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### PROBLEMS FOR SEPTEMBER

Professor E.J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3
no later than October 31, 2001.

Notes. A unit cube (tetrahedron) is a cube (tetrahedron) all of whose side lengths are 1.

90.
Let $m$ be a positive integer, and let $f\left(m\right)$ be the smallest value of $n$ for which the following statement is true:
given any set of $n$ integers, it is always possible to find a subset of $m$ integers whose sum is divisible by $m$
Determine $f\left(m\right)$.
[Comment. This problem is being reposed, as no one submitted a complete solution to this problem the first time around. Can you conjecture what $f\left(m\right)$ is? It is not hard to give a lower bound for this function. One approach is to try to relate $f\left(a\right)$ and $f\left(b\right)$ to $f\left(\mathrm{ab}\right)$ and reduce the problem to considering the case that $m$ is prime; this give access to some structure that might help.]

103.
Determine a value of the parameter $\theta$ so that

$f\left(x\right)\equiv \mathrm{cos}{}^{2}x+\mathrm{cos}{}^{2}\left(x+\theta \right)-\mathrm{cos}x\mathrm{cos}\left(x+\theta \right)$

is a constant function of $x$.

104.
Prove that there exists exactly one sequence $\left\{{x}_{n}\right\}$ of positive integers for which

${x}_{1}=1 , {x}_{2}>1 , {x}_{n+1}^{3}+1={x}_{n}{x}_{n+2}$

for $n\ge 1$.

105.
Prove that within a unit cube, one can place two regular unit tetrahedra that have no common point.

106.
Find all pairs $\left(x,y\right)$ of positive real numbers for which the least value of the function

$f\left(x,y\right)=\frac{{x}^{4}}{{y}^{4}}+\frac{{y}^{4}}{{x}^{4}}-\frac{{x}^{2}}{{y}^{2}}-\frac{{y}^{2}}{{x}^{2}}+\frac{x}{y}+\frac{y}{x}$

is attained. Determine that minimum value.

107.
Given positive numbers ${a}_{i}$ with ${a}_{1}<{a}_{2}<\dots <{a}_{n}$, for which permutation $\left({b}_{1},{b}_{2},\dots ,{b}_{n}\right)$ of these numbers is the product

$\underset{i=1}{\overset{n}{\Pi }}\left({a}_{i}+\frac{1}{{b}_{i}}\right)$

maximized?

108.
Determine all real-valued functions $f\left(x\right)$ of a real variable $x$ for which

$f\left(\mathrm{xy}\right)=\frac{f\left(x\right)+f\left(y\right)}{x+y}$

for all real $x$ and $y$ for which $x+y\ne 0$.

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