Navvye/Olympiad-Problems-CSV huggingface dataset:
1972 Vietnam National Olympiad
Let $\alpha$ be an arbitrary angle and let $x = cos\alpha, y = cosn\alpha$ ($n \in Z$).

i) Prove that to each value $x \in [-1, 1]$ corresponds one and only one value of $y$.

Thus we can write $y$ as a function of $x, y = T_n(x)$.

Compute $T_1(x), T_2(x)$ and prove that $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$.

From this it follows that $T_n(x)$ is a polynomial of degree $n$.

ii) Prove that the polynomial $T_n(x$) has $n$ distinct roots in $[-1, 1]$.
1972 Vietnam National Olympiad
$ABC$ is a triangle. $U$ is a point on the line $BC$. $I$ is the midpoint of $BC$. The line through $C$ parallel to $AI$ meets the line $AU$ at $E$. The line through $E$ parallel to $BC$ meets the line $AB$ at $F$. The line through $E$ parallel to $AB$ meets the line $BC$ at $H$. The line through $H$ parallel to $AU$ meets the line $AB$ at $K$. The lines $HK$ and $FG$ meet at $T. V$ is the point on the line $AU$ such that $A$ is the midpoint of $UV$. Show that $V, T$ and $I$ are collinear.
1972 Vietnam National Olympiad
Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$  in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.
1971 Vietnam National Olympiad
Consider positive integers $m <n,p < q$ such that $(m, n) = 1, (p, q) = 1$ and satisfy the condition that if $\frac{m}{n}= tan\alpha$ and $\frac{p}{q} = tan\beta$, then $\alpha + \beta = 45^o$.

i) Given $m, n$, find $p, q$.

ii) Given $n, q$, find $m, p$.

ii) Given $m, q$, find $n, p$.
1971 Vietnam National Olympiad
$ABCDA'B'C'D'$ is a cube (with $ABCD$ and $A'B'C'D'$ faces, and $AA', BB', CC', DD'$ edges). $L$ is a line which intersects or is parallel to the lines $AA', BC$ and $DB'$. $L$ meets the line $BC$ at $M$ (which may be the point at infinity). Let $m = |BM|$. The plane $MAA'$ meets the line $B'C'$ at $E$. Show that $|B'E| = m$. The plane $MDB'$ meets the line $A'D'$ at $F$.

Show that $|D'F| = m$.

Hence or otherwise show how to construct the point $P$ at the intersection of $L$ and the plane $A'B'C'D'$.

Find the distance between $P$ and the line $A'B'$ and the distance between $P$ and the line $A'D'$ in terms of $m$.

Find a relation between these two distances that does not depend on $m$.

Find the locus of $M$.

Let $S$ be the envelope of the line $L$ as $M$ varies. Find the intersection of $S$ with the faces of the cube.
2013 Albania Team Selection Test
Find the 3-digit number whose ratio with the sum of its digits it's minimal.
2013 Albania Team Selection Test
Let $a,b,c,d$  be positive real numbers such that $abcd=1$.Find with proof that $x=3 $ is the minimal value for which the following inequality holds:

$$a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}$$
2013 Albania Team Selection Test
Solve the function $f: \Re \to \Re$:

$$ f( x^{3} )+ f(y^{3}) = (x+y)(f(x^{2} )+f(y^{2} )-f(xy))$$
2013 Albania Team Selection Test
It is given a triangle $ABC$ whose circumcenter is $O$ and orthocenter $H$.

If $AO=AH$ find the angle $\hat{BAC}$ of that triangle.
2013 Albania Team Selection Test
Let $k$ be a natural number.Find all the couples of natural numbers $(n,m)$ such that :

$(2^k)!=2^n*m$
2012 Albania Team Selection Test
Find the greatest value of the expression $$ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} $$ where $x$, $y$, $z$ are nonnegative real numbers such that $x+y+z=1$.
2012 Albania Team Selection Test
It is given an acute triangle  $ABC$ , $AB \neq AC$  where the feet of altitude from $A$ its $H$.  In the extensions of the sides $AB$ and $AC$  (in the direction of $B$ and $C$) we take the points $P$ and $Q$ respectively such that $HP=HQ$ and the points $B,C,P,Q$ are concyclic.

