Absolute Value (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).

Absolute Value: core idea

Piecewise contests.

**(Proof track.)** Give a complete justification. Find all real solutions to $|x-1|+|x-5|=10$ ; enter their sum.

Hint: Break the real line at the expression's zeroes and justify each interval. Answer: 6.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. Find the minimum of $|2x-1|+|2x-9|$ .
2. **(Proof track.)** Give a complete justification. How many integers satisfy $|x^2-9|<10$ ?
3. **(Proof track.)** Give a complete justification. Find the maximum of $|x-3|-|x+1|$ over real $x$ .
4. **(Proof track.)** Give a complete justification. Solve $|x+4|=2|x-1|$ ; enter the larger solution.
5. **(Proof track.)** Give a complete justification. Find the sum of solutions of $|x^2-4|=x$ .
6. **(Proof track.)** Give a complete justification. For $0\le x\le10$ , find $\max |x-2||x-8|$ .
7. **(Proof track.)** Give a complete justification. Find the least $c$ such that $|x-2|+|x-7|\le c$ for all $x\in[0,10]$ .
8. **(Proof track.)** Give a complete justification. How many real roots has $|x|=x^2-2$ ?
9. **(Proof track.)** Give a complete justification. Find $\min_{x\in\mathbb R}(|x|+|x-1|+|x-4|+|x-10|)$ .