You met a genie. He gets $150$ magic lamps out, which are numbered from $1$ to $150$. You have to colour each lamp red or blue. After colouring, the genie will count the number of triples $T$ of magic lamps which are numbered $a,b,c$, with $a<b<c$, the magic lamps numbered $a,c$ are red and the magic lamp numbered $b$ is blue. Then the genie will give you $T$ grams of gold.

 

What should you do to maximise the amount of gold you get? You have to prove that it is the maximum.

You met a genie. He gets $150$ magic lamps out, which are numbered from $1$ to $150$. You have to colour each lamp red or blue. After colouring, the genie will count the number of triples $T$ of magic lamps which are numbered $a,b,c$, with $a<b<c$, the magic lamps numbered $a,c$ are red and the magic lamp numbered $b$ is blue. Then the genie will give you $T$ grams of gold.

What should you do to maximise the amount of gold you get? You have to prove that it is the maximum.