# IMO 1994 Q1 #

Let `m`

and `n`

be two positive integers.
Let `a₁, a₂, ..., aₘ`

be `m`

different numbers from the set `{1, 2, ..., n}`

such that for any two indices `i`

and `j`

with `1 ≤ i ≤ j ≤ m`

and `aᵢ + aⱼ ≤ n`

,
there exists an index `k`

such that `aᵢ + aⱼ = aₖ`

.
Show that `(a₁+a₂+...+aₘ)/m ≥ (n+1)/2`

# Sketch of solution #

We can order the numbers so that `a₁ ≤ a₂ ≤ ... ≤ aₘ`

.
The key idea is to pair the numbers in the sum and show that `aᵢ + aₘ₊₁₋ᵢ ≥ n+1`

.
Indeed, if we had `aᵢ + aₘ₊₁₋ᵢ ≤ n`

, then `a₁ + aₘ₊₁₋ᵢ, a₂ + aₘ₊₁₋ᵢ, ..., aᵢ + aₘ₊₁₋ᵢ`

would be `m`

elements of the set of `aᵢ`

's all larger than `aₘ₊₁₋ᵢ`

, which is impossible.