Find the ratio $\tfrac{HP}{HA}$.
2012 Albania Team Selection Test
It is given the equation $x^4-2ax^3+a(a+1)x^2-2ax+a^2=0$.

a) Find the greatest value of $a$, such that this equation has at least one real root.

b) Find all the values of $a$, such that the equation has at least one real root.
2012 Albania Team Selection Test
Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that

$$\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.$$
2012 Albania Team Selection Test
Let $f:\mathbb R^+ \to \mathbb R^+$ be a function such that: $$

	x,y > 0 \qquad f(x+f(y)) = yf(xy+1).

$$

a) Show that $(y-1)*(f(y)-1) \le 0$ for $y>0$.

b) Find all such functions that require the given condition.
2011 Albania Team Selection Test
The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.
2011 Albania Team Selection Test
The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.
2011 Albania Team Selection Test
In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$.



(a) Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$.



(b) If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.
2011 Albania Team Selection Test
Find all prime numbers  p such that $2^p+p^2 $ is also a prime number.
2011 Albania Team Selection Test
The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20$ students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of $4,7$ or $10$ students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?
2010 Albania Team Selection Test
$ABC$  is an acute angle triangle such that $AB>AC$ and $\hat{BAC}=60^{\circ}$. Let's denote by $O$ the center of the circumscribed circle of the triangle and $H$ the intersection of altitudes of this triangle. Line $OH$ intersects $AB$ in point $P$ and $AC$ in point $Q$. Find the value of the ration $\frac{PO}{HQ}$.
2010 Albania Team Selection Test
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$,

$(1+f(x)f(y))f(x+y)=f(x)+f(y)$.
2010 Albania Team Selection Test
One point of the plane is called $rational$ if both coordinates are rational and $irrational$ if both coordinates are irrational. Check whether the following statements are true or false:

a) Every point of the plane is in a line that can be defined by $2$ rational points.

b) Every point of the plane is in a line that can be defined by $2$ irrational points.



This maybe is not algebra so sorry if I putted it in the wrong category!
2010 Albania Team Selection Test
With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different  prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?
2010 Albania Team Selection Test
a) Let's consider a finite number of big circles of a sphere that do not pass all from a point. Show that there exists such a point that is found only in two of the circles. (With big circle we understand the circles with radius equal to the radius of the sphere.)

b) Using the result of part $a)$ show that, for a set of $n$ points in a plane, that are not all in a line, there exists a line that passes through only two points of the given set.
2009 Albania Team Selection Test
An equilateral triangle has inside it a point with distances 5,12,13 from the vertices . Find its side.
2009 Albania Team Selection Test
Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $  with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$
2009 Albania Team Selection Test
Two people play a game as follows: At the beginning both of them have one point and in every move, one of them can double it's points, or when the other have more point than him, subtract to him his points. Can the two competitors have 2009 and 2002 points respectively? What about 2009 and 2003? Generally  which couples of points can they have?
2009 Albania Team Selection Test
Find all the natural numbers $m,n$ such that $1+5 \cdot 2^m=n^2$.
2017 BMO TST
Given $n$ numbers different from $0$, ($n \in \mathbb{N}$) which are arranged randomly. We do the following operation: Choose some consecutive numbers in the given order and change their sign (i.e. $x \rightarrow -x$). What is the minimum number of operations needed, in order to make all the numbers positive for any given initial configuration of the $n$ numbers?
2017 BMO TST
Given a random positive integer $N$. Prove that there exist infinitely many positive integers $M$ whose none of its digits is $0$ and such that the sum of the digits of $N \cdot M$ is same as sum of digits $M$.
2017 BMO TST
Find all functions $f : \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that : $f(x)f(y)f(z)=9f(z+xyf(z))$, where $x$, $y$, $z$, are three positive real numbers.
2017 BMO TST
The incircle of $ \triangle A_{0}B_{0}C_{0}$, meets legs $B_{0}C_{0}$, $C_{0}A_{0}$, $A_{0}B_{0}$, respectively on points  $A$, $B$, $C$, and the incircle of $ \triangle ABC$, with center $I$, meets legs $BC$, $CA$, $AB$, on points $A_{1}$, $B_{1}$, $C_{1}$, respectively. We write with $ \sigma (ABC)$, and $ \sigma (A_{1}B_{1}C_{1})$ the areas of $ \triangle ABC$, and $ \triangle A_{1}B_{1}C_{1}$ respectively. Prove that if $ \sigma (ABC)=2 \sigma (A_{1}B_{1}C_{1})$, then lines $AA_{0}$, $BB_{0}$, $CC_{0}$ are concurrent.
2017 BMO TST
Given a set $A$ which contains $n$ elements. For any two distinct subsets $A_{1}$, $A_{2}$ of the given set $A$, we fix the number of elements of $A_1 \cap A_2$. Find the sum of all the numbers obtained in the described way.
2014 BMO TST
Prove that for $n\ge 2$ the following inequality holds:

