136- The center of a circle is on the first-quadrant angle bisector. If this circle passes through point $A(6,3)$ and is tangent to the line $y = 2x$, what is its radius?
(1) $\sqrt{5}$ (2) $\sqrt{6}$ (3) $2\sqrt{2}$ (4) $\sqrt{10}$

129. Two circles with centers $O$ and $O'$ are externally tangent. A circle with diameter $\overline{OO'}$ is drawn with external common tangency to these two circles. What is the position of the two circles?

(1) Intersecting

(2) Tangent

(3) External

(4) Indeterminate

136- Circles $C$ and $C'$ are tangent at point $(0,1)$ and their common internal tangent lines with respect to circle $C$ are equidistant from point $(-3, 2)$. If circle $C'$ with radius $\sqrt{5}$ passes through $(-3,2)$, what is the center of circle $C'$?
(1) $(-1, 3)$ (2) $(-1, 2)$
(3) $(1, -2)$ (4) $(1, -1)$

125. In a square with side 2 units, a circle with center at one vertex and radius $2.5$ units cuts two sides of the square. What is the distance from the nearest vertex of the square to the intersection points?
(1) $\dfrac{1}{\mathfrak{f}}$ (2) $\dfrac{1}{\mathfrak{r}}$ (3) $\dfrac{\sqrt{\mathfrak{r}}}{\mathfrak{r}}$ (4) $\dfrac{\sqrt{\mathfrak{r}}}{\mathfrak{r}}$

136- A circle passing through point $(-9, -2)$ is tangent to both coordinate axes. What is the radius of the larger circle?
(1) $14$ (2) $15$ (3) $17$ (4) $19$

128. Quadrilateral $ABCD$ is inscribed in a circle. If $AB$ is the farthest chord and $BC$ is the closest chord to the center of this circle, which relationship between the angles cannot hold?

• [(1)] $\hat{D} > \hat{C}$
• [(2)] $\hat{B} > \hat{C}$
• [(3)] $\hat{A} > \hat{B}$
• [(4)] $\hat{B} > \hat{D}$

135. For which value of $a$, the angle between the tangent line to the circle $x^2 + y^2 - 2x + y = 1$ and the line $3x + 2y = a$ at their intersection point is $90°$?
(1) $2$ (2) $3$ (3) $4$ (4) $5$

129. In the figure below, $AD$ is tangent to the circle with center $O$, and $OH$ is perpendicular to $AC$. If $\widehat{DBC} = 2\widehat{DAC}$, how many times is angle $\widehat{COH}$ equal to angle $\widehat{DAC}$?
\begin{minipage}{0.45\textwidth} [Figure: Circle with center $O$, tangent line $AD$, points $B$, $C$, $H$ marked] \end{minipage} \begin{minipage}{0.45\textwidth} \begin{flushright} (1) $2.5$
(2) $3$
(3) $3.5$
(4) $4$ \end{flushright} \end{minipage}

130. Two circles with radii $4$ and $8$ are internally tangent at point $A$. A chord $BC$ of the large circle is tangent to the small circle, and the line through the center of the small circle parallel to the radical axis passes through point $P$. What is $PB \times PC$?
(1) $24$ (2) $32$ (3) $36$ (4) $48$
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126. An isosceles trapezoid, under which condition can be inscribed in a circle?

1. [(1)] Two diameters perpendicular to each other
2. [(2)] One of the bases of the trapezoid equals one of the legs
3. [(3)] The line connecting the midpoints of the two legs passes through the intersection of the diameters
4. [(4)] The length of the segment connecting the midpoints of the two legs equals one of the legs

133- The common chord of circle $C$ with equation $x^2 + y^2 - 4x + y^2 = 6$ is tangent to the first region of circle $C$. If the point $(-1, 4)$ lies on it, the equation of the common chord is which of the following?
(1) $x^2 + y^2 - y + 3x = 6$ (2) $x^2 + y^2 + 3y - x = 6$
(3) $x^2 + y^2 - 2y + x = 6$ (4) $x^2 + y^2 - 3y - x = 6$

131- In the figure below, segment $AC$ equals chord $AB$. Which of the following is necessarily true?
[Figure: Circle with points A, B, C, D where AC is a chord equal to chord AB]

• [(1)] $BC = BA$
• [(2)] $BD = AC$
• [(3)] $BC = BD$
• [(4)] $DA = DC$

132- Which of the following quadrilaterals can be inscribed in a circle with diameter $(x+2)$?
[Figure: Trapezoid with sides labeled x, y, 4, 9 and angle $60^\circ$]

• [(1)] $\sqrt{51}$
• [(2)] $\sqrt{55}$
• [(3)] $\sqrt{57}$
• [(4)] $\sqrt{59}$

133- The smallest circle passing through the two points $A(2,5)$ and $B(-4,1)$ intersects the $x$-axis at what length?

