Navvye/Olympiad-Problems-CSV huggingface dataset:
2010 International Zhautykov Olympiad
A rectangle formed by the lines of checkered paper is divided into figures of three kinds: isosceles right triangles  (1) with base of two units, squares  (2) with unit side, and parallelograms (3)  formed by two sides and two diagonals of unit squares (figures may be oriented in any way). Prove that the number of figures of the third kind is even.



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2010 International Zhautykov Olympiad
Positive integers $1,2,...,n$  are written on а blackboard ($n >2$ ). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number 97 remains. Find the least $n$  for which it is possible.
2010 International Zhautykov Olympiad
In every vertex of a regular $n$ -gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $[\frac{n}{2}]$  positions clockwise from its initial position.
2010 International Zhautykov Olympiad
Let $ABC$ arbitrary triangle ($AB \neq BC \neq AC \neq AB$) And O,I,H it's circum-center, incenter and ortocenter (point of intersection altitudes). Prove, that

1) $\angle OIH > 90^0$(2 points)

2)$\angle OIH >135^0$(7 points)



balls for 1) and 2) not additive.
2009 International Zhautykov Olympiad
Find all pairs of integers $ (x,y)$, such that

$$ x^2 - 2009y + 2y^2 = 0

$$
2009 International Zhautykov Olympiad
Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality:

$$ x+af(y)\leq y+f(f(x))

$$

for all $ x,y\in\mathbb{R}$
2009 International Zhautykov Olympiad
For a convex hexagon $ ABCDEF$ with an area $ S$, prove that:

$$ AC\cdot(BD+BF-DF)+CE\cdot(BD+DF-BF)+AE\cdot(BF+DF-BD)\geq 2\sqrt{3}S

$$
2009 International Zhautykov Olympiad
On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y = x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y = 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i = 1,2,3,4$.

Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.
2009 International Zhautykov Olympiad
Given a quadrilateral $ ABCD$ with $ \angle B=\angle D=90^{\circ}$. Point $ M$ is chosen on segment $ AB$ so taht $ AD=AM$. Rays $ DM$ and $ CB$ intersect at point $ N$. Points $ H$ and $ K$ are feet of perpendiculars from points $ D$ and $ C$ to lines $ AC$ and $ AN$, respectively.

Prove that $ \angle MHN=\angle MCK$.
2009 International Zhautykov Olympiad
In a checked $ 17\times 17$ table, $ n$ squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least $ 6$ of the squares in some line are black, then one can paint all the squares of this line in black.

Find the minimal value of $ n$ such that for some initial arrangement of $ n$ black squares one can paint all squares of the table in black in some steps.
2008 International Zhautykov Olympiad
Points $ K,L,M,N$ are repectively the midpoints of sides $ AB,BC,CD,DA$ in a convex quadrliateral $ ABCD$.Line $ KM$ meets dioganals $ AC$ and $ BD$ at points $ P$ and $ Q$,respectively.Line $ LN$ meets dioganals $ AC$ and $ BD$ at points $ R$ and $ S$,respectively.

Prove that if $ AP\cdot PC=BQ\cdot QD$,then $ AR\cdot RC=BS\cdot SD$.
2008 International Zhautykov Olympiad
A polynomial $ P(x)$ with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable $ x$) with integer coefficients.For example,the polynomials $ x^3 - 1$ and $ 9x^3 - 3x^2 + 3x + 7 = (x - 1)^3 + (2x)^3 + 2^3$ are good.

a)Is the polynomial $ P(x) = 3x + 3x^7$ good?

b)Is the polynomial $ P(x) = 3x + 3x^7 + 3x^{2008}$ good?

Justify your answers.
2008 International Zhautykov Olympiad
Let $ A = \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$ , $ 1\leq a_i\leq i + 1$ for each $ i = 1,2\dots,8\}$.A subset $ X\subset A$ is called sparse if for each two distinct elements $ (a_1,\dots,a_8)$,$ (b_1,\dots,b_8)\in X$,there exist at least three indices $ i$,such that $ a_i\neq b_i$.

