Navvye/Olympiad-Problems-CSV huggingface dataset:
2017 Abels Math Contest (Norwegian MO) Final
Nils has a telephone number with eight different digits.

He has made $28$ cards with statements of the type “The digit $a$ occurs earlier than the digit $b$ in my telephone number” – one for each pair of digits appearing in his number.

How many cards can Nils show you without revealing his number?
2017 Abels Math Contest (Norwegian MO) Final
In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up.

Every other time, Niels picks a triangle and writes $\times$ in it, and every other time, Henrik picks a triangle where he writes a $o$. If one of the players gets four in a row in some direction (see figure), he wins the game.

Determine whether one of the players can force a victory.
2017 Abels Math Contest (Norwegian MO) Final
Let $a > 0$ and $0 < \alpha <\pi$ be given. Let $ABC$ be a triangle with $BC = a$ and $\angle BAC = \alpha$ , and call the cicumcentre $O$, and the orthocentre $H$. The point $P$ lies on the ray from $A$ through $O$. Let $S$ be the mirror image of $P$ through $AC$, and $T$ the mirror image of $P$ through $AB$. Assume that $SATH$ is cyclic. Show that the length $AP$ depends only on $a$ and $\alpha$.
2018 Abels Math Contest (Norwegian MO) Final
For an odd number n, we write $n!! = n\cdot (n-2)...3 \cdot 1$.

How many different residues modulo $1000$ do you get from $n!!$  for  $n= 1, 3, 5, …$?
2018 Abels Math Contest (Norwegian MO) Final
The circumcentre of a triangle $ABC$ is called $O$.

The points $A',B'$ and $C'$ are the reflections of $O$ in $BC, CA$, and $AB$, respectively.

Show that the three lines $AA' , BB'$, and $CC'$ meet in a common point.
2018 Abels Math Contest (Norwegian MO) Final
Find all polynomials $P$ such that $P(x)+3P(x+2)=3P(x+1)+P(x+3)$ for all real numbers $x$.
2018 Abels Math Contest (Norwegian MO) Final
Find all real functions  $f$ defined on the real numbers except zero, satisfying

$f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$
2018 Abels Math Contest (Norwegian MO) Final
Find all polynomials $P$ such that

$P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$

$=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$

for all real numbers $x$.
2019 Abels Math Contest (Norwegian MO) Final
You have an $n \times n$ grid of empty squares. You place a cross in all the squares, one at a time. When you place a cross in an empty square, you receive  $i+j$ points if there were $i$ crosses in the same row and $j$ crosses in the same column before you placed the new cross. Which are the possible total scores you can get?
2019 Abels Math Contest (Norwegian MO) Final
$find$ all non negative integers $m$, $n$ such that $mn-1$ divides $n^3-1$
2019 Abels Math Contest (Norwegian MO) Final
Three circles are pairwise tangent, with none of them lying inside another.

The centres of the circles are the corners of a triangle with circumference $1$.

What is the smallest possible value for the sum of the areas of the circles?
2019 Abels Math Contest (Norwegian MO) Final
The diagonals of a convex quadrilateral $ABCD$ intersect at $E$. The triangles $ABE, BCE, CDE$ and $DAE$ have centroids $K,L,M$ and $N$, and orthocentres  $Q,R,S$ and $T$. Show that the quadrilaterals $QRST$ and $LMNK$ are similar.
2020 Abels Math Contest (Norwegian MO) Final
In how many ways can the circles be coloured using three colours, so that no two circles connected by a line segment have the same colour?
2020 Abels Math Contest (Norwegian MO) Final
A round table has room for n diners ( $n\ge  2$). There are napkins in three different colours. In how many ways can the napkins be placed, one for each seat, so that no two neighbours get napkins of the same colour?
2020 Abels Math Contest (Norwegian MO) Final
Find all natural numbers  $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$

