Hendrata's posts

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**Polynomial and divisibility**

Find an integer polynomial $P(x)$ with the lowest degree that satisfies the following:

$P(x)$ is divisible by $x^2+x+1$

$P(x)+1$ is divisible by $x^{2017} -1$ Solution Let $Q(x) = x^2+x+1$ and $R(x) = x^{2016} + \dots + 1$. It's easy to show that $Q$ and ...

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**Dancing partners**

In a party with $n$ boys and $n$ girls, each boy dances with some girls and vice versa. (No pair of boy and girl dance more than once). There are $n^2-n+1$ dances that occurred throughout the night. Show that we can pair each boy with each girl so that ever...

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**Tournament, number of dominated players**

In a tournament with $n$ people, each player plays against each other player exactly once, with each game results in a win or a loss (no draw). If we choose $A$ a set of 5 players at random, let $X_3$ denote the number of players not in $A$ that lose to exa...

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**Various double counting problems**

These problems can all be solved by the double counting principle, that is, to count the same object two different ways and assert that those must match. Suppose $n > 4$. Let $A_1, A_2, \dots ,A_k$ each be a set of $n$ integers, where $k \leq 4^{n-1} / 3^n$...

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**Paintings and colors**

In a museum there are $n$ paintings, each of which was painted with at least 3 colors. The total number of colors from all paintings is $m << n$. The average number of colors per painting is $s$. If we choose $k$ paintings at random, what's the expected num...

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**Spanning set**

Let $S = \{0,1,2,\dots,10\}$. A set of numbers $A \subset S$ is called "spanning" if the set $\{ x, x+1, x+2, x+3 \mod 11 | x \in A \} = S$. And a set of numbers $B \subset S$ is called "potentially spanning" if there is an integer $k$ such that the set $\{...

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**Spanning set**

Let $S = \{0,1,2,\dots,10\}$.

A set of numbers $A \subset S$ is called "spanning" if the set $\{ x, x+1, x+2, x+3 \mod 11 | x \in A \} = S$.

And a set of numbers $B \subset S$ is called "potentially spanning" if there is an integer $k$ such that the set $...

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**Various double counting problems**

These problems can all be solved by the double counting principle, that is, to count the same object two different ways and assert that those must match.

In a class there are $N$ students, and we choose comittees among them. Each committee has $n$ students...

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**Paintings and colors**

In a museum there are $n$ paintings, each of which was painted with at least 3 colors. The total number of colors from all paintings is $m << n$. The average number of colors per painting is $s$. If we choose $k$ paintings at random, what's the expected num...

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