62nd Ukrainian National Mathematical Olympiad, Third Round, First Tour

Question

You are given 5 distinct positive integers. Can their arithmetic mean be: a) exactly 3 times larger than their largest common divisor; b) exactly 2 times larger than their largest common divisor;

Accepted Answer

a) It's enough to provide an example of such 5 integers. One example is the set

1, 2, 3, 4, 5

, whose arithmetic mean is

3

, and the largest common divisor is

1

.

b) Suppose that such numbers

a_{1}, a_{2}, a_{3}, a_{4}

and

a_{5}

exist, let

d

be their largest common divisor, then these 5 integers can be rewritten as

a_{i} = d b_{i}

,

i = 1, 5

and the equality in the statement is rewritten as:

\frac{a _{1} + \dots + a _{5}}{5} = 2 d \Leftrightarrow d (b_{1} + \dots + b_{5}) = 10 d \Leftrightarrow b_{1} + \dots + b_{5} = 10.

Clearly numbers

b_{1}, \dots, b_{5}

are distinct, so their smallest possible sum is

1 + 2 + 3 + 4 + 5 = 15 > 10

, this contradiction completes the proof.