Found problems: 77
1986 IMO Shortlist, 17
Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.
1970 Bulgaria National Olympiad, Problem 4
Let $\delta_0=\triangle A_0B_0C_0$ be a triangle. On each of the sides $B_0C_0$, $C_0A_0$, $A_0B_0$, there are constructed squares in the halfplane, not containing the respective vertex $A_0,B_0,C_0$ and $A_1,B_1,C_1$ are the centers of the constructed squares. If we use the triangle $\delta_1=\triangle A_1B_1C_1$ in the same way we may construct the triangle $\delta_2=\triangle A_2B_2C_2$; from $\delta_2=\triangle A_2B_2C_2$ we may construct $\delta_3=\triangle A_3B_3C_3$ and etc. Prove that:
(a) segments $A_0A_1,B_0B_1,C_0C_1$ are respectively equal and perpendicular to $B_1C_1,C_1A_1,A_1B_1$;
(b) vertices $A_1,B_1,C_1$ of the triangle $\delta_1$ lies respectively over the segments $A_0A_3,B_0B_3,C_0C_3$ (defined by the vertices of $\delta_0$ and $\delta_1$) and divide them in ratio $2:1$.
[i]K. Dochev[/i]
1969 Swedish Mathematical Competition, 6
Given $3n$ points in the plane, no three collinear, is it always possible to form $n$ triangles (with vertices at the points), so that no point in the plane lies in more than one triangle?
1986 IMO, 2
Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.
1989 Bulgaria National Olympiad, Problem 1
In triangle $ABC$, point $O$ is the center of the excircle touching the side $BC$, while the other two excircles touch the sides $AB$ and $AC$ at points $M$ and $N$ respectively. A line through $O$ perpendicular to $MN$ intersects the line $BC$ at $P$. Determine the ratio $AB/AC$, given that the ratio of the area of $\triangle ABC$ to the area of $\triangle MNP$ is $2R/r$, where $R$ is the circumradius and $r$ the inradius of $\triangle ABC$.
1971 Bulgaria National Olympiad, Problem 4
It is given a triangle $ABC$. Let $R$ be the radius of the circumcircle of the triangle and $O_1,O_2,O_3$ be the centers of excircles of the triangle $ABC$ and $q$ is the perimeter of the triangle $O_1O_2O_3$. Prove that $q\le6R\sqrt3$. When does equality hold?
1975 Czech and Slovak Olympiad III A, 5
Let a square $\mathbf P=P_1P_2P_3P_4$ be given in the plane. Determine the locus of all vertices $A$ of isosceles triangles $ABC,AB=BC$ such that the vertices $B,C$ are points of the square $\mathbf P.$
1997 Brazil Team Selection Test, Problem 1
In an isosceles triangle $ABC~(AC=BC)$, let $O$ be its circumcenter, $D$ the midpoint of $AC$ and $E$ the centroid of $DBC$. Show that $OE$ is perpendicular to $BD$.
2000 Croatia National Olympiad, Problem 2
Two squares $ACXE$ and $CBDY$ are constructed in the exterior of an acute-angled triangle $ABC$. Prove that the intersection of the lines $AD$ and $BE$ lies on the altitude of the triangle from $C$.
1986 IMO Longlists, 14
Given a point $P_0$ in the plane of the triangle $A_1A_2A_3$. Define $A_s=A_{s-3}$ for all $s\ge4$. Construct a set of points $P_1,P_2,P_3,\ldots$ such that $P_{k+1}$ is the image of $P_k$ under a rotation center $A_{k+1}$ through an angle $120^o$ clockwise for $k=0,1,2,\ldots$. Prove that if $P_{1986}=P_0$, then the triangle $A_1A_2A_3$ is equilateral.
1953 Moscow Mathematical Olympiad, 247
Inside a convex $1000$-gon, $500$ points are selected so that no three of the $1500$ points — the ones selected and the vertices of the polygon — lie on the same straight line. This $1000$-gon is then divided into triangles so that all $1500$ points are vertices of the triangles, and so that these triangles have no other vertices.
How many triangles will there be?
1997 Croatia National Olympiad, Problem 4
On the sides of a triangle $ABC$ are constructed similar triangles $ABD,BCE,CAF$ with $k=AD/DB=BE/EC=CF/FA$ and $\alpha=\angle ADB=\angle BEC=\angle CFA$. Prove that the midpoints of the segments $AC,BC,CD$ and $EF$ form a parallelogram with an angle $\alpha$ and two sides whose ratio is $k$.
2009 Sharygin Geometry Olympiad, 4
Given regular $17$-gon $A_1 ... A_{17}$. Prove that two triangles formed by lines $A_1A_4, A_2A_{10}, A_{13}A_{14}$ and $A_2A_3, A_4A_6 A_{14}A_{15} $ are equal.