Note that we do not consider $0$ to be a natural number.
2020 Abels Math Contest (Norwegian MO) Final
Assume that $a$ and $b$ are natural numbers with $a \ge b$ so that $ \sqrt{a+\sqrt{a^2-b^2}}$ is a natural number. Show that $a$ and $b$ have the same parity.
2020 Abels Math Contest (Norwegian MO) Final
Show that the equation $x^2 \cdot (x - 1)^2 \cdot (x - 2)^2 \cdot ... \cdot (x -  1008)^2 \cdot (x- 1009)^2 =  c$ has $2020$ real solutions, provided $0 < c <\frac{(1009 \cdot1007 \cdot  ... \cdot 3\cdot 1)^4}{2^{2020}}$ .
2020 Abels Math Contest (Norwegian MO) Final
The midpoint of the side $AB$ in the triangle $ABC$ is called $C'$. A point on the side $BC$ is called $D$, and $E$ is the point of intersection of $AD$ and $CC'$. Assume that $AE/ED = 2$. Show that $D$ is the midpoint of $BC$.
2020 Abels Math Contest (Norwegian MO) Final
The triangle $ABC$ has a right angle at $A$. The centre of the circumcircle is called $O$, and the base point of the normal from $O$ to $AC$ is called $D$. The point $E$ lies on $AO$ with $AE = AD$. The angle bisector of $\angle CAO$ meets $CE$ in $Q$. The lines $BE$ and $OQ$ intersect in $F$. Show that the lines $CF$ and $OE$ are parallel.
2021 Abels Math Contest (Norwegian MO) Final
A $3n$-table is a table with three rows and $n$ columns containing all the numbers $1, 2, …, 3n$. Such a table is called tidy  if the $n$ numbers in the first row appear in ascending order from left to right, and the three numbers in each column appear in ascending order from top to bottom. How many tidy $3n$-tables exist?
2021 Abels Math Contest (Norwegian MO) Final
Pål has more chickens than he can manage to keep track of. Therefore, he keeps an index card for each chicken. He keeps the cards in ten boxes, each of which has room for $2021$ cards.

Unfortunately, Pål is quite disorganized, so he may lose some of his boxes. Therefore, he makes several copies of each card and distributes them among different boxes, so that even if he can only find seven boxes, no matter which seven, these seven boxes taken together will contain at least one card for each of his chickens.

What is the largest number of chickens Pål can keep track of using this system?
2021 Abels Math Contest (Norwegian MO) Final
Show that for all $n\ge 3$ there are $n$ different positive integers $x_1,x_2, ...,x_n$ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= 1.$$
2021 Abels Math Contest (Norwegian MO) Final
If $a_1,\cdots,a_n$ and $b_1,\cdots,b_n$ are real numbers satisfying $a_1^2+\cdots+a_n^2 \le 1$ and $b_1^2+\cdots+b_n^2 \le 1$ , show that:

$$(1-(a_1^2+\cdots+a_n^2))(1-(b_1^2+\cdots+b_n^2)) \le (1-(a_1b_1+\cdots+a_nb_n))^2$$
2021 Abels Math Contest (Norwegian MO) Final
For which integers $0 \le k \le 9$ do there exist positive integers $m$ and $n$ so that the number $3^m + 3^n + k$ is a perfect square?
2021 Abels Math Contest (Norwegian MO) Final
We say that a set $S$ of natural numbers is synchronous  provided that the digits of $a^2$ are the same (in occurence and numbers, if differently ordered) for all numbers $a$ in $S$. For example, $\{13, 14, 31\}$ is synchronous, since we find $\{13^2, 14^2, 31^2\} = \{169, 196, 961\}$. But $\{119, 121\}$ is not synchronous, for even though $119^2 = 14161$ and $121^2 = 14641$ have the same digits, they occur in different numbers. Show that there exists a synchronous set containing $2021$ different natural numbers.
2021 Abels Math Contest (Norwegian MO) Final
A tetrahedron $ABCD$ satisfies $\angle BAC=\angle CAD=\angle DAB=90^o$. Show that the areas of its faces satisfy the equation $area(BAC)^2 + area(CAD)^2 + area(DAB)^2 = area(BCD)^2$.

