Olympiad problems

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 $2^{N-\lfloor \frac{log(min\{N,k\})}{log 2} \rfloor}$

2016 Brazil National Olympiad, 1

Tags: geometry , incenter , Parallel Lines , Brazilian Math Olympiad 2016

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, 4

Tags: Brazilian Math Olympiad 2016 , combinatorics

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, 5

Tags: Brazilian Math Olympiad 2016 , algebra , polynomial

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 $n \geq 1$. [list='a'] [*]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 $n$. [/list]

2016 Brazil National Olympiad, 2

Tags: combinatorics , combinatorial geometry , Brazilian Math Olympiad , Brazilian Math Olympiad 2016 , number theory

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, 6

Tags: Brazilian Math Olympiad 2016 , geometry , circumcircle , angle bisector

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.