If I buy a investment property that is cheaper than the home value,
Do I still need to put down 20%?
Say the value of the house is $150K, if I buy the house for 80K
Do I still need to put 20% down if the house has a lot of equity?
Typically a conventional lender will require around 20% down irrespective of purchase price. Other options such as private or hard money lenders may have different terms.
Even Hard Money Lenders will usually want you to have some skin in the game. Even if it's just 10% of purchase. Even if as-is value is much greater than the purchase price.
I would be cautious of any HML promising to finance 100% of purchase, because as the saying goes - there ain't no such thing as a free lunch.