(N.Beluhov)
2006 VTRMC, Problem 6
In the diagram below, $BP$ bisects $\angle ABC$, $CP$ bisects $\angle BCA$, and $PQ$ is perpendicular to $BC$. If $BQ\cdot QC=2PQ^2$, prove that $AB+AC=3BC$.
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOC8zL2IwZjNjMDAxNWEwMTc1ZGNjMTkwZmZlZmJlMGRlOGRhYjk4NzczLnBuZw==&rn=VlRSTUMgMjAwNi5wbmc=[/img]
1997 Brazil Team Selection Test, Problem 1
In an isosceles triangle $ABC~(AC=BC)$, let $O$ be its circumcenter, $D$ the midpoint of $AC$ and $E$ the centroid of $DBC$. Show that $OE$ is perpendicular to $BD$.
1965 Bulgaria National Olympiad, Problem 3
In the triangle $ABC$, angle bisector $CD$ intersects the circumcircle of $ABC$ at the point $K$.
(a) Prove the equalities:
$$\frac1{ID}-\frac1{IK}=\frac1{CI},\enspace\frac{CI}{ID}-\frac{ID}{DK}=1$$where $I$ is the center of the inscribed circle of triangle $ABC$.
(b) On the segment $CK$ some point $P$ is chosen whose projections on $AC,BC,AB$ respectively are $P_1,P_2,P_3$. The lines $PP_3$ and $P_1P_2$ intersect at a point $M$. Find the locus of $M$ when $P$ moves around segment $CK$.
2023 Romanian Master of Mathematics Shortlist, C1
Determine all integers $n \geq 3$ for which there exists a con guration of $n$ points in the plane, no three collinear, that can be labelled $1$ through $n$ in two different ways, so that the following
condition be satis fied: For every triple $(i,j,k), 1 \leq i < j < k \leq n$, the triangle $ijk$ in one labelling has the same orientation as the triangle labelled $ijk$ in the other, except for $(i,j,k) = (1,2,3)$.
2013 Singapore Senior Math Olympiad, 1
In the Triangle ABC AB>AC, the extension of the altitude AD with D lying inside BC intersects the circum-circle of the Triangle ABC at P. The circle through P and tangent to BC at D intersects the circum-circle of Triangle ABC at Q distinct from P with PQ=DQ. Prove that AD=BD-DC
2016 AMC 12/AHSME, 8
What is the area of the shaded region of the given $8 \times 5$ rectangle?
[asy]
size(6cm);
defaultpen(fontsize(9pt));
draw((0,0)--(8,0)--(8,5)--(0,5)--cycle);
filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8));
label("$1$",(1/2,5),dir(90));
label("$7$",(9/2,5),dir(90));
label("$1$",(8,1/2),dir(0));
label("$4$",(8,3),dir(0));
label("$1$",(15/2,0),dir(270));
label("$7$",(7/2,0),dir(270));
label("$1$",(0,9/2),dir(180));
label("$4$",(0,2),dir(180));
[/asy]
$\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf{(D)}\ 6\dfrac{1}{2} \qquad \textbf{(E)}\ 8$
1990 French Mathematical Olympiad, Problem 5
In a triangle $ABC$, $\Gamma$ denotes the excircle corresponding to $A$, $A',B',C'$ are the points of tangency of $\Gamma$ with $BC,CA,AB$ respectively, and $S(ABC)$ denotes the region of the plane determined by segments $AB',AC'$ and the arc $C'A'B'$ of $\Gamma$.
Prove that there is a triangle $ABC$ of a given perimeter $p$ for which the area of $S(ABC)$ is maximal. For this triangle, give an approximate measure of the angle at $A$.
1990 Bulgaria National Olympiad, Problem 5
Given a circular arc, find a triangle of the smallest possible area which covers the arc so that the endpoints of the arc lie on the same side of the triangle.
2015 China Team Selection Test, 4
Prove that : For each integer $n \ge 3$, there exists the positive integers $a_1<a_2< \cdots <a_n$ , such that for $ i=1,2,\cdots,n-2 $ , With $a_{i},a_{i+1},a_{i+2}$ may be formed as a triangle side length , and the area of the triangle is a positive integer.
2003 Spain Mathematical Olympiad, Problem 3
The altitudes of the triangle ${ABC}$ meet in the point ${H}$. You know that ${AB = CH}$. Determine the value of the angle $\widehat{BCA}$.
2021 ISI Entrance Examination, 6
If a given equilateral triangle $\Delta$ of side length $a$ lies in the union of five equilateral triangles of side length $b$, show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$.
1969 Bulgaria National Olympiad, Problem 4
Find the sides of a triangle if it is known that the inscribed circle meets one of its medians in two points and these points divide the median into three equal segments and the area of the triangle is equal to $6\sqrt{14}\text{ cm}^2$.