Found problems: 1546
1988 IMO Longlists, 21
Let "AB" and $CD$ be two perpendicular chords of a circle with centre $O$ and radius $r$ and let $X,Y,Z,W$ denote the cyclical order of the four parts into which the disc is thus divided. Find the maximum and minimum of the quantity \[ \frac{A(X) + A(Z)}{A(Y) + A(W)}, \] where $A(U)$ denotes the area of $U.$
2013 India Regional Mathematical Olympiad, 1
Let $ABC$ be an acute-angled triangle. The circle $\Gamma$ with $BC$ as diameter intersects $AB$ and $AC$ again at $P$ and $Q$, respectively. Determine $\angle BAC$ given that the orthocenter of triangle $APQ$ lies on $\Gamma$.
2011 Sharygin Geometry Olympiad, 5
Given triangle $ABC$. The midperpendicular of side $AB$ meets one of the remaining sides at point $C'$. Points $A'$ and $B'$ are defined similarly. Find all triangles $ABC$ such that triangle $A'B'C'$ is regular.
2003 China Team Selection Test, 1
There are $n$($n\geq 3$) circles in the plane, all with radius $1$. In among any three circles, at least two have common point(s), then the total area covered by these $n$ circles is less than $35$.
2006 China Team Selection Test, 3
$\triangle{ABC}$ can cover a convex polygon $M$.Prove that there exsit a triangle which is congruent to $\triangle{ABC}$ such that it can also cover $M$ and has one side line paralel to or superpose one side line of $M$.
2007 All-Russian Olympiad, 6
A line, which passes through the incentre $I$ of the triangle $ABC$, meets its sides $AB$ and $BC$ at the points $M$ and $N$ respectively. The triangle $BMN$ is acute. The points $K,L$ are chosen on the side $AC$ such that $\angle ILA=\angle IMB$ and $\angle KC=\angle INB$. Prove that $AM+KL+CN=AC$.
[i]S. Berlov[/i]
2001 IberoAmerican, 2
The incircle of the triangle $\triangle{ABC}$ has center at $O$ and it is tangent to the sides $BC$, $AC$ and $AB$ at the points $X$, $Y$ and $Z$, respectively. The lines $BO$ and $CO$ intersect the line $YZ$ at the points $P$ and $Q$, respectively.
Show that if the segments $XP$ and $XQ$ has the same length, then the triangle $\triangle ABC$ is isosceles.
1988 IMO Longlists, 82
The triangle $ABC$ has a right angle at $C.$ The point $P$ is located on segment $AC$ such that triangles $PBA$ and $PBC$ have congruent inscribed circles. Express the length $x = PC$ in terms of $a = BC, b = CA$ and $c = AB.$
2003 China Team Selection Test, 1
Triangle $ABC$ is inscribed in circle $O$. Tangent $PD$ is drawn from $A$, $D$ is on ray $BC$, $P$ is on ray $DA$. Line $PU$ ($U \in BD$) intersects circle $O$ at $Q$, $T$, and intersect $AB$ and $AC$ at $R$ and $S$ respectively. Prove that if $QR=ST$, then $PQ=UT$.
2011 Uzbekistan National Olympiad, 2
Let triangle ABC with $ AB=c$ $AC=b$ $BC=a$ $R$ circumradius, $p$ half peremetr of $ABC$.
I f $\frac{acosA+bcosB+ccosC}{asinA+bsinB+csinC}=\frac{p}{9R}$ then find all value of $cosA$.
2006 China Western Mathematical Olympiad, 3
In $\triangle PBC$, $\angle PBC=60^{o}$, the tangent at point $P$ to the circumcircle$g$ of $\triangle PBC$ intersects with line $CB$ at $A$. Points $D$ and $E$ lie on the line segment $PA$ and $g$ respectively, satisfying $\angle DBE=90^{o}$, $PD=PE$. $BE$ and $PC$ meet at $F$. It is known that lines $AF,BP,CD$ are concurrent.
a) Prove that $BF$ bisect $\angle PBC$
b) Find $\tan \angle PCB$
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 $\angle PXI+\angle PYI=90^{\circ }.$
1979 IMO Longlists, 41
Prove the following statement: There does not exist a pyramid with square base and congruent lateral faces for which the measures of all edges, total area, and volume are integers.
2007 Canada National Olympiad, 2
You are given a pair of triangles for which two sides of one triangle are equal in length to two sides of the second triangle, and the triangles are similar, but not necessarily congruent. Prove that the ratio of the sides that correspond under the similarity is a number between $ \frac {1}{2}(\sqrt {5} \minus{} 1)$ and $ \frac {1}{2}(\sqrt {5} \plus{} 1)$.
