Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 Brazil National Olympiad
Determine the smallest real number $C$ such that the inequality $$ C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16}  $$ holds for all positive real numbers $x_1,x_2,x_3,x_4,x_5$.
2005 Brazil National Olympiad
A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$.
2005 Brazil National Olympiad
We have four charged batteries, four uncharged batteries and a radio which needs two charged batteries to work.



Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists in putting two batteries in the radio and check if the radio works or not.
2005 Brazil National Olympiad
Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid.



Prove that these three Euler lines pass through one common point.



Remark. The Fermat point $F$ is also known as the first Fermat point or the first Toricelli point of triangle $ABC$.



Floor van Lamoen
2005 Brazil National Olympiad
Given positive integers $a,c$ and integer $b$, prove that there exists a positive integer $x$ such that

$$ a^x + x \equiv b \pmod c, $$

that is, there exists a positive integer $x$ such that $c$ is a divisor of $a^x + x - b$.
2004 Brazil National Olympiad
Let $ABCD$ be a convex quadrilateral. Prove that the incircles of the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a point in common if, and only if, $ABCD$ is a rhombus.
2004 Brazil National Olympiad
Determine all values of $n$ such that it is possible to divide a triangle in $n$ smaller triangles such that there are not three collinear vertices and such that each vertex belongs to the same number of segments.
2004 Brazil National Olympiad
Let $x_1, x_2, ..., x_{2004}$ be a sequence of integer numbers such that $x_{k+3}=x_{k+2}+x_{k}x_{k+1}$, $\forall  1 \le k \le 2001$. Is it possible that more than half of the elements are negative?
2004 Brazil National Olympiad
Consider all the ways of writing exactly ten times each of the numbers $0, 1, 2, \ldots , 9$ in the squares of a $10 \times 10$ board.

Find the greatest integer $n$ with the property that there is always a row or a column with $n$ different numbers.
2004 Brazil National Olympiad
Consider the sequence $(a_n)_{n\in \mathbb{N}}$ with $a_0=a_1=a_2=a_3=1$ and $a_na_{n-4}=a_{n-1}a_{n-3} + a^2_{n-2}$. Prove that all the terms of this sequence are integer numbers.
2004 Brazil National Olympiad
Let $a$ and $b$ be real numbers. Define $f_{a,b}\colon R^2\to R^2$ by $f_{a,b}(x;y)=(a-by-x^2;x)$. If $P=(x;y)\in R^2$, define $f^0_{a,b}(P) = P$ and $f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))$ for all nonnegative integers $k$.



The set $per(a;b)$ of the periodic points of $f_{a,b}$ is the set of points $P\in R^2$ such that $f_{a,b}^n(P) = P$ for some positive integer $n$.



Fix $b$. Prove that the set $A_b=\{a\in R \mid per(a;b)\neq \emptyset\}$ admits a minimum. Find this minimum.
2003 Brazil National Olympiad
Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$.
2003 Brazil National Olympiad
Let $S$ be a set with $n$ elements. Take a positive integer $k$. Let $A_1, A_2, \ldots, A_k$ be any distinct subsets of $S$. For each $i$ take $B_i = A_i$ or $B_i = S - A_i$. Find the smallest $k$ such that we can always choose $B_i$ so that $\bigcup_{i=1}^k B_i = S$, no matter what the subsets $A_i$ are.
2003 Brazil National Olympiad
$ABCD$ is a rhombus. Take points $E$, $F$, $G$, $H$ on sides $AB$, $BC$, $CD$, $DA$ respectively so that $EF$ and $GH$ are tangent to the incircle of $ABCD$. Show that $EH$ and $FG$ are parallel.
2003 Brazil National Olympiad
Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.
2003 Brazil National Olympiad
Let $f(x)$ be a real-valued function defined on the positive reals such that



(1) if $x < y$, then $f(x) < f(y)$,



(2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$.



