Navvye/Olympiad-Problems-CSV huggingface dataset:
2017 Brazil National Olympiad
4. We see, in Figures 1 and 2, examples of lock screens from a cellphone that only works with a password that is not typed but drawn with straight line segments. Those segments form a polygonal line with vertices in a lattice. When drawing the pattern that corresponds to a password, the finger can't lose contact with the screen. Every polygonal line corresponds to a sequence of digits and this sequence is, in fact, the password. The tracing of the polygonal obeys the following rules:



i. The tracing starts at some of the detached points which correspond to the digits from $1$ to $9$ (Figure 3).



ii. Each segment of the pattern must have as one of its extremes (on which we end the tracing of the segment) a point that has not been used yet.



iii. If a segment connects two points and contains a third one (its middle point), then the corresponding digit to this third point is included in the password. That does not happen if this point/digit has already been used.



iv. Every password has at least four digits.



Thus, every polygonal line is associated to a sequence of four or more digits, which appear in the password in the same order that they are visited. In Figure 1, for instance, the password is 218369, if the first point visited was $2$. Notice how the segment connecting the points associated with $3$ and $9$ includes the points associated to digit $6$. If the first visited point were the $9$, then the password would be $963812$. If the first visited point were the $6$, then the password would be $693812$. In this case, the $6$ would be skipped, because it can't be repeated. On the other side, the polygonal line of Figure 2 is associated to a unique password.



Determine the smallest $n (n \geq 4)$ such that, given  any subset of $n$ digits from $1$ to $9$, it's possible to elaborate a password that involves exactly those digits in some order.
2017 Brazil National Olympiad
5. In triangle $ABC$, let $r_A$ be the line that passes through the midpoint of $BC$ and is perpendicular to the internal bisector of $\angle{BAC}$. Define $r_B$ and $r_C$ similarly. Let $H$ and $I$ be the orthocenter and incenter of $ABC$, respectively. Suppose that the three lines $r_A$, $r_B$, $r_C$ define a triangle. Prove that the circumcenter of this triangle is the midpoint of $HI$.
2017 Brazil National Olympiad
6. Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is integer.
2016 Brazil National Olympiad
Let $ABC$ be a triangle.

$r$ and $s$ are the angle bisectors of $\angle ABC$ and $\angle BCA$, respectively.

The points $E$ in $r$ and $D$ in $s$ are such that $AD \| BE$ and $AE \| CD$.

The lines $BD$ and $CE$ cut each other at $F$.

$I$ is the incenter of $ABC$.



Show that if $A,F,I$ are collinear, then $AB=AC$.
2016 Brazil National Olympiad
Find the smallest number \(n\) such that any set of \(n\) ponts in a Cartesian plan, all of them with integer coordinates, contains two poitns such that the square of its mutual distance is a multiple of \(2016\).
2016 Brazil National Olympiad
Let it \(k\) be a fixed positive integer. Alberto and Beralto play the following game:

Given an initial number \(N_0\) and starting with Alberto, they alternately do the following operation: change the number \(n\) for a number \(m\) such that \(m < n\) and \(m\) and \(n\) differ, in its base-2 representation, in exactly \(l\) consecutive digits for some \(l\) such that \(1 \leq l \leq k\).

If someone can't play, he loses.



We say a non-negative integer \(t\) is a winner number when the gamer who receives the number \(t\) has a winning strategy, that is, he can choose the next numbers in order to guarrantee his own victory, regardless the options of the other player.

Else, we call it loser.



Prove that, for every positive integer \(N\), the total of non-negative loser integers lesser than \(2^N\) is \(2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}\)
2016 Brazil National Olympiad
What is the greatest number of positive integers lesser than or equal to 2016 we can choose such that it doesn't have two of them differing by 1,2, or 6?
2016 Brazil National Olympiad
Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials

\(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\).



Show that exists a  real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\).

Find how many integers \(m\) are such that \(P_n(m)<0\) for infinite positive integers \(n\).
2016 Brazil National Olympiad
Lei it \(ABCD\) be a non-cyclical, convex quadrilateral, with no parallel sides.

