Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Olympic Revenge
Let $n$ a positive integer. In a $2n\times 2n$ board,  $1\times n$ and $n\times 1$ pieces are arranged without overlap.

Call an arrangement maximal if it is impossible to put a new piece in the board without overlapping the previous ones.

Find the least $k$ such that there is a maximal arrangement that uses $k$ pieces.
2014 Olympic Revenge
Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that

$$n \mid a^{f(n)}-1.$$

Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.
2013 Olympic Revenge
Let $n$ to be a positive integer. A family $\wp$ of intervals $[i, j]$ with $0 \le i < j \le n$ and $i$, $j$ integers is considered happy if, for any $I_1 = [i_1, j_1] \in \wp$ and $I_2 = [i_2, j_2] \in \wp$ such that $I_1 \subset I_2$, we have $i_1 = i_2$ or $j_1 = j_2$.



Determine the maximum number of elements of a happy family.
2013 Olympic Revenge
Let $ABC$ to be an acute triangle. Also, let $K$ and $L$ to be the two intersections of the perpendicular from $B$ with respect to side $AC$ with the circle of diameter $AC$, with $K$ closer to $B$ than $L$. Analogously, $X$ and $Y$ are the two intersections of the perpendicular from $C$ with respect to side $AB$ with the circle of diamter $AB$, with $X$ closer to $C$ than $Y$. Prove that the intersection of $XL$ and $KY$ lies on $BC$.
2013 Olympic Revenge
Let $a,b,c,d$ to be non negative real numbers satisfying $ab+ac+ad+bc+bd+cd=6$. Prove that



$$\dfrac{1}{a^2+1} + \dfrac{1}{b^2+1} + \dfrac{1}{c^2+1} + \dfrac{1}{d^2+1} \ge 2$$
2013 Olympic Revenge
Find all triples $(p,n,k)$ of positive integers, where $p$ is a Fermat's Prime, satisfying $$p^n + n = (n+1)^k$$.



Observation: a Fermat's Prime is a prime number of the form $2^{\alpha} + 1$, for $\alpha$ positive integer.
2013 Olympic Revenge
Consider $n$ lamps clockwise numbered from $1$ to $n$ on a circle.



Let $\xi$ to be a configuration where $0 \le \ell \le n$ random lamps are turned on. A cool procedure consists in perform, simultaneously, the following operations: for each one of the $\ell$ lamps which are turned on, we verify the number of the lamp; if $i$ is turned on, a signal of range $i$ is sent by this lamp, and it will be received only by the next $i$ lamps which follow $i$, turned on or turned off, also considered clockwise. At the end of the operations we verify, for each lamp, turned on or turned off, how many signals it has received. If it was reached by an even number of signals, it remains on the same state(that is, if it was turned on, it will be turned on; if it was turned off, it will be turned off). Otherwise, it's state will be changed.



The example in attachment, for $n=4$, ilustrates a configuration where lamps $2$ and $4$  are initially turned on. Lamp $2$ sends signal only for the lamps $3$ e $4$, while lamp $4$ sends signal for lamps $1$, $2$, $3$ e $4$. Therefore, we verify that lamps $1$ e $2$ received only one signal, while lamps $3$ e $4$ received two signals. Therefore, in the next configuration, lamps $1$ e $4$ will be turned on, while lamps $2$ e $3$ will be turned off.



Let $\Psi$ to be the set of all $2^n$ possible configurations, where $0 \le \ell \le n$ random lamps are turned on. We define a function $f: \Psi \rightarrow \Psi$ where, if $\xi$ is a configuration of lamps, then $f(\xi)$ is the configurations obtained after we perform the cool procedure described above.



