Found problems: 2023
2014 Contests, 3
Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.
1995 Canada National Olympiad, 3
Define a boomerang as a quadrilateral whose opposite sides do not intersect and one of whose internal angles is greater than $180^{\circ}$. Let $C$ be a convex polygon with $s$ sides. The interior region of $C$ is the union of $q$ quadrilaterals, none of whose interiors overlap each other. $b$ of these quadrilaterals are boomerangs. Show that $q\ge b+\frac{s-2}{2}$.
2007 QEDMO 4th, 10
Let $ ABC$ be a triangle.
The $ A$-excircle of triangle $ ABC$ has center $ O_{a}$ and touches the side $ BC$ at the point $ A_{a}$.
The $ B$-excircle of triangle $ ABC$ touches its sidelines $ AB$ and $ BC$ at the points $ C_{b}$ and $ A_{b}$.
The $ C$-excircle of triangle $ ABC$ touches its sidelines $ BC$ and $ CA$ at the points $ A_{c}$ and $ B_{c}$.
The lines $ C_{b}A_{b}$ and $ A_{c}B_{c}$ intersect each other at some point $ X$.
Prove that the quadrilateral $ AO_{a}A_{a}X$ is a parallelogram.
[i]Remark.[/i] The $ A$[i]-excircle[/i] of a triangle $ ABC$ is defined as the circle which touches the segment $ BC$ and the extensions of the segments $ CA$ and $ AB$ beyound the points $ C$ and $ B$, respectively. The center of this circle is the point of intersection of the interior angle bisector of the angle $ CAB$ and the exterior angle bisectors of the angles $ ABC$ and $ BCA$.
Similarly, the $ B$-excircle and the $ C$-excircle of triangle $ ABC$ are defined.
[hide="Source of the problem"][i]Source of the problem:[/i] Theorem (88) in: John Sturgeon Mackay, [i]The Triangle and its Six Scribed Circles[/i], Proceedings of the Edinburgh Mathematical Society 1 (1883), pages 4-128 and drawings at the end of the volume.[/hide]
2010 ELMO Shortlist, 6
Let $ABC$ be a triangle with circumcircle $\Omega$. $X$ and $Y$ are points on $\Omega$ such that $XY$ meets $AB$ and $AC$ at $D$ and $E$, respectively. Show that the midpoints of $XY$, $BE$, $CD$, and $DE$ are concyclic.
[i]Carl Lian.[/i]
1992 Baltic Way, 20
Let $ a\le b\le c$ be the sides of a right triangle, and let $ 2p$ be its perimeter. Show that
\[ p(p \minus{} c) \equal{} (p \minus{} a)(p \minus{} b) \equal{} S,
\]
where $ S$ is the area of the triangle.
2005 ITAMO, 1
Let $ABC$ be a right angled triangle with hypotenuse $AC$, and let $H$ be the foot of the altitude from $B$ to $AC$. Knowing that there is a right-angled triangle with side-lengths $AB, BC, BH$, determine all the possible values ​​of $\frac{AH}{CH}$
2004 Turkey MO (2nd round), 5
The excircle of a triangle $ABC$ corresponding to $A$ touches the lines $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. The excircle corresponding to $B$ touches $BC,CA,AB$ at $A_2,B_2,C_2$, and the excircle corresponding to $C$ touches $BC,CA,AB$ at $A_3,B_3,C_3$, respectively. Find the maximum possible value of the ratio of the sum of the perimeters of $\triangle A_1B_1C_1$, $\triangle A_2B_2C_2$ and $\triangle A_3B_3C_3$ to the circumradius of $\triangle ABC$.
2024 ITAMO, 2
We are given a unit square in the plane. A point $M$ in the plane is called [i]median [/i]if there exists points $P$ and $Q$ on the boundary of the square such that $PQ$ has length one and $M$ is the midpoint of $PQ$.
Determine the geometric locus of all median points.
2007 Junior Balkan Team Selection Tests - Romania, 2
Consider a convex quadrilateral $ABCD$. Denote $M, \ N$ the points of tangency of the circle inscribed in $\triangle ABD$ with $AB, \ AD$, respectively and $P, \ Q$ the points of tangency of the circle inscribed in $\triangle CBD$ with the sides $CD, \ CB$, respectively. Assume that the circles inscribed in $\triangle ABD, \ \triangle CBD$ are tangent. Prove that:
a) $ABCD$ is circumscriptible.
b) $MNPQ$ is cyclic.
c) The circles inscribed in $\triangle ABC, \ \triangle ADC$ are tangent.
2007 All-Russian Olympiad Regional Round, 8.1
In a convex quadrilateral. eight segments are drawn, each of them connects a vertex with the midpoint of some opposite side. Seven of these segments have the same length $ a$. Prove that the eight one is also of length $ a$.
