Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that: For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that: $$m+a_1b+a_2b^2+\cdots+a_nb^n$$Is a perfect power. Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$

Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$. Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$. Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$. (Hungary)

Given a positive integer $n$, let $M$ be the set of all points in space with integer coordinates $(a, b, c)$ such that $0 \le a, b, c \le n$. A frog must go to the point $(0, 0, 0)$ to the point $(n, n, n)$ according to the following rules: $\bullet$ The frog can only jump to points of M. $\bullet$ In each jump, the frog can go from point $(a, b, c)$ to one of the following points: $(a + 1, b, c)$, $(a, b + 1, c)$, $(a, b, c + 1)$, or $(a, b, c - 1)$. $\bullet$ The frog cannot pass through the same point more than once. In how many different ways can the frog achieve its goal?

Find all functions $\,f: {\mathbb{N}}\rightarrow {\mathbb{N}}\,$ such that\[f(a)^{bf(b^2)}\le a^{f(b)^3}\hspace{0.2in}\text{for all}\,a,b\in \mathbb{N}. \]

You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.

Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.