Determine all values of $n$ for which $f$ is bijective.
2012 Olympic Revenge
Let $a$ and $b$ real numbers. Let $f:[a,b] \rightarrow \mathbb{R}$ a continuous function. We say that f is "smp" if $[a,b]=[c_0,c_1]\cup[c_1,c_2]...\cup[c_{n-1},c_n]$ satisfying $c_0<c_1...<c_n$ and for each $i\in\{0,1,2...n-1\}$:

$c_i<x<c_{i+1} \Rightarrow f(c_i)<f(x)<f(c_{i+1})$

or

$c_i>x>c_{i+1} \Rightarrow f(c_i)>f(x)>f(c_{i+1})$



Prove that if $f:[a,b] \rightarrow \mathbb{R}$ is continuous such that for each $v\in\mathbb{R}$ there are only finitely many $x$ satisfying $f(x)=v$, then $f$ is "smp".
2012 Olympic Revenge
We define $(x_1, x_2, \ldots , x_n) \Delta (y_1, y_2, \ldots , y_n) = \left( \sum_{i=1}^{n}x_iy_{2-i}, \sum_{i=1}^{n}x_iy_{3-i}, \ldots , \sum_{i=1}^{n}x_iy_{n+1-i} \right)$, where the indices are taken modulo $n$.



Besides this, if $v$ is a vector, we define $v^k = v$, if $k=1$, or $v^k = v \Delta v^{k-1}$, otherwise.



Prove that, if $(x_1, x_2, \ldots , x_n)^k = (0, 0, \ldots , 0)$, for some natural number $k$, then $x_1 = x_2 = \ldots = x_n = 0$.
2012 Olympic Revenge
Let $G$ be a finite graph. Prove that one can partition $G$ into two graphs $A \cup B=G$ such that if we erase all edges conecting a vertex from $A$ to a vertex from $B$, each vertex of the new graph has even degree.
2012 Olympic Revenge
Say that two sets of positive integers $S, T$ are $\emph{k-equivalent}$ if the sum of the $i$th powers of elements of $S$ equals the sum of the $i$th powers of elements of $T$, for each $i= 1, 2, \ldots, k$.  Given $k$, prove that there are infinitely many numbers $N$ such that $\{1,2,\ldots,N^{k+1}\}$ can be divided into $N$ subsets, all of which are $k$-equivalent to each other.
2012 Olympic Revenge
Let $x_1,x_2,\ldots ,x_n$ positive real numbers. Prove that:

$$\sum_{cyc} \frac{1}{x_i^3+x_{i-1}x_ix_{i+1}} \le \sum_{cyc} \frac{1}{x_ix_{i+1}(x_i+x_{i+1})}$$
2012 Olympic Revenge
Let $ABC$ be an scalene triangle and $I$ and $H$ its incenter, ortocenter respectively.

The incircle touchs $BC$, $CA$ and $AB$ at $D,E$ an $F$. $DF$ and $AC$ intersects at $K$ while $EF$ and $BC$ intersets at $M$.

Shows that $KM$ cannot be paralel to $IH$.



PS1: The original problem without the adaptation apeared at the Brazilian Olympic Revenge 2011 but it was incorrect.

PS2:The Brazilian Olympic Revenge is a competition for teachers, and the problems are created by the students.



Sorry if I had some English mistakes here.
2011 Olympic Revenge
Let $p, q, r, s, t \in \mathbb{R}^{*}_{+}$ satisfying:

i) $p^2 + pq + q^2 = s^2$

ii) $q^2 + qr + r^2 = t^2$

iii) $r^2 + rp + p^2 = s^2 - st + t^2$

Prove that

$$\frac{s^2 - st + t^2}{s^2t^2} = \frac{r^2}{q^2t^2} + \frac{p^2}{q^2s^2} - \frac{pr}{q^2ts}$$
2011 Olympic Revenge
Let $p$ be a fixed prime. Determine all the integers $m$, as function of $p$, such that there exist $a_1, a_2, \ldots, a_p \in \mathbb{Z}$ satisfying

$$m \mid a_1^p + a_2^p + \cdots + a_p^p - (p+1).$$
2011 Olympic Revenge
Let $E$ to be an infinite set of congruent ellipses in the plane, and $r$ a fixed line. It is known that each line parallel to $r$ intersects at least one ellipse belonging to $E$. Prove that there exist infinitely many triples of ellipses belonging to $E$, such that there exists a line that intersect the triple of ellipses.
2011 Olympic Revenge
Let $ABCD$ to be a quadrilateral inscribed in a circle $\Gamma$. Let $r$ and $s$ to be the tangents to $\Gamma$ through $B$ and $C$, respectively, $M$ the intersection between the lines $r$ and $AD$ and $N$ the intersection between the lines $s$ and $AD$. After all, let $E$ to be the intersection between the lines $BN$ and $CM$, $F$ the intersection between the lines $AE$ and $BC$ and $L$ the midpoint of $BC$. Prove that the circuncircle of the triangle $DLF$ is tangent to $\Gamma$.
2011 Olympic Revenge
Let $n \in \mathbb{N}$ and $z \in \mathbb{C}^{*}$. Prove that



