Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 Bosnia And Herzegovina - Regional Olympiad
In triangle $ABC$ holds $\angle ACB = 90^{\circ}$, $\angle BAC = 30^{\circ}$ and $BC=1$. In triangle $ABC$ is inscribed equilateral triangle (every side of a triangle $ABC$ contains one vertex of inscribed triangle). Find the least possible value of side of inscribed equilateral triangle
2009 Bosnia And Herzegovina - Regional Olympiad
For given positive integer $n$ find all quartets $(x_1,x_2,x_3,x_4)$ such that $x_1^2+x_2^2+x_3^2+x_4^2=4^n$
2009 Bosnia And Herzegovina - Regional Olympiad
There are $n$ positive integers on the board. We can add only positive integers $c=\frac{a+b}{a-b}$, where $a$ and $b$ are numbers already writted on the board.

$a)$ Find minimal value of $n$, such that with adding numbers with described method, we can get any positive integer number written on the board

$b)$ For such $n$, find numbers written on the board at the beginning
2009 Bosnia And Herzegovina - Regional Olympiad
What is the minimal value of $\sqrt{2x+1}+\sqrt{3y+1}+\sqrt{4z+1}$, if $x$, $y$ and $z$ are nonnegative real numbers such that $x+y+z=4$
2009 Bosnia And Herzegovina - Regional Olympiad
Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers
2009 Bosnia And Herzegovina - Regional Olympiad
Let $ABC$ be an equilateral triangle such that length of its altitude is $1$. Circle with center on the same side of line $AB$ as point $C$ and radius $1$ touches side $AB$. Circle rolls on the side $AB$. While the circle is rolling, it constantly intersects sides $AC$ and $BC$. Prove that length of an arc of the circle, which lies inside the triangle, is constant
2008 Bosnia And Herzegovina - Regional Olympiad
Squares $ BCA_{1}A_{2}$ , $ CAB_{1}B_{2}$ , $ ABC_{1}C_{2}$ are outwardly drawn on sides of triangle $ \triangle ABC$. If $ AB_{1}A'C_{2}$ , $ BC_{1}B'A_{2}$  , $ CA_{1}C'B_{2}$ are parallelograms then prove that:



(i) Lines $ BC$ and $ AA'$ are orthogonal.



(ii)Triangles $ \triangle ABC$ and $ \triangle A'B'C'$ have common centroid
2008 Bosnia And Herzegovina - Regional Olympiad
For arbitrary reals $ x$, $ y$ and $ z$ prove the following inequality:



$ x^{2} + y^{2} + z^{2} - xy - yz - zx \geq \max \{\frac {3(x - y)^{2}}{4} , \frac {3(y - z)^{2}}{4} , \frac {3(y - z)^{2}}{4} \}$
2008 Bosnia And Herzegovina - Regional Olympiad
Let $ b$ be an even positive integer. Assume that there exist integer $ n > 1$ such that $ \frac {b^{n} - 1}{b - 1}$ is perfect square.

Prove that $ b$ is divisible by 8.
2008 Bosnia And Herzegovina - Regional Olympiad
Given are two disjoint sets $ A$ and $ B$ such that their union is $ \mathbb N$. Prove that for all positive integers $ n$ there exist different numbers $ a$ and $ b$, both greater than $ n$, such that either $ \{ a,b,a + b \}$ is contained in $ A$ or $ \{ a,b,a + b \}$ is contained in $ B$.
2008 Bosnia And Herzegovina - Regional Olympiad
Given is an acute angled triangle $ \triangle ABC$ with side lengths $ a$, $ b$ and $ c$ (in an usual way) and circumcenter $ O$. Angle bisector of angle $ \angle BAC$ intersects circumcircle at points $ A$ and $ A_{1}$. Let $ D$ be projection of point $ A_{1}$ onto line $ AB$, $ L$ and $ M$ be midpoints of $ AC$ and $ AB$ , respectively.



