A class with $2N$ students took a quiz, on which the possible scores were $0,1,\dots,10$. Each of these scores occurred at least once, and the average score was exactly $7.4$. Show that the class can be divided into two groups of $N$ students in such a way that the average score for each group was exactly $7.4$
I do not know, I am Clueless
Initially I assumed that there are $a_0$ $0's$,$a_1$ $1's$ $ \dots $ $a_{10}$ $10's$. Where $a_0+ a_1 \dots a_{10}=N$ each $a_i>0$ Further we have Average 7.4 but I do not know what to do next
Let $a_1,\dots,a_{2N}$ be the scores in nondecreasing order, and define the sums $$s_i = \sum_{j=i+1}^{i+N} a_i \text{ for } i=0,\dots,N$$.
Then $$s_0 \leq \cdots \leq s_{N}$$
and
$$s_0 + s_{N} = \sum_{j=1}^{2N} a_i = 7.4(2N)$$,
so $$s_0 \leq 7.4N \leq s_N$$.
Let $i$ be the largest index for which $$s_i \leq 7.4N$$;
note that we cannot have $i = N$, as otherwise $s_0 = s_N = 7.4N$ and hence $a_1 = \cdots = a_{2N} = 7.4$, contradiction.
Then $$7.4N - s_i < s_{i+1} - s_i = a_{i+N+1} - a_i$$ and so $$ a_i < s_i + a_{i+N+1} - 7.4N \leq a_{i+N+1}; $$
since all possible scores occur, this means that we can find $N$ scores with sum $7.4N$ by taking $$a_i, \dots, a_{i+N+1}$$ and omitting one occurrence of the value $$s_i + a_{i+N+1} - 7.4N$$.