Let $p$ be an odd prime number. How many $p$-element subsets of $\{1,2,3,4, \ldots, 2p\}$ have sums divisible by $p$?

This is IMO 1995 Q6.

I shall provide 2 solutions, one using only standard number theory and another using generating functions.

Solution 1: Clearly $\{1, 2, \ldots , p\}$ and $\{p+1, p+2, \ldots , 2p\}$ are 2 such $p$-element subsets with sum of elements divisible by $p$.

Consider all other $p$-element subsets of $\{1, 2, \ldots , 2p\}$. There are $\dbinom{2p}{p}-2$ of them. For a subset with $k$ elements in $\{1, 2, \ldots , p\}$ and $p-k$ elements in $\{p+1, p+2, \ldots , 2p\}$, where $1 \leq k \leq p-1$, we can take any $1 \leq i \leq p-1$, add $i$ to each of the $k$ elements in $\{1, 2, \ldots , p\}$, then take the resulting $k$ numbers $\pmod{p}$ to keep them in $\{1, 2, \ldots, p\}$. This gives us $(p-1)$ more $p$-element subsets, so that we have $p$ $p$-element subsets with sum of elements giving different remainder $\pmod{p}$.

It is now clear that the $\dbinom{2p}{p}-2$ subsets can be partitioned into groups of $p$ subsets, where each group has exactly one subset with sum of elements divisible by $p$. This gives $\dfrac{\dbinom{2p}{p}-2}{p}$.

Finally, combining gives the total number as $2+\dfrac{\dbinom{2p}{p}-2}{p}$.

Solution 2: We proceed using generating functions. Consider the generating function $f(x, y)=\prod\limits_{m=1}^{2p}{(1+x^my)}$. In its expansion, the coefficient of $x^ay^b$ is the number of $b$-element subsets with sum of elements equal to $a$.

Suppose we sum all the coefficients of the terms where the power of $x$ and the power of $y$ are both divisible by $p$. This will give the number of $kp$ element subsets with sum of elements divisible by $p$. To get the number of $p$-element subsets with sum divisible by $p$, we would have to subtract the number of $0$-element subsets and $2p$-element subset with sum divisible by $p$, which is $2$.

To get the required sum, we average over all $p$th roots of unity for both $x, y$ (and subtract $2$ to get the final answer):

\begin{align} &-2+\frac{1}{p^2}\sum_{k=0}^{p-1}{\sum_{j=0}^{p-1}{f(e^{\frac{2i\pi j}{p}}, e^{\frac{2i\pi k}{p})}}} \\ = &-2+\frac{1}{p^2}\sum_{k=0}^{p-1}{\sum_{j=0}^{p-1}{\prod_{m=1}^{2p}{(1+(e^{\frac{2i\pi j}{p}})^m(e^{\frac{2i\pi k}{p}}))}}} \\ = &-2+\frac{1}{p^2}\sum_{k=0}^{p-1}{\left((1+e^{\frac{2i\pi k}{p}})^{2p}+\sum_{j=1}^{p-1}{\prod_{m=1}^{2p}{(1+(e^{\frac{2i\pi j}{p}})^m(e^{\frac{2i\pi k}{p}}))}}\right)} \\ = &-2+\frac{1}{p^2}\sum_{k=0}^{p-1}{\left((1+e^{\frac{2i\pi k}{p}})^{2p}+(p-1)\prod_{m=1}^{2p}{(1+e^{\frac{2i\pi m}{p}})}\right)} \\ = &-2+\frac{1}{p^2}\sum_{k=0}^{p-1}{\left((1+e^{\frac{2i\pi k}{p}})^{2p}\right)}+\frac{1}{p}\left((p-1)\prod_{m=1}^{2p}{(1+e^{\frac{2i\pi m}{p}})}\right) \\ = &-2+\frac{1}{p}\left(2+\binom{2p}{p}\right)+\frac{1}{p}\left((p-1)[(-1)^p((-1)^p-1)]^2\right) \\ = & \frac{\dbinom{2p}{p}+2p-2}{p} \end{align}

I remember an elegant complex number proof from an Olympiad book (apparently this solution was given by a Bulgarian contestant who won an award for creativity).

Let $\omega$ be a $p$-th root of unity. Consider the sum: $$F(\omega) = \sum_{1 \leq i_1 < i_2 ... < i_p \leq 2p} \omega^{i_1}\omega^{i_2}...\omega^{i_p}.$$

We observe that $\omega, \omega^{2}, ... , \omega^{2p}$ are the roots of $(x^p - 1)^2 = x^{2p} - 2x^p + 1$. By the relation between elementary symmetric functions of roots and coefficients, we get $F(\omega) = 2$.

But we can write $\omega^{i_1}\omega^{i_2}...\omega^{i_p} = \omega^{i_1+i_2+...+i_p}$ and read the powers of $\omega$ modulo $p$. So another way to write the sum is $$F(\omega) = \sum_{0 \leq k \leq p-1}a_k \omega ^k.$$

Therefore equating the two different ways of writing the sum, we get: $$(a_o - 2) + a_1 \omega + a_2 \omega^2 ... + a_{p-1} \omega^{p-1} = 0.$$

But this means $$a_0 - 2 = a_1 = a_2 = ... = a_{p-1}.$$ Clearly $\sum_k a_k$ counts the total number of terms in the summation $F(\omega)$ which is $\binom{2p}{p}$.

Therefore $$\binom{2p}{p} - 2 = (a_0 - 2) + a_1 + a_2 + ...+ a_{p-1} = p(a_0 - 2).$$

Thus we get $$a_0 = \frac{\binom{2p}{p} - 2}{p} + 2.$$

Note that this solves the problem since $a_0$ counts the number of terms in the sum with the property $i_1 + i_2 + ... + i_p = 0$ modulo $p$.