.
2021 Abels Math Contest (Norwegian MO) Final
The tangent at $C$ to the circumcircle of triangle $ABC$ intersects the line through $A$ and $B$ in a point $D$. Two distinct points $E$ and $F$ on the line through $B$ and $C$ satisfy $|BE| = |BF | =\frac{||CD|^2 - |BD|^2|}{|BC|}$. Show that either $|ED| = |CD|$ or $|FD| = |CD|$.
2021 Olympic Revenge
Let $a$, $b$, $c$, $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$. Prove that:



$$\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.$$

Proposed by Zhang Yanzong and Song Qing
2021 Olympic Revenge
Evan is a $n$-dimensional being that lives in a house formed by the points of $\mathbb{Z}_{\geq 0}^n$. His room is the set of points in which coordinates are all less than or equal to $2021$.  Evan's room has been infested with bees, so he decides to flush them out through $\textit{captures}$. A $\textit{capture}$ can be performed by eliminating a bee from point $ (a_1, a_2, \ldots, a_n) $ and replacing it with $ n $ bees, one in each of the points: $$ (a_1 + 1, a_2 , \ldots, a_n), (a_1, a_2 + 1, \ldots, a_n), \ldots, (a_1, a_2, \ldots, a_n + 1) $$However, two bees can never occupy the same point in the house. Determine, for every $ n $, the greatest value $ A (n) $ of bees such that, for some initial arrangement of these bees in Evan's room, he is able to accomplish his goal with a finite amount of $\textit{captures}$.
2021 Olympic Revenge
Let $I, C, \omega$ and $\Omega$ be the incenter, circumcenter, incircle and circumcircle, respectively, of the scalene triangle  $XYZ$ with $XZ > YZ > XY$. The incircle $\omega$ is tangent to the sides $YZ, XZ$ and $XY$ at the points $D, E$ and $F$. Let $S$ be the point on $\Omega$ such that $XS, CI$ and $YZ$ are concurrent. Let $(XEF) \cap \Omega = R$, $(RSD) \cap (XEF) = U$, $SU \cap CI = N$, $EF \cap YZ = A$, $EF \cap CI = T$ and $XU \cap YZ = O$.



Prove that $NARUTO$ is cyclic.
2021 Olympic Revenge
On a chessboard, Po controls a white queen and plays, in alternate turns, against an invisible black king (there are only those two pieces on the board). The king cannot move to a square where he would be in check, neither capture the queen. Every time the king makes a move, Po receives a message from beyond that tells which direction the king has moved (up, right, up-right, etc). His goal is to make the king unable to make a movement.



Can Po reach his goal with at most $150$ moves, regardless the starting position of the pieces?
2021 Olympic Revenge
Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.
2020 Olympic Revenge
Let $n$ be a positive integer and $a_1, a_2, \dots, a_n$ non-zero real numbers. What is the least number of non-zero coefficients that the polynomial $P(x) = (x - a_1)(x - a_2)\cdots(x - a_n)$ can have?
2020 Olympic Revenge
For a positive integer $n$, we say an $n$-shuffling is a bijection $\sigma: \{1,2, \dots , n\} \rightarrow \{1,2, \dots , n\}$ such that there exist exactly two elements $i$ of $\{1,2, \dots , n\}$ such that $\sigma(i) \neq i$.



Fix some three pairwise distinct $n$-shufflings $\sigma_1,\sigma_2,\sigma_3$. Let $q$ be any prime, and let $\mathbb{F}_q$ be the integers modulo $q$. Consider all functions $f:(\mathbb{F}_q^n)^n\to\mathbb{F}_q$ that satisfy, for all integers $i$ with $1 \leq i \leq n$ and all $x_1,\ldots x_{i-1},x_{i+1}, \dots ,x_n, y, z\in\mathbb{F}_q^n$, $$f(x_1, \ldots ,x_{i-1}, y, x_{i+1}, \ldots , x_n) +f(x_1, \ldots ,x_{i-1}, z, x_{i+1}, \ldots , x_n) = f(x_1, \ldots ,x_{i-1}, y+z, x_{i+1}, \ldots , x_n), $$and that satisfy, for all $x_1,\ldots,x_n\in\mathbb{F}_q^n$ and all $\sigma\in\{\sigma_1,\sigma_2,\sigma_3\}$, $$f(x_1,\ldots,x_n)=-f(x_{\sigma(1)},\ldots,x_{\sigma(n)}).$$

For a given tuple $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$, let $g(x_1,\ldots,x_n)$ be the number of different values of $f(x_1,\ldots,x_n)$ over all possible functions $f$ satisfying the above conditions.