$\left | n\textrm{e}^{z} - \sum_{j=1}^{n}\left (1+\frac{z}{j^2}\right )^{j^2}\right | < \frac{1}{3}\textrm{e}^{|z|}\left (\frac{\pi|z|}{2}\right)^2$.
2010 Olympic Revenge
Prove that the number of ordered triples $(x, y, z)$ such that $(x+y+z)^2  \equiv axyz \mod{p}$, where $gcd(a, p) = 1$ and $p$ is prime is $p^2 + 1$.
2010 Olympic Revenge
Joaquim, José and João participate of the worship of triangle $ABC$. It is well known that $ABC$ is a random triangle, nothing special. According to the dogmas of the worship, when they form a triangle which is similar to $ABC$, they will get immortal. Nevertheless, there is a condition: each person must represent a vertice of the triangle. In this case, Joaquim will represent vertice $A$, José vertice $B$ and João will represent vertice $C$. Thus, they must form a triangle which is similar to $ABC$, in this order.



Suppose all three points are in the Euclidean Plane. Once they are very excited to become immortal, they act in the following way: in each instant $t$, Joaquim, for example, will move with constant velocity $v$ to the point in the same semi-plan determined by the line which connects the other two points, and which would create a triangle similar to $ABC$ in the desired order. The other participants act in the same way.

If the velocity of all of them is same, and if they initially have a finite, but sufficiently large life, determine if they can get immortal.



Observation: Initially, Joaquim, José and João do not represent three collinear points in the plane
2010 Olympic Revenge
Prove that there exists a set $S$ of lines in the three dimensional space satisfying the following conditions:



$i)$ For each point $P$ in the space, there exist a unique line of $S$ containing $P$.



$ii)$ There are no two lines of $S$ which are parallel.
2010 Olympic Revenge
Let $a_n$ and $b_n$ to be two sequences defined as below:



$i)$ $a_1 = 1$



$ii)$ $a_n + b_n = 6n - 1$



$iii)$ $a_{n+1}$ is the least positive integer different of $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$.



Determine $a_{2009}$.
2010 Olympic Revenge
Secco and Ramon are drunk in the real line over the integer points $a$ and $b$, respectively. Our real line is a little bit special, though: the interval $(-\infty, 0)$ is covered by a sea of lava. Being aware of this fact, and also because they are drunk, they decided to play the following game: initially they choose an integer number $k>1$ using a fair dice as large as desired, and therefore they start the game. In the first round, each player writes the point $h$ for which it wants to go.



After that, they throw a coin: if the result is heads, they go to the desired points; otherwise, they go to the points $2g - h$, where $g$ is the point where each of the players were in the precedent round (that is, in the first round $g = a$ for Secco and $g = b$ for Ramon). They repeat this procedure in the other rounds, and the game finishes when some of the player is over a point exactly $k$ times bigger than the other (if both of the player end up in the point $0$, the game finishes as well).



Determine, in values of $k$, the initial values $a$ and $b$ such that Secco and Ramon has a winning strategy to finish the game alive.



Observation: If any of the players fall in the lave, he dies and both of them lose the game
2010 Olympic Revenge
Let $ABC$ to be a triangle and $\Gamma$ its circumcircle. Also, let $D, F, G$ and $E$, in this order, on the arc $BC$ which does not contain $A$ satisfying $\angle BAD = \angle CAE$ and $\angle BAF = \angle CAG$. Let $D`, F`, G`$ and $E`$ to be the intersections of $AD, AF, AG$ and $AE$ with $BC$, respectively. Moreover, $X$ is the intersection of $DF`$ with $EG`$, $Y$ is the intersection of $D`F$ with $E`G$, $Z$ is the intersection of $D`G$ with $E`F$ and $W$ is the intersection of $EF`$ with $DG`$.



