Problem 1344

1. Define Variables and Given Information:
- Let $N = 2014$ be the total number of students.
- Let $n$ be the number of students who took both exams.
- Let $a$ be the total score of all students who took only one exam.
- Let $M$ be the total score of all students who took both exams.
- Let $m$ be the total score of the worse exams of the students who took both exams.

We are given:
- The average score of all exam papers is 20.
- The average grade of all students is 14.

2. Formulate Equations:
- The total number of exam papers is $N + n$.
- The total score of all exam papers is $20(N + n)$.
- The total grade of all students is $14N$.

Therefore, we have:
$$M + a = 14N$$
$$M + m + a = 20(N + n)$$

3. Subtract the First Equation from the Second:
$$(M + m + a) - (M + a) = 20(N + n) - 14N$$
$$m = 20N + 20n - 14N$$
$$m = 6N + 20n$$

4. Establish Inequalities:
- Since $m \leq 40n$ (because the worst score for each student who took both exams is at most 40):
$$6N + 20n \leq 40n$$
$$6N \leq 20n$$
$$n \geq \frac{3N}{10}$$
$$n \geq \frac{3 \times 2014}{10} = 604.2$$
Since $n$ must be an integer:
$$n \geq 605$$

5. **Upper Bound for $n$:**
- Since $M \geq m$ and $a \geq 0$:
$$14N = M + a \geq 6N + 20n$$
$$14N \geq 6N + 20n$$
$$8N \geq 20n$$
$$n \leq \frac{8N}{20}$$
$$n \leq \frac{8 \times 2014}{20} = 805.6$$
Since $n$ must be an integer:
$$n \leq 805$$

6. **Compute $G - L$:**
$$G = 805$$
$$L = 605$$
$$G - L = 805 - 605 = 200$$