Philippine Mathematical Olympiad - Area Stage

Solution:

Let the two integers be $x$ and $y$.

Let $S = x + y$ (sum), $P = x y$ (product).

We are given:

$P + S = 95$ $\quad$ (1)

$P - S = 59$ $\quad$ (2)

Add (1) and (2):

$P + S + P - S = 95 + 59$

$2P = 154$

$P = 77$

Now substitute $P = 77$ into (1):

$77 + S = 95$

$S = 18$

So $x + y = 18$, $x y = 77$.

The integers are the roots of the equation:

$t^2 - (x + y)t + x y = 0$

$t^2 - 18 t + 77 = 0$

Solve for $t$:

$t = \frac{18 \pm \sqrt{18^2 - 4 \times 77}}{2}$

$= \frac{18 \pm \sqrt{324 - 308}}{2}$

$= \frac{18 \pm \sqrt{16}}{2}$

$= \frac{18 \pm 4}{2}$

So $t = \frac{18 + 4}{2} = 11$ or $t = \frac{18 - 4}{2} = 7$

Therefore, the integers are $7$ and $11$.