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1. The quadratic $x^2 + 5x + c$ has roots in the form of $x = \frac{-5 \pm \sqrt{c}}{2}$. What is the value of $c$?

 

2. The quadratic $x^2+440x+440^2$ can be written in the form $(x+b)^2+c$, where $b$ and $c$ are constants. What is $\frac{c}{b}$?'

 

3. Trapezoid $WXYZ$ is inscribed in a circle, with $WX \parallel YZ$. If arc $YZ$ is $30$ degrees, arc $WZ$ is $t^2 + 7t$ degrees, and arc $XY$ is $60 - 4t$ degrees, find arc $WTX$. THE ANSWER IS NOT 146, 236, OR 44

 

4. In cyclic quadrilateral $PQRS,$
\[\frac{\angle P}{2} = \frac{\angle Q}{3} = \frac{\angle R}{4}.\]Find the largest angle of quadrilateral $PQRS,$ in degrees. THE ANSWER IS NOT 135.

 

5. In rectangle $PQRS$, $PS = 3$ and $PQ = 4$. Let $M$ be the midpoint of $\overline{PQ}$, and let $X$ be the point such that $MS = MX$ and $\angle MSX = 77^\circ$, as shown below. Find $\angle XRS$, in degrees.

 

6. A regular dodecagon $P_1 P_2 P_3 \dotsb P_{12}$ is inscribed in a circle with radius $1.$ Compute
\[(P_1 P_2)^2 + (P_1 P_3)^2 + \dots + (P_{11} P_{12})^2.\]  THE ANSWER IS NOT 432.

 

7. Circles $S$ and $T$ have radii $1,$ and intersect at $A$ and $B$. The distance between their centers is $\sqrt{2}$. Let $P$ be a point on major arc $AB$ of circle $S$, and let $\overline{PA}$ and $\overline{PB}$ intersect circle $T$ again at $C$ and $D$, respectively. Show that $\overline{CD}$ is a diameter of circle $T$.

 Dec 25, 2019
 #1
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These are all AoPS homework problems.  We should not help students cheat on their homework.

 Dec 25, 2019
 #2
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Yah, too lazy to even get rid of the LaTeX.

 Dec 25, 2019

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