Pick $(x_1,\ldots,x_n)\in(\mathbb{F}_q^n)^n$ uniformly at random, and let $\varepsilon(q,\sigma_1,\sigma_2,\sigma_3)$ be the expected value of $g(x_1,\ldots,x_n)$. Finally, let $$\kappa(\sigma_1,\sigma_2,\sigma_3)=-\lim_{q \to \infty}\log_q\left(-\ln\left(\frac{\varepsilon(q,\sigma_1,\sigma_2,\sigma_3)-1}{q-1}\right)\right).$$

Pick three pairwise distinct $n$-shufflings $\sigma_1,\sigma_2,\sigma_3$ uniformly at random from the set of all $n$-shufflings. Let $\pi(n)$ denote the expected value of $\kappa(\sigma_1,\sigma_2,\sigma_3)$. Suppose that $p(x)$ and $q(x)$ are polynomials with real coefficients such that $q(-3) \neq 0$ and such that $\pi(n)=\frac{p(n)}{q(n)}$ for infinitely many positive integers $n$. Compute $\frac{p\left(-3\right)}{q\left(-3\right)}$.
2020 Olympic Revenge
Let $ABC$ be a triangle and $\omega$ its circumcircle. Let $D$ and $E$ be the feet of the angle bisectors relative to $B$ and $C$, respectively. The line $DE$ meets $\omega$ at $F$ and $G$. Prove that the tangents to $\omega$ through $F$ and $G$ are tangents to the excircle of  $\triangle ABC$ opposite to $A$.
2020 Olympic Revenge
Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.
2020 Olympic Revenge
Let $n$ be a positive integer. Given $n$ points in the plane, prove that it is possible to draw an angle with measure $\frac{2\pi}{n}$ with vertex as each one of the given points, such that any point in the plane is covered by at least one of the angles.
2019 Olympic Revenge
Let $ABC$ be a scalene acute-angled triangle and $D$ be the point on its circumcircle such that $AD$ is a symmedian of triangle $ABC$. Let $E$ be the reflection of $D$ about $BC$, $C_0$ the reflection of $E$ about $AB$ and $B_0$ the reflection of $E$ about $AC$. Prove that the lines $AD$, $BB_0$ and $CC_0$ are concurrent if and only if $\angle BAC = 60^{\circ}.$
2019 Olympic Revenge
Prove that there exist infinitely many positive integers $n$ such that the greatest prime divisor of $n^2+1$ is less than $n \cdot \pi^{-2019}.$
2019 Olympic Revenge
Let $\Gamma$ be a circle centered at $O$ with radius $R$. Let $X$ and $Y$ be points on $\Gamma$ such that $XY<R$. Let $I$ be a point such that $IX = IY$ and $XY = OI$. Describe how to construct with ruler and compass a triangle which has circumcircle $\Gamma$, incenter $I$ and Euler line $OX$. Prove that this triangle is unique.
2019 Olympic Revenge
A regular icosahedron is a regular solid of $20$ faces, each in the form of an equilateral triangle, with $12$ vertices, so that each vertex is in $5$ edges.

Twelve indistinguishable candies are glued to the vertices of a regular icosahedron (one at each vertex), and four of these twelve candies are special. André and Lucas want to together create a strategy for the following game:

• First, André is told which are the four special sweets and he must remove exactly four sweets that are not special from the icosahedron and leave the solid on a table, leaving afterwards without communicating with Lucas.

• Later, Sponchi, who wants to prevent Lucas from discovering the special sweets, can pick up the icosahedron from the table and rotate it however he wants.

• After Sponchi makes his move, he leaves the room, Lucas enters and he must determine the four special candies out of the eight that remain in the icosahedron.

