AMO-Bench 2025 Problem 41

Answer

Let $m$ be an odd integer with $m \geq 3$ . Construct a positive integer $x$ such that for some positive integers $a$ and $b$ , the sequence of consecutive integers $a+1, a+2, \ldots, a+b$ contains exactly 20232024 terms $y$ satisfying $y \equiv -1 \pmod{m}$ and $y^2 \equiv 1 \pmod{x}$ , and exactly 2023 terms $z$ satisfying $z \equiv 1 \pmod{m}$ and $z^2 \equiv 1 \pmod{x}$ .

Dataset text AMO-Bench. Open licence.

Show the answer

Let $x=\prod^{n}_{j=1}p_j$ , where $p_1,p_2,...$ is a sequence of primes such that $p_1=2$ , and for any positive integer $i \geq 2$ , $p_i\equiv 1 ( mod~m*\prod^{i-1}_{j=1}p_j)$ and where $n$ satisfies $n>1+log_{2}(20232024)$

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.

Add a solution

Solutions

Similar problems