A new theorem on quadratic residues modulo primes

Let $p>3$ be a prime, and let $(frac{cdot}p)$ be the Legendre symbol. Let $binmathbb Z$ and $varepsilonin{pm 1}$. We mainly prove that $$left|left{N_p(a,b): 1<a<p text{and} left(frac apright)=varepsilonright}right|=frac{3-(frac{-1}p)}2,$$ where $N_p(a,b)$ is the number of positive integers $x<p/2$ with ${x^2+b}_p>{ax^2+b}_p$, and ${m}_p$ with $minmathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.