Find all polynomials f(x) with integer coefficients and leading coefficient equal to 1, for which f(0)=2010 and for each irrational x, f(x) is also irrational.
Problem
Source: I International Festival of Young Mathematicians Sozopol 2010, Theme for 10-12 grade
Tags: algebra, polynomial
14.12.2019 07:18
I claim the only answer is f(x)=x+2010. Note that, x∈Qc⟹f(x)∈Qc, that is, x∈Q⟺f(x)∈Q (as f(x)∈Z[x]). Now, let f(x)=∑nk=0akxk with an=1 and a0=2010. Take a large prime p>∑nk=0|ak|. Now note that, f(x)→∞ as x→∞. In particular, f(x)=p+2010 has a solution, which happens to be a rational. By the rational root theorem, the only such solutions are x∈{±1,±p}. From the choice of p, f(x)=±1 is impossible. Also, by taking p large, −p case is also not possible. Hence, x=p, that is, f(p)=p+2010. Now since p is arbitrary, we have that f(p)=p+2010 holds infinitely often, that is, f(x)−(x+2010) has infinitely many distinct roots, so it must be a zero polynomial. This yields the answer above. Punchline: Any f(x)∈Q[x] with f−1(Q)⊆Q is necessarily linear. This is originally a problem from an old Iranian olympiad.
14.12.2019 07:24
grupyorum wrote: I claim the only answer is f(x)=x+2010. Note that, x∈Qc⟹f(x)∈Qc, that is, x∈Q⟺f(x)∈Q (as f(x)∈Z[x]). Now, let f(x)=∑nk=0akxk with an=1 and a0=2010. Take a large prime p>∑nk=0|ak|. Now note that, f(x)→∞ as x→∞. In particular, f(x)=p+2010 has a solution, which happens to be a rational. By the rational root theorem, the only such solutions are x∈{±1,±p}. From the choice of p, f(x)=±1 is impossible. Also, by taking p large, −p case is also not possible. Hence, x=p, that is, f(p)=p+2010. Now since p is arbitrary, we have that f(p)=p+2010 holds infinitely often, that is, f(x)−(x+2010) has infinitely many distinct roots, so it must be a zero polynomial. This yields the answer above. Punchline: Any f(x)∈Q[x] with f−1(Q)⊆Q is necessarily linear. This is originally a problem from an old Iranian olympiad. I think even cx+2010 works where c is integer. @below oh sorry
14.12.2019 07:28
Read the question carefully. Leading coefficient is one.