Olympiad problems

2020 Brazil Undergrad MO, Problem 6

Tags: function , college contests , quadratics , roots , Fixed point , Brazilian Undergrad MO , Brazilian Undergrad MO 2020

Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times). a) Find the number of distinct real roots of the equation $f^{3}(x) = x$ b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$

2020 Brazil Undergrad MO, Problem 1

Tags: limits , calculus , Brazilian Undergrad MO , Brazilian Undergrad MO 2020 , geometry

Let $R > 0$, be an integer, and let $n(R)$ be the number um triples $(x, y, z) \in \mathbb{Z}^3$ such that $2x^2+3y^2+5z^2 = R$. What is the value of $\lim_{ R \to \infty}\frac{n(1) + n(2) + \cdots + n(R)}{R^{3/2}}$?

2020 Brazil Undergrad MO, Problem 3

Tags: linear algebra , matrix , Brazilian Undergrad MO , Brazilian Undergrad MO 2020 , number theory

Let $\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}$ be the finite field with $13$ elements (with sum and product modulus $13$). Find how many matrix $A$ of size $5$ x $5$ with entries in $\mathbb{F}_{13}$ exist such that $$A^5 = I$$ where $I$ is the identity matrix of order $5$

2020 Brazil Undergrad MO, Problem 5

Tags: Brazilian Undergrad MO 2020 , games , combinatorics , college contests , Combinatorial games , Game Theory

Let $N$ a positive integer. In a spaceship there are $2 \cdot N$ people, and each two of them are friends or foes (both relationships are symmetric). Two aliens play a game as follows: 1) The first alien chooses any person as she wishes. 2) Thenceforth, alternately, each alien chooses one person not chosen before such that the person chosen on each turn be a friend of the person chosen on the previous turn. 3) The alien that can't play in her turn loses. Prove that second player has a winning strategy [i]if, and only if[/i], the $2 \cdot N$ people can be divided in $N$ pairs in such a way that two people in the same pair are friends.

2020 Brazil Undergrad MO, Problem 2

Tags: Brazilian Undergrad MO 2020 , number theory , Sequences , Brazilian Undergrad MO

For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is fibonatic when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not fibonatic integers.

2020 Brazil Undergrad MO, Problem 4

Tags: linear algebra , college contest , Brazilian Undergrad MO 2020 , Matrix algebra

For each of the following, provide proof or a counterexample: a) Every $2\times2$ matrix with real entries can we written as the sum of the squares of two $2\times2$ matrices with real entries. b) Every $3\times3$ matrix with real entries can we written as the sum of the squares of two $3\times3$ matrices with real entries.