Author Archives: kevin

MATHCOUNTS 2016-2017 – 240 Circle $P$ is internally tangent to circle $O$ at $A$, as shown. $\overline{AC}$ and $\overline{BE}$ intersect at $F$, which is also the point of tangency between $\overline{BE}$ and circle $P$. $\overline{AD}$ and $\overline{BE}$ are diameters of … Continue reading →

MATHCOUNTS 2016-2017 – 233 Circle O, shown here with chords AB and BE, has secants AC and DE that intersect at X. If \(m\angle{ABE}=35^\circ\) and \(m\angle{AXE}=15^\circ\), what is the measure of \(\overparen{CD\ }\)? Click here for the solutions. Solution 1 … Continue reading →

MASSCOUNTS 2016-2017 – 232 In Circle A, shown here, $\overleftrightarrow{BD}$ is tangent to the circle at B, and major $\overparen{BC\ }$ has measure $230^\circ$. What is $m\angle{CBD}$? Click here for the solution. Solution Draw lines $\overline{AB}$ and $\overline{AC}$ $$\begin{align} &\because … Continue reading →

MATHCOUNTS 2016-2017 – 231 Regular nonagon ABCDEFGHI is inscribed in a circle, as shown. What is $m\angle{AHC}$? Click here to show the solutions. Solution 1 Draw line $\overline{AC}$. $$ \begin{align} &\because\ ABCDEFGHI\ \text{is a regular nonagon} \\ &\therefore \triangle{ABC}, \triangle{AIH}, … Continue reading →