Suppose that \( X \) is a discrete random variable on the set \( \{ a_1, a_2, \dots \} \) with \( P(X=a_i) = p_i \). Define the discrete entropy
\[
H(X) = -\sum_{n=1}^{\infty} p_i \log p_i.
\]
Find constants \( C_1, C_2 \geq 0 \) such that
\[
e^{2H(X)} \leq C_1 Var(X) + C_2
\]
holds for any \( X \).
Category Archives: problem
2019-08 Group action
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.
2019-07 An inquality
Suppose that \( f: \mathbb{R} \to \mathbb{R} \) is differentiable and \( \max_{ x \in \mathbb{R}} |f(x)| = M < \infty \). Prove that \[ \int_{-\infty}^{\infty} (|f'|^2 + |f|^2) \geq 2M^2. \]
2019-06 Simple but not too simple integration
Compute the following integral \[ \int_{0}^{\pi/2} \log{ (2 \cos{x} )} dx \].
2019-05 Convergence with primes
Let \( p_n \) be the \(n\)-th prime number, \( p_1 = 2, p_2 = 3, p_3 = 5, \dots \). Prove that the following series converges:
\[
\sum_{n=1}^{\infty} \frac{1}{p_n} \prod_{k=1}^n \frac{p_k -1}{p_k}.
\]
2019-04 Food distribution at a dinner party
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?
2019-03 Simple spectrum
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.
2019-02 Simplification of an expression with factorials
For any positive integers m and n, show that
\[ C_{n,m} = \frac{(mn)!}{(m!)^n n!} \] is an integer.
2019-01 Equilateral polygon
Suppose that \( \Pi \) is a closed polygon in the plane. If \( \Pi \) is equilateral \( k \)-gon, and if \( A \) is the area of \( \Pi \), and \( L \) the length of its boundary, prove that
\[
\frac{A}{L^2} \leq \frac{1}{4k} \cot \frac{\pi}{k} \leq \frac{1}{4\pi}.
\]
2018-23 Game of polynomials
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.