Category Archives: problem

2016-19 Zeta function

Let
\[
P(k) = \sum_{i_1=1}^{\infty} \dots \sum_{i_k=1}^{\infty} \frac{1}{i_1 \dots i_k (i_1 + \dots + i_k)}
\]
for a positive integer \( k \). Find \( \zeta(k+1) / P(k) \), where \( \zeta \) is the Riemann-zeta function.

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2016-18 Partitions with equal sums

Suppose that we have a list of \(2n+1\) integers such that whenever we remove any one of them, the remaining can be partitioned into two lists of \(n\) integers with the same sum. Prove that all \(2n+1\) integers are equal.

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2016-17 Integral with two variables

Set \[ L(z,w)=\int_{-2}^2\int_{-2}^2 ( \log(z-x)-\log(z-y))( \log(w-x)-\log(w-y))Q(x,y) dx dy, \]
for \(z,w\in \mathbb{C}\setminus(-\infty, 2] \), where \[ Q(x,y)= \frac{4-xy}{(x-y)^2\sqrt{4-x^2}\sqrt{4-y^2}}. \]
Prove that \[ L(z,w)=2\pi^2 \log \left[ \frac{(z+R(z))(w+R(w))}{2(zw-4+R(z)R(w))} \right], \]
where \(R(z)=\sqrt{z^2-4}\) with branch cut \([-2,2]\).

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2016-13 How to divide camels

A rich old man had \( k \) sons and \( N \) camels in the herd. The will of the father stated that his \( r \)-th son should receive \( 1/ N_r\) of his camels for \( r = 1, 2, \dots, k\). Since \( N+1 \) is a common multiple of \( N_1, N_2, \dots, N_k \), the sons could not divided \( N \) camels as their father wished. The sons visited a wise man to solve the issue. The wise man listened about the will, and he brought his own camel, which he added to the herd. The herd was then divided up according to the old man’s wishes. The wise man then took back the one camel that remained, which was his own. For given \( k \), find the maximal number of camels \( N \equiv N(k) \) for which there is a solution to the problem where \( N_1, N_2, \dots, N_k\) are positive integers.

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