Navvye/Olympiad-Problems-CSV huggingface dataset:
2000 Bosnia and Herzegovina Team Selection Test
It is given triangle $ABC$ such that $\angle ABC = 3 \angle CAB$. On side $AC$ there are two points $M$ and $N$ in order $A - N - M - C$ and $\angle CBM = \angle MBN = \angle NBA$. Let $L$ be an arbitrary point on side $BN$ and $K$ point on $BM$ such that $LK \mid \mid AC$. Prove that lines $AL$, $NK$ and $BC$ are concurrent
1999 Bosnia and Herzegovina Team Selection Test
Let $a$, $b$ and $c$ be lengths of sides of triangle $ABC$. Prove that at least one of the equations $$x^2-2bx+2ac=0$$$$x^2-2cx+2ab=0$$$$x^2-2ax+2bc=0$$does not have real solutions
1999 Bosnia and Herzegovina Team Selection Test
Prove the inequality $$\frac{a^2}{b+c-a}+\frac{b^2}{a+c-b}+\frac{c^2}{a+b-c} \geq 3\sqrt{3}R$$in triangle $ABC$ where $a$, $b$ and $c$ are sides of triangle and $R$ radius of circumcircle of $ABC$
1999 Bosnia and Herzegovina Team Selection Test
Let $f : [0,1] \rightarrow \mathbb{R}$ be injective function such that $f(0)+f(1)=1$. Prove that exists $x_1$, $x_2 \in [0,1]$, $x_1 \neq x_2$ such that $2f(x_1)<f(x_2)+\frac{1}{2}$. After that state at least one generalization of this result
1999 Bosnia and Herzegovina Team Selection Test
Let angle bisectors of angles $\angle BAC$ and $\angle ABC$ of triangle $ABC$ intersect sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let points $F$ and $G$ be foots of perpendiculars from point $C$ on lines $AD$ and $BE$, respectively. Prove that $FG \mid \mid AB$
1999 Bosnia and Herzegovina Team Selection Test
For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that

$a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$

$b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$

where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$
1999 Bosnia and Herzegovina Team Selection Test
It is given polynomial $$P(x)=x^4+3x^3+3x+p, (p \in \mathbb{R})$$$a)$ Find $p$ such that there exists polynomial with imaginary root $x_1$ such that $\mid x_1 \mid =1$ and $2Re(x_1)=\frac{1}{2}\left(\sqrt{17}-3\right)$

$b)$ Find all other roots of polynomial $P$

$c)$ Prove that does not exist positive integer $n$ such that $x_1^n=1$
1998 Bosnia and Herzegovina Team Selection Test
Let $P_1$, $P_2$, $P_3$, $P_4$ and $P_5$ be five different points which are inside $D$ or on the border of figure $D$. Let $M=min\left\{P_iP_j \mid i \neq j\right\}$ be minimal distance between different points $P_i$. For which configuration of points $P_i$, value $M$ is at maximum, if :

$a)$ $D$ is unit square

$b)$ $D$ is equilateral triangle with side equal $1$

$c)$ $D$ is unit circle, circle with radius $1$
1998 Bosnia and Herzegovina Team Selection Test
For positive real numbers $x$, $y$ and $z$ holds $x^2+y^2+z^2=1$. Prove that $$\frac{x}{1+x^2}+\frac{y}{1+y^2}+\frac{z}{1+z^2} \leq \frac{3\sqrt{3}}{4}$$
1998 Bosnia and Herzegovina Team Selection Test
Angle bisectors of angles by vertices $A$, $B$ and $C$ in triangle $ABC$ intersect opposing sides in points $A_1$, $B_1$ and $C_1$, respectively. Let $M$ be an arbitrary point on one of the lines $A_1B_1$, $B_1C_1$ and $C_1A_1$. Let $M_1$, $M_2$ and $M_3$ be orthogonal projections of point $M$ on lines $BC$, $CA$ and $AB$, respectively. Prove that one of the lines $MM_1$, $MM_2$ and $MM_3$ is equal to sum of other two
1998 Bosnia and Herzegovina Team Selection Test
Circle $k$  with radius $r$ touches the line $p$ in point $A$. Let $AB$ be a dimeter of circle and $C$ an arbitrary point of circle distinct from points $A$ and $B$. Let $D$ be a foot of perpendicular from point $C$ to line $AB$. Let $E$ be a point on extension of line $CD$, over point $D$, such that $ED=BC$. Let tangents on circle from point $E$ intersect line $p$ in points $K$ and $N$. Prove that length of $KN$ does not depend from $C$
1998 Bosnia and Herzegovina Team Selection Test
Let $a$, $b$ and $c$ be integers such that $$bc+ad=1$$$$ac+2bd=1$$Prove that $a^2+c^2=2b^2+2d^2$
1998 Bosnia and Herzegovina Team Selection Test
Sequence of integers $\{u_n\}_{n \in \mathbb{N}_0}$ is given as: $u_0=0$, $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for all $n \in \mathbb{N}_0$

