3. Determine all integers greater than 1 by which some fraction of the form

$$
\frac{3 p-q}{5 p+2 q}
$$

can be divided, where $p$ and $q$ are coprime integers.

The written part of the school round in category A takes place

## on Tuesday, December 2, 2008

so that it starts in the morning and the contestants have 4 hours of pure time to solve the problems. For each problem, a contestant can earn 6 points, and a successful solver is a student who scores 10 points or more. Allowed aids are writing and drawing supplies, school MF tables, and calculators without a graphical display. These details will be communicated to the students before the start of the competition.

## 58th Mathematical Olympiad

## Solutions to the problems of the written part of the school round in category $A$
3. A fraction can be reduced by an integer $d>1$ if and only if the number $d$ is a common divisor of the numerator and the denominator of the considered fraction. Let us assume, therefore, that $d \mid 3 p-q$ and at the same time $d \mid 5 p+2 q$, where $p$ and $q$ are coprime integers. By adding suitable multiples of the binomials $3 p-q$ and $5 p+2 q$, we get

$$
2(3 p-q)+(5 p+2 q)=11 p \quad \text { and } \quad 3(5 p+2 q)-5(3 p-q)=11 q
$$

Since both numbers $3 p-q$ and $5 p+2 q$ are multiples of the number $d$ by assumption, the constructed numbers $11 p$ and $11 q$ are also its multiples. In other words, the number $d$ is a common divisor of the numbers $11 p$ and $11 q$. However, the numbers $p$ and $q$ are coprime and the number 11 is a prime number, so the numbers $11 p$ and $11 q$ have only one common divisor greater than 1, and that is the number 11. Therefore, it must be $d=11$.

The solution is not yet complete: we must show that the number 11 can indeed reduce some of the considered fractions. How, then, to find a pair of coprime numbers $p$ and $q$ such that $11 \mid 3 p-q$ and at the same time $11 \mid 5 p+2 q$? With a little patience, we can find such values of $p$ and $q$ by trial substitution; however, it is sufficient to write the system of equations

$$
3 p-q=11 m \quad \text { and } \quad 5 p+2 q=11 n
$$

find its solution $(p, q)=(2 m+n, 3 n-5 m)$ and then conveniently substitute: we will certainly get a pair of coprime numbers $p$ and $q$ when $q=3 n-5 m=1$, i.e., for example, for $n=2$ and $m=1$, when $(p, q)=(4,1)$ and the considered fraction is $11 / 22$.

Answer. The only integer greater than 1 by which some of the given fractions can be reduced is the number 11.

For a complete solution, award 6 points, of which 1 point for converting to the relations $d \mid 3 p-q$ and $d \mid 5 p+2 q$, another 2 points for deriving the relations $d \mid 11 p$ and $d \mid 11 q$, another point for the conclusion $d=11$ and the remaining 2 points for providing a suitable example of a pair of coprime numbers $p$ and $q$.


[^0]:    $1 \mathrm{~V}$ in the given solution, the equality $|C U|=c|\cos \gamma| / \sin \gamma$ was derived using standard notation, where $U$ is the intersection of the altitudes of triangle $A B C$.