Prove that $X, Y$ and $A$ are collinear, such as $W, Z$ and $A$. Moreover, prove that $\angle BAX = \angle CAZ$.
2009 Olympic Revenge
Given a scalene triangle $ABC$ with circuncenter $O$ and circumscribed circle $\Gamma$. Let $D, E ,F$ the midpoints of $BC, AC, AB$. Let $M=OE \cap AD$, $N=OF \cap AD$ and $P=CM \cap BN$. Let $X=AO \cap PE$, $Y=AP \cap OF$. Let $r$ the tangent of $\Gamma$  through $A$. Prove that $r, EF, XY$ are concurrent.
2009 Olympic Revenge
Prove that $\int_{0}^{\frac{\pi}{2}} arctg (1 - \sin^2x\cos^2x)dx = \frac{\pi^2}{4} - \pi arctg\sqrt{\frac{\sqrt{2}-1}{2}}$
2009 Olympic Revenge
Let $ABC$ to be a triangle with incenter $I$. $\omega_{A}$, $\omega_{B}$ and $\omega_{C}$ are the incircles of the triangles $BIC$, $CIA$ and $AIB$, repectively. After all, $T$ is the tangent point between $\omega_{A}$ and $BC$. Prove that the other internal common tangent to $\omega_{B}$ and $\omega_{C}$ passes through the point $T$.
2009 Olympic Revenge
Let $d_i(k)$ the number of divisors of $k$ greater than $i$.

Let $f(n)=\sum_{i=1}^{\lfloor \frac{n^2}{2} \rfloor}d_i(n^2-i)-2\sum_{i=1}^{\lfloor \frac{n}{2} \rfloor}d_i(n-i)$.

Find all $n \in N$ such that $f(n)$ is a perfect square.
2009 Olympic Revenge
Thin and Fat eat a pizza of $2n$ pieces. Each piece contains a distinct amount of olives between $1$ and $2n$. Thin eats the first piece, and the two players alternately eat a piece neighbor of an eaten piece. However, neither Thin nor Fat like olives, so they will choose pieces that minimizes the total amount of olives they eat. For each arrangement  $\sigma$ of the olives, let $s(\sigma)$ the minimal amount of olives that Thin can eat, considering that both play in the best way possible. Let $S(n)$ the maximum of $s(\sigma)$, considering all arrangements.

$a)$ Prove that $n^2-1+\lfloor \frac{n}{2} \rfloor \le S(n) \le n^2+\lfloor \frac{n}{2} \rfloor$

$b)$ Prove that $S(n)=n^2-1+\frac{n}{2}$ for each even n.
2009 Olympic Revenge
Let $a, n \in \mathbb{Z}^{*}_{+}$. $a$ is defined inductively in the base $n$-recursive. We first write $a$ in the base $n$, e.g., as a sum of terms of the form $k_tn^t$, with $0 \le k_t < n$. For each exponent $t$, we write $t$ in the base $n$-recursive, until all the numbers in the representation are less than $n$. For instance,



$1309 = 3^6 + 2.3^5 + 1.3^4 + 1.3^2 + 1.3 + 1$



$ = 3^{2.3} + 2.3^{3+2} + 1.3^{3+1} + 1.3^2 + 1$



Let $x_1 \in \mathbb{Z}$ arbitrary. We define $x_n$ recursively, as following: if $x_{n-1} > 0$, we write $x_{n-1}$ in the base $n$-recursive and we replace all the numbers $n$ for $n+1$ (even the exponents!), so we obtain the successor of $x_n$. If $x_{n-1} = 0$, then $x_n = 0$.