$$\frac{1}{n+1}\left(1+\frac{1}{3}+\ldots +\frac{1}{2n-1}\right) >\frac{1}{n}\left(\frac{1}{2}+\ldots+\frac{1}{2n}\right).$$
2014 BMO TST
Solve the following equation in $\mathbb{R}$:

$$\left(x-\frac{1}{x}\right)^\frac{1}{2}+\left(1-\frac{1}{x}\right)^\frac{1}{2}=x.$$
2014 BMO TST
From the point $P$ outside a circle $\omega$ with center $O$ draw the tangents $PA$ and $PB$ where $A$ and $B$ belong to $\omega$.In a random point $M$ in the chord $AB$ we draw the perpendicular to $OM$, which intersects $PA$ and $PB$ in $C$ and $D$. Prove that $M$ is the midpoint $CD$.
2014 BMO TST
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.
2014 BMO TST
Find all non-negative integers $k,n$ which satisfy $2^{2k+1} + 9\cdot 2^k+5=n^2$.
2010 BMO TST
a) Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number?

b) Prove that every prime factor of  $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j+1$ where $ j$ is a natural number.
2010 BMO TST
Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2-ax+1=0$ we build the sequence with $ S_{n}=x_{1}^n + x_{2}^n$.

a)Prove that the sequence $ \frac{S_{n}}{S_{n+1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing.

b)Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}+\cdots +\frac{S_{n}}{S_{n+1}}>n-1$
2010 BMO TST
Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB=a$ and $ CD=c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.
2010 BMO TST
Let's consider the inequality $ a^3+b^3+c^3<k(a+b+c)(ab+bc+ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number.

a) Prove the inequality for $ k=1$.

b) Find the smallest value of $ k$ such that the inequality holds for all triangles.
2009 BMO TST
Given the equation $x^4-x^3-1=0$

(a) Find the number of its real roots.

(b) We denote by $S$ the sum of the real roots and by $P$ their product. Prove that $P< - \frac{11}{10}$ and $S> \frac {6}{11}$.
2009 BMO TST
Let $C_{1}$ and $C_{2}$ be concentric circles, with $C_{2}$ in the interior of $C_{1}$. From a point $A$ on $C_{1}$, draw the tangent $AB$ to $C_{2}$ $(B \in C_{2})$. Let $C$ be the second point of intersection of $AB$ and $C_{1}$,and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects $C_{2}$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.



This question is taken from Mathematical Olympiad Challenges , the 9-th exercise in 1.3 Power of a Point.
2009 BMO TST
For the give functions in $\mathbb{N}$:

(a) Euler's $\phi$ function ($\phi(n)$- the number of natural numbers smaller than $n$ and coprime with $n$);

(b) the $\sigma$ function such that the $\sigma(n)$ is the sum of natural divisors of $n$.

solve the equation $\phi(\sigma(2^x))=2^x$.
2009 BMO TST
Find all the polynomials $P(x)$ of a degree $\leq n$ with real non-negative coefficients such that $P(x) \cdot P(\frac{1}{x}) \leq [P(1)]^2$ , $ \forall x>0$.
2014 Argentina Cono Sur TST
A positive integer $N$ is written on a board. In a step, the last digit $c$ of the number on the board is erased, and after this, the remaining number $m$ is erased and replaced with $|m-3c|$ (for example, if the number $1204$ is on the board, after one step, we will replace it with $120 - 3 \cdot 4 = 108$).

We repeat this until the number on the board has only one digit. Find all positive integers $N$ such that after a finite number of steps, the remaining one-digit number is $0$.
2014 Argentina Cono Sur TST
The numbers $1$ through $9$ are written on a $3 \times 3$ board, without repetitions. A valid operation is to choose a row or a column of the board, and replace its three numbers $a, b, c$ (in order, i.e., the first number of the row/column is $a$, the second number of the row/column is $b$, the third number of the row/column is $c$) with either the three non-negative numbers $a-x, b-x, c+x$ (in order) or with the three non-negative numbers $a+x, b-x, c-x$ (in order), where $x$ is a real positive number which may vary in each operation .



a) Determine if there is a way of getting all $9$ numbers on the board to be the same, starting with the following board:



$\begin{array}{|c|c|c|c|c|c|c|c|} \hline 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \\ \hline \end{array}$



b) For all posible configurations such that it is possible to get all $9$ numbers to be equal to a number $m$ using the valid operations, determine the maximum value of $m$.
2014 Argentina Cono Sur TST
All diagonals of a convex pentagon are drawn, dividing it in one smaller pentagon and $10$ triangles. Find the maximum number of triangles with the same area that may exist in the division.
2014 Argentina Cono Sur TST
Find all pairs of positive prime numbers $(p,q)$ such that



