# Thread: a old german contest problem

1. ## a old german contest problem

3.(1977.)How many pairs of numbers p,and q from which are smaller than 100 and for which has a rational solution exist?

Biggest problem is that I don't know when this type of equation does have a rational solution?

I do not know how to start?

2. there can be infinitely many solutions

but,I'm not sure

3. Originally Posted by myro111
3.(1977.)How many pairs of numbers p,and q from which are smaller than 100 and for which has a rational solution exist?

Biggest problem is that I don't know when this type of equation does have a rational solution?

I do not know how to start?
Step 1. It follows from the rational root theorem that any rational solution x of this equation must in fact be an integer.

Step 2. If p and q are natural numbers then x must be negative (if x is positive then the left side of the equation will be positive, not zero). So let $\displaystyle y=-x$. Then the equation becomes $\displaystyle y^5+py=q$, and p, q, y are all positive integers, with p and q less than 100.

Step 3. If $\displaystyle y\geqslant3$ then $\displaystyle y^5\geqslant3^5=243>100$, which is larger than q. Therefore y = 1 or 2.

Step 4. Now all you have to do is to count how many pairs (p,q) there are when y=1 and when y=2.