Found problems: 1546
Indonesia MO Shortlist - geometry, g2.3
For every triangle $ ABC$, let $ D,E,F$ be a point located on segment $ BC,CA,AB$, respectively. Let $ P$ be the intersection of $ AD$ and $ EF$. Prove that:
\[ \frac{AB}{AF}\times DC\plus{}\frac{AC}{AE}\times DB\equal{}\frac{AD}{AP}\times BC\]
2007 Germany Team Selection Test, 3
A point $ P$ in the interior of triangle $ ABC$ satisfies
\[ \angle BPC \minus{} \angle BAC \equal{} \angle CPA \minus{} \angle CBA \equal{} \angle APB \minus{} \angle ACB.\]
Prove that \[ \bar{PA} \cdot \bar{BC} \equal{} \bar{PB} \cdot \bar{AC} \equal{} \bar{PC} \cdot \bar{AB}.\]
2007 Cono Sur Olympiad, 2
Let $ABCDE$ be a convex pentagon that satisfies all of the following:[list]
[*]There is a circle $\Gamma$ tangent to each of the sides.
[*]The lengths of the sides are all positive integers.
[*]At least one of the sides of the pentagon has length $1$.
[*]The side $AB$ has length $2$.[/list]
Let $P$ be the point of tangency of $\Gamma$ with $AB$.[list]
(a) Determine the lengths of the segments $AP$ and $BP$.
(b) Give an example of a pentagon satisfying the given conditions.[/list]
2010 Contests, 2
Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively.
[b]a)[/b] Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$.
[b]b)[/b] Prove that $S$ lies on a fix line.
1989 IMO Longlists, 30
Let $ ABC$ be an equilateral triangle. Let $ D,E, F,M,N,$ and $ P$ be the mid-points of $ BC, CA, AB, FD, FB,$ and $ DC$ respectively.
[b](a)[/b] Show that the line segments $ AM,EN,$ and $ FP$ are concurrent.
[b](b)[/b] Let $ O$ be the point of intersection of $ AM,EN,$ and $ FP.$ Find $ OM : OF : ON : OE : OP : OA.$
1999 APMO, 5
Let $S$ be a set of $2n+1$ points in the plane such that no three are collinear and no four concyclic. A circle will be called $\text{Good}$ if it has 3 points of $S$ on its circumference, $n-1$ points in its interior and $n-1$ points in its exterior.
Prove that the number of good circles has the same parity as $n$.
2011 Czech and Slovak Olympiad III A, 5
In acute triangle ABC, which is not equilateral, let $P$ denote the foot of the altitude from $C$ to side $AB$; let $H$ denote the orthocenter; let $O$ denote the circumcenter; let $D$ denote the intersection of line $CO$ with $AB$; and let $E$ denote the midpoint of $CD$. Prove that line $EP$ passes through the midpoint of $OH$.
2014 Bosnia Herzegovina Team Selection Test, 1
Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$
1970 IMO Longlists, 13
Each side of an arbitrary $\triangle ABC$ is divided into equal parts, and lines parallel to $AB,BC,CA$ are drawn through each of these points, thus cutting $\triangle ABC$ into small triangles. Points are assigned a number in the following manner:
$(1)$ $A,B,C$ are assigned $1,2,3$ respectively
$(2)$ Points on $AB$ are assigned $1$ or $2$
$(3)$ Points on $BC$ are assigned $2$ or $3$
$(4)$ Points on $CA$ are assigned $3$ or $1$
Prove that there must exist a small triangle whose vertices are marked by $1,2,3$.
1998 IberoAmerican Olympiad For University Students, 4
Four circles of radius $1$ with centers $A,B,C,D$ are in the plane in such a way that each circle is tangent to two others. A fifth circle passes through the center of two of the circles and is tangent to the other two.
Find the possible values of the area of the quadrilateral $ABCD$.