Example:



$x_1 = 2^{2^{2} + 2 + 1} + 2^{2+1} + 2 + 1$



$\Rightarrow x_2 = 3^{3^{3} + 3 + 1} + 3^{3+1} + 3$



$\Rightarrow x_3 = 4^{4^{4} + 4 + 1} + 4^{4+1} + 3$



$\Rightarrow x_4 = 5^{5^{5} + 5 + 1} + 5^{5+1} + 2$



$\Rightarrow x_5 = 6^{6^{6} + 6 + 1} + 6^{6+1} + 1$



$\Rightarrow x_6 = 7^{7^{7} + 7 + 1} + 7^{7+1}$



$\Rightarrow x_7 = 8^{8^{8} + 8 + 1} + 7.8^8 + 7.8^7 + 7.8^6 + ... + 7$



$.$



$.$



$.$



Prove that $\exists N : x_N = 0$.
2007 Olympic Revenge
Let $a$, $b$, $n$ be positive integers with $a,b > 1$ and $\gcd(a,b) = 1$. Prove that $n$ divides $\phi\left(a^{n}+b^{n}\right)$.
2007 Olympic Revenge
Let $a, b, c \in \mathbb{R}$ with $abc = 1$. Prove that

$$a^{2}+b^{2}+c^{2}+{1\over a^{2}}+{1\over b^{2}}+{1\over c^{2}}+2\left(a+b+c+{1\over a}+{1\over b}+{1\over c}\right) \geq 6+2\left({b\over a}+{c\over b}+{a\over c}+{c\over a}+{c\over b}+{b\over c}\right)$$
2007 Olympic Revenge
The triangles $BCD$ and $ACE$ are externally constructed to sides $BC$ and $CA$ of a triangle $ABC$ such that $AE = BD$ and $\angle BDC+\angle AEC = 180^\circ$. Let $F$ be a point on segment $AB$ such that ${AF\over FB}={CD\over CE}$. Prove that ${DE\over CD+CE}={EF\over BC}={FD\over AC}$.
2007 Olympic Revenge
Let $A_{1}A_{2}B_{1}B_{2}$ be a convex quadrilateral. At adjacent vertices $A_{1}$ and $A_{2}$ there are two Argentinian cities. At adjacent vertices $B_{1}$ and $B_{2}$ there are two Brazilian cities. There are $a$ Argentinian cities and $b$ Brazilian cities in the quadrilateral interior, no three of which collinear. Determine if it's possible, independently from the cities position, to build straight roads, each of which connects two Argentinian cities ou two Brazilian cities, such that:



$\bullet$ Two roads does not intersect in a point which is not a city;

$\bullet$ It's possible to reach any Argentinian city from any Argentinian city using the roads; and

$\bullet$ It's possible to reach any Brazilian city from any Brazilian city using the roads.



If it's always possible, construct an algorithm that builds a possible set of roads.
2007 Olympic Revenge
Find all functions $f\colon R \to R$ such that

$$f\left(x^{2}+yf(x)\right) = f(x)^{2}+xf(y)$$

for all reals $x,y$.
2007 Olympic Revenge
Mediovagio is a computer game that consists in a $3 \times 3$ table in which each of the nine cells has a integer number from $1$ to $n$. When one clicks a cell, the numbers in the clicked cell and in the cells that share an edge with it are increased by $1$ and the sum is evaluated${}\bmod n$. Determine the values of $n$ for which it's possible, with a finite number of clicks, obtain any combination of numbers from an given initial combination.



EDIT: I corrected the statement.
2005 Olympic Revenge
Let $S=\{1,2,3,\ldots,n\}$, $n$ an odd number. Find the parity of number of permutations $\sigma : S \Rightarrow S$ such that the sequence defined by $$a(i)=|\sigma(i)-i|$$ is monotonous.
2005 Olympic Revenge
Let $\Gamma$ be a circumference, and $A,B,C,D$ points of $\Gamma$ (in this order).



$r$ is the tangent to $\Gamma$ at point A.

$s$ is the tangent to $\Gamma$ at point D.



Let $E=r \cap BC,F=s \cap BC$.

Let $X=r \cap s,Y=AF \cap DE,Z=AB \cap CD$

Show that the points $X,Y,Z$ are collinear.



Note: assume the existence of all above points.
2005 Olympic Revenge
Find all functions $f: R \rightarrow R$ such that

$$f(x+yf(x))+f(xf(y)-y)=f(x)-f(y)+2xy$$

for all $x,y \in R$
2005 Olympic Revenge
Let A be a symmetric matrix such that the sum of elements of any row is zero.

