MathDB

3

Part of 2024 Caucasus Mathematical Olympiad

Problems(2)

Product of 10 factorials is 10-th power

Source: Caucasus MO 2024, Juniors P3

3/15/2024
Given 1010 positive integers with a sum equal to 10001000. The product of their factorials is a 1010-th power of an integer. Prove that all these numbers are equal.
number theory
Product of n and its reverse equals 888...8

Source: Caucasus MO 2024, Seniors P3

3/15/2024
Let nn be a dd-digit (i.e., having dd digits in its decimal representation) positive integer not divisible by 1010. Writing all the digits of nn in reverse order, we obtain the number n′n'. Determine if it is possible that the decimal representation of the product n⋅n′n\cdot n' consists of digits 88 only, if (a) d=9998d = 9998; (b) d=9999?d = 9999?
number theory