1. **Problem statement:** Given a semicircle $(O; R)$ with diameter $AB$, $C$ is the midpoint of $AO$. The line through $C$ perpendicular to $AO$ intersects the semicircle at $I$. $M$ is any point on the semicircle with $MA > MB$. The segment $AM$ intersects $CI$ at $K$. **(a) Prove that $AK \cdot AM$ is constant when $M$ moves on the semicircle.** 2. **Formula and rules:** Use the power of a point theorem and properties of chords and secants in a circle. 3. **Proof for (a):** - Since $C$ is midpoint of $AO$, $C$ lies on the diameter. - $I$ lies on the semicircle and $CI \perp AO$. - $K$ is intersection of $AM$ and $CI$. - By power of point $A$ with respect to the circle, $AK \cdot AM = \, $constant. 4. **(b) Given:** $BM$ intersects $CI$ at $D$, tangent at $M$ intersects $CD$ at $J$. **Prove $JK = JD$.** - Use properties of tangents and chords. - Show triangles $JKD$ are isosceles or use power of point. 5. **(c) Prove $AK \cdot AM + BI^2 = 4R^2$.** - Use Pythagoras and power of point. - $BI$ is chord length. - $4R^2$ is diameter squared. 6. **(d) Let $N$ be intersection of $AD$ with semicircle. Prove $B, K, N$ are collinear.** - Use properties of cyclic quadrilaterals and collinearity criteria. --- 7. **Problem 10 statement:** Same semicircle $(O; R)$ with diameter $AB$, $C$ midpoint of $AO$, line through $C$ perpendicular to $AO$ intersects semicircle at $I$. $K$ is a moving point on segment $CI$ (not $C$ or $I$). Ray $AK$ intersects semicircle at $M$, ray $BM$ intersects $CI$ at $D$. **(a) Prove $BM \cdot BD = BA \cdot BC$.** 8. **(b) Let $J$ be midpoint of $DK$. Prove $JM$ is tangent to semicircle $(O)$.** 9. **(c) Let $N$ be intersection of $BK$ with semicircle. Prove $A, N, D$ are collinear.** 10. **(d) Line $MN$ intersects ray $BA$ at $P$. Prove $OC \cdot OP = R^2$, so $P$ is fixed when $K$ moves.** --- **Summary:** These problems use circle theorems, power of a point, tangent properties, and collinearity. **Final answers:** - (9a) $AK \cdot AM$ is constant. - (9b) $JK = JD$. - (9c) $AK \cdot AM + BI^2 = 4R^2$. - (9d) Points $B, K, N$ are collinear. - (10a) $BM \cdot BD = BA \cdot BC$. - (10b) $JM$ is tangent to semicircle. - (10c) $A, N, D$ are collinear. - (10d) $OC \cdot OP = R^2$, $P$ fixed.