3. Annemiek and Bart each wrote three different positive integers on a piece of paper. It turns out that there is exactly one number that appears on both of their papers. Furthermore, if you take two different numbers from Annemiek's paper and add them together, the result is always a number on Bart's paper. One of the three numbers on Annemiek's paper is her favorite number, and if you multiply that by 3, you also get a number on Bart's paper. On Bart's paper, his favorite number is 25.

What is Annemiek's favorite number?
B3. Suppose we call the three numbers on Annemieke's note $a, b$, and $c$ and assume that $a<b<c$. Then the three numbers on Bart's note, in order of size, are $a+b, a+c$, and $b+c$. The last two numbers are greater than $c$ and thus greater than any number on Annemieke's note. The number that appears on both notes must therefore be $a+b$ and that must be equal to $c$.

Annemieke thus has the numbers $a, b$, and $a+b$, and Bart has $a+b, 2a+b$, and $a+2b$. If we now look at $3b$ and $3(a+b)=3a+3b$, we see that these numbers are both greater than all the numbers on Bart's note. Annemieke's favorite number, which appears on Bart's note when multiplied by 3, is therefore $a$.

The number $3a$ is thus one of the numbers $a+b, 2a+b, a+2b$ on Bart's note, and since $a$ and $b$ are different, it must be the number $a+b$. In other words: $b=2a$. Bart thus has the numbers $3a$, $4a$, and $5a$. The only one of these three numbers that can be the number 25 is $5a$. We thus see that $a=5$ and that is Annemieke's favorite number.