The lines \(AB\) and \(CD\) meet in \(E\).

Let it \(M \not= E\) be the intersection of circumcircles of \(ADE\) and \(BCE\).

The internal angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(I\).

The external angle bisectors of \(ABCD\) form an convex, cyclical quadrilateral with circumcenter \(J\).



Show that \(I,J,M\) are colinear.
2015 Brazil National Olympiad
Let $\triangle ABC$ be an acute-scalene triangle, and let $N$ be the center of the circle wich pass trough the feet of altitudes. Let $D$ be the intersection of tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$. Prove that $A$, $D$ and $N$ are collinear iff $\measuredangle BAC = 45º$.
2015 Brazil National Olympiad
Consider $S=\{1, 2, 3, \cdots, 6n\}$, $n>1$. Find the largest $k$ such that the following statement is true: every subset $A$ of $S$ with $4n$ elements has at least $k$ pairs $(a,b)$, $a<b$ and $b$ is divisible by $a$.
2015 Brazil National Olympiad
Given a natural $n>1$ and its prime fatorization $n=p_1^{\alpha 1}p_2^{\alpha_2} \cdots p_k^{\alpha_k}$, its false derived is defined by $$f(n)=\alpha_1p_1^{\alpha_1-1}\alpha_2p_2^{\alpha_2-1}...\alpha_kp_k^{\alpha_k-1}.$$Prove that there exist infinitely many naturals $n$ such that $f(n)=f(n-1)+1$.
2015 Brazil National Olympiad
Let $n$ be a integer and let $n=d_1>d_2>\cdots>d_k=1$ its positive divisors.

a) Prove that $$d_1-d_2+d_3-\cdots+(-1)^{k-1}d_k=n-1$$iff $n$ is prime or $n=4$.

b) Determine the three positive integers such that $$d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4.$$
2015 Brazil National Olympiad
Is that true that there exist a polynomial $f(x)$ with rational coefficients, not all integers, with degree $n>0$, a polynomial $g(x)$, with integer coefficients, and a set $S$ with $n+1$ integers such that $f(t)=g(t)$ for all $t \in S$?
2015 Brazil National Olympiad
Let $\triangle ABC$ be a scalene triangle and $X$, $Y$ and $Z$ be points on the lines $BC$, $AC$ and $AB$, respectively, such that $\measuredangle AXB = \measuredangle BYC = \measuredangle CZA$. The circumcircles of $BXZ$ and $CXY$ intersect at $P$. Prove that $P$ is on the circumference which diameter has ends in the ortocenter $H$ and in the baricenter $G$ of $\triangle ABC$.
2014 Brazil National Olympiad
Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.
2014 Brazil National Olympiad
Find all integers $n$, $n>1$, with the following property: for all $k$, $0\le k < n$, there exists a multiple of $n$ whose digits sum leaves a remainder of $k$ when divided by $n$.
2014 Brazil National Olympiad
Let $N$ be an integer, $N>2$. Arnold and Bernold play the following game: there are initially $N$ tokens on a pile. Arnold plays first and removes $k$ tokens from the pile, $1\le k < N$. Then Bernold removes $m$ tokens from the pile, $1\le m\le 2k$ and so on, that is, each player, on its turn, removes a number of tokens from the pile that is between $1$ and twice the number of tokens his opponent took last. The player that removes the last token wins.



For each value of $N$, find which player has a winning strategy and describe it.
2014 Brazil National Olympiad
The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as