$a)$ Find $u_{1998}$

$b)$ If $p$ is a positive integer and $m=(2^p-1)^2$, find $u_m$
1997 Bosnia and Herzegovina Team Selection Test
Solve system of equation $$8(x^3+y^3+z^3)=73$$$$2(x^2+y^2+z^2)=3(xy+yz+zx)$$$$xyz=1$$in set $\mathbb{R}^3$
1997 Bosnia and Herzegovina Team Selection Test
In isosceles triangle $ABC$ with base side $AB$, on side $BC$ it is given point $M$. Let $O$ be a circumcenter and $S$ incenter of triangle $ABC$. Prove that $$ SM \mid \mid AC \Leftrightarrow OM \perp BS$$
1997 Bosnia and Herzegovina Team Selection Test
It is given function  $f : A \rightarrow \mathbb{R}$, $(A\subseteq \mathbb{R})$ such that $$f(x+y)=f(x)\cdot f(y)-f(xy)+1; (\forall x,y \in A)$$If $f : A \rightarrow \mathbb{R}$, $(\mathbb{N} \subseteq A\subseteq \mathbb{R})$ is solution of given functional equation, prove that: $$f(n)=\begin{cases}

\frac{c^{n+1}-1}{c-1} \text{, } \forall n \in \mathbb{N}, c \neq 1 \\
n+1 \text{, } \forall n \in \mathbb{N}, c = 1

\end{cases}$$where $c=f(1)-1$

$a)$ Solve given functional equation for $A=\mathbb{N}$

$b)$ With $A=\mathbb{Q}$, find all functions $f$ which are solutions of the given functional equation and also $f(1997) \neq f(1998)$
1997 Bosnia and Herzegovina Team Selection Test
$a)$ In triangle $ABC$ let $A_1$, $B_1$ and $C_1$ be touching points of incircle $ABC$ with $BA$, $CA$ and $AB$, respectively. Let $l_1$, $l_2$ and $l_3$ be lenghts of arcs $ B_1C_1$, $A_1C_1$, $B_1A_1$ of incircle $ABC$, respectively, which does not contain points $A_1$, $B_1$ and $C_1$, respectively.

Does the following inequality hold: $$ \frac{a}{l_1}+\frac{b}{l_2}+\frac{c}{l_3} \geq \frac{9\sqrt{3}}{\pi}$$

$b)$ Tetrahedron $ABCD$ has three pairs of equal opposing sides. Find length of height of tetrahedron in function od lengths of sides
1997 Bosnia and Herzegovina Team Selection Test
$a)$ Prove that for all positive integers $n$ exists a set $M_n$ of positive integers with exactly $n$ elements and:



$i)$ Arithmetic mean of arbitrary non-empty subset of $M_n$ is integer

$ii)$ Geometric mean of arbitrary non-empty subset of $M_n$ is integer

$iii)$ Both arithmetic mean and geometry mean of arbitrary non-empty subset of $M_n$ is integer



$b)$ Does there exist infinite set $M$ of positive integers such that  arithmetic mean of arbitrary non-empty subset of $M$ is integer
1997 Bosnia and Herzegovina Team Selection Test
Let $k$, $m$ and $n$ be integers such that $1<n \leq m-1 \leq k$. Find maximum size of subset $S$ of set $\{1,2,...,k\}$ such that sum of any $n$ different elements from $S$ is not:

$a)$ equal to $m$,

$b)$ exceeding $m$
1996 Bosnia and Herzegovina Team Selection Test
$a)$ Let $a$, $b$ and $c$ be positive real numbers. Prove that for all positive integers $m$ holds: $$(a+b)^m+(b+c)^m+(c+a)^m \leq 2^m(a^m+b^m+c^m)$$$b)$ Does inequality $a)$ holds for

$1)$ arbitrary real numbers $a$, $b$ and $c$

$2)$ any integer $m$
1996 Bosnia and Herzegovina Team Selection Test
$a)$ Let $m$ and $n$ be positive integers. If $m>1$ prove that $ n \mid \phi(m^n-1)$ where $\phi$ is Euler function