Determine if there is a strategy for which Lucas can always properly discover the four special sweets.
2019 Olympic Revenge
Define $f: \mathbb{N} \rightarrow \mathbb{N}$ by $$f(n) = \sum \frac{(1+\sum_{i=1}^{n} t_i)!}{(1+t_1) \cdot \prod_{i=1}^{n} (t_i!) }$$where the sum runs through all $n$-tuples such that $\sum_{j=1}^{n}j \cdot t_j=n$ and $t_j \ge 0$ for all $1 \le j \le n$.

Given a prime $p$ greater than $3$, prove that $$\sum_{1 \le i < j <k \le p-1 } \frac{f(i)}{i \cdot j \cdot k} \equiv \sum_{1 \le i < j <k \le p-1 } \frac{2^i}{i \cdot j \cdot k} \pmod{p}.$$
2018 Olympic Revenge
Let $(F_{n})_{n\geq1}$ the Fibonacci sequence. Find all $n \in \mathbb{N}$ such that for every $k=0,1,...,F_{n}$



$$ {F_{n}\choose k} \equiv (-1)^{k} \ (mod \ F_{n}+1) $$
2018 Olympic Revenge
Let $\triangle ABC$ a scalene triangle with incenter $I$, circumcenter $O$ and circumcircle $\Gamma$. The incircle of $\triangle ABC$ is tangent to $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. The line $AI$ meet $EF$ and $\Gamma$ at $N$ and $M\neq A$, respectively. $MD$ meet $\Gamma$ at $L\neq M$ and $IL$ meet $EF$ at $K$. The circumference of diameter $MN$ meet $\Gamma$ at $P\neq M$. Prove that $AK, PN$ and $OI$ are concurrent.
2018 Olympic Revenge
In a mathematical challenge, positive real numbers $a_{1}\geq a_{2} \geq ... \geq a_{n}$ and an initial sequence of positive real numbers $(b_{1}, b_{2},...,b_{n+1})$ are given to Secco. Let $C$ a non-negative real number. In a sequence $(x_{1},x_{2},...,x_{n+1})$, consider the following operation:

Subtract $1$ of some $x_{j}$, $j \in \{1,2,...,n+1\}$, add $C$ to $x_{n+1}$ and replace $(x_{1},x_{2},...,x_{j-1})$ for $(x_{1}+a_{\sigma (1)}, x_{2}+a_{\sigma (2)}, ..., x_{j-1}+a_{\sigma (j-1)})$, where $\sigma$ is a permutation of $(1,2,...,j-1)$.

Secco's goal is to make all terms of sequence $(b_{k})$ negative after a finite number of operations. Find all values of $C$, depending of $a_{1}, a_{2},..., a_{n}, b_{1}, b_{2}, ..., b_{n+1}$, for which Secco can attain his goal.
2018 Olympic Revenge
Let $\triangle ABC$ an acute triangle of incenter $I$ and incircle $\omega$. $\omega$ is tangent to $BC, CA$ and $AB$ at points $T_{A}, T_{B}$ and $T_{C}$, respectively. Let $l_{A}$ the line through $A$ and parallel to $BC$ and define $l_{B}$ and $l_{C}$ analogously. Let $L_{A}$ the second intersection point of $AI$ with the circumcircle of $\triangle ABC$ and define $L_{B}$ and $L_{C}$ analogously. Let $P_{A}=T_{B}T_{C}\cap l_{A}$ and define $P_{B}$ and $P_{C}$ analogously. Let $S_{A}=P_{B}T_{B}\cap P_{C}T_{C}$ and define $S_{B}$ and $S_{C}$ analogously. Prove that $S_{A}L_{A}, S_{B}L_{B}, S_{C}L_{C}$ are concurrent.
2018 Olympic Revenge
Let $p$ a positive prime number and $\mathbb{F}_{p}$ the set of integers $mod \ p$. For $x\in \mathbb{F}_{p}$, define $|x|$ as the cyclic distance of $x$ to $0$, that is, if we represent $x$ as an integer between $0$ and $p-1$, $|x|=x$ if $x<\frac{p}{2}$, and $|x|=p-x$ if $x>\frac{p}{2}$ . Let $f: \mathbb{F}_{p} \rightarrow \mathbb{F}_{p}$ a function such that for every $x,y \in \mathbb{F}_{p}$