Show that $f(x) < 0$ for some value of $x$.
2003 Brazil National Olympiad
A graph $G$ with $n$ vertices is called cool if we can label each vertex with a different positive integer not greater than $\frac{n^2}{4}$ and find a set of non-negative integers $D$ so that there is an edge between two vertices iff the difference between their labels is in $D$. Show that if $n$ is sufficiently large we can always find a graph with $n$ vertices which is not cool.
2002 Brazil National Olympiad
Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.
2002 Brazil National Olympiad
$ABCD$ is a cyclic quadrilateral and $M$ a point on the side $CD$ such that $ADM$ and $ABCM$ have the same area and the same perimeter. Show that two sides of $ABCD$ have the same length.
2002 Brazil National Olympiad
The squares of an $m\times n$ board are labeled from $1$ to $mn$ so that the squares labeled $i$ and $i+1$ always have a side in common. Show that for some $k$ the squares $k$ and $k+3$ have a side in common.
2002 Brazil National Olympiad
For any non-empty subset $A$ of $\{1, 2, \ldots , n\}$ define $f(A)$ as the largest element of $A$ minus the smallest element of $A$. Find $\sum f(A)$ where the sum is taken over all non-empty subsets of $\{1, 2, \ldots , n\}$.
2002 Brazil National Olympiad
A finite collection of squares has total area $4$. Show that they can be arranged to cover a square of side $1$.
2002 Brazil National Olympiad
Show that we cannot form more than $4096$ binary sequences of length $24$ so that any two differ in at least $8$ positions.
2001 Brazil National Olympiad
Show that for any $a,b,c$ positive reals,



$$ (a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)}  $$
2001 Brazil National Olympiad
Given $a_0 > 1$, the sequence $a_0, a_1, a_2, ...$ is such that for all $k > 0$, $a_k$ is the smallest integer greater than $a_{k-1}$ which is relatively prime to all the earlier terms in the sequence.

Find all $a_0$ for which all terms of the sequence are primes or prime powers.
2001 Brazil National Olympiad
$ABC$ is a triangle

$E, F$ are points in $AB$, such that $AE = EF = FB$



$D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$

$AD$ is perpendicular to $CF$.

The angle CFA is the triple of angle BDF. ($3\angle BDF = \angle CFA$)



Determine the ratio $\frac{DB}{DC}$.





%Edited!%
2001 Brazil National Olympiad
A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)
2001 Brazil National Olympiad
An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic.
2001 Brazil National Olympiad
A one-player game is played as follows: There is a bowl at each integer on the $Ox$-axis. All the bowls are initially empty, except for that at the origin, which contains $n \geq 2$ stones. A move is either



(A) to remove two stones from a bowl and place one in each of the two adjacent bowls, or



(B) to remove a stone from each of two adjacent bowls and to add one stone to the bowl immediately to their left.



Show that only a finite number of moves can be made and that the final position (when no more moves are possible) is independent of the moves made (for a given $n$).
2000 Brazil National Olympiad
A rectangular piece of paper has top edge $AD$. A line $L$ from $A$ to the bottom edge makes an angle $x$ with the line $AD$. We want to trisect $x$. We take $B$ and $C$ on the vertical ege through $A$ such that $AB = BC$. We then fold the paper so that $C$ goes to a point $C'$ on the line $L$ and $A$ goes to a point $A'$ on the horizontal line through $B$. The fold takes $B$ to $B'$. Show that $AA'$ and $AB'$ are the required trisectors.
2000 Brazil National Olympiad
Let $s(n)$ be the sum of all positive divisors of $n$, so $s(6) = 12$. We say $n$ is almost perfect if $s(n) = 2n - 1$. Let $\mod(n, k)$ denote the residue of $n$ modulo $k$ (in other words, the remainder of dividing $n$ by $k$). Put $t(n) = \mod(n, 1) + \mod(n, 2) + \cdots + \mod(n, n)$.



