## Subject II (7 p)

Find four natural numbers, knowing that their sum is 55, the first number is half of the second, the third number is the arithmetic mean of the first two, and the fourth number is double the difference between the second and the third.

Mathematics Gazette
## Subject II

Find four natural numbers, knowing that their sum is 55, the first number is half of the second, the third number is the arithmetic mean of the first two, and the fourth number is double the difference between the second and the third.

Gazeta Matematica

## Solution

Let the numbers be denoted by a, b, c, d. We have the following relationships: $\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}=55, \mathrm{a}=\mathrm{b}: 2, \mathrm{c}=(\mathrm{a}+\mathrm{b}): 2$ and $\mathrm{d}=2$ (b-c).......... $.1 p$

We substitute $\mathrm{b}=2 \mathrm{a}$ into the third relation and obtain $\mathrm{c}=3 \mathrm{a}: 2$, which we also substitute into the last relation. $.2 p$

$d=2(b-c)=2(2 a-3 a: 2)=4 a-3 a=a$ $1 p$

We have all unknowns expressed in terms of $a$ and substitute them into their sum $\mathrm{a}+2 \mathrm{a}+3 \mathrm{a}: 2+\mathrm{a}=55$. Multiplying by 2, we get $11 a=110$, from which $a=10$. . $.2 p$

The other numbers are $b=20, c=15$ and $d=10$. $1 \mathrm{p}$