Problem 3

Let

n>3

be an integer. Choose three numbers from

\{1,2,\ldots,n\}

. Using each once, together with addition, multiplication, and parentheses, form all possible combinations. (a) Show that if all three chosen numbers are greater than

n/2

, their values are all distinct. (b) Let

p

be a prime with

p\le \sqrt{n}

. Show that the number of choices whose smallest number is

p

and whose combination values are not all distinct is exactly the number of positive divisors of

p-1

Step 3 of 7: Compare the two chains

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(y+z)x-(y+zx)=(x-1)y>0,\qquad (x+z)y-(x+yz)=(y-1)x>0

Detailed analysis

The first inequality places $(y+z)x$ above $y+zx$ , and the second places $(x+z)y$ above $x+yz$ . Therefore every cross-comparison is settled except possibly $x+yz$ versus $(y+z)x$ ; all other values are strictly ordered.