Prove or disprove that if all elements of an infinite group G has order less than n for some positive integer n, then G is finitely generated.

Show that for any triangle T and any Jordan curve C in the Euclidean plane, there exists a triangle inscribed in C which is similar to T.

Suppose we choose a point on the unit circle in the plane at random with the uniform probability measure on the circle. When we choose n points in that way, what is the probability of the n-gon obtained as the convex hull of the chosen points has the area bigger than \( \pi/2 \) in terms of n?

Let \(S_g\) denote the closed orientable connected surface of genus \(g\). Suppose we glue triangles along the edges so that the resulting space is \(S_g\) and the intersection of any two triangles are either empty or a single edge. Let \( n(g) \) be the minimum number of triangles one needs to make \(S_g\) while satisfying the above rule. What are \( n(1), n(2), n(3) \)? Does the limit \( \lim_{g \to \infty} n(g)/g \) exist?

Prove or disprove that a surjective homomorphism from a finitely generated abelian group to itself is an isomorphism.

Say there are n points. For each pair of points, we add an edge with probability 1/3. Let \(P_n\) be the probability of the resulting graph to be connected (meaning any two vertices can be joined by an edge path). What can you say about the limit of \(P_n\) as n tends to infinity?

In the problem 2019-08 (https://mathsci.kaist.ac.kr/pow/2019/2019-08-group-action/), we considered a group G acting by isometries on a proper geodesic metric space X properly discontinuously and cocompactly. Such an action is called a geometric action. The conclusion was that a geometric action leads to that G is finitely generated.

Would this conclusion still hold in the case the space X is not necessarily proper?

We say a metric space complete if every Cauchy sequence converges.

Let (X, d) be a metric space. Show that there exists an isometric imbedding from X to a complete metric space Y so that the image of X in Y is dense.

Either find an example of a group which is expressed as the union of two proper subgroups or prove that such a group cannot exist.

Let \(f = X^n + a_{n-1}X^{n-1} + \dots + a_0\in \mathbb{Z}[X]\) be a polynomial with integer coefficients, and let \(m\in \mathbb{Z}\).

Consider the sequence \[f_0,f_1,f_2,\dots \]

where \(f_0:=m\), and \(f_i:=f(f_{i-1})\) for all \(i\ge 1\).

Let \(S:=\{p\in \mathbb{P}: p \text{ divides } f_i \text{ for some } i\ge 0\}\) be the set of prime divisors of the sequence \(f_0,f_1,f_2,\dots\). 

Assume that \(S\) is finite, but \(\{f_i\mid i\ge 0\}\) is infinite. Show that \(f=X^n\).