Find the maximal possible number of elements in a sparse subset of set $ A$.
2008 International Zhautykov Olympiad
For each positive integer $ n$,denote by $ S(n)$ the sum of all digits in decimal representation of $ n$.

Find all positive integers $ n$,such that $ n=2S(n)^3+8$.
2008 International Zhautykov Olympiad
Let $ A_1A_2$ be the external tangent line to the nonintersecting cirlces $ \omega_1(O_1)$ and $ \omega_2(O_2)$,$ A_1\in\omega_1$,$ A_2\in\omega_2$.Points $ K$ is the midpoint of $ A_1A_2$.And $ KB_1$ and $ KB_2$ are tangent lines to $ \omega_1$ and $ \omega_2$,respectvely($ B_1\neq A_1$,$ B_2\neq A_2$).Lines $ A_1B_1$ and $ A_2B_2$ meet in point $ L$,and lines $ KL$ and $ O_1O_2$ meet in point $ P$.

Prove that points $ B_1,B_2,P$ and $ L$ are concyclic.
2008 International Zhautykov Olympiad
Let $ a, b, c$ be positive integers for which $ abc = 1$. Prove that

$ \sum \frac{1}{b(a+b)} \ge \frac{3}{2}$.
2007 International Zhautykov Olympiad
There are given $111$ coins and a $n\times n$ table divided into unit cells. This coins are placed inside the unit cells (one unit cell may contain one coin, many coins, or may be empty), such that the difference between the number of coins from two neighbouring cells (that have a common edge) is $1$. Find the maximal $n$ for this to be possible.
2007 International Zhautykov Olympiad
Let $ABCD$ be a convex quadrilateral, with $\angle BAC=\angle DAC$ and $M$ a point inside such that $\angle MBA=\angle MCD$ and $\angle MBC=\angle MDC$. Show that the angle $\angle ADC$ is equal to $\angle BMC$ or $\angle AMB$.
2007 International Zhautykov Olympiad
Show that there are an infinity of positive integers $n$ such that $2^{n}+3^{n}$ is divisible by $n^{2}$.
2007 International Zhautykov Olympiad
Does there exist a function $f: \mathbb{R}\rightarrow\mathbb{R}$ such that $f(x+f(y))=f(x)+\sin y$, for all reals $x,y$ ?
2007 International Zhautykov Olympiad
The set of positive nonzero real numbers are partitioned into three mutually disjoint non-empty subsets $(A\cup B\cup C)$.

a) show that there exists a triangle of side-lengths $a,b,c$, such that $a\in A, b\in B, c\in C$.

b) does it always happen that there exists a right triangle with the above property ?
2007 International Zhautykov Olympiad
Let $ABCDEF$ be a convex hexagon and it`s diagonals have one common point $M$. It is known that the circumcenters of triangles $MAB,MBC,MCD,MDE,MEF,MFA$ lie on a circle.

Show that the quadrilaterals $ABDE,BCEF,CDFA$ have equal areas.
2006 International Zhautykov Olympiad
Solve in positive integers the equation

$$ n = \varphi(n) + 402 ,

$$

where $ \varphi(n)$ is the number of positive integers less than $ n$ having no common prime factors with $ n$.
2006 International Zhautykov Olympiad
Let $ ABC$ be a triangle and $ K$ and $ L$ be two points on $ (AB)$, $ (AC)$ such that $ BK = CL$ and let $ P = CK\cap BL$. Let the parallel through $ P$ to the interior angle bisector of $ \angle BAC$ intersect $ AC$ in $ M$. Prove that $ CM = AB$.
2006 International Zhautykov Olympiad
Let $ m\geq n\geq 4$ be two integers. We call a $ m\times n$ board filled with 0's or 1's good if



1) not all the numbers on the board are 0 or 1;



2) the sum of all the numbers in $ 3\times 3$ sub-boards is the same;



3) the sum of all the numbers in $ 4\times 4$ sub-boards is the same.



Find all $ m,n$ such that there exists a good $ m\times n$ board.
2006 International Zhautykov Olympiad
In a pile you have 100 stones. A partition of the pile in $ k$ piles is good if:



1) the small piles have different numbers of stones;



2) for any partition of one of the small piles in 2 smaller piles, among the $ k + 1$ piles you get 2 with the same number of stones (any pile has at least 1 stone).