$p^5+p^3+2=q^2-q$
2014 Argentina Cono Sur TST
In an acute triangle $ABC$, let $D$ be a point in $BC$ such that $AD$ is the angle bisector of $\angle{BAC}$. Let $E \neq B$ be the point of intersection of the circumcircle of triangle $ABD$ with the line perpendicular to $AD$ drawn through $B$. Let $O$ be the circumcenter of triangle $ABC$. Prove that $E$, $O$, and $A$ are collinear.
2014 Argentina Cono Sur TST
$120$ bags with $100$ coins are placed on the floor. One bag has coins that weigh $9$ grams, the other bags have coins that weigh $10$ grams. One may place some coins (not necessarily from the same bag) on a weighing scale, but it will only properly display the weight if it is less than $1000$ grams. Determine the minimum number of times that the weighing scale may be used in order to identify the bag that has the $9$-gram coins.
2013 Argentina Cono Sur TST
$2000$ people are standing on a line. Each one of them is either a liar, who will always lie, or a truth-teller, who will always tell the truth. Each one of them says: "there are more liars to my left than truth-tellers to my right". Determine, if possible, how many people from each class are on the line.
2013 Argentina Cono Sur TST
If $ x\neq1$, $ y\neq1$, $ x\neq y$ and

$$ \frac{yz-x^{2}}{1-x}=\frac{xz-y^{2}}{1-y}$$

show that both fractions are equal to $ x+y+z$.
2013 Argentina Cono Sur TST
$1390$ ants are placed near a line, such that the distance between their heads and the line is less than $1\text{cm}$ and the distance between the heads of two ants is always larger than $2\text{cm}$. Show that there is at least one pair of ants such that the distance between their heads is at least $10$ meters (consider the head of an ant as point).
2013 Argentina Cono Sur TST
Show that  the number $\begin{matrix} \\ N= \end{matrix} \underbrace{44 \ldots 4}_{n} \underbrace{88 \ldots 8}_{n} - 1\underbrace{33 \ldots3 }_{n-1}2$ is a perfect square for all positive integers $n$.
2013 Argentina Cono Sur TST
Let $ABC$ be an equilateral triangle and $D$ a point on side $AC$. Let $E$ be a point on $BC$ such that $DE \perp BC$, $F$ on $AB$ such that $EF \perp AB$, and $G$ on $AC$ such that $FG \perp AC$. Lines $FG$ and $DE$ intersect in $P$. If $M$ is the midpoint of $BC$, show that $BP$ bisects $AM$.
2013 Argentina Cono Sur TST
Let $m\geq 4$ and $n\geq 4$. An integer is written on each cell of a $m \times n$ board. If each cell has a number equal to the arithmetic mean of some pair of numbers written on its neighbouring cells, determine the maximum amount of distinct numbers that the board may have.



Note: two neighbouring cells share a common side.
2012 Argentina Cono Sur TST
Sofía colours $46$ cells of a $9 \times 9$ board red. If Pedro can find a $2 \times 2$ square from the board that has $3$ or more red cells, he wins; otherwise, Sofía wins. Determine the player with the winning strategy.
2012 Argentina Cono Sur TST
Find all four-element sets of positive integers $\{w,x,y,z\}$ such that $w^x+w^y=w^z$.
2012 Argentina Cono Sur TST
$16$ people sit around a circular table. After some time, they all stand up and sit down in either the chair they were previously sitting on or on a chair next to it. Determine the number of ways that this can be done.



Note: two or more people cannot sit on the same chair.
2012 Argentina Cono Sur TST
Determine the number of positive integers $n \leq 1000$ such that the sum of the digits of $5n$ and the sum of the digits of $n$ are the same.
2012 Argentina Cono Sur TST
Let $ABC$ be a triangle, and $K$ and $L$ be points on $AB$ such that $\angle ACK = \angle KCL = \angle LCB$. Let $M$ be a point in $BC$ such that $\angle MKC = \angle BKM$. If $ML$ is the angle bisector of $\angle KMB$, find $\angle MLC$.
2012 Argentina Cono Sur TST
A large number of rocks are placed on a table. On each turn, one may remove some rocks from the table following these rules: on the first turn, only one rock may be removed, and on every subsequent turn, one may remove either twice as many rocks or the same number of rocks as they have discarded on the previous turn. Determine the minimum number of turns required to remove exactly $2012$ rocks from the table.
2009 Argentina Iberoamerican TST
In the vertexes of a regular $ 31$-gon there are written the numbers from $ 1$ to $ 31$, ordered increasingly, clockwise oriented.