$b)$ Prove that number of elements in sequence $1,2,...,n$ $(n \in \mathbb{N})$, which greatest common divisor with $n$ is $d$, is $\phi\left(\frac{n}{d}\right)$
1996 Bosnia and Herzegovina Team Selection Test
Let $M$ be a point inside quadrilateral $ABCD$ such that $ABMD$ is parallelogram. If $\angle CBM = \angle CDM$ prove that $\angle ACD = \angle BCM$
1996 Bosnia and Herzegovina Team Selection Test
Solve the functional equation $$f(x+y)+f(x-y)=2f(x)\cos{y}$$where $x,y \in \mathbb{R}$ and $f : \mathbb{R} \rightarrow \mathbb{R}$
1996 Bosnia and Herzegovina Team Selection Test
Group of $10$ people are buying books. We know the following:

$i)$ Every person bought four different books

$ii)$ Every two persons bought at least one book common for both of them

Taking in consideration book which was bought by maximum number of people, determine minimal value of that number
1996 Bosnia and Herzegovina Team Selection Test
Let $a$ and $b$ be two integers which are coprime and let $n$ be one variable integer. Determine probability that number of solutions $(x,y)$, where $x$ and $y$ are nonnegative integers, of equation $ax+by=n$ is $\left\lfloor \frac{n}{ab} \right\rfloor + 1$
2018 Brazil Team Selection Test
Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.
2018 Brazil Team Selection Test
Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:



Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell.

Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell.



At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$.



Proposed by Warut Suksompong, Thailand
2018 Brazil Team Selection Test
Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:



In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.

In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are

two (not necessarily distinct) numbers from the first line.

In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.



Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.
2018 Brazil Team Selection Test
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both

$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$are integers.
2018 Brazil Team Selection Test
Let $H$ be the orthocenter of the triangle $ABC$. Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$, respectively. Assume that $H$ lies inside the quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of the triangles $BMH$ and $CNH$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $MK$ and $NL$ and let $J$ be the incenter of triangle $MHN$. Prove that $F J = F A$.
2018 Brazil Team Selection Test
Let $f(x)$ and $g(x)$ be given by

$f(x) = \frac{1}{x} + \frac{1}{x-2} + \frac{1}{x-4} + \cdots + \frac{1}{x-2018}$

$g(x) = \frac{1}{x-1} + \frac{1}{x-3} + \frac{1}{x-5} + \cdots + \frac{1}{x-2017}$.



Prove that $|f(x)-g(x)| >2$ for any non-integer real number $x$ satisfying $0 < x < 2018$.
2018 Brazil Team Selection Test
A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied:



(i) All the squares are congruent.

(ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares.

(iii) Each square touches exactly three other squares.



How many positive integers $n$ are there with $2018\leq n \leq 3018$, such that there exists a collection of $n$ squares that is tri-connected?
2018 Brazil Team Selection Test
Let $ABC$ be an equilateral triangle. From the vertex $A$ we draw a ray towards the interior of the triangle such that the ray reaches one of the sides of the triangle. When the ray reaches a side, it then bounces off following the law of reflection, that is, if it arrives with a directed angle $\alpha$, it leaves with a directed angle $180^{\circ}-\alpha$. After $n$ bounces, the ray returns to $A$ without ever landing on any of the other two vertices. Find all possible values of $n$.
2018 Brazil Team Selection Test
Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.
2018 Brazil Team Selection Test
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
2018 Brazil Team Selection Test
Given a set $S$ of positive real numbers, let $$\Sigma (S) = \Bigg\{ \sum_{x \in A} x : \emptyset \neq A \subset S \Bigg\}.$$be the set of all the sums of elements of non-empty subsets of $S$. Find the least constant $L> 0$ with the following property: for every integer greater than $1$ and every set $S$ of $n$ positive real numbers, it is possible partition $\Sigma(S)$ into $n$ subsets $\Sigma_1,\ldots, \Sigma_n$ so that the ratio between the largest and smallest element of each $\Sigma_i$ is at most $L$.
2018 Brazil Team Selection Test
Let $n \ge 1$ be an integer. For each subset $S \subset \{1, 2, \ldots , 3n\}$, let $f(S)$ be the sum of the elements of $S$, with $f(\emptyset) = 0$. Determine, as a function of $n$, the sum $$\sum_{\mathclap{\substack{S \subset \{1,2,\ldots,3n\}\\
                              3 \mid f(S)}}} f(S)$$where $S$ runs through all subsets of $\{1, 2,\ldots, 3n\}$ such that $f(S)$ is a multiple of $3$.
2018 Brazil Team Selection Test
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.
2018 Brazil Team Selection Test
A convex  quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.
2018 Brazil Team Selection Test
In a triangle $ABC$, points $H, L, K$ are chosen on the sides $AB, BC, AC$, respectively, so that $CH \perp AB$, $HL \parallel AC$ and $HK \parallel BC$. In the triangle $BHL$, let $P, Q$ be the feet of the heights from the vertices $B$ and $H$. In the triangle $AKH$, let $R, S$ be the feet of the heights from the vertices $A$ and $H$. Show that the four points $P, Q, R, S$ are collinear.
2018 Brazil Team Selection Test
Prove: there are polynomials $S_1, S_2, \ldots$ in the variables $x_1, x_2, \ldots,y_1,  y_2,\ldots$ with integer coefficients satisfying, for every integer $n \ge 1$, $$\sum_{d \mid n} d \cdot S_d ^{n/d}=\sum_{d \mid n} d \cdot (x_d ^{n/d}+y_d ^{n/d}) \quad (*)$$Here, the sums run through the positive divisors $d$ of $n$.