Find the maximum and minimal values of $ k$ for which this is possible.
2006 International Zhautykov Olympiad
Let $ a,b,c,d$ be real numbers with sum 0. Prove the inequality:

$$ (ab + ac + ad + bc + bd + cd)^2 + 12\geq 6(abc + abd + acd + bcd).

$$
2006 International Zhautykov Olympiad
Let $ ABCDEF$ be a convex hexagon such that $ AD = BC + EF$, $ BE = AF + CD$, $ CF = DE + AB$. Prove that:

$$ \frac {AB}{DE} = \frac {CD}{AF} = \frac {EF}{BC}.

$$
2005 International Zhautykov Olympiad
The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table?
2005 International Zhautykov Olympiad
Let $ m,n$ be integers such that $ 0\le m\le 2n$. Then prove that the number $ 2^{2n + 2} + 2^{m + 2} + 1$ is perfect square iff $ m = n$.
2005 International Zhautykov Olympiad
Let $ A$ be a set of $ 2n$ points on the plane such that no three points are collinear. Prove that for any distinct two points $ a,b\in A$ there exists a line that partitions $ A$ into two subsets each containing $ n$ points and such that $ a,b$ lie on different sides of the line.
2005 International Zhautykov Olympiad
For the positive real numbers $ a,b,c$ prove the inequality

$$ \frac {c}{a + 2b} + \frac {a}{b + 2c} + \frac {b}{c + 2a}\ge1.

$$
2005 International Zhautykov Olympiad
Let the circle $ (I; r)$ be inscribed in the triangle $ ABC$. Let $ D$ be the point of contact of this circle with $ BC$. Let $ E$ and $ F$ be the midpoints of $ BC$ and $ AD$, respectively. Prove that the three points $ I$, $ E$, $ F$ are collinear.
2005 International Zhautykov Olympiad
Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 + 4$, $ p|q^2 + 4$.
2005 International Zhautykov Olympiad
Prove that the equation $ x^{5} + 31 = y^{2}$ has no integer solution.
2005 International Zhautykov Olympiad
Let $ r$ be a real number such that the sequence $ (a_{n})_{n\geq 1}$ of positive real numbers satisfies the equation $ a_{1} + a_{2} + \cdots + a_{m + 1} \leq r a_{m}$ for each positive integer $ m$. Prove that $ r \geq 4$.
2005 International Zhautykov Olympiad
Let SABC be a regular triangular pyramid. Find the set of all points $ D (D! = S)$ in the space satisfing the equation $ |cos ASD - 2cosBSD - 2 cos CSD| = 3$.
2005 International Zhautykov Olympiad
For the positive real numbers $ a,b,c$ prove that $$ \frac c{a + 2b} + \frac d{b + 2c} + \frac a{c + 2d} + \frac b{d + 2a} \geq \frac  43.$$
2005 International Zhautykov Olympiad
The inner point $ X$ of a quadrilateral is observable from the side $ YZ$ if the perpendicular to the line $ YZ$ meet it in the colosed interval $ [YZ].$ The inner point of a quadrilateral is a $ k-$point if it is observable from the exactly $ k$ sides of the quadrilateral. Prove that if a convex quadrilateral has a 1-point then it has a $ k-$point for each $ k=2,3,4.$
2005 International Zhautykov Olympiad
Find all prime numbers $ p,q < 2005$ such that $ q | p^{2} + 8$  and $ p|q^{2} + 8.$
2021 Lusophon Mathematical Olympiad
Juca has decided to call all positive integers with 8 digits as $sextalternados$ if it is a multiple of 30 and its consecutive digits have different parity. At the same time, Carlos decided to classify all $sextalternados$ that are multiples of 12 as $super sextalternados$.



a) Show that $super sextalternados$ numbers don't exist.



b) Find the smallest $sextalternado$ number.
2021 Lusophon Mathematical Olympiad
Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2\times4$ rectangle instead of $2\times3$ like in chess). In this movement, it doesn't land on the squares between the beginning square and the final square it lands on.



