Let \(G\) be a group. A topology on \(G\) is said to be a group topology if the map \(\mu: G \times G \to G\) defined by \(\mu(g, h) = g^{-1}h\) is continuous with respect to this topology where \(G \times G\) is equipped with the product topology. A group equipped with a group topology is called a topological group. When we have two topologies \(T_1, T_2\) on a set S, we write \(T_1 \leq T_2\) if \(T_2\) is finer than \(T_1\), which gives a partial order on the set of topologies on a given set. Prove or disprove the following statement: for a give group \(G\), there exists a unique minimal group topology on \(G\) (minimal with respect to the partial order we described above) so that \(G\) is a Hausdorff space?

Let \(G\) be a group acting by isometries on a proper geodesic metric space \(X\). Here \(X\) being proper means that every closed bounded subset of \(X\) is compact. Suppose this action is proper and cocompact,. Here, the action is said to be proper if for all compact subsets \(B \subset X\), the set \[\{g \in G | g(B) \cap B \neq \emptyset \}\] is finite. The quotient space \(X/G\) is obtained from \(X\) by identifying any two points \(x, y\) if and only if there exists \(g \in G\) such that \(gx = y\), and equipped with the quotient topology. Then the action of \(G\) on \(X\) is said to be cocompact if \(X/G\) is compact. Under these assumptions, show that \(G\) is finitely generated.

Compute the following integral  \[ \int_{0}^{\pi/2} \log{ (2 \cos{x} )} dx \].

Ten mathematicians sit at a round table. Each has a certain amount of food. At each full
minute, every mathematician divides his share of food into two equal parts and hands
it out to the two people seated closest to him in counter-clockwise direction. How will
the food be distributed at the end of a long evening? Does the answer change if instead
every mathematician shares his food with the two people sitting immediately next to
him?

Suppose that \( T \) is an \( N \times N \) matrix
\[
T = \begin{pmatrix}
a_1 & b_1 & 0 & \cdots & 0 \\
b_1 & a_2 & b_2 & \ddots & \vdots \\
0 & b_2 & a_3 & \ddots & 0 \\
\vdots & \ddots & \ddots & \ddots & b_{N-1} \\
0 & \cdots & 0 & b_{N-1} & a_N
\end{pmatrix}
\]
with \( b_i > 0 \) for \( i =1, 2, \dots, N-1 \). Prove that \( T \) has \( N \) distinct eigenvalues.

For any positive integers m and n, show that

\[ C_{n,m} = \frac{(mn)!}{(m!)^n n!} \] is an integer.