Show that $n$ is almost perfect if and only if $t(n) = t(n-1)$.
2000 Brazil National Olympiad
Define $f$ on the positive integers by $f(n) = k^2 + k + 1$, where $n=2^k(2l+1)$ for some $k,l$ nonnegative integers.

Find the smallest $n$ such that $f(1) + f(2) + ... + f(n) \geq 123456$.
2000 Brazil National Olympiad
An infinite road has traffic lights at intervals of 1500m. The lights are all synchronised and are alternately green for $\frac 32$ minutes and red for 1 minute. For which $v$ can a car travel at a constant speed of $v$m/s without ever going through a red light?
2000 Brazil National Olympiad
Let $ X$ the set of all sequences $ \{a_1, a_2,\ldots , a_{2000}\}$, such that each of the first 1000 terms is 0, 1 or 2, and each of the remaining terms is 0 or 1. The distance between two members $ a$ and $ b$ of $ X$ is defined as the number of $ i$ for which $ a_i$ and $ b_i$ are different.



Find the number of functions $ f : X \to X$ which preserve the distance.
2000 Brazil National Olympiad
Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get?
1999 Brazil National Olympiad
Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$, while $BD$ and $CE$ meet at $Q$. Find the area of $APQD$.
1999 Brazil National Olympiad
Show that, if $\sqrt{2}$ is written in decimal notation, there is at least one nonzero digit at the interval of 1,000,000-th and 3,000,000-th digits.
1999 Brazil National Olympiad
How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?
1999 Brazil National Olympiad
On planet Zork there are some cities. For every city there is a city at the diametrically opposite point. Certain roads join the cities on Zork. If there is a road between cities $P$ and $Q$, then there is also a road between the cities $P'$ and $Q'$ diametrically opposite to $P$ and $Q$. In plus, the roads do not cross each other and for any two cities $P$ and $Q$ it is possible to travel from $P$ to $Q$.



The prices of Kriptonita in Urghs (the planetary currency) in two towns connected by a road differ by at most 100. Prove that there exist two diametrically opposite cities in which the prices of Kriptonita differ by at most 100 Urghs.
1999 Brazil National Olympiad
There are $n$ football teams in Tumbolia. A championship is to be organised in which each team plays against every other team exactly once. Ever match takes place on a sunday and each team plays at most one match each sunday. Find the least possible positive integer $m_n$ for which it is possible to set up a championship lasting $m_n$ sundays.
1999 Brazil National Olympiad
Given any triangle $ABC$, show how to construct $A'$ on the side $AB$, $B'$ on the side $BC$ and $C'$ on the side $CA$, such that $ABC$ and $A'B'C'$ are similar (with $\angle A = \angle A', \angle B = \angle B', \angle C = \angle C'$) and $A'B'C'$ has the least possible area.
1998 Brazil National Olympiad
15 positive integers, all less than 1998(and no one equal to 1), are relatively prime (no pair has a common factor > 1).

Show that at least one of them must be prime.
1998 Brazil National Olympiad
Let $ABC$ be a triangle. $D$ is the midpoint of $AB$, $E$ is a point on the side $BC$ such that $BE = 2 EC$ and $\angle ADC = \angle BAE$. Find $\angle BAC$.
1998 Brazil National Olympiad
Two players play a game as follows: there $n > 1$ rounds and $d \geq 1$ is fixed. In the first round A picks a positive integer $m_1$, then B picks a positive integer $n_1 \not = m_1$. In round $k$ (for $k = 2, \ldots , n$), A picks an integer $m_k$ such that $m_{k-1} < m_k \leq m_{k-1} + d$. Then B picks an integer $n_k$ such that $n_{k-1} < n_k \leq n_{k-1} + d$. A gets $\gcd(m_k,n_{k-1})$ points and B gets $\gcd(m_k,n_k)$ points. After $n$ rounds, A wins if he has at least as many points as B, otherwise he loses.



