Track / Stage 3 / 143 of 260 #783 of 2604

Problem 783

AMC 10/12, early questions

Number theory Difficulty 3.7 Multiple choice Olimpiadi della Matematica · Italy

Anna, Bianca, Carla and Diana are four women of different ages (in years). Their ages are prime numbers whose sum is $240$ . Knowing that none of them is more than $70$ years old, what is the age of the youngest?

Pick one

Official solution

Solution:

The answer is (B). Let $a < b < c < d$ be the four prime numbers corresponding to the ages. Then
$60 = \frac{a + b + c + d}{4} < \max \{a, b, c, d\} = d < 70.$
So we have only two possibilities: $d = 61$ or $d = 67$ . If $d$ were $61$ , then $a < b < c < 60$ so that
$240 = a + b + c + d < 3 \cdot 59 + 61 < 240$
and this is impossible. Otherwise $d = 67$ and since $a < b < c < d$ we have $a \leq 53$ , $b \leq 59$ , $c \leq 61$ . On the other hand
$240 = a + b + c + d \leq 53 + 59 + 61 + 67 = 240,$
so that $(a, b, c, d) = (53, 59, 61, 67)$ is the only solution.

Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty, ordering) added by this project.