Search found 15 matches
- Sat Dec 12, 2020 5:01 pm
- Forum: Geometry
- Topic: Iranian Geometry Olympiad 2020 (Elementary) P5
- Replies: 0
- Views: 6670
Iranian Geometry Olympiad 2020 (Elementary) P5
We say two vertices of a simple polygon are visible from each other if either they are adjacent, or the segment joining them is completely inside the polygon (except two endpoints that lie on the boundary). Find all positive integers $n$ such that there exists a simple polygon with $n$ vertices in w...
- Sat Dec 12, 2020 5:00 pm
- Forum: Geometry
- Topic: Iranian Geometry Olympiad 2020 (Elementary) P4
- Replies: 1
- Views: 5704
Iranian Geometry Olympiad 2020 (Elementary) P4
Let $P$ be an arbitrary point in the interior of triangle $ABC$. Lines $BP$ and $CP$ intersect $AC$ and $AB$ at $E$ and $F$, respectively. Let $K$ and $L$ be the midpoints of the segments $BF$ and $CE$, respectively. Let the lines through $L$ and $K$ parallel to $CF$ and $BE$ intersect $BC$ at $S$ a...
- Sat Dec 12, 2020 4:59 pm
- Forum: Geometry
- Topic: Iranian Geometry Olympiad 2020 (Elementary) P3
- Replies: 0
- Views: 6655
Iranian Geometry Olympiad 2020 (Elementary) P3
According to the figure, three equilateral triangles with side lengths $a$, $b$, $c$ have one common vertex and do not have any other common point. The lengths $x$, $y$ and $z$ are defined as in the figure. Prove that $3(x + y + z) \gt 2(a + b + c)$.
- Sat Dec 12, 2020 4:57 pm
- Forum: Geometry
- Topic: Iranian Geometry Olympiad 2020 (Elementary) P2
- Replies: 0
- Views: 6778
Iranian Geometry Olympiad 2020 (Elementary) P2
A parallelogram $ABCD$ is given $(AB \ne BC)$. Points $E$ and $G$ are chosen on the line $CD$ such that $AC$ is the angle bisector of both angles $\angle EAD$ and $\angle BAG$. The line $BC$ intersects $AE$ and $AG$ at $F$ and $H$, respectively. Prove that the line $FG$ passes through the midpoint o...
- Sat Dec 12, 2020 4:54 pm
- Forum: Geometry
- Topic: Iranian Geometry Olympiad 2020 (Elementary) P1
- Replies: 1
- Views: 5564
Iranian Geometry Olympiad 2020 (Elementary) P1
By a fold of a polygon-shaped paper, we mean drawing a segment on the paper and folding the paper along that. Suppose that a paper with the following figure is given. We cut the paper along the boundary of the shaded region to get a polygon-shaped paper. Start with this shaded polygon and make a rec...
- Sat Dec 12, 2020 4:45 pm
- Forum: Geometry
- Topic: Iranian Geometry Olympiad 2020 (Advanced) P5
- Replies: 0
- Views: 6668
Iranian Geometry Olympiad 2020 (Advanced) P5
Consider an acute-angled triangle $ABC (AC \gt AB)$ with its orthocenter $H$ and circumcircle $\Gamma$. Points $M$ and $P$ are the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ meets $\Gamma$ again at $X$ and point $N$ lies on the line $BC$ so that $NX$ is tangent to $\Gamma$....
- Sat Dec 12, 2020 4:35 pm
- Forum: Geometry
- Topic: Iranian Geometry Olympiad 2020 (Advanced) P4
- Replies: 0
- Views: 6495
Iranian Geometry Olympiad 2020 (Advanced) P4
Convex circumscribed quadrilateral $ABCD$ with incenter $I$ is given such that its incircle is tangent to $AD$, $DC$, $CB$, and $BA$ at $K$, $L$, $M$, and $N$. Lines $AD$ and $BC$ meet at $E$ and lines $AB$ and $CD$ meet at $F$. Let $KM$ intersects $AB$ and $CD$ at $X$ and $Y$ , respectively. Let $L...
- Sat Dec 12, 2020 4:29 pm
- Forum: Geometry
- Topic: Iranian Geometry Olympiad 2020 (Advanced) P3
- Replies: 0
- Views: 6224
Iranian Geometry Olympiad 2020 (Advanced) P3
Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most $2\sqrt{2}$ times the sum of their radii(radius). (A line separate...
- Sat Dec 12, 2020 4:24 pm
- Forum: Geometry
- Topic: Iranian Geometry Olympiad 2020 (Advanced) P2
- Replies: 0
- Views: 6145
Iranian Geometry Olympiad 2020 (Advanced) P2
Let $ABC$ be an acute-angled triangle with its incenter $I$. Suppose that $N$ is the midpoint of the arc $BAC$ of the circumcircle of triangle $ABC$, and $P$ is a point such that $ABPC$ is a parallelogram. Let $Q$ be the reflection of $A$ over $N$, and $R$ the projection of $A$ on $QI$. Show that th...
- Sat Dec 12, 2020 4:22 pm
- Forum: Geometry
- Topic: Iranian Geometry Olympiad 2020 (Advanced) P1
- Replies: 0
- Views: 6253
Iranian Geometry Olympiad 2020 (Advanced) P1
Let $M$, $N$, and $P$ be the midpoints of sides $BC$, $AC$, and $AB$ of triangle $ABC$, respectively. $E$ and $F$ are two points on the segment $BC$ so that $\angle NEC =\frac{1}{2}\angle AMB$ and $\angle PFB = \frac{1}{2}\angle AMC$. Prove that $AE = AF$.