to some sequence $\beta$ which contains a subsequence such that your size is(at most) equal to $h$ and your sum is exactly equal to $n$.
2016 Rioplatense Mathematical Olympiad, Level 3
Ana and Beto play against each other. Initially, Ana chooses a non-negative integer $N$ and announces it to Beto. Next Beto writes a succession of $2016$ numbers, $1008$ of them equal to $1$ and $1008$ of them equal to $-1$. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to $N$, Ana wins. If not, Beto wins. Determine all values of $N$ for which Ana can ensure victory, no matter how Beto plays.
2016 Rioplatense Mathematical Olympiad, Level 3
Determine all positive integers $n$ for which there are positive real numbers $x,y$   and  $z$  such that $\sqrt x +\sqrt y +\sqrt z=1$ and $\sqrt{x+n} +\sqrt{y+n} +\sqrt{z+n}$ is an integer.
2016 Rioplatense Mathematical Olympiad, Level 3
Let  $A B C$ be an acute-angled triangle of circumcenter $O$ and orthocenter $H$. Let $M$ be the midpoint of  $BC, N$ be the  symmetric  of $H$ with respect to $A, P$ be the midpoint of  $NM$ and $X$ be a point on the line A H such that $MX$ is parallel to $CH$. Prove that $BX$ and $OP$ are perpendicular.
2016 Rioplatense Mathematical Olympiad, Level 3
Let $c > 1$ be a real number. A function $f: [0 ,1 ] \to R$ is called c-friendly if $f(0) = 0, f(1) = 1$ and $|f(x) -f(y)| \le c|x - y|$ for all the numbers $x ,y \in [0,1]$. Find the maximum of the expression $|f(x) - f(y)|$ for all c-friendly functions  $f$ and for all the numbers $x,y \in [0,1]$.
2016 Rioplatense Mathematical Olympiad, Level 3
Initially one have the number $0$ in each cell of the table $29 \times 29$. A moviment is when one choose a sub-table $5 \times 5$ and add $+1$ for every cell of this sub-table. Find the maximum value of $n$, where after $1000$ moviments, there are $4$ cells such that your centers are vertices of a square and the sum of this $4$ cells is at least $n$.

Note: A square does not, necessarily, have your sides parallel with the sides of the table.
2016 Rioplatense Mathematical Olympiad, Level 3
When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112 ....$ Denote $A_k$ the number formed by the first $k$ digits of this sequence . Prove that for all positive integer $n$ there is a positive integer $m$ which simultaneously verifies the following three conditions:

(i) $n$ divides $A_m$,

(ii) $n$ divides $m$,

(iii) $n$ divides the sum of the digits of $A_m$.
2015 Rioplatense Mathematical Olympiad, Level 3
Let $ABC$ be a triangle and $P$ a point on the side $BC$. Let $S_1$ be the circumference with center $B$ and radius $BP$ that cuts the side $AB$ at $D$ such that $D$ lies between $A$ and $B$. Let $S_2$ be the circumference with center $C$ and radius $CP$ that cuts the side $AC$ at $E$ such that $E$ lies between $A$ and $C$. Line $AP$ cuts $S_1$ and $S_2$ at $X$ and $Y$ different from $P$, respectively. We call $T$ the point of intersection of $DX$ and $EY$. Prove that $\angle BAC+ 2 \angle DTE=180$
2015 Rioplatense Mathematical Olympiad, Level 3
Let $a , b , c$ positive integers, coprime. For each whole number $n \ge 1$, we denote by $s ( n )$  the number of elements in the set $\{ a , b , c \}$ that divide $n$. We consider $k_1< k_2< k_3<...$ .the sequence of all positive integers that are divisible by some element of  $\{ a , b , c \}$.  Finally we define the characteristic sequence of $( a , b , c )$ like the succession $ s ( k_1) , s ( k_2) , s ( k_3) , .... $ .