For each $(n, d)$ which player has a winning strategy?
1998 Brazil National Olympiad
Two players play a game as follows. The first player chooses two non-zero integers A and B. The second player forms a quadratic with A, B and 1998 as coefficients (in any order). The first player wins iff the equation has two distinct rational roots. Show that the first player can always win.
1998 Brazil National Olympiad
Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, $$ f(2f(x)) = x + 1998 . $$
1998 Brazil National Olympiad
Two mathematicians, lost in Berlin, arrived on the corner of Barbarossa street with Martin Luther street and need to arrive on the corner of Meininger street with Martin Luther street. Unfortunately they don't know which direction to go along Martin Luther Street to reach Meininger Street nor how far it is, so they must go fowards and backwards along Martin Luther street until they arrive on the desired corner. What is the smallest value for a positive integer $k$ so that they can be sure that if there are $N$ blocks between Barbarossa street and Meininger street then they can arrive at their destination by walking no more than $kN$ blocks (no matter what $N$ turns out to be)?
1997 Brazil National Olympiad
Given $R, r > 0$. Two circles are drawn radius $R$, $r$ which meet in two points. The line joining the two points is a distance $D$ from the center of one circle and a distance $d$ from the center of the other. What is the smallest possible value for $D+d$?
1997 Brazil National Olympiad
Let $A$ be a set of $n$ non-negative integers. We say it has property $\mathcal P$ if the set $\{x + y \mid x, y \in A\}$ has $\binom{n}{2}$ elements. We call the largest element of $A$ minus the smallest element, the diameter of $A$. Let $f(n)$ be the smallest diameter of any set $A$ with property $\mathcal P$. Show that $n^2 \leq 4 f(n) < 4 n^3$.



Comment(If you have some amount of time, try a best estimative for $f(n)$, such that $f(p)<2p^2$ for prime $p$).
1997 Brazil National Olympiad
a) Show that there are no functions $f, g: \mathbb R \to \mathbb R$ such that $g(f(x)) = x^3$ and $f(g(x)) = x^2$ for all $x \in \mathbb R$.



b) Let $S$ be the set of all real numbers greater than 1. Show that there are functions $f, g : S \to S$ satsfying the condition above.
1997 Brazil National Olympiad
Let $V_n=\sqrt{F_n^2+F_{n+2}^2}$, where $F_n$ is the Fibonacci sequence

($F_1=F_2=1,F_{n+2}=F_{n+1}+F_{n}$)



Show that $V_n,V_{n+1},V_{n+2}$ are the sides of a triangle with area $1/2$
1997 Brazil National Olympiad
Let $f(x)= x^2-C$ where $C$ is a rational constant.

Show that exists only finitely many rationals $x$ such that $\{x,f(x),f(f(x)),\ldots\}$ is finite
1997 Brazil National Olympiad
$f$ is a plane map onto itself such that points at distance 1 are always taken at point at distance 1.

Show that $f$ preserves distances.
1996 Brazil National Olympiad
Show that there exists infinite triples $(x,y,z) \in N^3$ such that $x^2+y^2+z^2=3xyz$.
1996 Brazil National Olympiad
Does there exist a set of $n > 2, n < \infty$ points in the plane such that no three are collinear and the circumcenter of any three points of the set is also in the set?
1996 Brazil National Olympiad
Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$. For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$. Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$.
1996 Brazil National Olympiad
$ABC$ is acute-angled. $D$ s a variable point on the side BC. $O_1$ is the circumcenter of $ABD$, $O_2$ is the circumcenter of $ACD$, and $O$ is the circumcenter of $AO_1O_2$. Find the locus of $O$.
1996 Brazil National Olympiad
There are $n$ boys $B_1, B_2, ... , B_n$ and $n$ girls $G_1, G_2, ... , G_n$. Each boy ranks the girls in order of preference, and each girl ranks the boys in order of preference. Show that we can arrange the boys and girls into n pairs so that we cannot find a boy and a girl who prefer each other to their partners.