We are allowed to perform an operation which consists in taking any three vertexes, namely the ones who have written $ a$,$ b$, and $ c$ and change them into $ c$, $ a-\frac{1}{10}$ and $ b+\frac{1}{10}$ respectively ( $ a$ becomes $ c$, $ b$ becomes $ a-\frac{1}{10}$ and $ c$ turns into $ b+\frac{1}{10}$

Prove that after applying several operations we can reach the state in which the numbers in the vertexes are the numbers from $ 1$ to $ 31$, ordered increasingly,anti-clockwise oriented.
2009 Argentina Iberoamerican TST
There are $ m + 1$ horizontal lines and $ m$ vertical lines on the plane so that $ m(m + 1)$ intersections are made.

A mark is placed at one of the $ m$ points of the lowest horizontal line.

2 players play the game of the following rules on this lines and points.



1. Each player moves a mark from a point to a point along the lines in turns.

2. The segment is erased after a mark moved along it.

3. When a player cannot make a move, then he loses.



Prove that the lead always wins the game.





PS I haven't found a student who solved it. There can be no one.
2009 Argentina Iberoamerican TST
Let $ ABC$ be an isosceles triangle with $ AC = BC.$ Its incircle touches $ AB$ in $ D$ and $ BC$ in $ E.$ A line distinct of $ AE$ goes through $ A$ and intersects the incircle  in $ F$ and $ G.$ Line $ AB$ intersects line $ EF$ and $ EG$ in $ K$ and $ L,$ respectively. Prove that $ DK = DL.$
2009 Argentina Iberoamerican TST
Find all positive integers $ (x,y)$ such that $ \frac{y^2x}{x+y}$ is a prime number
2009 Argentina Iberoamerican TST
Let $ a$ and $ k$ be positive integers. Let $ a_i$ be the sequence defined by

$ a_1 = a$ and

$ a_{n + 1} = a_n + k\pi(a_n)$

where

$ \pi(x)$ is the product of the digits of $ x$ (written in base ten)

Prove that we can choose $ a$ and $ k$ such that the infinite sequence $ a_i$ contains exactly $ 100$ distinct terms
2009 Argentina Iberoamerican TST
Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.
2008 Argentina Iberoamerican TST
We have $ 100$ equal cubes. Player $ A$ has to paint the faces of the cubes, each white or black, such that every cube has at least one face of each colour, at least $ 50$ cubes have more than one black face and at least $ 50$ cubes have more than one white face .



Player $ B$ has to place the coloured cubes in a table in a way that their bases form the frame that surrounds a $ 40*12$ rectangle. There are some faces that can not been seen because they are overlapped with other faces or based on the table, we call them invisible faces. On the other hand, the ones which can be seen are called visible faces. Prove that player $ B$ can always place the cubes in such a way that the number of visible faces is the the same as the number of invisible faces, despite the initial colouring of player $ A$



Note: It is easy to see that in the configuration, each cube has three visible faces and three invisible faces
2008 Argentina Iberoamerican TST
Two circunmferences $ \Gamma_1$ $ \Gamma_2$ intersect at $ A$ and $ B$

$ r_1$ is the tangent  from $ A$ to $ \Gamma_1$  and $ r_2$ is the tangent from $ B$ to $ \Gamma_2$

$ r_1 \cap r_2=C$

$ T= r_1 \cap \Gamma_2$ ($ T \neq A$)

We consider a point $ X$ in $ \Gamma_1$ which is distinct from $ A$ and $ B$.

$ XA \cap \Gamma_2 =Y$ ($ Y \neq A$)

$ YB \cap XC=Z$

Prove that $ TZ \parallel XY$
2008 Argentina Iberoamerican TST
Show that exists a sequence of $ 100$ terms such that:

1)Every term is a perfect square

2) every term is greater than the one before it ( it is strictly increasing)

3)Every two terms of the sequence are relative prime

4) The average between two consecutive terms is also a perfect square



Daniel
2008 Argentina Iberoamerican TST
Find all integers $ x$ such that $ x(x+1)(x+7)(x+8)$ is a perfect square

It's a nice problem ...hope you enjoy it!



Daniel
2008 Argentina Iberoamerican TST
The plane is divided into regions by $ n \ge 3$ lines, no two of which are parallel, and no three of which are concurrent. Some regions are coloured , in such a way that no two coloured regions share a common segment or half-line  of their borders. Prove that the number of coloured regions is at most $ \frac{n(n+1)}{3}$