Citation needed.
Just a casual web search for your claim seems to be yielding the opposite conclusion.
My anecdotal experience from moving around a lot (over half a dozen public schools from kindergarten to 12th grade) also contradicts your claim. Competition at richer schools has definitely given me an extra edge. The phrase "competition at poor schools" is also oxymoronic. What competition? Competition in classmates ignoring teachers and fights constantly erupting?
The gulf between poor schools and rich schools is ridiculous. Some schools offer several Advanced Placement courses, several International Baccalaureate courses, and partnerships with multiple nearby colleges for college courses. Students at poorer schools don't even know what AP and IB courses are. You don't know competition until the top 10% of your graduating class are either all going to Ivy League and prestigious colleges or all already have credit for college courses before even applying to college.
And I was just talking about the insane competition from fellow students. School resources and personnel are another matter. State-of-the-art SMART Boards. Laptops. Tablets. Multiple fields for multiple sports. Paid field trips. Study abroad. A robust student exchange program. Multiple year foreign language programs in around half a dozen languages. Teachers with multiple/advanced degrees. Several guidance counselors. Multiple vice/assistant principals. Multiple coaches.
I am genuinely curious what edge you think a poor school gave you.
How so? Are they curving grades? Smaller class sizes because of more drop-outs? More teacher attention because you cared more than the 'poor' kids?
Are tests not standardized that class rank matters? When I first entered university here, grade average was all that mattered. Certain grades even guarantee university acceptance...
The fact the US hasn't figured out how to offer a consistent level of K-12 education is, quite frankly, mind-boggling (along with the lack of health care, and many other things we take for granted in the rest of the 'western' world).
Don't believe everything you read.