$$P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.$$

Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.
2014 Brazil National Olympiad
There is an integer in each cell of a $2m\times 2n$ table. We define the following operation: choose three cells forming an L-tromino (namely, a cell $C$ and two other cells sharing a side with $C$, one being horizontal and the other being vertical) and sum $1$ to each integer in the three chosen cells. Find a necessary and sufficient condition, in terms of $m$, $n$ and the initial numbers on the table, for which there exists a sequence of operations that makes all the numbers on the table equal.
2014 Brazil National Olympiad
Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$. Circle $\omega_A$ is externally tangent to $\omega$ and tangent to sides $AB$ and $AC$ at $A_1$ and $A_2$, respectively. Let $r_A$ be the line $A_1A_2$. Define $r_B$ and $r_C$ in a similar fashion. Lines $r_A$, $r_B$ and $r_C$ determine a triangle $XYZ$. Prove that the incenter of $XYZ$, the circumcenter of $XYZ$ and $I$ are collinear.
2013 Brazil National Olympiad
Let $\Gamma$ be a circle and $A$ a point outside $\Gamma$. The tangent lines to $\Gamma$ through $A$ touch $\Gamma$ at $B$ and $C$. Let $M$ be the midpoint of $AB$. The segment $MC$ meets $\Gamma$ again at $D$ and the line $AD$ meets $\Gamma$ again at $E$. Given that $AB=a$, $BC=b$, compute $CE$ in terms of $a$ and $b$.
2013 Brazil National Olympiad
Arnaldo and Bernaldo play the following game: given a fixed finite set of positive integers $A$ known by both players, Arnaldo picks a number $a \in A$ but doesn't tell it to anyone. Bernaldo thens pick an arbitrary positive integer $b$ (not necessarily in $A$). Then Arnaldo tells the number of divisors of $ab$. Show that Bernaldo can choose $b$ in a way that he can find out the number $a$ chosen by Arnaldo.
2013 Brazil National Olympiad
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that $$f(x+y) \left(f(x) + f(y)\right) = f(xy)$$ for all non-zero reals $x, y$ such that $x+y \neq 0$.
2013 Brazil National Olympiad
Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$.



P.S.: This is basically the same problem as
2013 Brazil National Olympiad
Let $x$ be an irrational number between 0 and 1 and $x = 0.a_1a_2a_3\cdots$ its decimal representation. For each $k \ge 1$, let $p(k)$ denote the number of distinct sequences $a_{j+1} a_{j+2} \cdots a_{j+k}$ of $k$ consecutive digits in the decimal representation of $x$. Prove that $p(k) \ge k+1$ for every positive integer $k$.
2013 Brazil National Olympiad
The incircle of triangle $ABC$ touches sides $BC, CA$ and $AB$ at points $D, E$ and $F$, respectively. Let $P$ be the intersection of lines $AD$ and $BE$. The reflections of $P$ with respect to $EF, FD$ and $DE$ are $X,Y$ and $Z$, respectively. Prove that lines $AX, BY$ and $CZ$ are concurrent at a point on line $IO$, where $I$ and $O$ are the incenter and circumcenter of triangle $ABC$.
2012 Brazil National Olympiad
In a culturing of bacteria, there are two species of them: red and blue bacteria.



When two red bacteria meet, they transform into one blue bacterium.

When two blue bacteria meet, they transform into four red bacteria.

When a red and a blue bacteria meet, they transform into three red bacteria.



Find, in function of the amount of blue bacteria and the red bacteria initially in the culturing,

all possible amounts of bacteria, and for every possible amount, the possible amounts of red and blue bacteria.
2012 Brazil National Olympiad
$ABC$ is a non-isosceles triangle.

$T_A$ is the tangency point of incircle of $ABC$ in the side $BC$ (define $T_B$,$T_C$ analogously).

$I_A$ is the ex-center relative to the side BC (define $I_B$,$I_C$ analogously).

$X_A$ is the mid-point of $I_BI_C$ (define $X_B$,$X_C$ analogously).



Show that $X_AT_A$,$X_BT_B$,$X_CT_C$ meet in a common point, colinear with the incenter and circumcenter of $ABC$.
2012 Brazil National Olympiad
Find the least non-negative integer $n$ such that exists a non-negative integer $k$ such that the last 2012 decimal digits of $n^k$ are all $1$'s.
2012 Brazil National Olympiad
There exists some integers $n,a_1,a_2,\ldots,a_{2012}$ such that