(i) Prove that $ AD=\frac{1}{2}(b+c)$



(ii) If triangle $ \triangle ABC$ is an acute angled prove that $ A_{1}D=OM+OL$
2008 Bosnia And Herzegovina - Regional Olympiad
IF $ a$, $ b$ and $ c$ are positive reals such that $ a^{2}+b^{2}+c^{2}=1$ prove the inequality:



$$ \frac{a^{5}+b^{5}}{ab(a+b)}+ \frac {b^{5}+c^{5}}{bc(b+c)}+\frac {c^{5}+a^{5}}{ca(a+b)}\geq 3(ab+bc+ca)-2.$$
2008 Bosnia And Herzegovina - Regional Olympiad
Prove that equation $ p^{4}+q^{4}=r^{4}$ does not have solution in set of prime numbers.
2008 Bosnia And Herzegovina - Regional Olympiad
$ n$ points (no three being collinear) are given in a plane. Some points are connected and they form $ k$ segments. If no three of these segments form triangle ( equiv. there are no three points, such that each two of them are connected) prove that $ k \leq \left \lfloor \frac {n^{2}}{4}\right\rfloor$
2008 Bosnia And Herzegovina - Regional Olympiad
Two circles with centers $ S_{1}$ and $ S_{2}$ are externally tangent at point $ K$. These circles are also internally tangent to circle $ S$  at points $ A_{1}$ and $ A_{2}$, respectively. Denote by $ P$one of the intersection points of $ S$ and common tangent to $ S_{1}$ and $ S_{2}$ at $ K$.Line $ PA_{1}$ intersects $ S_{1}$ at $ B_{1}$ while $ PA_{2}$ intersects $ S_{2}$ at $ B_{2}$.

Prove that $ B_{1}B_{2}$ is common tangent of circles $ S_{1}$ and $ S_{2}$.
2008 Bosnia And Herzegovina - Regional Olympiad
If $ a$, $ b$ and $ c$ are positive reals prove inequality:



$$ \left(1+\frac{4a}{b+c}\right)\left(1+\frac{4b}{a+c}\right)\left(1+\frac{4c}{a+b}\right) > 25.$$
2008 Bosnia And Herzegovina - Regional Olympiad
Find all positive integers $ a$ and $ b$ such that $ \frac{a^{4}+a^{3}+1}{a^{2}b^{2}+ab^{2}+1}$ is an integer.
2008 Bosnia And Herzegovina - Regional Olympiad
A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?
2008 Bosnia And Herzegovina - Regional Olympiad
Given are three pairwise externally tangent circles $ K_{1}$ , $ K_{2}$ and $ K_{3}$. denote by $ P_{1}$ tangent point of $ K_{2}$ and $ K_{3}$ and by $ P_{2}$ tangent point of $ K_{1}$ and $ K_{3}$.



Let $ AB$ ($ A$ and $ B$ are different from tangency points) be a diameter of circle $ K_{3}$. Line $ AP_{2}$ intersects circle $ K_{1}$ (for second time) at point $ X$  and line $ BP_{1}$ intersects circle $ K_{2}$(for second time) at $ Y$.



If $ Z$ is intersection point of lines $ AP_{1}$ and $ BP_{2}$ prove that points $ X$, $ Y$ and $ Z$ are collinear.
2008 Bosnia And Herzegovina - Regional Olympiad
Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that:

$i)$ $a(0)=0$

$ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$.

If exists prove:

$a)$ $a(k)\geq a(k-1)$

$b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.
2020 Brazil National Olympiad
Prove that there are positive integers $a_1, a_2,\dots, a_{2020}$ such that



$$\dfrac{1}{a_1}+\dfrac{1}{2a_2}+\dfrac{1}{3a_3}+\dots+\dfrac{1}{2020a_{2020}}=1.$$
2020 Brazil National Olympiad
For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is fibonatic when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not fibonatic integers.
2020 Brazil National Olympiad
Let $r_A,r_B,r_C$ rays from point $P$. Define circles $w_A,w_B,w_C$ with centers $X,Y,Z$ such that $w_a$ is tangent to $r_B,r_C , w_B$ is tangent to $r_A, r_C$ and $w_C$ is tangent to $r_A,r_B$. Suppose $P$ lies inside triangle $XYZ$, and let $s_A,s_B,s_C$ be the internal tangents to circles $w_B$ and $w_C$; $w_A$ and $w_C$; $w_A$ and $w_B$ that do not contain rays $r_A,r_B,r_C$ respectively. Prove that $s_A, s_B, s_C$ concur at a point $Q$, and also that $P$ and $Q$ are isotomic conjugates.

