Problem 1

Prove that the following assertion is true for

n=3

and

n=5

, and false for every other natural number

n>2

: for arbitrary real numbers

a_1,a_2,\ldots,a_n

(a_1-a_2)(a_1-a_3)\cdots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots(a_2-a_n)+\cdots+(a_n-a_1)(a_n-a_2)\cdots(a_n-a_{n-1})\ge0

Step 5 of 6: Check the middle term for n=5

(a_3-a_1)(a_3-a_2)(a_3-a_4)(a_3-a_5)\ge0

Detailed analysis

The third term has two nonpositive factors, $a_3-a_1$ and $a_3-a_2$ , and two nonnegative factors, $a_3-a_4$ and $a_3-a_5$ . Hence it is nonnegative.