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I want to partition the set of positive integers $\{1,2,3,\ldots\}$ into subsets such that each subset satisfies the property that if a number $x$ is in it, $2x$ is not.
What is the minimum possible number of subsets that I can get?
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You and some friends would like to explore a labyrinth from its entrance to its exit:
What is the minimum number of people that need to enter the labyrinth for there to be a strategy to guarantee that all can come out at the exit in a finite amount of time?
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You are given the following information about the number of possible tic-tac-toe games:
How many distinct tic-tac-toe games are there?
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$62208\big(a^{7}+b^{7}+c^{7}+d^{7}\big)^{2}\le M\big(a^{2}+b^{2}+c^{2}+d^{2}\big)^{7}$
What is the smallest positive integer $M$ such that this inequality holds true for all $a,b,c,d$ satisfying $a,b,c,d\in \mathbb{R}$ and $a+b+c+d=0?$
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$x_{1}$, $x_{2}$, and $x_{3}$ are chosen randomly and independently on the interval $[0,2\pi]$ and let $f(x)=\sin(x)$.
What is the average area of the triangle created with the three vertices below (to two significant figures)?
$P_{1}=\big( x_{1} , f( x_{1})\big),\quad P_{2}=\big( x_{2} , f( x_{2})\big),\quad P_{3}=( x_{3} , f( x_{3})\big)$
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