This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 6

2021 Brazil Undergrad MO, Problem 4
Tags: real analysis , limits , mean , Exponents , Brazilian Undergrad MO 2021
For every positive integeer $n>1$, let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$. Find $$lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n}$$

2021 Brazil Undergrad MO, Problem 6
Tags: combinatorics , Combinatorics of words , college contests , Brazilian Undergrad MO 2021
We recursively define a set of [i]goody pairs[/i] of words on the alphabet $\{a,b\}$ as follows: - $(a,b)$ is a goody pair; - $(\alpha, \beta) \not= (a,b)$ is a goody pair if and only if there is a goody pair $(u,v)$ such that $(\alpha, \beta) = (uv,v)$ or $(\alpha, \beta) = (u,uv)$ Show that if $(\alpha, \beta)$ is a good pair then there exists a palindrome $\gamma$ (possibly empty) such that $\alpha\beta = a \gamma b$

2021 Brazil Undergrad MO, Problem 5
Tags: linear algebra , eigenvalues , matrix , Brazilian Undergrad MO 2021
Find all triplets $(\lambda_1,\lambda_2,\lambda_3) \in \mathbb{R}^3$ such that there exists a matrix $A_{3 \times 3}$ with all entries being non-negative reals whose eigenvalues are $\lambda_1,\lambda_2,\lambda_3$.

2021 Brazil Undergrad MO, Problem 1
Tags: Brazilian Undergrad MO , Matrix algebra , Square matrix , linear algebra , Brazilian Undergrad MO 2021
Consider the matrices like $$M= \left( \begin{array}{ccc} a & b & c \\ c & a & b \\ b & c & a \end{array} \right)$$ such that $det(M) = 1$. Show that a) There are infinitely many matrices like above with $a,b,c \in \mathbb{Q}$ b) There are finitely many matrices like above with $a,b,c \in \mathbb{Z}$

2021 Brazil Undergrad MO, Problem 3
Tags: Brazilian Undergrad MO 2021 , real analysis , powers , irrational number , approximation , Perfect Square
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 of the form $m^2-k$ com $m \in \mathbb{Z}$ for every integer $n>N$.

2021 Brazil Undergrad MO, Problem 2
Tags: functions , real analysis , function , Brazilian Undergrad MO 2021
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ from $C^2$ (id est, $f$ is twice differentiable and $f''$ is continuous.) such that for every real number $t$ we have $f(t)^2=f(t \sqrt{2})$.