Prove that if the characteristic sequences of $( a , b , c )$ and $( a', b', c')$ are equal, then $a = a', b = b'$ and $c=c'$
2015 Rioplatense Mathematical Olympiad, Level 3
We say an integer number $n \ge 1$ is conservative, if the smallest prime divisor of $(n!)^n+1$ is at most $n+2015$. Decide if the number of conservative numbers is infinite or not.
2015 Rioplatense Mathematical Olympiad, Level 3
You have a $9 \times 9$ board with white squares. A tile can be moved from one square to another neighbor (tiles that share one side). If we paint some squares of black, we say that such coloration is good  if there is a white square where we can place a chip that moving through white squares can return to the initial square having passed through at least $3$ boxes, without passing the same square twice.

Find the highest possible value of $k$ such that any form of painting $k$ squares of black are a good  coloring.
2015 Rioplatense Mathematical Olympiad, Level 3
For a positive integer number $n$ we denote $d(n)$ as the greatest common divisor of the binomial coefficients $\dbinom{n+1}{n} , \dbinom{n+2}{n} ,..., \dbinom{2n}{n}$.

Find all possible values of $d(n)$
2015 Rioplatense Mathematical Olympiad, Level 3
Let $A B C$ be an acut-angles triangle of incenter $I$, circumcenter $O$ and inradius $r.$ Let  $\omega$ be  the inscribed circle of the triangle $A B C$. $A_1$ is the point of ω such that $A IA_1O$ is a convex trapezoid of bases $A O$ and  $IA_1$. Let $\omega_1$ be the circle of radius $r$ which goes through $A_1$, tangent to the line $A B$ and is different from $\omega$ . Let $\omega_2$ be the circle of radius $r$ which goes through $A_1$, is tangent to the line $A C$ and is different from $\omega$ . Circumferences $\omega_1$ and $\omega_2$ they are cut at points $A_1$ and $A_2$. Similarly are defined points  $B_2$ and  $C_2$. Prove that the lines $A A_2, B B_2$ and $CC2$ they are concurrent.
2014 Rioplatense Mathematical Olympiad, Level 3
Let $n \ge 3$ be a positive integer.  Determine, in terms of $n$, how many triples of sets $(A,B,C)$ satisfy the conditions:

$\bullet$ $A, B$ and $C$ are pairwise disjoint , that is, $A \cap  B = A \cap  C= B \cap C= \emptyset$.

$\bullet$ $A \cup B \cup C= \{ 1 , 2 , ... , n \}$.

$\bullet$ The sum of the elements of $A$, the sum of the elements of $B$ and the sum of the elements of $C$ leave the same remainder when divided by $3$.



Note: One or more of the sets may be empty.
2014 Rioplatense Mathematical Olympiad, Level 3
El Chapulín observed that the number $2014$  has an unusual property. By placing its eight positive divisors in increasing order, the fifth divisor is equal to three times the third minus $4$. A number of eight divisors with this unusual property is called the red number . How many red numbers  smaller than $2014$ exist?
2014 Rioplatense Mathematical Olympiad, Level 3
Kiko and Ñoño play with a rod of length $2n$ where $n \le 3$  is an integer. Kiko cuts the rod in $ k \le 2n$ pieces of whole lengths. Then Ñoño has to arrange these pieces so that they form a hexagon of equal opposite sides and equal angles. The pieces can not be split and they all have to be used. If Ñoño achieves his goal, he wins, in any other case, Kiko wins. Determine which victory can be secured based on $k$.
2014 Rioplatense Mathematical Olympiad, Level 3
A pair (a,b) of positive integers is Rioplatense if it is true that $b + k$ is a multiple of $a + k$ for all $k \in\{ 0 , 1 , 2 , 3 , 4 \}$. Prove that there is an infinite set $A$ of positive integers such that for any two elements $a$ and $b$ of $A$, with $a < b$, the pair $(a,b)$ is Rioplatense.
2014 Rioplatense Mathematical Olympiad, Level 3
In the segment $A C$ a point $B$ is taken. Construct circles $T_1, T_2$ and $T_3$ of diameters $A B, BC$ and $AC$ respectively. A line that passes through $B$ cuts  $T_3$ in the points $P$ and $Q$, and the circles $T_1$ and  $T_2$ respectively at points $R$ and $S$. Prove that $PR = Q S$.
2014 Rioplatense Mathematical Olympiad, Level 3
Let $n \in  N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge  a_2\ge  a_3\ge  2$  have sum $n$ and they satisfy $1 + 2 + ... + a_1\le  \frac{1}{3}( 1 + 2 + ... + n ) $ and  $1 + 2 + ... + (a_1+ a_2) \le  \frac{2}{3}( 1 + 2 + ... + n )$.

Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .
2013 Rioplatense Mathematical Olympiad, Level 3
Let $a,b,c,d$ be real positive numbers such that $a^2+b^2+c^2+d^2 = 1$. Prove that $(1-a)(1-b)(1-c)(1-d) \geq abcd$.
2013 Rioplatense Mathematical Olympiad, Level 3
Let $ABCD$ be a square, and let $E$ and $F$ be points in $AB$ and $BC$ respectively such that $BE=BF$. In the triangle $EBC$, let N be the foot of the altitude relative to $EC$. Let $G$ be the intersection between $AD$ and the extension of the previously mentioned altitude. $FG$ and $EC$ intersect at point $P$, and the lines $NF$ and $DC$ intersect at point $T$. Prove that the line $DP$ is perpendicular to the line $BT$.
2013 Rioplatense Mathematical Olympiad, Level 3
A division of a group of people into various groups is called $k$-regular if the number of groups is less or equal to $k$ and two people that know each other are in different groups.

Let $A$, $B$, and $C$ groups of people such that there are is no person in $A$ and no person in $B$ that know each other. Suppose that the group $A \cup C$ has an $a$-regular division and the group $B \cup C$ has a $b$-regular division.

For each $a$ and $b$, determine the least possible value of $k$ for which it is guaranteed that the group $A \cup B \cup C$ has a $k$-regular division.
2013 Rioplatense Mathematical Olympiad, Level 3
Two players $A$ and $B$ play alternatively in a convex polygon with $n \geq 5$ sides. In each turn, the corresponding player has to draw a diagonal that does not cut inside the polygon previously drawn diagonals. A player loses if after his turn, one quadrilateral is formed such that its two diagonals are not drawn. $A$ starts the game.

For each positive integer $n$, find a winning strategy for one of the players.
2013 Rioplatense Mathematical Olympiad, Level 3
Find all positive integers $n$ for which there exist two distinct numbers of $n$ digits, $\overline{a_1a_2\ldots a_n}$ and  $\overline{b_1b_2\ldots b_n}$, such that the number of $2n$ digits $\overline{a_1a_2\ldots a_nb_1b_2\ldots b_n}$ is divisible by $\overline{b_1b_2\ldots b_na_1a_2\ldots a_n}$.
2013 Rioplatense Mathematical Olympiad, Level 3
Let $ABC$ be an acute scalene triangle, $H$ its orthocenter and $G$ its geocenter. The circumference with diameter $AH$ cuts the circumcircle of $BHC$ in $A'$ ($A' \neq H$). Points $B'$ and $C'$ are defined similarly. Show that the points $A'$, $B'$, $C'$, and $G$ lie in one circumference.
2012 Rioplatense Mathematical Olympiad, Level 3
An integer $n$ is called apocalyptic if the addition of $6$ different positive divisors of $n$ gives $3528$. For example, $2012$ is apocalyptic, because it has six divisors, $1$, $2$, $4$, $503$, $1006$ and $2012$, that add up to $3528$.



