Is it true that for any positive integer $n>1$, there exists an infinite arithmetic progression $M_n$ of positive integers, such that for any $m \in M_n$, the number $n^m-1$ is not a perfect power (a positive integer is a perfect power if it is of the form $a^b$ for positive integers $a, b>1$)?

Given is a triangle $ABC$ and the points $M, P$ lie on the segments $AB, BC$, respectively, such that $AM=BC$ and $CP=BM$. If $AP$ and $CM$ meet at $O$ and $2\angle AOM=\angle ABC$, find the measure of $\angle ABC$.

Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb{R}^{+}$, such that $$f(af(b)+a)(f(bf(a))+a)=1$$for any positive reals $a, b$.

Do there exist $2024$ non-zero reals $a_1, a_2, \ldots, a_{2024}$, such that $$\sum_{i=1}^{2024}(a_i^2+\frac{1}{a_i^2})+2\sum_{i=1}^{2024} \frac{a_i} {a_{i+1}}+2024=2\sum_{i=1}^{2024}(a_i+\frac{1}{a_i})?$$

Let $\mathcal{F}$ be a family of $4$-element subsets of a set of size $5^m$, where $m$ is a fixed positive integer. If the intersection of any two sets in $\mathcal{F}$ does not have size exactly $2$, find the maximal value of $|\mathcal{F}|$.

Given is a triangle $ABC$ and a circle $\omega$ with center $I$ that touches $AB, AC$ and meets $BC$ at $X, Y$. The line through $I$ perpendicular to $BC$ meets the line through $A$ parallel to $BC$ at $Z$. Show that the circumcircles of $\triangle XYZ$ and $\triangle ABC$ are tangent to each other.