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Hendrata Dharmawan
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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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