Show that all elements in the main diagonal of cofator matrix of A are equal.
2005 Olympic Revenge
Find all sets X of points in a plane, not all collinear, such that:

For any two distinct circumferences, each contains three points of X, its intersection points are points of X.
2005 Olympic Revenge
Zé Roberto and Humberto are playing the Millenium Game!

There are 30 empty boxes in a queue, and each box have a capacity of one blue stome.

Each player takes a blue stone and places it in a box (and it is a move).

The winner is who, in its move, obtain three full consecutive boxes.



If Zé Roberto is the first player, who has the winner strategy?
2004 Olympic Revenge
$ABC$ is a triangle and $D$ is an internal point such that  $\angle DAB=\angle  DBC =\angle DCA$. $O_a$ is the circumcenter of $DBC$. $O_b$ is the circumcenter of $DAC$. $O_c$ is the circumcenter of $DAB$. Show that if the area of $ABC$ and $O_aO_bO_c$ are equal then $ABC$ is equilateral.
2004 Olympic Revenge
If $a,b,c,x$ are positive reals, show that

$$\frac{a^{x+2}+1}{a^xbc+1}+\frac{b^{x+2}+1}{b^xac+1}+\frac{c^{x+2}+1}{c^xab+1}\geq 3$$
2004 Olympic Revenge
$ABC$ is a triangle and $\omega$ its incircle. Let $P,Q,R$ be the intersections with $\omega$ and the sides $BC,CA,AB$ respectively. $AP$ cuts $\omega$ in $P$ and $X$. $BX,CX$ cut $\omega$ in $M,N$ respectively. Show that $MR,NQ,AP$ are parallel or concurrent.
2004 Olympic Revenge
Find all functions $f:R \rightarrow R$ such that for any reals $x,y$, $f(x^2+y)=f(x)f(x+1)+f(y)+2x^2y$.
2004 Olympic Revenge
$a_0 = a_1 = 1$ and ${a_{n+1} . a_{n-1}} = a_n . (a_n + 1)$ for all positive integers n.



prove that $a_n$ is one integer for all positive integers n.
2004 Olympic Revenge
For any natural $n$, $f(n)$ is the number of labeled digraphs with $n$ vertices such that for any vertex the number if in-edges is equal to the number of out-edges and the total of (in+out) edges is even. Let $g(n)$ be the odd-analogous of $f(n)$. Find $g(n)-f(n)$ with proof .



original formulationDado $n$ natural, seja $f(n)$ o número de grafos rotulados direcionados com $n$ vértices de modo que em cada vértice o número de arestas que chegam é igual ao número de

arestas que saem e o número de arestas total do grafo é par  . Defina $g(n)$ analogamente trocando "par" por "ímpar" na definição acima. Calcule $f(n) - g (n)$.



(Observação: Um grafo rotulado direcionado é um par $G = (V, E)$ onde $V = \{1, 2, …, n\}$ e $E$ é um subconjunto de $V^2 -\{(i, i); 0 < i < n + 1\}$).
2003 Olympic Revenge
Let $ABC$ be a triangle with circumcircle $\Gamma$. $D$ is the midpoint of arc $BC$ (this arc does not contain $A$). $E$ is the common point of $BC$ and the perpendicular bisector of $BD$. $F$ is the common point of $AC$ and the parallel to $AB$ containing $D$. $G$ is the common point of $EF$ and $AB$. $H$ is the common point of $GD$ and $AC$. Show that $GAH$ is isosceles.
2003 Olympic Revenge
Let $x_n$ the sequence defined by any nonnegatine integer $x_0$ and $x_{n+1}=1+\prod_{0 \leq i \leq n}{x_i}$

Show that there exists prime  $p$ such that $p\not|x_n$ for any $n$.
2003 Olympic Revenge
Let $ABC$ be a triangle with $\angle BAC =60^\circ$. $A'$ is the symmetric point of $A$ wrt $\overline{BC}$. $D$ is the point in $\overline{AC}$ such that $\overline{AB}=\overline{AD}$. $H$ is the orthocenter of triangle $ABC$. $l$ is the external angle bisector of $\angle BAC$. $\{M\}=\overline{A'D}\cap l$,$\{N\}=\overline{CH} \cap l$. Show that $\overline{AM}=\overline{AN}$.
2003 Olympic Revenge
In the Mobius Planet (a plane and infinite planet!,	in a similar manner to the $N \times N$ lattice), the Supreme King Mobius is planning to construct a water reservoir. There are some restrictions to this project:

1. There exists only $k < \infty$ bricks.

2. These bricks will delimit a closed finite area.

What is the maximum area of this resevoir in function of $k$?
2003 Olympic Revenge
Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number.