PS: The rays can be lines and the problem is still true.
2020 Brazil National Olympiad
Let $ABC$ be a triangle. The ex-circles touch sides $BC, CA$ and $AB$ at points $U, V$ and $W$, respectively. Be $r_u$ a straight line that passes through $U$ and is perpendicular to $BC$, $r_v$ the straight line that passes through $V$ and is perpendicular to $AC$ and $r_w$ the straight line that passes through W and is perpendicular to $AB$. Prove that the lines $r_u$, $r_v$ and $r_w$ pass through the same point.
2020 Brazil National Olympiad
Let $n$ and $k$ be positive integers with $k$ $\le$ $n$. In a group of $n$ people, each one or always

speak the truth or always lie. Arnaldo can ask questions for any of these people

provided these questions are of the type: “In set $A$, what is the parity of people who speak to

true? ”, where $A$ is a subset of size $ k$ of the set of $n$ people. The answer can only

be “$even$” or “$odd$”.

a) For which values of $n$ and $k$ is it possible to determine which people speak the truth and

which people always lie?

b)  What is the minimum number of questions required to determine which people

speak the truth and which people always lie, when that number is finite?
2020 Brazil National Olympiad
Let $f (x) = 2x^2 + x - 1$, $f^0(x) = x$ and $f^{n + 1}(x) = f (f^n(x))$ for all real $x$ and $n \ge 0$ integer .

(a) Determine the number of real distinct solutions of the equation of $f^3(x) = x$.

(b) Determine, for each integer $n \ge 0$, the number of real distinct solutions of the equation $f^n(x) = 0$.
2020 Brazil National Olympiad
Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$.
2020 Brazil National Olympiad
The following sentece is written on a board:





The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions.





Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?
2020 Brazil National Olympiad
Consider an inifinte sequence $x_1, x_2,\dots$ of positive integers such that, for every integer $n\geq 1$:



If $x_n$ is even, $x_{n+1}=\dfrac{x_n}{2}$;

If $x_n$ is odd, $x_{n+1}=\dfrac{x_n-1}{2}+2^{k-1}$, where $2^{k-1}\leq x_n<2^k$.



