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Omni-MATH / For each prime p, a polynomial P(x) with rational coefficients is called p-good if and only if there exist three integers a,b, and c such that 0≤a<b<c<(p)/(3) and p divides all the…

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For each prime pp, a polynomial P(x)P(x) with rational coefficients is called pp-good if and only if there exist three integers a,ba, b, and cc such that 0≤a<b<c<p30 \leq a<b<c<\frac{p}{3} and pp divides all the numerators of P(a)P(a), P(b)P(b), and P(c)P(c), when written in simplest form. Compute the number of ordered pairs (r,s)(r, s) of rational numbers such that the polynomial x3+10x2+rx+sx^{3}+10x^{2}+rx+s is pp-good for infinitely many primes pp.
Plain-text mathematical notation (without MathML)
For each prime p, a polynomial P(x) with rational coefficients is called p-good if and only if there exist three integers a,b, and c such that 0≤a<b<c<(p)/(3) and p divides all the numerators of P(a), P(b), and P(c), when written in simplest form. Compute the number of ordered pairs (r,s) of rational numbers such that the polynomial x³+10x²+rx+s is p-good for infinitely many primes p.
Original LaTeX notation
For each prime $p$, a polynomial $P(x)$ with rational coefficients is called $p$-good if and only if there exist three integers $a, b$, and $c$ such that $0 \leq a<b<c<\frac{p}{3}$ and $p$ divides all the numerators of $P(a)$, $P(b)$, and $P(c)$, when written in simplest form. Compute the number of ordered pairs $(r, s)$ of rational numbers such that the polynomial $x^{3}+10x^{2}+rx+s$ is $p$-good for infinitely many primes $p$.

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Source and history

task_id
87ab21a0-0a1f-5764-acfd-7dc719727b7c
task_key
test--87ab21a0-0a1f-5764-acfd-7dc719727b7c
task_revision_id
1
config / split
/ test

Official source

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