Found problems: 65
2018 Brazil National Olympiad, 3
Let $k$, $n$ be fixed positive integers. In a circular table, there are placed pins numbered successively with the numbers $1, 2 \dots, n$, with $1$ and $n$ neighbors. It is known that pin $1$ is golden and the others are white. Arnaldo and Bernaldo play a game, in which a ring is placed initially on one of the pins and at each step it changes position. The game begins with Bernaldo choosing a starting pin for the ring, and the first step consists of the following: Arnaldo chooses a positive integer $d$ any and Bernaldo moves the ring $d$ pins clockwise or counterclockwise (positions are considered modulo $n$, i.e., pins $x$, $y$ equal if and only if $n$ divides $x-y$). After that, the ring changes its position according to one of the following rules, to be chosen at every step by Arnaldo:
[b]Rule 1:[/b] Arnaldo chooses a positive integer $d$ and Bernaldo moves the ring $d$ pins clockwise or counterclockwise.
[b]Rule 2:[/b] Arnaldo chooses a direction (clockwise or counterclockwise), and Bernaldo moves the ring in the chosen direction in $d$ or $kd$ pins, where $d$ is the size of the last displacement performed.
Arnaldo wins if, after a finite number of steps, the ring is moved to the golden pin. Determine, as a function of $k$, the values of $n$ for which Arnaldo has a strategy that guarantees his victory, no matter how Bernaldo plays.
2018 Brazil Undergrad MO, 20
What is the largest number of points that can exist on a plane so that each distance between any two of them is an odd integer?
Brazil L2 Finals (OBM) - geometry, 2013.5
Let ABC be a scalene triangle and AM is the median relative to side BC. The diameter circumference AM intersects for the second time the side AB and AC at points P and Q, respectively, both different from A. Assuming that PQ is parallel to BC, determine the angle measurement <BAC.
Any solution without trigonometry?
1988 Brazil National Olympiad, 2
Show that, among all triangles whose vertices are at distances 3,5,7 respectively from a given point P, the ones with largest area have P as orthocenter.
([i]You can suppose, without demonstration, the existence of a triangle with maximal area in this question.[/i])
2019 Brazil National Olympiad, 3
Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the
vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre
2017 Brazil National Olympiad, 4.
[b]4.[/b] 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]i.[/i] The tracing starts at some of the detached points which correspond to the digits from $1$ to $9$ (Figure 3).
[i]ii.[/i] 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.
[i]iii.[/i] 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.
[i]iv.[/i] 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.
1990 Brazil National Olympiad, 1
Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.
2021 Brazil National Olympiad, 3
Find all positive integers \(k\) for which there is an irrational \(\alpha>1\) and a positive integer \(N\) such that \(\left\lfloor\alpha^{n}\right\rfloor\) is a perfect square minus \(k\) for every integer \(n\) with \(n>N\).
2018 Brazil Undergrad MO, 23
How many prime numbers $ p $ the number $ p ^ 3-4 p + 9 $ is a perfect square
1988 Brazil National Olympiad, 3
Find all functions $f:\mathbb{N}^* \rightarrow \mathbb{N}$ such that
[list]
[*] $f(x \cdot y) = f(x) + f(y)$
[*] $f(30) = 0$
[*] $f(x)=0$ always when the units digit of $x$ is $7$
[/list]
1979 Brazil National Olympiad, 2
The remainder on dividing the polynomial $p(x)$ by $x^2 - (a+b)x + ab$ (where $a \not = b$) is $mx + n$. Find the coefficients $m, n$ in terms of $a, b$. Find $m, n$ for the case $p(x) = x^{200}$ divided by $x^2 - x - 2$ and show that they are integral.
2019 Brazil National Olympiad, 1
An eight-digit number is said to be 'robust' if it meets both of the following conditions:
(i) None of its digits is $0$.
(ii) The difference between two consecutive digits is $4$ or $5$.
Answer the following questions:
(a) How many are robust numbers?
(b) A robust number is said to be 'super robust' if all of its digits are distinct. Calculate the sum of all
the super robust numbers.