Determine the smaller possible value of $x_1$ for which $2020$ is in the sequence.
2020 Brazil National Olympiad
A positive integer is brazilian if the first digit and the last digit are equal. For instance, $4$ and $4104$ are brazilians, but $10$ is not brazilian. A brazilian number is superbrazilian if it can be written as sum of two brazilian numbers. For instance, $101=99+2$ and $22=11+11$ are superbrazilians, but $561=484+77$ is not superbrazilian, because $561$ is not brazilian. How many $4$-digit numbers are superbrazilians?
2020 Brazil National Olympiad
Let $ABC$ be a triangle and $M$ the midpoint of $AB$. Let circumcircles of triangles $CMO$ and $ABC$ intersect at $K$ where $O$ is the circumcenter of $ABC$. Let $P$ be the intersection of lines $OM$ and $CK$. Prove that $\angle{PAK} = \angle{MCB}$.
2020 Brazil National Olympiad
Let $k$ be a positive integer. Arnaldo and Bernaldo play a game in a table $2020\times 2020$, initially all the cells are empty. In each round a player chooses a empty cell and put one red token or one blue token, Arnaldo wins if in some moment, there are $k$ consecutive cells in the same row or column with tokens of same color, if all the cells have a token and there aren't $k$ consecutive cells(row or column) with same color, then Bernaldo wins. If the players play alternately and Arnaldo goes first, determine for which values of $k$, Arnaldo has the winning strategy.
2019 Brazil National Olympiad
Let $\omega_1$ and $\omega_2$ be two circles with centers $C_1$ and $C_2$, respectively, which intersect at two points $P$ and $Q$. Suppose that the circumcircle of triangle $PC_1C_2$ intersects $\omega_1$ at $A \neq P$ and $\omega_2$ at $B \neq P$. Suppose further that $Q$ is inside the triangle $PAB$. Show that $Q$ is the incenter of triangle $PAB$.
2019 Brazil National Olympiad
Given are the real line and the two unique marked points $0$ and $1$. We can perform as many times as we want the following operation: we take two already marked points $a$ and $b$ and mark the reflection of $a$ over $b$. Let $f(n)$ be the minimum number of operations needed to mark on the real line the number $n$ (which is the number at a distance $\left| n\right|$ from $0$ and it is on the right of $0$ if $n>0$ and on the left of $0$ if $n<0$). For example, $f(0)=f(1)=0$ and $f(-1)=f(2)=1$. Find $f(n)$.
2019 Brazil National Olympiad
Let $\mathbb{R}_{>0}$ be the set of the positive real numbers. Find all functions $f:\mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that $$f(xy+f(x))=f(f(x)f(y))+x$$for all positive real numbers $x$ and $y$.
2019 Brazil National Olympiad
Prove that for every positive integer $m$ there exists a positive integer $n_m$ such that for every positive integer $n \ge n_m$, there exist positive integers $a_1, a_2, \ldots, a_n$ such that $$\frac{1}{a_1^m}+\frac{1}{a_2^m}+\ldots+\frac{1}{a_n^m}=1.$$
2019 Brazil National Olympiad
(a) Prove that given constants $a,b$ with $1<a<2<b$, there is no partition of the set of positive integers into two subsets $A_0$ and $A_1$ such that: if $j \in \{0,1\}$ and $m,n$ are in $A_j$, then either $n/m <a$ or $n/m>b$.

(b) Find all pairs of real numbers $(a,b)$ with $1<a<2<b$ for which the following property holds: there exists a partition of the set of positive integers into three subsets $A_0, A_1, A_2$ such that if $j \in \{0,1,2\}$ and $m,n$ are in $A_j$, then either $n/m <a$ or $n/m>b$.
2019 Brazil National Olympiad
Let $A_1A_2A_3A_4A_5$ be a convex, cyclic pentagon with $\angle A_i + \angle A_{i+1} >180^{\circ}$ for all $i \in \{1,2,3,4,5\}$ (all indices modulo $5$ in the problem). Define $B_i$ as the intersection of lines $A_{i-1}A_i$ and $A_{i+1}A_{i+2}$, forming a star. The circumcircles of triangles $A_{i-1}B_{i-1}A_i$ and $A_iB_iA_{i+1}$ meet again at $C_i \neq A_i$, and the circumcircles of triangles $B_{i-1}A_iB_i$ and $B_iA_{i+1}B_{i+1}$ meet again at $D_i \neq B_i$. Prove that the ten lines $A_iC_i, B_iD_i$, $i \in \{1,2,3,4,5\}$, have a common point.
2019 Brazil National Olympiad
An eight-digit number is said to be 'robust' if it meets both of the following conditions:

(i) None of its digits is $0$.

(ii) The difference between two consecutive digits is $4$ or $5$.



Answer the following questions:



(a) How many are robust numbers?

(b) A robust number is said to be 'super robust' if all of its digits are distinct. Calculate the sum of all

the super robust numbers.
2019 Brazil National Olympiad
Let $a, b$ and $k$ be positive integers with $k> 1$ such that

$lcm (a, b) + gcd (a, b) = k (a + b)$.

Prove that $a + b \geq 4k$
2019 Brazil National Olympiad
Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the

vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre
2019 Brazil National Olympiad
Let $ ABC $ be an acutangle triangle and $ D $ any point on the $ BC $ side. Let $ E $ be the symmetrical of $ D $ in

$ AC $ and $ F $ is the symmetrical $ D $ relative to $ AB $. $ A $ straight $ ED $ intersects straight $ AB $ at $ G $, while straight $ F D $ intersects the line $ AC $ in $ H $. Prove that the points $ A, E, F, G$ and $ H $ are on the same circumference.
2019 Brazil National Olympiad
In the picture below, a white square is surrounded by four black squares and three white squares. They are surrounded by seven black squares.



