MathLabs

Problem 3

Let n>3n>3 be an integer. Choose three numbers from {1,2,…,n}\{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/2n/2, their values are all distinct. (b) Let pp be a prime with p≤np\le \sqrt{n}. Show that the number of choices whose smallest number is pp and whose combination values are not all distinct is exactly the number of positive divisors of p−1p-1.
Step 2 of 7: Order the expressions into two increasing chains
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x+y+z<z+xy<y+zx<x+yz,(y+z)x<(x+z)y<(x+y)z<xyzx+y+z<z+xy<y+zx<x+yz,\qquad (y+z)x<(x+z)y<(x+y)z<xyz
Detailed analysis

For 1<m<n1<m<n and t>1t>1, use mn>m+nmn>m+n, tn+m−(tm+n)=(t−1)(n−m)>0tn+m-(tm+n)=(t-1)(n-m)>0, and (t+m)n−(t+n)m=t(n−m)>0(t+m)n-(t+n)m=t(n-m)>0. Applying these inequalities to x<y<zx<y<z gives the two displayed chains.