2020 Brazil National Olympiad, 1
Let $ABC$ be an acute triangle and $AD$ a height. The angle bissector of $\angle DAC$ intersects $DC$ at $E$. Let $F$ be a point on $AE$ such that $BF$ is perpendicular to $AE$. If $\angle BAE=45º$, find $\angle BFC$.
2017 Brazil National Olympiad, 6.
[b]6.[/b] 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.
2020 Brazil National Olympiad, 3
Consider an inifinte sequence $x_1, x_2,\dots$ of positive integers such that, for every integer $n\geq 1$:
[list] [*]If $x_n$ is even, $x_{n+1}=\dfrac{x_n}{2}$;
[*]If $x_n$ is odd, $x_{n+1}=\dfrac{x_n-1}{2}+2^{k-1}$, where $2^{k-1}\leq x_n<2^k$.[/list]
Determine the smaller possible value of $x_1$ for which $2020$ is in the sequence.
2018 Brazil Undergrad MO, 1
An equilateral triangle is cut as shown in figure 1 and the parts are used to form figure 2. What is the shape of figure 2?
2018 Brazil Undergrad MO, 15
A real number $ to $ is randomly and uniformly chosen from the $ [- 3,4] $ interval. What is the probability that all roots of the polynomial $ x ^ 3 + ax ^ 2 + ax + 1 $ are real?
1989 Brazil National Olympiad, 5
A tetrahedron is such that the center of the its circumscribed sphere is inside the tetrahedron.
Show that at least one of its edges has a size larger than or equal to the size of the edge of a regular tetrahedron inscribed in this same sphere.
2018 Brazil Undergrad MO, 21
Consider $ p (x) = x ^ n + a_ {n-1} x ^ {n-1} + ... + a_ {1} x + 1 $ a polynomial of positive real coefficients, degree $ n \geq 2 $ e with $ n $ real roots. Which of the following statements is always true?
a) $ p (2) <2 (2 ^ {n-1} +1) $ (b) $ p (1) <3 $ c) $ p (1)> 2 ^ n $ d) $ p (3 ) <3 (2 ^ {n-1} -2) $
1979 Brazil National Olympiad, 5
[list=i]
[*] ABCD is a square with side 1. M is the midpoint of AB, and N is the midpoint of BC. The lines CM and DN meet at I. Find the area of the triangle CIN.
[*] The midpoints of the sides AB, BC, CD, DA of the parallelogram ABCD are M, N, P, Q respectively. Each midpoint is joined to the two vertices not on its side. Show that the area outside the resulting 8-pointed star is $\frac{2}{5}$ the area of the parallelogram.
[*] ABC is a triangle with CA = CB and centroid G. Show that the area of AGB is $\frac{1}{3}$ of the area of ABC.
[*] Is (ii) true for all convex quadrilaterals ABCD?
[/list]
2021 Brazil National Olympiad, 2
Let \(n\) be a positive integer. On a \(2 \times 3 n\) board, we mark some squares, so that any square (marked or not) is adjacent to at most two other distinct marked squares (two squares are adjacent when they are distinct and have at least one vertex in common, i.e. they are horizontal, vertical or diagonal neighbors; a square is not adjacent to itself).
(a) What is the greatest possible number of marked square?
(b) For this maximum number, in how many ways can we mark the squares? configurations that can be achieved through rotation or reflection are considered distinct.
2020 Brazil National Olympiad, 2
The following sentece is written on a board:
[center]The equation $x^2-824x+\blacksquare 143=0$ has two integer solutions.[/center]
Where $\blacksquare$ represents algarisms of a blurred number on the board. What are the possible equations originally on the board?
2016 Brazil National Olympiad, 3
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 [i]winner[/i] 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 [i]loser[/i].
Prove that, for every positive integer \(N\), the total of non-negative loser integers lesser than \(2^N\) is \(2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}\)
1979 Brazil National Olympiad, 4
Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it.
2019 Brazil National Olympiad, 5
In the picture below, a white square is surrounded by four black squares and three white squares. They are surrounded by seven black squares.
[img]https://i.stack.imgur.com/Dalmm.png[/img]
What is the maximum number of white squares that can be surrounded by $ n $ black squares?