2007 Germany Team Selection Test, 3
A point $ P$ in the interior of triangle $ ABC$ satisfies
\[ \angle BPC \minus{} \angle BAC \equal{} \angle CPA \minus{} \angle CBA \equal{} \angle APB \minus{} \angle ACB.\]
Prove that \[ \bar{PA} \cdot \bar{BC} \equal{} \bar{PB} \cdot \bar{AC} \equal{} \bar{PC} \cdot \bar{AB}.\]
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.
2010 Contests, 1
The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$
2013 Dutch IMO TST, 2
Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.
2001 India National Olympiad, 5
$ABC$ is a triangle. $M$ is the midpoint of $BC$. $\angle MAB = \angle C$, and $\angle MAC = 15^{\circ}$. Show that $\angle AMC$ is obtuse. If $O$ is the circumcenter of $ADC$, show that $AOD$ is equilateral.
2010 Sharygin Geometry Olympiad, 10
Let three lines forming a triangle $ABC$ be given. Using a two-sided ruler and drawing at most eight lines construct a point $D$ on the side $AB$ such that $\frac{AD}{BD}=\frac{BC}{AC}.$
1985 IMO Longlists, 27
Let $O$ be a point on the oriented Euclidean plane and $(\mathbf i, \mathbf j)$ a directly oriented orthonormal basis. Let $C$ be the circle of radius $1$, centered at $O$. For every real number $t$ and non-negative integer$ n$ let $M_n$ be the point on $C$ for which $\langle \mathbf i , \overrightarrow{OM_n} \rangle = \cos 2^n t.$ (or $\overrightarrow{OM_n} =\cos 2^n t \mathbf i +\sin 2^n t \mathbf j$).
Let $k \geq 2$ be an integer. Find all real numbers $t \in [0, 2\pi)$ that satisfy
[b](i)[/b] $M_0 = M_k$, and
[b](ii)[/b] if one starts from $M0$ and goes once around $C$ in the positive direction, one meets successively the points $M_0,M_1, \dots,M_{k-2},M_{k-1}$, in this order.
1981 USAMO, 1
The measure of a given angle is $\frac{180^{\circ}}{n}$ where $n$ is a positive integer not divisible by $3$. Prove that the angle can be trisected by Euclidean means (straightedge and compasses).
Estonia Open Senior - geometry, 2007.1.2
Three circles with centres A, B, C touch each other pairwise externally, and touch circle c from inside. Prove that if the centre of c coincideswith the orthocentre of triangle ABC, then ABC is equilateral.
2012 Baltic Way, 12
Let $P_0$, $P_1$, $\dots$, $P_8 = P_0$ be successive points on a circle and $Q$ be a point inside the polygon $P_0 P_1 \dotsb P_7$ such that $\angle P_{i - 1} QP_i = 45^\circ$ for $i = 1$, $\dots$, 8. Prove that the sum
\[\sum_{i = 1}^8 P_{i - 1} P_i^2\]
is minimal if and only if $Q$ is the centre of the circle.
2010 Slovenia National Olympiad, 5
For what positive integers $n \geq 3$ does there exist a polygon with $n$ vertices (not necessarily convex) with property that each of its sides is parallel to another one of its sides?
1991 IMTS, 2
Show that every triangle can be dissected into nine convex nondegenrate pentagons.
2008 Vietnam National Olympiad, 2
Given a triangle with acute angle $ \angle BEC,$ let $ E$ be the midpoint of $ AB.$ Point $ M$ is chosen on the opposite ray of $ EC$ such that $ \angle BME \equal{} \angle ECA.$ Denote by $ \theta$ the measure of angle $ \angle BEC.$ Evaluate $ \frac{MC}{AB}$ in terms of $ \theta.$
1999 Mongolian Mathematical Olympiad, Problem 1
In a convex quadrilateral $ABCD$, ${\angle}ABD=65^\circ$,${\angle}CBD=35^\circ$, ${\angle}ADC=130^\circ$ and $BC=AB$.Find the angles of $ABCD$.
2000 France Team Selection Test, 2
$A,B,C,D$ are points on a circle in that order. Prove that $|AB-CD|+|AD-BC| \ge 2|AC-BD|$.