For example, the first two polynomials are $S_1 = x_1 + y_1$ and $S_2 = x_2 + y_2 - x_1y_1$, which verify identity

$(*)$ for $n = 2$: $S_1^2 + 2S_2 = (x_1^2 + y_1^2) + 2 \cdot(x_2 + y_2)$.
2018 Brazil Team Selection Test
An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ Shiny if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have

$$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$Find the largest constant $K = K(n)$ such that

$$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.
1997 Brazil Team Selection Test
Let $ABC$ be a triangle and $L$ its circumscribed circle. The internal bisector of angle $A$ meets $BC$ at point $P$. Let $L_1$ be the circle tangent to $AP,BP$ and $L$. Similarly, let $L_2$ be the circle tangent to $AP,CP$ and $L$. Prove that the tangency points of $L_1$ and $L_2$ with $AP$ coincide.
1997 Brazil Team Selection Test
We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.
1997 Brazil Team Selection Test
Find all positive integers $x>1, y$ and primes $p,q$ such that $p^{x}=2^{y}+q^{x}$
1997 Brazil Team Selection Test
Prove that it is impossible to arrange the numbers $1,2,\ldots,1997$ around a circle in such a way that, if $x$ and $y$ are any two neighboring numbers, then $499\le|x-y|\le997$.
1997 Brazil Team Selection Test
Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly.



(a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$.

(b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.
1997 Brazil Team Selection Test
In an isosceles triangle $ABC~(AC=BC)$, let $O$ be its circumcenter, $D$ the midpoint of $AC$ and $E$ the centroid of $DBC$. Show that $OE$ is perpendicular to $BD$.
1997 Brazil Team Selection Test
Prove that any group of people can be divided into two disjoint groups $A$ and $B$ such that any member from $A$ has at least half of his acquaintances in $B$ and any member from $B$ has at least half of his acquaintances in $A$ (acquaintance is reciprocal).
1997 Brazil Team Selection Test
Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.
1997 Brazil Team Selection Test
Consider an $N\times N$ matrix, where $N$ is an odd positive integer, such that all its entries are $-1,0$ or $1$. Consider the sum of the numbers in every line and every column. Prove that at least two of the $2N$ sums are equal.
1997 Brazil Team Selection Test
Consider an infinite strip, divided into unit squares. A finite number of nuts is placed in some of these squares. In a step, we choose a square $A$ which has more than one nut and take one of them and put it on the square on the right, take another nut (from $A$) and put it on the square on the left. The procedure ends when all squares has at most one nut. Prove that, given the initial configuration, any procedure one takes will end after the same number of steps and with the same final configuration.
1998 Brazil Team Selection Test
Let N be a positive integer greater than 2. We number the vertices

of a regular 2n-gon clockwise with the numbers 1, 2, . . . ,N,−N,−N +

1, . . . ,−2,−1. Then we proceed to mark the vertices in the following way.

In the first step we mark the vertex 1. If ni is the vertex marked in the

i-th step, in the i+1-th step we mark the vertex that is |ni| vertices away

from vertex ni, counting clockwise if ni is positive and counter-clockwise

if ni is negative. This procedure is repeated till we reach a vertex that has

already been marked. Let $f(N)$ be the number of non-marked vertices.

(a) If $f(N) = 0$, prove that 2N + 1 is a prime number.