Find how many finite non-empty sets $S\in [p] \times [p]$ are such that 	$$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$
2003 Olympic Revenge
Find all functions $f:R^{*} \rightarrow R$ such that $f(x)\not = x$ and

$$ f(y(f(x)-x))=\frac{f(x)}{y}-\frac{f(y)}{x} $$for any $x,y \not = 0$.
2003 Olympic Revenge
Let $X$ be a subset of $R_{+}^{*}$ with $m$ elements.

Find $X$ such that the number of subsets with the same sum is maximum.
2002 Olympic Revenge
Show that there is no function \(f:\mathbb{N}^* \rightarrow \mathbb{N}^*\) such that \(f^n(n)=n+1\) for all \(n\) (when \(f^n\) is the \(n\)th iteration of \(f\))
2002 Olympic Revenge
\(ABCD\) is a inscribed quadrilateral.

\(P\) is the intersection point of its diagonals.

\(O\) is its circumcenter.

\(\Gamma\) is the circumcircle of \(ABO\).

\(\Delta\) is the circumcircle of \(CDO\).

\(M\) is the midpoint of arc \(AB\) on \(\Gamma\) who doesn't contain \(O\).

\(N\) is the midpoint of arc \(CD\) on \(\Delta\) who doesn't contain \(O\).



Show that \(M,N,P\) are collinear.
2002 Olympic Revenge
Show that if $x,y,z,w$ are positive reals, then



$$

\frac{3}{2}\sqrt{(x^2+y^2)(w^2+z^2)} + \sqrt{(x^2+w^2)(y^2+z^2) - 3xyzw} \geq (x+z)(y+w)

$$
2002 Olympic Revenge
Find all pairs \((m,n)\) of positive integers such that there exists a polyhedron, with all faces being regular polygons, such that each vertex of the polyhedron is the vertex of exactly three faces, two of them having \(m\) sides, and the other having \(n\) sides.
2002 Olympic Revenge
In a "Hanger Party", the guests are initially dressed. In certain moments, the host chooses a guest, and the chosen guest and all his friends will wear its respective clothes if they are naked, and undress it if they are dressed.



It is possible that, in some moment, the guests are naked, independent of their mutual friendships? (Suppose friendship is reciprocal.)
2002 Olympic Revenge
Let \(p\) a prime number, and \(N\) the number of matrices \(p \times p\)



$$\begin{array}{cccc}

a_{11} & a_{12} & \ldots & a_{1p}\\
a_{21} & a_{22} & \ldots & a_{2p}\\
\vdots & \vdots & \ddots & \vdots \\
a_{p1} & a_{p2} & \ldots & a_{pp} 

\end{array}$$



such that \(a_{ij} \in \{0,1,2,\ldots,p\} \) and if \(i \leq i^\prime\) and \(j \leq j^\prime\), then \(a_{ij} \leq a_{i^\prime j^\prime}\).



Find \(N \pmod{p}\).
2002 Olympic Revenge
Show that



$$A_n=\prod_{j=0}^{n-1}\cfrac{(3j+1)!}{(n+j)!}$$



is an integer, for any positive integer \(n\).
2015 Paraguay Mathematical Olympiad
Alexa wrote the first $16$ numbers of a sequence:

$$1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …$$Then she continued following the same pattern, until she had $2015$ numbers in total.

What was the last number she wrote?
2015 Paraguay Mathematical Olympiad
Consider all sums that add up to $2015$. In each sum, the addends are consecutive positive integers, and all sums have less than $10$ addends. How many such sums are there?