Practice (40)
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Solve in positive integers the equation $x^2 + y^2 = z^4$, where $\gcd(x,y)=1$ and $x$ is even.
Compute the value of $\sin{18^\circ}$ using regular geometry.
Let $F$ be a point inside $\triangle{ABC}$ such that $\angle{CAF} = \angle{FAB} = \angle{FBC} = \angle{FCA}$, show that the lengths of three sides form a geometric sequence.
Suppose the point $F$ is inside a square $ABCD$ such that $BF=1$, $FA=2$, and $FD=3$, as shown. Find the measurement of $\angle{BFA}$.
The vertex $E$ of square $EFGH$ is at the center of square $ABC$D. The length of a side of $ABCD$ is 1 and the length of a side of $EFGH$ is 2. Side $EF$ intersects $CD$ at $I$ and $EH$ intersects $AD$ at $J$. If angle $EID = 60^\circ$, what is the area of quadrilateral $EIDJ$?
A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet?
Points $E$ and $F$ are located on square $ABCD$ so that $\triangle BEF$ is equilateral. What is the ratio of the area of $\triangle DEF$ to that of $\triangle ABE$?
Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is $\frac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.)
Square $ABCD$ has side length $2$. A semicircle with diameter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}$ at $E$. What is the length of $\overline{CE}$?
Circles $A$, $B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius $1$ and passes through the center of $D$. What is the radius of circle $B$?
Three pairwise-tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
Show that the three medians of a triangle intersect at one point.
In $\triangle{ABC}$, let $AD$, $BE$, and $CF$ be the three altitudes as shown. If $AB=6$, $BC=5$, and $EF=3$, what is the length of $BE$?
Let $\triangle{ABC}$ be an acute triangle. If the distance between the vertex $A$ and the orthocenter $H$ is equal to the radius of its circumcircle, find the measurement of $\angle{A}$.
Let $AD$ be the altitude in $\triangle{ABC}$ from the vertex $A$. If $\angle{A}=45^\circ$, $BD=3$, $DC=2$, find the area of $\triangle{ABC}$.
Let $O$ be the centroid of $\triangle{ABC}$. Line $\mathcal{l}$ passes $O$ and intersects $AB$ and $AC$ at $P$ and $Q$, respectively. Point $D$, $E$, and $F$ are on the line $l$ such that $AD\perp l$, $BE \perp l$, and $CF\perp l$. Show that $AD = BE+CF$.
Let $O$ be the incenter of $\triangle{ABC}$. Connect $AO$, $BO$, and $CO$ and extends so that they intersect with $\triangle{ABC}$'s circumcircle at $D$, $E$, and $F$, respectively. Let $DE$ intersect $AC$ at $G$, and $DF$ intersects $AB$ at $H$. Show that $G$, $H$ and $O$ are collinear.
Let $ABC$ be an acute triangle. Circle $O$ passes its vertex $B$ and $C$, and intersects $AB$ and $AC$ at $D$ and $E$, respectively. If $O$'s radius equals the radius of $\triangle{ADE}$'s circumcircle, then the circle $O$ must passes $\triangle{ABC}$'s
(A) incenter (B) circumcenter (C) centroid (D) orthocenter.
Let $K$ is an arbitrary point inside $\triangle{ABC}$, and $D$, $E$, and $F$ be the centroids of $\triangle{ABK}$, $\triangle{BCK}$ and $\triangle{CAK}$, respectively. Find the value of $S_{\triangle{ABC}} : S_{\triangle{DEF}}$.
Let $I$ be the incenter of $\triangle{ABC}$. $AI$, $BI$, and $CI$ intersect $\triangle{ABC}$'s circumcircle at $D$, $E$, and $F$, respectively. Show that $EF \perp AD$
In $\triangle{ABC}$, $AB=AC$. Extending $CA$ to an arbitrary point $P$. Extending $AB$ to point $Q$ such that $AP=BQ$. Let $O$ be the circumcenter of $\triangle{ABC}$. Show that $A$, $P$, $Q$, and $O$ concyclic.
In triangle ABC, M is median of BC. O is incenter. AH is altitude. MO and AH intersect at E. Prove that AE equal to the radius of incircle
Let $G$ be the centroid of $\triangle{ABC}$, $L$ be a straight line. Prove that $$GG'=\frac{AA'+BB'+CC'}{3}$$ where $A'$, $B'$, $C'$ and $G'$ are the feet of perpendicular lines from $A$, $B$, $C$, and $G$ to $L$.
Let $A$, $B$, $C$ and $D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at the points $X$ and $Y$. The line $XY$ meets $BC$ at the point $Z$. Let $P$ be a point on the line $XY$ different from $Z$. The line $CP$ intersects the circle with diameter $AC$ at the points $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at the points $B$ and $N$. Prove that the lines $AM$, $DN$ and $XY$ are concurrent
(Euler Line) For any triangle $ABC$, show that the cicumcenter $O$, centroid $G$, and the orthocenter $H$ are collinear. Moreover, we have $OG:GH=1:2$.