A trip of the length $n$ of the knight is a sequence of $n$ squares $C1, C2, ..., Cn$ which are all distinct such that the knight starts at the $C1$ square and for each $i$ from $1$ to $n-1$ it can use the movement described before to go from the $Ci$ square to the $C(i+1)$.



Determine the greatest $N \in \mathbb{N}$ such that there exists a path of the knight with length $N$ on a $5\times5$ board.
2021 Lusophon Mathematical Olympiad
Let triangle $ABC$ be an acute triangle with $AB\neq AC$. The bisector of $BC$ intersects the lines $AB$ and $AC$ at points $F$ and $E$, respectively. The circumcircle of triangle $AEF$ has center $P$ and intersects the circumcircle of triangle $ABC$ at point $D$ with $D$ different to $A$.



Prove that the line $PD$ is tangent to the circumcircle of triangle $ABC$.
2021 Lusophon Mathematical Olympiad
Let $x_1, x_2, x_3, x_4, x_5\in\mathbb{R}^+$ such that



$x_1^2-x_1x_2+x_2^2=x_2^2-x_2x_3+x_3^2=x_3^2-x_3x_4+x_4^2=x_4^2-x_4x_5+x_5^2=x_5^2-x_5x_1+x_1^2$



Prove that $x_1=x_2=x_3=x_4=x_5$.
2021 Lusophon Mathematical Olympiad
There are 3 lines $r, s$ and $t$ on a plane. The lines $r$ and $s$ intersect perpendicularly at point $A$. the line $t$ intersects the line $r$ at point $B$ and the line $s$ at point $C$. There exist exactly 4 circumferences on the plane that are simultaneously tangent to all those 3 lines.



Prove that the radius of one of those circumferences is equal to the sum of the radius of the other three circumferences.
2021 Lusophon Mathematical Olympiad
A positive integer $n$ is called $omopeiro$ if there exists $n$ non-zero integers that are not necessarily distinct such that $2021$ is the sum of the squares of those $n$ integers. For example, the number $2$ is not an $omopeiro$, because $2021$ is not a sum of two non-zero squares, but $2021$ is an $omopeiro$, because $2021=1^2+1^2+ \dots +1^2$, which is a sum of $2021$ squares of the number $1$.



Prove that there exist more than 1500 $omopeiro$ numbers.



Note: proving that there exist at least 500 $omopeiro$ numbers is worth 2 points.
2020 Lusophon Mathematical Olympiad
In certain country, the coins have the following values: $2^0, 2^1, 2^2,\dots 2^{10}$. A cash machine has $1000$ coins of each value and give the money using each coin(of each value) at most once. The customers order all the positive integers:  $1,2,3,4,5,\dots$ (in this order) in coins.

a) Determine the first integer, such that the cash machine cannot provide.

b) In the moment that the first customer can not be attended, by the lack of coins, what are the coins which are not available in the cash machine?
2020 Lusophon Mathematical Olympiad
a) Find a pair(s) of integers $(x,y)$ such that:

$y^2=x^3+2017$

b) Prove that there isn't integers $x$ and $y$, with $y$ not divisible by $3$, such that:

$y^2=x^3-2017$
2020 Lusophon Mathematical Olympiad
Let $ABC$ be a triangle and on the sides we draw, externally, the squares $BADE, CBFG$ and $ACHI$. Determine the greatest positive real constant $k$ such that, for any triangle $\triangle ABC$, the following inequality is true:

