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Triangle Inequality (Hard)

Late AMC 10/12 through early AIME — dense computation.

Easy Medium Hard AIME / USAJMO / AMO

Triangle Inequality: core idea

Optimization.

Example problem: hint and answer

How many integer third sides are possible with sides $8,15$ ?

Hint: Apply strict triangle bounds, and use a dropped altitude when useful. Answer: 15.

Practice (9 problems — type your answers)

1. A triangle has sides $5,12,x$ and perimeter $30$ . Find $x$ .
2. Consecutive integer triangle sides have perimeter $42$ . Find longest.
3. How many integer isosceles triangles have perimeter $20$ ?
4. Triangle sides are $7,9,12$ . Find its largest integer altitude.
5. Two sides are $10,17$ . Find least integer third side.
6. An isosceles triangle has equal sides $13$ and base $10$ . Find area.
7. Sides are $x,x+3,2x-1$ . Find least integer $x$ making a triangle.
8. How many integer-sided triangles have perimeter $12$ ?
9. A $6$ - $8$ - $10$ triangle has what inradius?

Triangle Inequality Practice Problems — Hard