Found problems: 1546
2008 Sharygin Geometry Olympiad, 1
(B.Frenkin) Does a convex quadrilateral without parallel sidelines exist such that it can be divided into four congruent triangles?
1979 USAMO, 4
Show how to construct a chord $BPC$ of a given angle $A$ through a given point $P$ such that $\tfrac{1}{BP}\plus{} \tfrac{1}{PC}$ is a maximum.
[asy]
size(200);
defaultpen(linewidth(0.7));
pair A = origin, B = (5,0), C = (4.2,3), P = waypoint(B--C,0.65);
pair Bp = 1.3 * B, Cp = 1.2 * C;
draw(A--B--C--A^^Bp--A--Cp);
dot(P);
label("$A$",A,W);
label("$B$",B,S);
label("$C$",C,dir(B--C));
label("$P$",P,dir(A--P));
[/asy]
2006 Moldova National Olympiad, 10.7
Consider an octogon with equal angles and rational side lengths. Prove that it has a symmetry center.
2002 Bosnia Herzegovina Team Selection Test, 2
The vertices of the convex quadrilateral $ABCD$ and the intersection point $S$ of its diagonals are integer points in the plane. Let $P$ be the area of $ABCD$ and $P_1$ the area of triangle $ABS$. Prove that
\[\sqrt{P} \ge \sqrt{P_1}+\frac{\sqrt2}2\]
2002 All-Russian Olympiad, 3
Let $A^\prime$ be the point of tangency of the excircle of a triangle $ABC$ (corrsponding to $A$) with the side $BC$. The line $a$ through $A^\prime$ is parallel to the bisector of $\angle BAC$. Lines $b$ and $c$ are analogously defined. Prove that $a, b, c$ have a common point.
2007 Moldova National Olympiad, 11.3
$ABCDA_{1}B_{1}C_{1}D_{1}$ is a cube with side length $4a$. Points $E$ and $F$ are taken on $(AA_{1})$ and $(BB_{1})$ such that $AE=B_{1}F=a$. $G$ and $H$ are midpoints of $(A_{1}B_{1})$ and $(C_{1}D_{1})$, respectively.
Find the minimum value of the $CP+PQ$, where $P\in[GH]$ and $Q\in[EF]$.
2009 Mediterranean Mathematics Olympiad, 2
Let $ABC$ be a triangle with $90^\circ \ne \angle A \ne 135^\circ$. Let $D$ and $E$ be external points to the triangle $ABC$ such that $DAB$ and $EAC$ are isoscele triangles with right angles at $D$ and $E$. Let $F = BE \cap CD$, and let $M$ and $N$ be the midpoints of $BC$ and $DE$, respectively.
Prove that, if three of the points $A$, $F$, $M$, $N$ are collinear, then all four are collinear.
2012 China Team Selection Test, 2
Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.
1983 Federal Competition For Advanced Students, P2, 3
Let $ P$ be a point in the plane of a triangle $ ABC$. Lines $ AP,BP,CP$ respectively meet lines $ BC,CA,AB$ at points $ A',B',C'$. Points $ A'',B'',C''$ are symmetric to $ A,B,C$ with respect to $ A',B',C',$ respectively. Show that: $ S_{A''B''C''}\equal{}3S_{ABC}\plus{}4S_{A'B'C'}$.
1988 IMO Longlists, 70
$ABC$ is a triangle, with inradius $r$ and circumradius $R.$ Show that: \[ \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{B}{2} \right) + \sin \left( \frac{B}{2} \right) \cdot \sin \left( \frac{C}{2} \right) + \sin \left( \frac{C}{2} \right) \cdot \sin \left( \frac{A}{2} \right) \leq \frac{5}{8} + \frac{r}{4 \cdot R}. \]
2010 Indonesia MO, 2
Given an acute triangle $ABC$ with $AC>BC$ and the circumcenter of triangle $ABC$ is $O$. The altitude of triangle $ABC$ from $C$ intersects $AB$ and the circumcircle at $D$ and $E$, respectively. A line which passed through $O$ which is parallel to $AB$ intersects $AC$ at $F$. Show that the line $CO$, the line which passed through $F$ and perpendicular to $AC$, and the line which passed through $E$ and parallel with $DO$ are concurrent.
[i]Fajar Yuliawan, Bandung[/i]
1989 China Team Selection Test, 1
A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area.