$$ |f(x+y)-f(x)-f(y)|<100 $$Prove that exist $m \in \mathbb{F}_{p}$ such that for every $x \in \mathbb{F}_{p}$

$$ |f(x)-mx|<1000 $$
2017 Olympic Revenge
Prove that does not exist positive integers $a$, $b$ and $k$ such that $4abk-a-b$ is a perfect square.
2017 Olympic Revenge
Let $\triangle$$ABC$ a triangle with circumcircle $\Gamma$. Suppose there exist points $R$ and $S$ on sides $AB$ and $AC$, respectively, such that $BR=RS=SC$. A tangent line through $A$ to $\Gamma$ meet the line $RS$ at $P$. Let $I$ the incenter of triangle $\triangle$$ARS$. Prove that $PA=PI$
2017 Olympic Revenge
Let $n$ a positive integer. We call a pair $(\pi ,C)$ composed by a permutation $\pi$$:$ {$1,2,...n$}$\rightarrow${$1,2,...,n$} and a binary function $C:$ {$1,2,...,n$}$\rightarrow${$0,1$} "revengeful" if it satisfies the two following conditions:



$1)$For every $i$ $\in$ {$1,2,...,n$}, there exist $j$ $\in$ $S_{i}=${$i, \pi(i),\pi(\pi(i)),...$} such that $C(j)=1$.



$2)$ If $C(k)=1$, then $k$ is one of  the $v_{2}(|S_{k}|)+1$ highest elements of $S_{k}$, where $v_{2}(t)$ is the highest nonnegative integer such that $2^{v_{2}(t)}$ divides $t$, for every positive integer $t$.



Let $V$ the number of revengeful pairs and $P$ the number of partitions of $n$ with all parts powers of $2$.

Determine $\frac{V}{P}$.
2017 Olympic Revenge
Let $f:\mathbb{R}_{+}^{*}$$\rightarrow$$\mathbb{R}_{+}^{*}$ such that $f'''(x)>0$ for all $x$ $\in$ $\mathbb{R}_{+}^{*}$. Prove that:



$f(a^{2}+b^{2}+c^{2})+2f(ab+bc+ac)$ $\geq$ $f(a^{2}+2bc)+f(b^{2}+2ca)+f(c^{2}+2ab)$, for all $a,b,c$ $\in$ $\mathbb{R}_{+}^{*}$.
2016 Olympic Revenge
It is given the sequence defined by

$$\{a_{n+2}=6a_{n+1}-a_n\}_{n \in \mathbb{Z}_{>0}},a_1=1, a_2=7 \text{.}$$Find all $n$ such that there exists an integer $m$ for which $a_n=2m^2-1$.
2016 Olympic Revenge
Let $S$ a finite subset of $\mathbb{N}$. For every positive integer $i$, let $A_{i}$ the number of partitions of $i$ with all parts in $ \mathbb{N}-S$.

Prove that there exists $M\in  \mathbb{N}$ such that $A_{i+1}>A_{i}$ for all $i>M$.

($ \mathbb{N}$ is the set of positive integers)
2016 Olympic Revenge
Let $\Gamma$ a fixed circunference. Find all finite sets $S$ of points in $\Gamma$ such that:

For each point $P\in \Gamma$, there exists a partition of $S$ in sets $A$ and $B$

($A\cup B=S$, $A\cap B=\phi$) such that $\sum_{X\in A}PX = \sum_{Y\in B}PY$.
2016 Olympic Revenge
Let $\Omega$ and $\Gamma$ two circumferences such that $\Omega$ is in interior of $\Gamma$. Let $P$ a point on $\Gamma$.

