I can't remember the specifics, but it has indeed been proven that some measure of your brain capacity is born into you. Which makes sense since that's where all the things that allow you to be "smart" are grown. If anything is off or something beneficial is much more present then you have either a stupider or smarter baby respectively.
We are not born smart at all. Our brains have not started developing and it won't until data has been input. We start learning the day we are born. A mother's touch, or smile. Her songs and words. She teaches us to eat, speak, and walk. We progress through learning. We are not born that way. We are, however, a product of our environment.
People aren't born smart. They are not born dumb. Each person is born with specific characteristics that make them what they are. Becoming "smart" is nothing more than realizing your potential and working at perfecting it.
If you take 2 identical specimens, born on the exact same day, and place each of them in different households, under different circumstances, they will turn out to be two different people in the future.
I don't think anyone is born smarter than the next person. I think you are born with different characteristics, and one may be better at a particular thing than another. However, if you had two identical specimens, born on the exact same day, and give each specimen different upbringings you'll find that they are not equals 18/25/40 years down the road.
To become smart you need to recognize your abilities and work to perfect those abilities. You also need good parents who will support you and give you room to grow into those abilities.
I do not think we are born smart because we all need people teaching us to be smart. Someone who has been educated against a baby then the winner will obviously be the person who has been given a good education. You cannot be born knowing the answers to difficult maths problems however you can learn it. Some people learn at different speeds but at birth I believe our level of intelligence is the same.
By what standards? I believe Albert Einstein said, "If you measure a fish's intelligence by its ability to climb a tree, it will believe it is stupid its entire life."
Smart in IQ? That's one scale. Smart in academics? That's another scale. Smart in policy? That's again another scale. Smart in leadership? That's another scale. There's many more scales. The important thing to realize is that everyone is smart at something, but that is cultured through life. You aren't born smart in foreign policy; you may have a knack for understanding political relations, but you are eventually conditioned to understand and "be smart" with foreign policy.