Olympiad problems

2019 Brazil Undergrad MO, 3

Tags: calculus , system of equations , college contests , algebra , Brazilian Undergrad MO , Brazilian Undergrad MO 2019

Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove that the equations $2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}$ $x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}$ $x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2}$ have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.

2019 Brazil Undergrad MO, 1

Tags: college contests , Matrices , linear algebra , polynomial , Brazilian Undergrad MO 2019 , Brazilian Undergrad MO

Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$ with rational entries and size $ 2019 $ such that: a) $ A ^ 3 + 6A ^ 2-2I = 0 $? b) $ A ^ 4 + 6A ^ 3-2I = 0 $?

2019 Brazil Undergrad MO, Problem 5

Tags: Brazilian Undergrad MO 2019 , Brazilian Undergrad MO , number theory , combinatorics , counting , limit

Let $M, k>0$ integers. Let $X(M,k)$ the (infinite) set of all integers that can be factored as ${p_1}^{e_1} \cdot {p_2}^{e_2} \cdot \ldots \cdot {p_r}^{e_r}$ where each $p_i$ is not smaller than $M$ and also each $e_i$ is not smaller than $k$. Let $Z(M,k,n)$ the number of elements of $X(M,k)$ not bigger than $n$. Show that there are positive reals $c(M,k)$ and $\beta(M,k)$ such that $$\lim_{n \rightarrow \infty}{\frac{Z(M,k,n)}{n^{\beta(M,k)}}} = c(M,k)$$ and find $\beta(M,k)$