MathDB

2

Part of 2019 Saudi Arabia IMO TST

Problems(2)

product or number of all divisor of $sm, sn are equal

Source: 2019 Saudi Arabia IMO TST I p2 - variation of 2018 Shortlist N1

7/28/2020
Find all pair of integers (m,n)(m,n) and m≥nm \ge n such that there exist a positive integer ss and a) Product of all divisor of sm,snsm, sn are equal. b) Number of divisors of sm,snsm,sn are equal.
number theoryDivisors
f(n + 1)f(n + 2)... f(n + k) / f(1)f(2) ... f(k) \in Z

Source: 2019 Saudi Arabia IMO TST II p2

7/28/2020
Let non-constant polynomial f(x)f(x) with real coefficients is given with the following property: for any positive integer nn and kk, the value of expression f(n+1)f(n+2)...f(n+k)f(1)f(2)...f(k)∈Z\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in Z Prove that f(x)f(x) is divisible by xx
algebrapolynomialdivisible