# 2024-10 Supremum

Find
$\sup \left[ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \left( \sum_{i=n}^{\infty} x_i^2 \right)^{1/2} \Big/ \sum_{i=1}^{\infty} x_i \right],$
where the supremum is taken over all monotone decreasing sequences of positive numbers $$(x_i)$$ such that $$\sum_{i=1}^{\infty} x_i < \infty$$.

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# 2024-09 Integer sums

Find all positive numbers $$a_1,…,a_{5}$$ such that $$a_1^\frac{1}{n} + \cdots + a_{5}^\frac{1}{n}$$ is integer for every integer $$n\geq 1.$$

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# 2024-08 Determinants of 16 by 16 matricies

Let $$A$$ be a $$16 \times 16$$ matrix whose entries are either $$1$$ or $$-1$$. What is the maximum value of the determinant of $$A$$?

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# 2024-07 Limit of a sequence

For fixed positive numbers $$x_1, x_2, \dots, x_m$$, we define a sequence $$\{ a_n \}$$ by $$a_n = x_n$$ for $$n \leq m$$ and
$a_n = a_{n-1}^r + a_{n-2}^r + \dots + a_{n-k}^r$
for $$n > m$$, where $$r \in (0, 1)$$. Find $$\lim_{n \to \infty} a_n$$.

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# 2024-06 Limit of concave functions

Let $$f_n(t)$$, $$n=1,2…$$ be a sequence of concave functions on $$\mathbb{R}$$. Assume $$\liminf_{n\to\infty} f_n(t) \geq 2024\,t^{5}+3$$ for $$t\in [-1, 1]$$ and $$\lim_{n\to \infty} f_n(0) = 3$$. Suppose $$f_n'(0)$$ exist for $$n=1,2,…$$. Compute $$\lim_{n\to \infty} f_n'(0)$$.

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# 2024-05 Knotennullstelle

A complex number $$z \in S^1 \smallsetminus \{1\}$$ is called a Knotennullstelle if there exists a Laurent polynomial $$p(t) \in \mathbb{Z} [t,t^{-1}]$$ such that $$p(1) =\pm 1$$ and $$p(z)=0$$. Prove that the collection of all Knotennullstelle numbers is a discrete subset of $$\mathbb{C}$$.

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# 2024-04 Real random variable

Prove the following: There exists a bounded real random variable $$Z$$ such that
$E[Z] = 0, E[Z^2] = 1, E[Z^3] = x, E[Z^4] = y$
if and only if $$y \geq x^2 + 1$$. (Here, $$E$$ denotes the expectation.)

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# 2024-03 Roots of complex derivative

Let $$P(z) = z^3 + c_1 z^2 + c_2 z+ c_3$$ be a complex polynomial in $$\mathbb{C}$$. Its complex derivative is given by $$P’(z) = 3z^{2} +2c_1z+c_{2}.$$ Assume that there exist two points a, b in the open unit disc of complex plane such that P(a) = P(b) =0. Show that  there is a point w belonging to the line segment joining a and b such that  $${\rm Re} (P’(w)) = 0$$.

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# 2024-02 Well-mixed permutations

A permutation $$\phi \colon \{ 1,2, \ldots, n \} \to \{ 1,2, \ldots, n \}$$ is called a well-mixed if $$\phi (\{1,2, \ldots, k \}) \neq \{1,2, \ldots, k \}$$ for each $$k<n$$. What is the number of well-mixed permutations of $$\{ 1,2, \ldots, 15 \}$$?

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Suppose that we roll $$n$$ (6-sided, fair) dice. Let $$S_n$$ be the sum of their faces. Find all positive integers $$k$$ such that the probability that $$k$$ divides $$S_n$$ is $$1/k$$ for all $$n \geq 1$$.