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a!+b+c, b!+c+a and c!+a+b prime numbers

Source: Kosovo Mathematical Olympiad 2022, Grade 10, Problem 3

March 6, 2022

number theoryprime numbers

Problem Statement

Let

a,b

and

c

be positive integers such that

a!+b+c,b!+c+a

and

c!+a+b

are prime numbers. Show that

\frac{a+b+c+1}{2}

is also a prime number.