$$ n^2=\sum_{1 \leq i \leq 2012}{{a_i}^{p_i}}  $$



where $p_i$ is the i-th prime ($p_1=2,p_2=3,p_3=5,p_4=7,\ldots$) and $a_i>1$ for all $i$?
2012 Brazil National Olympiad
In how many ways we can paint a $N \times N$ chessboard using 4 colours such that squares with a common side are painted with distinct colors and every $2 \times 2$ square (formed with 4 squares in consecutive lines and columns) is painted with the four colors?
2012 Brazil National Olympiad
Find all surjective functions $f\colon (0,+\infty) \to (0,+\infty)$ such that $2x f(f(x)) = f(x)(x+f(f(x)))$ for all $x>0$.
2011 Brazil National Olympiad
We call a number pal if it doesn't have a zero digit and the sum of the squares of the digits is a perfect square. For example, $122$ and $34$ are pal but $304$ and $12$ are not pal. Prove that there exists a pal number with $n$ digits, $n > 1$.
2011 Brazil National Olympiad
33 friends are collecting stickers for a 2011-sticker album. A distribution of stickers among the 33 friends is incomplete when there is a sticker that no friend has. Determine the least $m$ with the following property: every distribution of stickers among the 33 friends such that, for any two friends, there are at least $m$ stickers both don't have, is incomplete.
2011 Brazil National Olympiad
Prove that, for all convex pentagons $P_1 P_2 P_3 P_4 P_5$ with area 1, there are indices $i$ and $j$ (assume $P_7 = P_2$ and $P_6 = P_1$) such that:



$$ \text{Area of} \ \triangle P_i P_{i+1} P_{i+2} \le \frac{5 - \sqrt 5}{10} \le \text{Area of} \ \triangle P_j P_{j+1} P_{j+2}$$
2011 Brazil National Olympiad
Do there exist $2011$ positive integers $a_1 < a_2 < \ldots < a_{2011}$ such that $\gcd(a_i,a_j) = a_j - a_i$ for any $i$, $j$ such that $1 \le i < j \le 2011$?
2011 Brazil National Olympiad
Let $ABC$ be an acute triangle and $H$ is orthocenter. Let $D$ be the intersection of $BH$ and $AC$ and $E$ be the intersection of $CH$ and $AB$. The circumcircle of $ADE$ cuts the circumcircle of $ABC$ at $F \neq A$. Prove that the angle bisectors of $\angle BFC$ and $\angle BHC$ concur at a point on $BC.$
2011 Brazil National Olympiad
Let $a_{1}, a_{2}, a_{3}, ... a_{2011}$ be nonnegative reals with sum $\frac{2011}{2}$, prove :



$|\prod_{cyc} (a_{n} - a_{n+1})| = |(a_{1} - a_{2})(a_{2} - a_{3})...(a_{2011}-a_{1})| \le \frac{3 \sqrt3}{16}.$
2010 Brazil National Olympiad
Find all functions $f$ from the reals into the reals such that $$ f(ab) = f(a+b) $$ for all irrational $a, b$.
2010 Brazil National Olympiad
Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.
2010 Brazil National Olympiad
What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak?

Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.
2010 Brazil National Olympiad
Let $ABCD$ be a convex quadrilateral, and $M$ and $N$ the midpoints of the sides $CD$ and $AD$, respectively. The lines perpendicular to $AB$ passing through $M$ and to $BC$ passing through $N$ intersect at point $P$. Prove that $P$ is on the diagonal $BD$ if and only if the diagonals $AC$ and $BD$ are perpendicular.
2010 Brazil National Olympiad
Determine all values of $n$ for which there is a set $S$ with $n$ points, with no 3 collinear, with the following property: it is possible to paint all points of $S$ in such a way that all angles determined by three points in $S$, all of the same color or of three different colors, aren't obtuse. The number of colors available is unlimited.
2010 Brazil National Olympiad
Find all pairs $(a, b)$ of positive integers such that



$$ 3^a = 2b^2 + 1. $$
2009 Brazil National Olympiad
Emerald writes $ 2009^2$ integers in a $ 2009\times 2009$ table, one number in each entry of the table. She sums all the numbers in each row and in each column, obtaining $ 4018$ sums. She notices that all sums are distinct. Is it possible that all such sums are perfect squares?
2009 Brazil National Olympiad
Let $ q = 2p+1$, $ p, q > 0$ primes. Prove that there exists a multiple of $ q$ whose digits sum in decimal base is positive and at most $ 3$.
2009 Brazil National Olympiad
There are $ 2009$ pebbles in some points $ (x,y)$ with both coordinates integer. A operation consists in choosing a point $ (a,b)$ with four or more pebbles, removing four pebbles from $ (a,b)$ and putting one pebble in each of the points

