Area Techniques (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).

Area Techniques: core idea

Area method.

**(Proof track.)** Give a complete justification. A triangle has sides $10,13,13$ . Find its inradius.

Hint: Justify the dissection or area formula you invoke. Answer: 4.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. A regular octagon has side $2$ . Find its area.
2. **(Proof track.)** Give a complete justification. A circle is inscribed in a $6$ - $8$ - $10$ triangle. Find its area.
3. **(Proof track.)** Give a complete justification. A square and a circle have equal perimeter $16$ . Find the difference of their areas.
4. **(Proof track.)** Give a complete justification. A triangle has two sides $7,9$ with included angle $120^\circ$ . Find area.
5. **(Proof track.)** Give a complete justification. A rhombus has side $10$ and one diagonal $12$ . Find area.
6. **(Proof track.)** Give a complete justification. Find area of a regular dodecagon inscribed in a radius- $4$ circle.
7. **(Proof track.)** Give a complete justification. A sector is $1/6$ of a circle and has area $24\pi$ . Find its arc length.
8. **(Proof track.)** Give a complete justification. A rectangle has diagonal $25$ and area $168$ . Find perimeter.
9. **(Proof track.)** Give a complete justification. A triangle's medians have what ratio of areas among the six small triangles? Enter the common ratio.