$[DEFGHI]\geq k\cdot [ABC]$

Note: $[X]$ denotes the area of polygon $X$.
2020 Lusophon Mathematical Olympiad
Let $ABC$ be an acute triangle. Its incircle touches the sides $BC$, $CA$ and $AB$ at the points $D$, $E$ and $F$, respectively. Let $P$, $Q$ and $R$ be the circumcenters of triangles $AEF$, $BDF$ and $CDE$, respectively. Prove that triangles $ABC$ and $PQR$ are similar.
2020 Lusophon Mathematical Olympiad
In how many ways can we fill the cells of a $4\times4$ grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is $2020$?
2020 Lusophon Mathematical Olympiad
Prove that $\lfloor{\sqrt{9n+7}}\rfloor=\lfloor{\sqrt{n}+\sqrt{n+1}+\sqrt{n+2}}\rfloor$ for all postive integer $n$.
2019 Lusophon Mathematical Olympiad
Find a way to write all the digits of $1$ to $9$ in a sequence and without repetition, so that the numbers determined

by any two consecutive digits of the sequence are divisible by $7$ or $13$.
2019 Lusophon Mathematical Olympiad
Prove that for every $n$ nonzero integer , there are infinite triples of nonzero integers $a, b$ and $c$ that satisfy the conditions:

1. $a + b + c = n$

2. $ax^2 + bx + c = 0$ has rational roots.
2019 Lusophon Mathematical Olympiad
Let $ABC$ be a triangle with $AC \ne  BC$. In triangle $ABC$, let $G$ be the centroid, $I$  the incenter and O Its  circumcenter. Prove that $IG$ is parallel to $AB$ if, and only if, $CI$ is perpendicular on $IO$.
2019 Lusophon Mathematical Olympiad
Find all the real numbers $a$ and $b$ that satisfy the relation

$2(a^2 + 1)(b^2 + 1) = (a + 1)(b + 1)(ab + 1)$
2019 Lusophon Mathematical Olympiad
a) Show that there are five integers $A, B, C, D$, and $E$ such that $2018 = A^5 + B^5 + C^5 + D^5 + E^5$



b) Show that there are no four integers $A, B, C$ and $D$ such that $2018 = A^5 + B^5 + C^5 + D^5$
2019 Lusophon Mathematical Olympiad
Two players Arnaldo and Betania play alternately, with Arnaldo being the first to play. Initially there are two piles of stones containing $x$ and $y$ stones respectively. In each play, it is possible to perform one of the following operations:

1. Choose two non-empty piles and take one stone from each pile.

2. Choose a pile with an odd amount of stones, take one of their stones and, if possible, split into two piles with the same amount of stones.

The player who cannot perform either of operations 1  and 2  loses.

Determine who has the winning strategy based on $x$ and $y$.
2018 Lusophon Mathematical Olympiad
Fill in the corners of the square, so that the sum of the numbers in each one of the $5$ lines of the square is the same and the sum of the four corners is $123$.
2018 Lusophon Mathematical Olympiad
In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$  be the point on the extension of $BA$ such that the triangle $ADE$  is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$  and $Q$ the point of  intersection of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square
2018 Lusophon Mathematical Olympiad
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$.
2018 Lusophon Mathematical Olympiad
Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.
2018 Lusophon Mathematical Olympiad
Determine the increasing geometric progressions, with three integer terms, such that the sum of these terms is $57$
2018 Lusophon Mathematical Olympiad
In a $3 \times 25$ board, $1 \times 3$ pieces are placed (vertically or horizontally) so that they occupy entirely $3$ boxes on the board and do not have a common point.

What is the maximum number of pieces that can be placed, and for that number, how many configurations are there?



original formulationNum tabuleiro 3 × 25 s˜ao colocadas pe¸cas 1 × 3 (na vertical ou na horizontal) de modo que ocupem inteiramente 3 casas do tabuleiro e n˜ao se toquem em nenhum ponto.

Qual ´e o n´umero m´aximo de pe¸cas que podem ser colocadas, e para esse n´umero,

quantas configura¸c˜oes existem?

source
2017 Lusophon Mathematical Olympiad
In a math test, there are easy and hard questions. The easy questions worth 3 points and the hard questions worth D points.

If all the questions begin to worth 4 points, the total punctuation of the test increases 16 points.

Instead, if we exchange the questions scores, scoring D points for the easy questions and 3 for the hard ones, the total punctuation of the test is multiplied by $\frac{3}{2}$.

