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For how many values of $a$ is it true that the line $y=x+a$ passes through the vertex of the parabola $y=x^{2}+a^{2}$? (2005 AMC 12B Problem, Question \#8) 
Overseas Competition->Knowledge Point->Calculation Modules
We see that the vertex of the quadratic function $y=x^{2}+a^{2}$ is $\left(0, a^{2}\right)$. The $y$-intercept of the line $y=x+a$ is $(0, a)$. We want to find the values (if any) such that $a=a^{2}$. Solving for $a$, the only values that satisfy this are 0 and 1, so the answer is $(\text{C}) 2$. 