Define points $A$ and $B$ distinct of $P$ on $\Gamma$ such that $PA$ and $PB$ are tangentes to $\Omega$.  Prove that when $P$

varies on $\Gamma$, the line $AB$ is tangent to a fixed circunference.
2016 Olympic Revenge
Let $T$ the set of the infinite sequences of integers. For two given elements in $T$:

$(a_{1},a_{2},a_{3},...)$ and $(b_{1},b_{2},b_{3},...)$, define the sum

$(a_{1},a_{2},a_{3},...)+(b_{1},b_{2},b_{3},...)=(a_{1}+b_{1},a_{2}+b_{2},a_{3}+b_{3},...)$.

Let $f: T\rightarrow$ $\mathbb{Z}$ a function such that:

i) If $x\in T$ has exactly one of your terms equal $1$ and all the others equal $0$, then $f(x)=0$.

ii)$f(x+y)=f(x)+f(y)$, for all $x,y\in T$.

Prove that $f(x)=0$ for all $x\in T$
2015 Olympic Revenge
For $n \in \mathbb{N}$, let $P(n)$ denote the product of distinct prime factors of $n$, with $P(1) = 1$. Show that for any $a_0 \in \mathbb{N}$, if we define a sequence $a_{k+1} = a_k + P(a_k)$ for $k \ge 0$, there exists some $k \in \mathbb{N}$ with $a_k/P(a_k) = 2015$.
2015 Olympic Revenge
Given $v = (a,b,c,d) \in \mathbb{N}^4$, let $\Delta^{1} (v) = (|a-b|,|b-c|,|c-d|,|d-a|)$ and $\Delta^{k} (v) = \Delta(\Delta^{k-1} (v))$ for $k > 1$. Define $f(v) = \min\{k \in \mathbb{N} : \Delta^k (v) = (0,0,0,0)\}$ and $\max(v) = \max\{a,b,c,d\}.$ Show that $f(v) < 1000\log \max(v)$ for all sufficiently large $v$ and $f(v) > 0.001 \log \max (v)$ for infinitely many $v$.
2015 Olympic Revenge
For every $n \in \mathbb{N}$, there exist integers $k$ such that $n | k$ and $k$ contains only zeroes and ones in its decimal representation. Let $f(n)$ denote the least possible number of ones in any such $k$. Determine whether there exists a constant $C$ such that $f(n) < C$ for all $n \in \mathbb{N}$.
2015 Olympic Revenge
Consider a game in the integer points of the real line, where an Angel tries to escape from a Devil. A positive integer $k$ is chosen, and the Angel and the Devil take turns playing. Initially, no point is blocked. The Angel, in point $A$, can move to any point $P$ such that $|AP| \le k$, as long as $P$ is not blocked. The Devil may block an arbitrary point. The Angel loses if it cannot move and wins if it does not lose in finitely many turns. Let $f(k)$ denote the least number of rounds the Devil takes to win. Prove that $$0.5 k \log_2 (k) (1 + o(1)) \le f(k) \le k \log_2(k) (1 +o(1)).$$

Note: $a(x) = b(x) (1+o(1))$ if $\lim_{x \to \infty} \frac{b(x)}{a(x)} = 1$.
2015 Olympic Revenge
Given a triangle $A_1 A_2 A_3$, let $a_i$ denote the side opposite to $A_i$, where indices are taken modulo 3. Let $D_1 \in a_1$. For $D_i \in A_i$, let $\omega_i$ be the incircle of the triangle formed by lines $a_i, a_{i+1}, A_iD_i$, and $D_{i+1} \in a_{i+1}$ with $A_{i+1} D_{i+1}$ tangent to $\omega_i$. Show that the set $\{D_i: i \in \mathbb{N}\}$ is finite.
2014 Olympic Revenge
Let $ABC$ an acute triangle and $\Gamma$ its circumcircle. The bisector of $BAC$ intersects $\Gamma$ at $M\neq A$. A line $r$ parallel to $BC$ intersects $AC$ at $X$ and $AB$ at $Y$. Also, $MX$ and $MY$ intersect $\Gamma$ again at $S$ and $T$, respectively.



If $XY$ and $ST$ intersect at $P$, prove that $PA$ is tangent to $\Gamma$.
2014 Olympic Revenge
$a)$ Let $n$ a positive integer. Prove that $gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}$.



$b)$ Prove that there are infinitely many positive integers $n$ such that $gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}$.