okx inequalities
JL
You can’t solve any of them, can you?
1.
Let x, y, z be distinct positive integers. Show that
x2 + y 2 + z 2 > xy + yz + xz + 2
2.
Prove that if a, b, c > 0, then
√
a+b+c
3
≥ abc
3
When does equality hold?
3.
Show that for any n ∈ Z+ ,
n ≤ mn − 1
where m ≥ 2 is an integer.
4.
Define
f (n) =
X
1≤i≤n
1
√
i
Prove that 87 < f (2022) < 90.
5.
Let n ≥ 3 be a positive integer. Show that
nn+1 > (n + 1)n
1