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$2f((x+y)^2)=f(x+y)+(f(x))^2+(4y-1)f(x)-2y+4y^2$

Source: 2024 Taiwan Mathematics Olympiad

February 4, 2024
algebrafunctional equationTaiwan

Problem Statement

Find all functions ff from real numbers to real numbers such that 2f((x+y)2)=f(x+y)+(f(x))2+(4y−1)f(x)−2y+4y22f((x+y)^2)=f(x+y)+(f(x))^2+(4y-1)f(x)-2y+4y^2 holds for all real numbers xx and yy.
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