1987 IMO Longlists, 57
The bisectors of the angles $B,C$ of a triangle $ABC$ intersect the opposite sides in $B', C'$ respectively. Prove that the straight line $B'C'$ intersects the inscribed circle in two different points.
2010 Contests, 1
A circle that passes through the vertex $A$ of a rectangle $ABCD$ intersects the side $AB$ at a second point $E$ different from $B.$ A line passing through $B$ is tangent to this circle at a point $T,$ and the circle with center $B$ and passing through $T$ intersects the side $BC$ at the point $F.$ Show that if $\angle CDF= \angle BFE,$ then $\angle EDF=\angle CDF.$
2004 Germany Team Selection Test, 2
Let two chords $AC$ and $BD$ of a circle $k$ meet at the point $K$, and let $O$ be the center of $k$. Let $M$ and $N$ be the circumcenters of triangles $AKB$ and $CKD$. Show that the quadrilateral $OMKN$ is a parallelogram.
2007 ISI B.Stat Entrance Exam, 4
Show that it is not possible to have a triangle with sides $a,b,$ and $c$ whose medians have length $\frac{2}{3}a, \frac{2}{3}b$ and $\frac{4}{5}c$.
2007 Bulgaria Team Selection Test, 1
Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$
2003 Iran MO (3rd Round), 24
$ A,B$ are fixed points. Variable line $ l$ passes through the fixed point $ C$. There are two circles passing through $ A,B$ and tangent to $ l$ at $ M,N$. Prove that circumcircle of $ AMN$ passes through a fixed point.
2010 China Western Mathematical Olympiad, 2
$AB$ is a diameter of a circle with center $O$. Let $C$ and $D$ be two different points on the circle on the same side of $AB$, and the lines tangent to the circle at points $C$ and $D$ meet at $E$. Segments $AD$ and $BC$ meet at $F$. Lines $EF$ and $AB$ meet at $M$. Prove that $E,C,M$ and $D$ are concyclic.
2009 International Zhautykov Olympiad, 3
For a convex hexagon $ ABCDEF$ with an area $ S$, prove that:
\[ AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S
\]
2007 China Team Selection Test, 1
When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.
2012 Mediterranean Mathematics Olympiad, 4
Let $O$ be the circumcenter,$R$ be the circumradius, and $k$ be the circumcircle of a triangle $ABC$ .
Let $k_1$ be a circle tangent to the rays $AB$ and $AC$, and also internally tangent to $k$.
Let $k_2$ be a circle tangent to the rays $AB$ and $AC$ , and also externally tangent to $k$. Let $A_1$ and $A_2$ denote the respective centers of $k_1$ and $k_2$.
Prove that:
$(OA_1+OA_2)^2-A_1A_2^2 = 4R^2.$
2011 Balkan MO Shortlist, G1
Let $ABCD$ be a convex quadrangle such that $AB=AC=BD$ (vertices are labelled in circular order). The lines $AC$ and $BD$ meet at point $O$, the circles $ABC$ and $ADO$ meet again at point $P$, and the lines $AP$ and $BC$ meet at the point $Q$. Show that the angles $COQ$ and $DOQ$ are equal.
2012 Serbia Team Selection Test, 3
Let $P$ and $Q$ be points inside triangle $ABC$ satisfying $\angle PAC=\angle QAB$ and $\angle PBC=\angle QBA$.
a) Prove that feet of perpendiculars from $P$ and $Q$ on the sides of triangle $ABC$ are concyclic.
b) Let $D$ and $E$ be feet of perpendiculars from $P$ on the lines $BC$ and $AC$ and $F$ foot of perpendicular from $Q$ on $AB$. Let $M$ be intersection point of $DE$ and $AB$. Prove that $MP\bot CF$.
1983 IMO Longlists, 42
Consider the square $ABCD$ in which a segment is drawn between each vertex and the midpoints of both opposite sides. Find the ratio of the area of the octagon determined by these segments and the area of the square $ABCD.$
2024 Dutch IMO TST, 2
Let $ABC$ be a triangle. A point $P$ lies on the segment $BC$ such that the circle with diameter $BP$ passes through the incenter of $ABC$. Show that $\frac{BP}{PC}=\frac{c}{s-c}$ where $c$ is the length of segment $AB$ and $2s$ is the perimeter of $ABC$.
2001 JBMO ShortLists, 10
A triangle $ABC$ is inscribed in the circle $\mathcal{C}(O,R)$. Let $\alpha <1$ be the ratio of the radii of the circles tangent to $\mathcal{C}$, and both of the rays $(AB$ and $(AC$. The numbers $\beta <1$ and $\gamma <1$ are defined analogously. Prove that $\alpha + \beta + \gamma =1$.