Found problems: 11
2006 Oral Moscow Geometry Olympiad, 6
Given triangle $ABC$ and points $P$. Let $A_1,B_1,C_1$ be the second points of intersection of straight lines $AP, BP, CP$ with the circumscribed circle of $ABC$. Let points $A_2, B_2, C_2$ be symmetric to $A_1,B_1,C_1$ wrt $BC,CA,AB$, respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.
(A. Zaslavsky)
2006 Mathematics for Its Sake, 2
The cevians $ AP,BQ,CR $ of the triangle $ ABC $ are concurrent at $ F. $ Prove that the following affirmations are equivalent.
$ \text{(i)} \overrightarrow{AP} +\overrightarrow{BQ} +\overrightarrow{CR} =0 $
$ \text{(ii)} F$ is the centroid of $ ABC $
[i]Doru Isac[/i]
2003 Paraguay Mathematical Olympiad, 4
Triangle $ABC$ is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/9/2/2013de890e438f5bf88af446692b495917b1ff.png[/img]
Indonesia Regional MO OSP SMA - geometry, 2004.2
Triangle $ABC$ is given. The points $D, E$, and $F$ are located on the sides $BC, CA$ and $AB$ respectively so that the lines $AD, BE$ and $CF$ intersect at point $O$. Prove that $\frac{AO}{AD} + \frac{BO}{BE} + \frac{CO}{ CF}=2$
2020-21 KVS IOQM India, 18
Let $D,E,F$ be points on the sides $BC,CA,AB$ of a triangle $ABC$, respectively. Suppose $AD, BE,CF$ are concurrent at $P$. If $PF/PC =2/3, PE/PB = 2/7$ and $PD/PA = m/n$, where $m, n$ are positive integers with $gcd(m, n) = 1$, find $m + n$.
2021 Sharygin Geometry Olympiad, 9.1
Three cevians concur at a point lying inside a triangle. The feet of these cevians divide the sides into six segments, and the lengths of these segments form (in some order) a geometric progression. Prove that the lengths of the cevians also form a geometric progression.
2010 Saudi Arabia BMO TST, 2
Consider a triangle $ABC$ and a point $P$ in its interior. Lines $PA$, $PB$, $PC$ intersect $BC$, $CA$, $AB$ at $A', B', C'$ , respectively. Prove that $$\frac{BA'}{BC}+ \frac{CB'}{CA}+ \frac{AC'}{AB}= \frac32$$ if and only if at least two of the triangles $PAB$, $PBC$, $PCA$ have the same area.
Kyiv City MO Seniors 2003+ geometry, 2008.10.4
Given a triangle $ABC $, $A {{A} _ {1}} $, $B {{B} _ {1}} $, $C {{C} _ {1}}$ - its chevians intersecting at one point. ${{A} _ {0}}, {{C} _ {0}} $ - the midpoint of the sides $BC $ and $AB$ respectively. Lines ${{B} _ {1}} {{C} _ {1}} $, ${{B} _ {1}} {{A} _ {1}} $and ${ {B} _ {1}} B$ intersect the line ${{A} _ {0}} {{C} _ {0}} $ at points ${{C} _ {2}} $ , ${{A} _ {2}} $ and ${{B} _ {2}} $, respectively. Prove that the point ${{B} _ {2}} $ is the midpoint of the segment ${{A} _ {2}} {{C} _ {2}} $.
(Eugene Bilokopitov)
2025 Kosovo National Mathematical Olympiad`, P4
Let $ABC$ be a given triangle. Let $A_1$ and $A_2$ be points on the side $BC$. Let $B_1$ and $B_2$ be points on the side $CA$. Let $C_1$ and $C_2$ be points on the side $AB$. Suppose that the points $A_1,A_2,B_1,B_2,C_1$ and $C_2$ lie on a circle. Prove that the lines $AA_1, BB_1$ and $CC_1$ are concurrent if and only if $AA_2, BB_2$ and $CC_2$ are concurrent.
Estonia Open Senior - geometry, 2003.2.4
Consider the points $D, E$ and $F$ on the respective sides $BC, CA$ and $AB$ of the triangle $ABC$ in a way that the segments $AD, BE$ and $CF$ have a common point $P$. Let $\frac{|AP|}{|PD|}= x,$ $\frac{|BP|}{|PE|}= y$ and $\frac{|CP|}{|PF|}= z$. Prove that $xyz - (x + y + z) = 2$.
2020 Macedonia Additional BMO TST, 1
Let $P$ and $Q$ be interior points in $\Delta ABC$ such that $PQ$ doesn't contain any vertices of $\Delta ABC$.
Let $A_1$, $B_1$, and $C_1$ be the points of intersection of $BC$, $CA$, and $AB$ with $AQ$, $BQ$, and $CQ$, respectively.
Let $K$, $L$, and $M$ be the intersections of $AP$, $BP$, and $CP$ with $B_1C_1$, $C_1A_1$, and $A_1B_1$, respectively.
Prove that $A_1K$, $B_1L$, and $C_1M$ are concurrent.