$$ (a,b-1),\ (a,b+1),\ (a-1,b),\ (a+1,b)$$



Show that after a finite number of operations each point will necessarily have at most three pebbles. Prove that the final configuration doesn't depend on the order of the operations.
2009 Brazil National Olympiad
Prove that there exists a positive integer $ n_0$ with the following property: for each integer $ n \geq n_0$ it is possible to partition a cube into $ n$ smaller cubes.
2009 Brazil National Olympiad
Let $ ABC$ be a triangle and $ O$ its circumcenter. Lines $ AB$ and $ AC$ meet the circumcircle of $ OBC$ again in $ B_1\neq B$ and $ C_1 \neq C$, respectively, lines $ BA$ and $ BC$ meet the circumcircle of $ OAC$ again in $ A_2\neq A$ and $ C_2\neq C$, respectively, and lines $ CA$ and $ CB$ meet the circumcircle of $ OAB$ in $ A_3\neq A$ and $ B_3\neq B$, respectively. Prove that lines $ A_2A_3$, $ B_1B_3$ and $ C_1C_2$ have a common point.
2009 Brazil National Olympiad
Let $ n > 3$ be a fixed integer and $ x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $ n$, all possible real values of

$$ {x_1\over x_n+x_1+x_2} + {x_2\over x_1+x_2+x_3} + {x_3\over x_2+x_3+x_4} + \cdots + {x_{n-1}\over x_{n-2}+x_{n-1}+x_n} + {x_n\over x_{n-1}+x_n+x_1}$$
2008 Brazil National Olympiad
A positive integer is dapper if at least one of its multiples begins with $ 2008$. For example, $ 7$ is dapper because $ 200858$ is a multiple of $ 7$ and begins with $ 2008$. Observe that $ 200858 = 28694\times 7$.



Prove that every positive integer is dapper.
2008 Brazil National Olympiad
Let $ S$ be a set of $ 6n$ points in a line. Choose randomly $ 4n$ of these points and paint them blue; the other $ 2n$ points are painted green. Prove that there exists a line segment that contains exactly $ 3n$ points from $ S$, $ 2n$ of them blue and $ n$ of them green.
2008 Brazil National Olympiad
Let $ x,y,z$ real numbers such that $ x + y + z = xy + yz + zx$. Find the minimum value of

$$ {x \over x^2 + 1} + {y\over y^2 + 1} + {z\over z^2 + 1}$$
2008 Brazil National Olympiad
Let $ ABCD$ be a cyclic quadrilateral and $ r$ and $ s$ the lines obtained reflecting $ AB$ with respect to the internal bisectors of $ \angle CAD$ and $ \angle CBD$, respectively. If $ P$ is the intersection of $ r$ and $ s$ and $ O$ is the center of the circumscribed circle of $ ABCD$, prove that $ OP$ is perpendicular to $ CD$.
2008 Brazil National Olympiad
Prove that for all integers $ a > 1$ and $ b > 1$ there exists a function $ f$ from the positive integers to the positive integers such that $ f(a\cdot f(n)) = b\cdot n$ for all $ n$ positive integer.
2008 Brazil National Olympiad
The venusian prophet Zabruberson sent to his pupils a $ 10000$-letter word, each letter being $ A$ or $ E$: the Zabrubic word. Their pupils consider then that for $ 1 \leq k \leq 10000$, each word comprised of $ k$ consecutive letters of the Zabrubic word is a prophetic word of length $ k$. It is known that there are at most $ 7$ prophetic words of lenght $ 3$. Find the maximum number of prophetic words of length $ 10$.
2007 Brazil National Olympiad
Let $ f(x) = x^2 + 2007x + 1$. Prove that for every positive integer $ n$, the equation $ \underbrace{f(f(\ldots(f}_{n\ {\rm times}}(x))\ldots)) = 0$ has at least one real solution.
2007 Brazil National Olympiad
Find the number of integers $ c$ such that $ -2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 + c$ is a multiple of $ 2^{2007}$.
2007 Brazil National Olympiad
Consider $ n$ points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least $ \lfloor n/2\rfloor$ elements.
2007 Brazil National Olympiad
$ 2007^2$ unit squares are arranged forming a $ 2007\times 2007$ table. Arnold and Bernold play the following game: each move by Arnold consists of taking four unit squares that forms a $ 2\times 2$ square; each move by Bernold consists of taking a single unit square. They play anternatively, Arnold being the first. When Arnold is not able to perform his move, Bernold takes all the remaining unit squares. The person with more unit squares in the end is the winner.