Find the smallest positive apocalyptic number.
2012 Rioplatense Mathematical Olympiad, Level 3
A rectangle is divided into $n^2$ smaller rectangle by $n - 1$ horizontal lines and $n - 1$ vertical lines. Between those rectangles there are exactly $5660$ which are not congruent. For what minimum value of $n$ is this possible?
2012 Rioplatense Mathematical Olympiad, Level 3
Let $T$ be a non-isosceles triangle and $n \ge 4$ an integer . Prove that you can divide $T$ in  $n$ triangles and draw in each of them an inner bisector so that those $n$ bisectors are parallel.
2012 Rioplatense Mathematical Olympiad, Level 3
Find all real numbers $x$, such that:

a) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2012x \rfloor = 2013$

b) $\lfloor x \rfloor + \lfloor 2x \rfloor +...+ \lfloor 2013x \rfloor = 2014$
2012 Rioplatense Mathematical Olympiad, Level 3
Let $a \ge 2$ and $n \ge 3$ be integers . Prove that one of the numbers $a^n+ 1 , a^{n + 1}+ 1 , ... , a^{2 n-2}+ 1$ does not share any odd divisor greater than $1$ with any of the other numbers.
2012 Rioplatense Mathematical Olympiad, Level 3
In each square of a $100 \times 100$ board there is written an integer. The allowed operation is to choose four squares that form the figure or any of its reflections or rotations, and add $1$ to each of the four numbers. The aim is, through operations allowed, achieving a board with the smallest possible number of different residues modulo $33$. What is the minimum number that can be achieved with certainty?
2011 Rioplatense Mathematical Olympiad, Level 3
Given a positive integer $n$, an operation consists of replacing $n$ with either $2n-1$, $3n-2$ or $5n-4$. A number $b$ is said to be a follower of number $a$ if $b$ can be obtained from $a$ using this operation multiple times. Find all positive integers $a < 2011$ that have a common follower with $2011$.
2011 Rioplatense Mathematical Olympiad, Level 3
Let $ABC$ an acute triangle and $H$ its orthocenter. Let $E$ and $F$ be the intersection of lines $BH$ and $CH$ with $AC$ and $AB$ respectively, and let $D$ be the intersection of lines $EF$ and $BC$. Let $\Gamma_1$ be the circumcircle of $AEF$, and $\Gamma_2$ the circumcircle of $BHC$. The line $AD$ intersects $\Gamma_1$ at point $I \neq A$. Let $J$ be the feet of the internal bisector of $\angle{BHC}$ and $M$ the midpoint of the arc $\stackrel{\frown}{BC}$ from $\Gamma_2$ that contains the point $H$. The line $MJ$ intersects $\Gamma_2$ at point $N \neq M$. Show that the triangles $EIF$ and $CNB$ are similar.
2011 Rioplatense Mathematical Olympiad, Level 3
Let $M$ be a map made of several cities linked to each other by flights. We say that there is a route between two cities if there is a nonstop flight linking these two cities. For each city to the $M$ denote by $M_a$ the map formed by the cities that have a route to and routes linking these cities together ( to not part of  $M_a$). The cities of $M_a$ are divided into two sets so that the number of routes linking cities of different sets is maximum; we call this number the cut of $M_a$. Suppose that for every  cut of $M_a$ , it is strictly less than two thirds of the number of routes $M_a$ . Show that for any coloring of the $M$ routes with two colors there are three cities of $M$ joined by three routes of the same color.
2011 Rioplatense Mathematical Olympiad, Level 3
We consider $\Gamma_1$ and $\Gamma_2$ two circles that intersect at points $P$ and $Q$ .  Let $A , B$ and $C$ be points on the circle $\Gamma_1$ and $D , E$ and $F$ points on the circle $\Gamma_2$ so that the lines $A E$ and $B D$ intersect at $P$ and the lines $A F$ and $C D$ intersect at $Q$. Denote   $M$ and $N$ the intersections of lines $A B$ and $D E$ and of lines $A C$ and $D F$ , respectively. Show that $A M D N$ is a parallelogram.