Triangle Inequality (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).

Triangle Inequality: core idea

Strict cases.

**(Proof track.)** Give a complete justification. How many integer triangles have sides $5,7,x$ ?

Hint: Start by proving every required strict inequality; then use the appropriate triangle theorem. Answer: 9.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. Find the largest integer $n$ for which $n,n+1,2n-3$ are triangle sides.
2. **(Proof track.)** Give a complete justification. How many scalene integer triangles have perimeter $15$ ?
3. **(Proof track.)** Give a complete justification. An isosceles triangle has perimeter $32$ and integer base. How many possible bases?
4. **(Proof track.)** Give a complete justification. A triangle has sides $9,40,41$ . Find its circumradius.
5. **(Proof track.)** Give a complete justification. Find the largest angle of a $7$ - $8$ - $9$ triangle, in degrees to nearest integer.
6. **(Proof track.)** Give a complete justification. How many integer $x$ make $4,x,13$ a triangle?
7. **(Proof track.)** Give a complete justification. A triangle has sides $10,10,12$ . Find its circumradius.
8. **(Proof track.)** Give a complete justification. Find the least perimeter of an integer-sided obtuse triangle with longest side $10$ .
9. **(Proof track.)** Give a complete justification. The sides of a triangle are $x,2x,3x-1$ . Find the least positive integer $x$ .