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.

Two players play a game with a polynomial with undetermined coefficients
\[
1 + c_1 x + c_2 x^2 + \dots + c_7 x^7 + x^8.
\]
Players, in turn, assign a real number to an undetermined coefficient until all coefficients are determined. The first player wins if the polynomial has no real zeros, and the second player wins if the polynomial has at least one real zero. Find who has the winning strategy.

Let \(f_1(x)=x^2+a_1x+b_1\) and \(f_2(x)=x^2+a_2x+b_2\) be polynomials with real coefficients. Prove or disprove that the following are equivalent.

(i) There exist two positive reals \(c_1, c_2\) such that \[ c_1f_1(x)+ c_2 f_2(x) > 0\] for all reals \(x\).

(ii) There  is no real \(x\) such that \( f_1(x)\le 0\) and \( f_2(x)\le 0\).