2008 Turkey Team Selection Test, 5
$ D$ is a point on the edge $ BC$ of triangle $ ABC$ such that $ AD\equal{}\frac{BD^2}{AB\plus{}AD}\equal{}\frac{CD^2}{AC\plus{}AD}$. $ E$ is a point such that $ D$ is on $ [AE]$ and $ CD\equal{}\frac{DE^2}{CD\plus{}CE}$. Prove that $ AE\equal{}AB\plus{}AC$.
1993 Baltic Way, 16
Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$, so that $|AB|=|BC|=|CD|=14\text{cm}$. Another line intersects the circles at $E,F$, respectively $G,H$ so that $|EF|=|FG|=|GH|=6\text{cm}$. Find the radius $r$.
2016 Bundeswettbewerb Mathematik, 2
A triangle $ABC$ with area $1$ is given. Anja and Bernd are playing the following game: Anja chooses a point $X$ on side $BC$. Then Bernd chooses a point $Y$ on side $CA$ und at last Anja chooses a point $Z$ on side $AB$. Also, $X,Y$ and $Z$ cannot be a vertex of triangle $ABC$. Anja wants to maximize the area of triangle $XYZ$ and Bernd wants to minimize that area.
What is the area of triangle $XYZ$ at the end of the game, if both play optimally?
2014 Indonesia MO Shortlist, G2
Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.
2005 MOP Homework, 5
Let $ABC$ be a triangle. Points $D$ and $E$ lie on sides $BC$ and $CA$, respectively, such that $BD=AE$. Segments $AD$ and $BE$ meet at $P$. The bisector of angle $BCA$ meet segments $AD$ and $BE$ at $Q$ and $R$, respectively. Prove that $\frac{PQ}{AD}=\frac{PR}{BE}$.
2007 Germany Team Selection Test, 3
Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove:
\[ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F
\]
When does equality occur?
2010 Postal Coaching, 5
A point $P$ lies on the internal angle bisector of $\angle BAC$ of a triangle $\triangle ABC$. Point $D$ is the midpoint of $BC$ and $PD$ meets the external angle bisector of $\angle BAC$ at point $E$. If $F$ is the point such that $PAEF$ is a rectangle then prove that $PF$ bisects $\angle BFC$ internally or externally.
2010 CHKMO, 3
Let $ \triangle ABC$ be a right-angled triangle with $ \angle C\equal{}90^\circ$. $ CD$ is the altitude from $ C$ to $ AB$, with $ D$ on $ AB$. $ \omega$ is the circumcircle of $ \triangle BCD$. $ \omega_1$ is a circle situated in $ \triangle ACD$, it is tangent to the segments $ AD$ and $ AC$ at $ M$ and $ N$ respectively, and is also tangent to circle $ \omega$.
(i) Show that $ BD\cdot CN\plus{}BC\cdot DM\equal{}CD\cdot BM$.
(ii) Show that $ BM\equal{}BC$.
2012 Mexico National Olympiad, 1
Let $\mathcal{C}_1$ be a circumference with center $O$, $P$ a point on it and $\ell$ the line tangent to $\mathcal{C}_1$ at $P$. Consider a point $Q$ on $\ell$ different from $P$, and let $\mathcal{C}_2$ be the circumference passing through $O$, $P$ and $Q$. Segment $OQ$ cuts $\mathcal{C}_1$ at $S$ and line $PS$ cuts $\mathcal{C}_2$ at a point $R$ diffferent from $P$. If $r_1$ and $r_2$ are the radii of $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively, Prove
\[\frac{PS}{SR} = \frac{r_1}{r_2}.\]
2007 Nordic, 2
Three given rectangles cover the sides of a triangle completely and each rectangle has a side parallel to a given line. Show that the rectangles also cover the interior of the triangle.
2010 Tournament Of Towns, 2
In a quadrilateral $ABCD$ with an incircle, $AB = CD; BC < AD$ and $BC$ is parallel to $AD$. Prove that the bisector of $\angle C$ bisects the area of $ABCD$.
2016 Indonesia TST, 3
Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$
2006 Estonia National Olympiad, 2
In a right triangle, the length of one side is a prime and the lengths of the other
side and the hypotenuse are integral. The ratio of the triangle perimeter and the incircle diameter is also an integer. Find all possible side lengths of the triangle.