Knowing that the number of easy questions is 9 times bigger the number of hard questions, find the number of questions in this test.
2017 Lusophon Mathematical Olympiad
Let ABCD be a parallelogram, E the midpoint of AD and F the projection of B on CE. Prove that the triangle ABF is isosceles.
2017 Lusophon Mathematical Olympiad
Determine all the positive integers with more than one digit, all distinct, such that the sum of its digits is equal to the product of its digits.
2017 Lusophon Mathematical Olympiad
Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where a, b, c, d are digits, with a > 0.
2017 Lusophon Mathematical Olympiad
The unit cells of a 5 x 5 board are painted with 5 colors in a way that every cell is painted by exactly one color and each color is used in 5 cells. Show that exists at least one line or one column of the board in which at least 3 colors were used.
2017 Lusophon Mathematical Olympiad
Let ABC be a scalene triangle. Consider points D, E, F on segments AB, BC, CA, respectively, such that $\overline{AF}$=$\overline{DF}$ and $\overline{BE}$=$\overline{DE}$.

Show that the circumcenter of ABC lies on the circumcircle of CEF.
2016 Lusophon Mathematical Olympiad
Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?
2016 Lusophon Mathematical Olympiad
The circle $\omega_1$ intersects the circle $\omega_2$ in the points $A$ and $B$, a tangent line to this circles intersects $\omega_1$ and $\omega_2$ in the points $E$ and $F$ respectively. Suppose that $A$ is inside of the triangle $BEF$, let $H$ be the orthocenter of $BEF$ and $M$ is the midpoint of $BH$. Prove that the centers of the circles $\omega_1$ and $\omega_2$ and the point $M$ are collinears.
2016 Lusophon Mathematical Olympiad
Suppose a real number $a$ is a root of a polynomial with integer coefficients $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$.

Let $G=|a_n|+|a_{n-1}|+...+|a_1|+|a_0|$. We say that $G$ is a gingado  of $a$.

For example, as $2$ is root of $P(x)=x^2+x-2, G=|1|+|1|+|-2|=4$, we say that $4$ is a  gingado of $2$.

What is the fourth largest real number $a$ such that $3$ is a gingado  of $a$?
2016 Lusophon Mathematical Olympiad
$8$ CPLP football teams competed in a championship in which each team played one and only time with each of the other teams. In football, each win is worth $3$ points, each draw is worth $1$ point and the defeated team does not score. In that championship four teams were in first place with $15$ points and the others four came in second with $N$  points each. Knowing that there were $12$ draws throughout the championship, determine $N$.
2016 Lusophon Mathematical Olympiad
A numerical sequence is called lusophone if it satisfies the following three conditions:

i) The first term of the sequence is number $1$.

ii) To obtain the next term of the sequence we can multiply the previous term by a positive prime number ($2,3,5,7,11, ...$) or add $1$.

(iii) The last term of the sequence is the number $2016$.

For example: $1\overset{{\times 11}}{\to}11 \overset{{\times 61}}{\to} 671 \overset{{+1}}{\to}672 \overset{{\times 3}}{\to}2016$

How many Lusophone sequences exist in which (as in the example above) the add $1$ operation was used exactly once and not multiplied twice by the same prime number?
2016 Lusophon Mathematical Olympiad
Source: Lusophon MO 2016

Prove that any positive power of $2$ can be written as:

$$5xy-x^2-2y^2$$where $x$ and $y$ are odd numbers.
2015 Lusophon Mathematical Olympiad
In a triangle $ABC, L$ and $K$ are the points of intersections of the angle bisectors of $\angle ABC$ and $\angle BAC$ with the segments $AC$ and $BC$, respectively. The segment $KL$ is angle bisector of $\angle AKC$, determine $\angle BAC$.
2015 Lusophon Mathematical Olympiad
Determine all ten-digit numbers whose decimal  $\overline{a_0a_1a_2a_3a_4a_5a_6a_7a_8a_9}$ is given by  such that for each integer $j$ with $0\le j \le 9, a_j$ is equal to the number of digits equal to $j$ in this representation.

That is: the first digit is equal to the amount of "$0$" in the writing of that number, the second digit is equal to the amount of "$1$" in the writing of that number, the third digit is equal to the amount of "$2$" in the writing of that number, ... , the tenth digit is equal to the number of "$9$" in the writing of that number.