For example if $(B_1, G_3)$ and $(B_4, G_7)$ are two of the pairs, then it must not be the case that $B_4$ prefers $G_3$ to $G_7$ and $G_3$ prefers $B_4$ to $B_1$.
1996 Brazil National Olympiad
Let p(x) be the polynomial $x^3 + 14x^2 - 2x + 1$. Let $p^n(x)$ denote $p(p^(n-1)(x))$. Show that there is an integer N such that $p^N(x) - x$ is divisible by 101 for all integers x.
1995 Brazil National Olympiad
$ABCD$ is a quadrilateral with a circumcircle centre $O$ and an inscribed circle centre $I$. The diagonals intersect at $S$. Show that if two of $O,I,S$  coincide, then it must be a square.
1995 Brazil National Olympiad
Find all real-valued functions on the positive integers such that $f(x + 1019) = f(x)$ for all $x$, and $f(xy) = f(x) f(y)$ for all $x,y$.
1995 Brazil National Olympiad
For any positive integer $ n>1$, let $ P\left(n\right)$ denote the largest prime divisor of $ n$. Prove that there exist infinitely many positive integers $ n$ for which

$$ P\left(n\right)<P\left(n+1\right)<P\left(n+2\right).$$
1995 Brazil National Olympiad
A regular tetrahedron has side $L$. What is the smallest $x$ such that the tetrahedron can be passed through a loop of twine of length $x$?
1995 Brazil National Olympiad
Show that no one $n$-th root of a rational (for $n$ a positive integer) can be a root of the polynomial $x^5 - x^4 - 4x^3 + 4x^2 + 2$.
1995 Brazil National Olympiad
$X$ has $n$ elements. $F$ is a family of subsets of $X$ each with three elements, such that any two of the subsets have at most one element in common. Show that there is a subset of $X$ with at least $\sqrt{2n}$ members which does not contain any members of $F$.
1994 Brazil National Olympiad
The edges of a cube are labeled from 1 to 12 in an arbitrary manner. Show that it is not possible to get the sum of the edges at each vertex the same.

Show that we can get eight vertices with the same sum if one of the labels is changed to 13.
1994 Brazil National Olympiad
Given any convex polygon, show that there are three consecutive vertices such that the polygon lies inside the circle through them.
1994 Brazil National Olympiad
We are given n objects of identical appearance, but different mass, and a balance which can be used to compare any two objects (but only one object can be placed in each pan at a time). How many times must we use the balance to find the heaviest object and the lightest object?
1994 Brazil National Olympiad
Let $a, b > 0$ be reals such that

$$ a^3=a+1\\ b^6=b+3a  $$



Show that $a>b$
1994 Brazil National Olympiad
Call a super-integer an infinite sequence of decimal digits: $\ldots d_n \ldots d_2d_1$.



(Formally speaking, it is the sequence $(d_1,d_2d_1,d_3d_2d_1,\ldots)$ )



Given two such super-integers $\ldots c_n \ldots c_2c_1$ and $\ldots d_n \ldots d_2d_1$, their product $\ldots p_n \ldots p_2p_1$ is formed by taking $p_n \ldots p_2p_1$ to be the last n digits of the product $c_n \ldots c_2c_1$ and $d_n \ldots d_2d_1$.

Can we find two non-zero super-integers with zero product?

(a zero super-integer has all its digits zero)
1994 Brazil National Olympiad
A triangle has semi-perimeter $s$, circumradius $R$ and inradius $r$. Show that it is right-angled iff $2R = s - r$.
1993 Brazil National Olympiad
The sequence $(a_n)_{n \in\mathbb{N}}$ is defined by $a_1 = 8, a_2 = 18, a_{n+2} = a_{n+1}a_{n}$. Find all terms which are perfect squares.
1993 Brazil National Olympiad
A real number with absolute value less than $1$ is written in each cell of an $n\times n$ array, so that the sum of the numbers in each $2\times 2$ square is zero. Show that for odd $n$ the sum of all the numbers is less than $n$.
1993 Brazil National Olympiad
Given a circle and its center $O$, a point $A$ inside the circle and a distance $h$, construct a triangle $BAC$ with $\angle BAC = 90^\circ$, $B$ and $C$ on the circle and the altitude from $A$ length $h$.
1993 Brazil National Olympiad
$ABCD$ is a convex quadrilateral with

$$\angle BAC = 30^\circ $$$$\angle CAD = 20^\circ$$$$\angle ABD = 50^\circ$$$$\angle DBC = 30^\circ$$