Is it possible to Bernold to win the game, no matter how Arnold play?
2007 Brazil National Olympiad
Let $ ABCD$ be a convex quadrangle, $ P$ the intersection of lines $ AB$ and $ CD$, $ Q$ the intersection of lines $ AD$ and $ BC$ and $ O$ the intersection of diagonals $ AC$ and $ BD$. Show that if $ \angle POQ= 90^\circ$ then $ PO$ is the bisector of $ \angle AOD$ and $ OQ$ is the bisector of $ \angle AOB$.
2007 Brazil National Olympiad
Given real numbers $ x_1 < x_2 < \ldots < x_n$ such that every real number occurs at most two times among the differences $ x_j - x_i$, $ 1\leq i < j \leq n$, prove that there exists at least $ \lfloor n/2\rfloor$ real numbers that occurs exactly one time among such differences.
2006 Brazil National Olympiad
Let $ABC$ be a triangle. The internal bisector of $\angle B$ meets $AC$ in $P$ and $I$ is the incenter of $ABC$. Prove that if $AP+AB = CB$, then $API$ is an isosceles triangle.
2006 Brazil National Olympiad
Let $n$ be an integer, $n \geq 3$. Let $f(n)$ be the largest number of isosceles triangles whose vertices belong to some set of $n$ points in the plane without three colinear points. Prove that there exists positive real constants $a$ and $b$ such that $an^{2}< f(n) < bn^{2}$ for every integer $n$, $n \geq 3$.
2006 Brazil National Olympiad
Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that

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

for every reals $x,y$.
2006 Brazil National Olympiad
A positive integer is bold iff it has $8$ positive divisors that sum up to $3240$. For example, $2006$ is bold because its $8$ positive divisors, $1$, $2$, $17$, $34$, $59$, $118$, $1003$ and $2006$, sum up to $3240$. Find the smallest positive bold number.
2006 Brazil National Olympiad
Let $P$ be a convex $2006$-gon. The $1003$ diagonals connecting opposite vertices and the $1003$ lines connecting the midpoints of opposite sides are concurrent, that is, all $2006$ lines have a common point. Prove that the opposite sides of $P$ are parallel and congruent.
2006 Brazil National Olympiad
Professor Piraldo takes part in soccer matches with a lot of goals and judges a match in his own peculiar way. A match with score of $m$ goals to $n$ goals, $m\geq n$, is tough when $m\leq f(n)$, where $f(n)$ is defined by $f(0) = 0$ and, for $n \geq 1$, $f(n) = 2n-f(r)+r$, where $r$ is the largest integer such that $r < n$ and $f(r) \leq n$.



Let $\phi ={1+\sqrt 5\over 2}$. Prove that a match with score of $m$ goals to $n$, $m\geq n$, is tough if $m\leq \phi n$ and is not tough if $m \geq \phi n+1$.
2005 Brazil National Olympiad
A natural number is a palindrome when one obtains the same number when writing its digits in reverse order. For example, $481184$, $131$ and $2$ are palindromes.



Determine all pairs $(m,n)$ of positive integers such that $\underbrace{111\ldots 1}_{m\ {\rm ones}}\times\underbrace{111\ldots 1}_{n\ {\rm ones}}$ is a palindrome.