Quadratics (AIME / USAJMO / AMO)

Proof-style AIME / USAJMO / AMO statements — develop a full argument; type only the final answer.

Proof track: treat each problem as a full olympiad writeup on paper. Here you only enter the final answer (integer or short expression).

Quadratics: core idea

Symmetric systems.

**(Proof track.)** Give a complete justification. If roots of $x^2-10x+m$ differ by $4$ , find $m$ .

Hint: Use root relations and discriminants, while justifying any domain restriction. Answer: 21.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. Find the minimum integer value of $x^2-8x+17$ over integer $x$ .
2. **(Proof track.)** Give a complete justification. For how many integers $k$ does $x^2+kx+20=0$ have integral roots?
3. **(Proof track.)** Give a complete justification. The roots of $x^2+ax+12$ are consecutive integers. Find the sum of possible $a$ .
4. **(Proof track.)** Give a complete justification. Find the positive root of $x^2+4x-45=0$ .
5. **(Proof track.)** Give a complete justification. If $x^2+y^2=17$ and $x+y=5$ , find $xy$ .
6. **(Proof track.)** Give a complete justification. Find the sum of all real roots of $x^4-5x^2+4=0$ .
7. **(Proof track.)** Give a complete justification. A parabola has vertex $(3,-4)$ and passes through $(1,0)$ . Find its $y$ -intercept.
8. **(Proof track.)** Give a complete justification. Find the discriminant of $3x^2-10x+7$ .
9. **(Proof track.)** Give a complete justification. Solve $\lfloor x^2\rfloor=8$ ; find the length of the solution set.