If the diagonals intersect at $P$, show that $PC = PD$.
1993 Brazil National Olympiad
Find at least one function $f: \mathbb R \rightarrow \mathbb R$ such that $f(0)=0$ and $f(2x+1) = 3f(x) + 5$ for any real $x$.
1992 Brazil National Olympiad
The equation $x^3+px+q=0$ has three distinct real roots. Show that $p<0$
1992 Brazil National Olympiad
Show that there is a positive integer n such that the first 1992 digits of $n^{1992}$ are 1.
1992 Brazil National Olympiad
Given positive real numbers $x_1, x_2, \ldots , x_n$ find the polygon $A_0A_1\ldots A_n$ with $A_iA_{i+1} = x_{i+1}$ and which has greatest area.
1992 Brazil National Olympiad
Let $d(n)=\sum_{0<d|n}{1}$. Show that, for any natural $n>1$,



$$ \sum_{2 \leq i \leq n}{\frac{1}{i}} \leq \sum{\frac{d(i)}{n}} \leq \sum_{1 \leq i \leq n}{\frac{1}{i}} $$
1992 Brazil National Olympiad
Given a set of n elements, find the largest number of subsets such that no subset is contained in any other
1992 Brazil National Olympiad
Find all 4-tuples $(a,b,c,n)$ of naturals such that



$n^a + n^b = n^c$
1992 Brazil National Olympiad
In a chess tournament each player plays every other player once. A player gets 1 point for a win, 0.5 point for a draw and 0 for a loss. Both men and women played in the tournament and each player scored the same total of points against women as against men. Show that the total number of players must be a square.
1991 Brazil National Olympiad
At a party every woman dances with at least one man, and no man dances with every woman. Show that there are men M and M' and women W and W' such that M dances with W, M' dances with W', but M does not dance with W', and M' does not dance with W.
1991 Brazil National Olympiad
$P$ is a point inside the triangle $ABC$. The line through $P$ parallel to $AB$ meets $AC$ $A_0$ and $BC$ at $B_0$. Similarly, the line through $P$ parallel to $CA$ meets $AB$ at $A_1$ and $BC$ at $C_1$, and the line through P parallel to BC meets $AB$ at $B_2$ and $AC$ at $C_2$. Find the point $P$ such that $A_0B_0 = A_1B_1 = A_2C_2$.
1991 Brazil National Olympiad
Given $k > 0$, the sequence $a_n$ is defined by its first two members and $$ a_{n+2} = a_{n+1} + \frac{k}{n}a_n  $$

a)For which $k$ can we write $a_n$ as a polynomial in $n$?



b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$? ($p,q$ are polynomials in $\mathbb R[X]$).
1991 Brazil National Olympiad
Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's).
1991 Brazil National Olympiad
$P_0 = (1,0), P_1 = (1,1), P_2 = (0,1), P_3 = (0,0)$.

$P_{n+4}$ is the midpoint of $P_nP_{n+1}$.

$Q_n$ is the quadrilateral $P_{n}P_{n+1}P_{n+2}P_{n+3}$.

$A_n$ is the interior of $Q_n$.



Find $\cap_{n \geq 0}A_n$.
1990 Brazil National Olympiad
Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.
1990 Brazil National Olympiad
There exists infinitely many positive integers such that

$a^3 + 1990b^3 = c^4$.
1990 Brazil National Olympiad
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$.