CN-TST 2023 Problem 23

Given a prime $p$ and a real number $\lambda \in (0,1)$ . Let $s$ and $t$ be positive integers such that $s \leqslant t < \frac{\lambda p}{12}$ . $S$ and $T$ are sets of $s$ and $t$ consecutive positive integers respectively, which satisfy $$\left| \left\{ (x,y) \in S \times T : kx \equiv y \pmod p \right\}\right| \geqslant 1 + \lambda s.$$ Prove that there exists integers $a$ and $b$ that $1 \leqslant a \leqslant \frac{1}{ \lambda}$ , $\left| b \right| \leqslant \frac{t}{\lambda s}$ and $ka \equiv b \pmod p$ .