Circle Geometry (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).

Circle Geometry: core idea

Advanced circle theorems.

**(Proof track.)** Give a complete justification. Two chords intersect with pieces $2,18$ and $6,x$ . Find $x$ .

Hint: Identify whether the theorem concerns arcs, angles, powers, or tangent lengths. Answer: 6.

No proof submission required — answers only.

1. **(Proof track.)** Give a complete justification. A tangent from $P$ is $15$ ; a secant has external segment $9$ . Find whole secant.
2. **(Proof track.)** Give a complete justification. An arc is $220^\circ$ . Find its corresponding minor inscribed angle.
3. **(Proof track.)** Give a complete justification. A radius- $10$ circle has a chord subtending $120^\circ$ . Find chord length.
4. **(Proof track.)** Give a complete justification. A cyclic quadrilateral has angles $x,2x,3x,4x$ . Find smallest angle.
5. **(Proof track.)** Give a complete justification. A triangle is inscribed in a circle with side $10$ opposite $30^\circ$ . Find radius.
6. **(Proof track.)** Give a complete justification. Two circles radii $3,4$ are externally tangent. Find center distance.
7. **(Proof track.)** Give a complete justification. A circle's diameter is $26$ . A point is $5$ from the center. Find its power.
8. **(Proof track.)** Give a complete justification. A $90^\circ$ arc has length $3\pi$ . Find circle area.
9. **(Proof track.)** Give a complete justification. Tangents at endpoints of a diameter meet at what angle?