MathDB

Problem 4

Part of 2002 Mongolian Mathematical Olympiad

Problems(2)

choose two natural numbers whose product is a square

Source: Mongolian MO 2002 Grade 10 P4

4/9/2021
Let there be 131131 given distinct natural numbers, each having prime divisors not exceeding 4242. Prove that one can choose four of them whose product is a perfect square.
number theory
Fermat-like nondivisibility

Source: Mongolian MO 2002 Teachers P4

4/9/2021
Let p≥5p\ge5 be a prime number. Prove that there exists a∈{1,2,…,p−2}a\in\{1,2,\ldots,p-2\} satisfying p2∤ap−1−1p^2\nmid a^{p-1}-1 and p2∤(a+1)p−1−1p^2\nmid(a+1)^{p-1}-1.
number theory