Problem 1320

Solution:

Let $n$ be the total number of students. Let $x$ be the number of students that study all three subjects. Let $a$ be the number of students that study Maths and Art but not Science. Let $b$ be the number of students that study Maths and Science but not Art. Let $c$ be the number of students that study Art and Science but not Maths.

So $a, b, c, x, n$ are non-negative real numbers with

$\cdot\ n \geq a + b + c + x$ (total number of students).

$\cdot\ x + a \geq 0.8 \times n$ (studying both Mathematics and Art),

$\cdot\ x + b \geq 0.8 \times n$ (studying both Mathematics and Science).

Adding all these together gives us $2x + a + b \geq 1.6 \times n$ and since $n \geq a + b + c + x$ we get:

$$2x + a + b \geq 1.6 \times (a + b + c + x).$$

Finally multiply both sides by $0.5$ and rearrange to get:

$$x \geq 0.8 \times (x + c) + 0.3 \times (a + b) \geq 0.8 \times (x + c).$$

Therefore $\frac{x}{x + c} \geq 0.8$ as required.

First note that if $x \geq 0.8n$ then $\frac{x}{x + c} \geq \frac{x}{n} \geq 0.8$.

Otherwise we have $x < 0.8n$ and so $n > \frac{5x}{4}$. Hence

$$\frac{x}{x + c} = \frac{x}{(a + b + c + x) - (a + x) - (b + x) + 2x}$$
$$\geq \frac{x}{n - 0.8n - 0.8n + 2x}$$
$$= \frac{x}{2x - 0.6n}$$
$$> \frac{x}{2x - 0.6 \times \frac{5x}{4}}$$
$$= 0.8.$$