What is the maximum number of white squares that can be surrounded by $ n $ black squares?
2019 Brazil National Olympiad
In the Cartesian plane, all points with both integer coordinates are painted blue. Blue colon

they are said to be mutually visible if the line segment connecting them has no other blue dots. Prove that

There is a set of $ 2019$ blue dots that are mutually visible two by two.
2018 Brazil National Olympiad
We say that a polygon $P$ is inscribed in another polygon $Q$ when all vertices of $P$ belong to perimeter of $Q$. We also say in this case that $Q$ is circumscribed to $P$. Given a triangle $T$, let $l$ be the maximum value of the side of a square inscribed in $T$ and $L$ be the minimum value of the side of a square circumscribed to $T$. Prove that for every triangle $T$ the inequality $L/l \ge 2$ holds and find all the triangles $T$ for which the equality occurs.
2018 Brazil National Olympiad
Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations.

a) Show that if the initial number written is $0$, then Azambuja cannot reach his goal.

b) Find all initial numbers for which Azambuja can achieve his goal.
2018 Brazil National Olympiad
Let $k$, $n$ be fixed positive integers. In a circular table, there are placed pins numbered successively with the numbers $1, 2 \dots, n$, with $1$ and $n$ neighbors. It is known that pin $1$ is golden and the others are white. Arnaldo and Bernaldo play a game, in which a ring is placed initially on one of the pins and at each step it changes position. The game begins with Bernaldo choosing a starting pin for the ring, and the first step consists of the following: Arnaldo chooses a positive integer $d$ any and Bernaldo moves the ring $d$ pins clockwise or counterclockwise (positions are considered modulo $n$, i.e., pins $x$, $y$ equal if and only if $n$ divides $x-y$). After that, the ring changes its position according to one of the following rules, to be chosen at every step by Arnaldo:



Rule 1: Arnaldo chooses a positive integer $d$ and Bernaldo moves the ring $d$ pins clockwise or counterclockwise.



Rule 2: Arnaldo chooses a direction (clockwise or counterclockwise), and Bernaldo moves the ring in the chosen direction in $d$ or $kd$ pins, where $d$ is the size of the last displacement performed.



Arnaldo wins if, after a finite number of steps, the ring is moved to the golden pin. Determine, as a function of $k$, the values of $n$ for which Arnaldo has a strategy that guarantees his victory, no matter how Bernaldo plays.
2018 Brazil National Olympiad
Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$, all belonging to the interval $[0, 1]$, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$, $p_2 = x_2x_3$, $\dots$ , $p_{2n} = x_{2n}x_1$. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?
2018 Brazil National Olympiad
Consider the sequence in which $a_1 = 1$ and $a_n$ is obtained by juxtaposing the decimal representation of $n$ at the end of the decimal representation of $a_{n-1}$. That is, $a_1 = 1$, $a_2 = 12$, $a_3 = 123$, $\dots$ , $a_9 = 123456789$, $a_{10} = 12345678910$ and so on. Prove that infinitely many numbers of this sequence are multiples of $7$.
2018 Brazil National Olympiad
Consider $4n$ points in the plane, with no three points collinear. Using these points as vertices, we form $\binom{4n}{3}$ triangles. Show that there exists a point $X$ of the plane that belongs to the interior of at least $2n^3$ of these triangles.
2018 Brazil National Olympiad
Every day from day 2, neighboring cubes (cubes with common faces) to red cubes also turn red and are numbered with the day number.
2018 Brazil National Olympiad
We say that a quadruple $(A,B,C,D)$ is dobarulho when $A,B,C$ are non-zero algorisms and $D$ is a positive integer such that:

$1.$ $A \leq 8$

$2.$ $D>1$

$3.$ $D$ divides the six numbers $\overline{ABC}$, $\overline{BCA}$, $\overline{CAB}$, $\overline{(A+1)CB}$, $\overline{CB(A+1)}$, $\overline{B(A+1)C}$.