• [(1)] $1\ ,\ -3$
• [(2)] $5\ ,\ -3$
• [(3)] $2\ ,\ -1$
• [(4)] $3\ ,\ -2$

134- Among the circles passing through the point $A(-4\ ,\ 1)$ and tangent to the lines $4x + 3y = 0$ and the $y$-axis, the one with the largest radius is:

• [(1)] $\dfrac{5}{3}$
• [(2)] $\dfrac{17}{9}$
• [(3)] $\dfrac{7}{3}$
• [(4)] $\dfrac{22}{9}$

152. In the figure below, line segment $AC$ is tangent to the circle. If $\dfrac{AC}{BC} = \sqrt{3}$, then what is $\dfrac{DB}{BC}$?
[Figure: Circle with points A, B, C, D, where AC is tangent to the circle at C]

• [(1)] $\sqrt{2}$
• [(2)] $\sqrt{3}$
• [(3)] $2$
• [(4)] $3$

153. According to the figure below, rectangle $ABCD$ is circumscribed about a circle with radius 3, and the circumference is $M\widehat{B}N = 120°$. What is the area of quadrilateral $OMNC$?
[Figure: Rectangle ABCD with inscribed circle centered at O, points M on AB and N inside, shaded region OMNC]

• [(1)] $\dfrac{27\sqrt{3}}{4}$
• [(2)] $\dfrac{9\sqrt{3}}{2}$
• [(3)] $\dfrac{27\sqrt{3}}{2}$
• [(4)] $9\sqrt{3}$

154. Suppose lines $x + y = 1$ and $x - y = 3$ are the diameters of a circle, and the line $4x + 3y + 5 = 0$ is tangent to it. What is the distance of point $M(4, -2)$ from the circle?

• [(1)] $\sqrt{3} - 1$
• [(2)] $\sqrt{3} - \sqrt{2}$
• [(3)] $\dfrac{\sqrt{7}}{2}$
• [(4)] $\sqrt{5} - 2$

155. Suppose the length of the radical axis of two circles with radii $1 - 6a$ and $6 - 2a^2$ equals 6 units. If the two circles have exactly one common tangent, what is the average of the possible values of $a$?

• [(1)] $3$
• [(2)] $\dfrac{13}{3}$
• [(3)] $6$
• [(4)] $7$

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140- For every $m$, the equation $y = 6$, $(m+1)x + (m-2)y = 6$ is the equation of a chord of circle $C$. If point $A(-1,1)$ lies on circle $C$, the circumference of circle $C$ is which?
\[ \text{(1)}\ 2\sqrt{3}\pi \qquad \text{(2)}\ 2\pi \qquad \text{(3)}\ 3\pi \qquad \text{(4)}\ 2\sqrt{7}\pi \]
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27. In a rectangle, lines from two opposite vertices are drawn perpendicular to a diagonal, and that diagonal is divided into three parts such that the middle part is twice each of the two side parts. The area of this rectangle is how many times the area of the smallest triangle formed inside the rectangle?
(1) $24$ (2) $16$ (3) $12$ (4) $8$

30. In the figure below, two tangent lines are drawn from point $A$. What is the radius of the circle?
[Figure: Two tangent lines drawn from external point $A$ to a circle, with segments labeled 9, 8, 7 and point $B$, $C$ marked]

• [(1)] $7/2\sqrt{2}$
• [(2)] $4/8\sqrt{5}$
• [(3)] $3/6\sqrt{2}$
• [(4)] $2/4\sqrt{5}$

36. Line $d$ has equation $y - x = 0$. A circle with center at the origin has a radius twice that of another circle. If line $d$ is tangent to the smaller circle with equation $x^2 + y^2 + 6x - 2y = r$, what is the product of the lengths of the chord(s) of intersection of the two circles?
(1) $\dfrac{5}{2}$ (2) $\dfrac{5}{4}$ (3) $\dfrac{65}{32}$ (4) $\dfrac{65}{64}$

Tangents are drawn to a given circle from a point on a given straight line, which does not meet the given circle. Prove that the locus of the mid-point of the chord joining the two points of contact of the tangents with the circle is a circle.

Let $A, B, C, D, E$ be the vertices of a regular pentagon inscribed in a circle of radius $r$. Let $F$ be the midpoint of side $AB$. Find the circumradius $AO$ in terms of the side length $x = AB$.