(b) Compute $f(1997)$.
1998 Brazil Team Selection Test
Let $\mathbb N$ be the set of positive integers. Find all functions defined on $\mathbb N$ and taking values on $\mathbb N$ satisfying, for all $n\in\mathbb N$,

$$f(n)+f(n+1)=f(n+2)f(n+3)-1998.$$
1998 Brazil Team Selection Test
Let $L$ be a circle with center $O$ and tangent to sides $AB$ and $AC$ of a triangle $ABC$ in points $E$ and $F$, respectively. Let the perpendicular from $O$ to $BC$ meet $EF$ at $D$. Prove that $A,D$ and $M$ are collinear, where $M$ is the midpoint of $BC$.
1998 Brazil Team Selection Test
Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that

$$k\le2\lfloor\sqrt n\rfloor.$$
1998 Brazil Team Selection Test
Let $ABC$ be an acute-angled triangle. Construct three semi-circles, each having a different side of ABC as diameter, and outside $ABC$. The perpendiculars dropped from $A,B,C$ to the opposite sides intersect these semi-circles in points $E,F,G$, respectively. Prove that the hexagon $AGBECF$ can be folded so as to form a pyramid having $ABC$ as base.
1998 Brazil Team Selection Test
There are $n\ge3$ integers around a circle. We know that for each of these numbers the ratio between the sum of its two neighbors and the number is a positive integer. Prove that the sum of the $n$ ratios is not greater than $3n$.
1998 Brazil Team Selection Test
Show that it is possible to color the points of $\mathbb Q\times\mathbb Q$ in two colors in such a way that any two points having distance $1$ have distinct colors.
1998 Brazil Team Selection Test
(a) Show that, for each positive integer $n$, the number of monic polynomials of degree $n$ with integer coefficients having all its roots on the unit circle is finite.

(b) Let $P(x)$ be a monic polynomial with integer coefficients having all its roots on the unit circle. Show that there exists a positive integer $m$ such that $y^m=1$ for each root $y$ of $P(x)$.
1998 Brazil Team Selection Test
Let $p$ be an odd prime integer and $k$ a positive integer not divisible by $p$, $1\le k<2(p+1)$, and let $N=2kp+1$. Prove that the following statements are equivalent:



(i) $N$ is a prime number

(ii) there exists a positive integer $a$, $2\le a<n$, such that $a^{kp}+1$ is divisible by $N$ and $\gcd\left(a^k+1,N\right)=1$.
1999 Brazil Team Selection Test
Find all positive integers n with the following property: There exists a positive integer $k$ and mutually distinct integers $x_1,x_2,\ldots,x_n$ such that the set $\{x_i+x_j\mid1\le i<j\le n\}$ is a set of distinct powers of $k$.
1999 Brazil Team Selection Test
If $a,b,c,d$ are Distinct Real no. such that



$a = \sqrt{4+\sqrt{5+a}}$



$b = \sqrt{4-\sqrt{5+b}}$



$c = \sqrt{4+\sqrt{5-c}}$



$d = \sqrt{4-\sqrt{5-d}}$



Then $abcd = $
1999 Brazil Team Selection Test
Let $BD$ and $CE$ be the bisectors of the interior angles $\angle B$ and $\angle C$, respectively ($D\in AC$, $E\in AB$). Consider the circumcircle of $ABC$ with center $O$ and the excircle corresponding to the side $BC$ with center $I_a$. These two circles intersect at points $P$ and $Q$.



(a) Prove that $PQ$ is parallel to $DE$.

(b) Prove that $I_aO$ is perpendicular to $DE$.
1999 Brazil Team Selection Test
Let Q+ and Z denote the set of positive rationals and the set of inte-

gers, respectively. Find all functions f : Q+ → Z satisfying the following

conditions:

(i) f(1999) = 1;

(ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+;

(iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.
1999 Brazil Team Selection Test
(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove

that $n$ is a power of $2$;

(b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such

that $2^n-1$ divides $m^2 + 9$.
1999 Brazil Team Selection Test
For a positive integer n, let $w(n)$ denote the number of distinct prime

divisors of n. Determine the least positive integer k such that

$2^{w(n)} \leq k \sqrt[4]{n}$

for all positive integers n.
1999 Brazil Team Selection Test
In a triangle $ABC$, the bisector of the angle at $A$ of a triangle $ABC$ intersects the segment $BC$ and the circumcircle of $ABC$ at points $A_1$ and $A_2$, respectively. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that

$$\frac{A_1A_2}{BA_2+CA_2}+\frac{B_1B_2}{CB_2+AB_2}+\frac{C_1C_2}{AC_2+BC_2}\ge\frac34.$$