Find all such quadruples.
2018 Brazil National Olympiad
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_a$ passes in the points $A$ and $H$ and is tangent to the circumcircle of $ABC$. Define $X_b, X_c$ analogously, let $O_a, O_b, O_c$ the symmetric of $O$ to the sides $BC, AC$ and $AB$, respectively. Prove that the lines $O_aX_a, O_bX_b, O_cX_c$ are concurrents.
2018 Brazil National Olympiad
a) In a $ XYZ$ triangle,  the incircle tangents the $ XY $ and $ XZ $ sides at the $ T $ and $ W $ points, respectively. Prove that: $$ XT = XW = \frac {XY + XZ-YZ} {2} $$Let $ ABC $ be a triangle and $ D $ is the foot of the relative height next to $ A. $ Are $ I $ and $ J $ the incentives from triangle $ ABD $ and $ ACD $, respectively. The circles of $ ABD $ and $ ACD $ tangency $ AD $ at points $ M $ and $ N $, respectively. Let $ P $ be the tangency point of the $ BC $ circle with the $ AB$ side.  The center circle $ A $ and radius $ AP $ intersect the height $ D $ at $ K. $

b) Show that the triangles $ IMK $ and $ KNJ $ are congruent

c) Show that the $ IDJK $ quad is inscritibed
2018 Brazil National Olympiad
One writes, initially, the numbers $1,2,3,\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\frac{ab}{f(a,b)}$, where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$, one will make this until remain two numbers $x, y$ with $x\geq y$. Find the maximum value of $x$.
2018 Brazil National Olympiad
Let $S(n)$ be the sum of digits of $n$. Determine all the pairs $(a, b)$ of positive integers, such that the expression

$S(an + b) - S(n)$ has a finite number of values, where $n$ is varying in the positive integers.
2017 Brazil National Olympiad
1. For each real number $r$ between $0$ and $1$ we can represent $r$ as an infinite decimal $r = 0.r_1r_2r_3\dots$ with $0 \leq r_i \leq 9$. For example, $\frac{1}{4} = 0.25000\dots$, $\frac{1}{3} = 0.333\dots$ and $\frac{1}{\sqrt{2}} = 0.707106\dots$.



a) Show that we can choose two rational numbers $p$ and $q$ between $0$ and $1$ such that, from their decimal representations $p = 0.p_1p_2p_3\dots$ and $q = 0.q_1q_2q_3\dots$, it's possible to construct an irrational number $\alpha = 0.a_1a_2a_3\dots$ such that, for each $i = 1, 2, 3, \dots$, we have $a_i = p_1$ or $a_1 = q_i$.



b) Show that there's a rational number $s = 0.s_1s_2s_3\dots$ and an irrational number $\beta = 0.b_1b_2b_3\dots$ such that, for all $N \geq 2017$, the number of indexes $1 \leq i \leq N$ satisfying $s_i \neq b_i$ is less than or equal to $\frac{N}{2017}$.
2017 Brazil National Olympiad
2. Let $n \geq 3$ be an integer. Prove that for all integers $k$, with $1 \leq k \leq \binom{n}{2}$, there exists a set $A$ with $n$ distinct positive integer elements such that the set $B = \{\gcd(x, y): x, y \in A, x \neq y \}$ (gotten from the greatest common divisor of all pairs of distinct elements from $A$) contains exactly $k$ distinct elements.
2017 Brazil National Olympiad
3. A quadrilateral $ABCD$ has the incircle $\omega$ and is such that the semi-lines $AB$ and $DC$ intersect at point $P$ and the semi-lines $AD$ and $BC$ intersect at point $Q$. The lines $AC$ and $PQ$ intersect at point $R$. Let $T$ be the point of $\omega$ closest from line $PQ$. Prove that the line $RT$ passes through the incenter of triangle $PQC$.