2008 Korea - Final Round, 1
Hexagon $ABCDEF$ is inscribed in a circle $O$.
Let $BD \cap CF = G, AC \cap BE = H, AD \cap CE = I$
Following conditions are satisfied.
$BD \perp CF , CI=AI$
Prove that $CH=AH+DE$ is equivalent to $GH \times BD = BC \times DE$
2002 Bosnia Herzegovina Team Selection Test, 2
Triangle $ABC$ is given in a plane. Draw the bisectors of all three of its angles. Then draw the line that connects the points where the bisectors of angles $ABC$ and $ACB$ meet the opposite sides of the triangle. Through the point of intersection of this line and the bisector of angle $BAC$, draw another line parallel to $BC$. Let this line intersect $AB$ in $M$ and $AC$ in $N$. Prove that $2MN = BM+CN$.
1996 Irish Math Olympiad, 5
Show how to dissect a square into at most five pieces in such a way that the pieces can be reassembled to form three squares of (pairwise) distinct areas.
1983 IMO Longlists, 29
Let $O$ be a point outside a given circle. Two lines $OAB, OCD$ through $O$ meet the circle at $A,B,C,D$, where $A,C$ are the midpoints of $OB,OD$, respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$ and show that the tangents at $A,D$ to the circle meet on the line $BC.$
2003 China Team Selection Test, 1
There are $n$($n\geq 3$) circles in the plane, all with radius $1$. In among any three circles, at least two have common point(s), then the total area covered by these $n$ circles is less than $35$.
1976 IMO Longlists, 2
Let $P$ be a set of $n$ points and $S$ a set of $l$ segments. It is known that:
$(i)$ No four points of $P$ are coplanar.
$(ii)$ Any segment from $S$ has its endpoints at $P$.
$(iii)$ There is a point, say $g$, in $P$ that is the endpoint of a maximal number of segments from $S$ and that is not a vertex of a tetrahedron having all its edges in $S$.
Prove that $l \leq \frac{n^2}{3}$
2009 APMO, 3
Let three circles $ \Gamma_1, \Gamma_2, \Gamma_3$, which are non-overlapping and mutually external, be given in the plane. For each point $ P$ in the plane, outside the three circles, construct six points $ A_1, B_1, A_2, B_2, A_3, B_3$ as follows: For each $ i \equal{} 1, 2, 3$, $ A_i, B_i$ are distinct points on the circle $ \Gamma_i$ such that the lines $ PA_i$ and $ PB_i$ are both tangents to $ \Gamma_i$. Call the point $ P$ exceptional if, from the construction, three lines $ A_1B_1, A_2 B_2, A_3 B_3$ are concurrent. Show that every exceptional point of the plane, if exists, lies on the same circle.
2003 Rioplatense Mathematical Olympiad, Level 3, 1
Inside right angle $XAY$, where $A$ is the vertex, is a semicircle $\Gamma$ whose center lies on $AX$ and that is tangent to $AY$ at the point $A$. Describe a ruler-and-compass construction for the tangent to $\Gamma$ such that the triangle enclosed by the tangent and angle $XAY$ has minimum area.
2006 Czech-Polish-Slovak Match, 6
Find out if there is a convex pentagon $A_1A_2A_3A_4A_5$ such that, for each $i = 1, \dots , 5$, the lines $A_iA_{i+3}$ and $A_{i+1}A_{i+2}$ intersect at a point $B_i$ and the points $B_1,B_2,B_3,B_4,B_5$ are collinear. (Here $A_{i+5} = A_i$.)
1993 Taiwan National Olympiad, 2
Let $E$ and $F$ are distinct points on the diagonal $AC$ of a parallelogram $ABCD$ . Prove that , if there exists a cricle through $E,F$ tangent to rays $BA,BC$ then there also exists a cricle through $E,F$ tangent to rays $DA,DC$.
2012 Korea - Final Round, 1
Let $ABC$ be an acute triangle. Let $ H $ be the foot of perpendicular from $ A $ to $ BC $. $ D, E $ are the points on $ AB, AC $ and let $ F, G $ be the foot of perpendicular from $ D, E $ to $ BC $. Assume that $ DG \cap EF $ is on $ AH $. Let $ P $ be the foot of perpendicular from $ E $ to $ DH $. Prove that $ \angle APE = \angle CPE $.