Found problems: 2023
2012 Pan African, 1
$AB$ is a chord (not a diameter) of a circle with centre $O$. Let $T$ be a point on segment $OB$. The line through $T$ perpendicular to $OB$ meets $AB$ at $C$ and the circle at $D$ and $E$. Denote by $S$ the orthogonal projection of $T$ onto $AB$ .
Prove that $AS \cdot BC = TE \cdot TD$.
2006 JBMO ShortLists, 10
Let $ ABCD$ be a trapezoid inscribed in a circle $ \mathcal{C}$ with $ AB\parallel CD$, $ AB\equal{}2CD$. Let $ \{Q\}\equal{}AD\cap BC$ and let $ P$ be the intersection of tangents to $ \mathcal{C}$ at $ B$ and $ D$. Calculate the area of the quadrilateral $ ABPQ$ in terms of the area of the triangle $ PDQ$.
2005 Baltic Way, 13
What the smallest number of circles of radius $\sqrt{2}$ that are needed to cover a rectangle
$(a)$ of size $6\times 3$?
$(b)$ of size $5\times 3$?
2006 Baltic Way, 14
There are $2006$ points marked on the surface of a sphere. Prove that the surface can be cut into $2006$ congruent pieces so that each piece contains exactly one of these points inside it.
1999 Mongolian Mathematical Olympiad, Problem 3
I couldn't solve this problem and the only solution I was able to find was very unnatural (it was an official solution, I think) and I couldn't be satisfied with it, so I ask you if you can find some different solutions. The problem is really great one!
If $M$ is the centroid of a triangle $ABC$, prove that the following inequality holds: \[\sin\angle CAM+\sin\angle CBM\leq\frac{2}{\sqrt3}.\] The equality occurs in a very strange case, I don't remember it.
1996 Greece National Olympiad, 2
Let $ ABC$ be an acute triangle, $ AD,BE,CZ$ its altitudes and $ H$ its orthocenter. Let $ AI,A \Theta$ be the internal and external bisectors of angle $ A$. Let $ M,N$ be the midpoints of $ BC,AH$, respectively. Prove that:
(a) $MN$ is perpendicular to $EZ$
(b) if $ MN$ cuts the segments $ AI,A \Theta$ at the points $ K,L$, then $ KL\equal{}AH$
2012 Indonesia TST, 2
Let $\omega$ be a circle with center $O$, and let $l$ be a line not intersecting $\omega$. $E$ is a point on $l$ such that $OE$ is perpendicular with $l$. Let $M$ be an arbitrary point on $M$ different from $E$. Let $A$ and $B$ be distinct points on the circle $\omega$ such that $MA$ and $MB$ are tangents to $\omega$. Let $C$ and $D$ be the foot of perpendiculars from $E$ to $MA$ and $MB$ respectively. Let $F$ be the intersection of $CD$ and $OE$. As $M$ moves, determine the locus of $F$.
2001 Irish Math Olympiad, 2
Let $ ABC$ be a triangle with sides $ BC\equal{}a, CA\equal{}b,AB\equal{}c$ and let $ D$ and $ E$ be the midpoints of $ AC$ and $ AB$, respectively. Prove that the medians $ BD$ and $ CE$ are perpendicular to each other if and only if $ b^2\plus{}c^2\equal{}5a^2$.
2008 Sharygin Geometry Olympiad, 3
(A.Zaslavsky, 8) A triangle can be dissected into three equal triangles. Prove that some its angle is equal to $ 60^{\circ}$.
2004 Romania National Olympiad, 4
Let $\displaystyle \left( P_n \right)_{n \geq 1}$ be an infinite family of planes and $\displaystyle \left( X_n \right)_{n \geq 1}$ be a family of non-void, finite sets of points such that $\displaystyle X_n \subset P_n$ and the projection of the set $\displaystyle X_{n+1}$ on the plane $\displaystyle P_n$ is included in the set $X_n$, for all $n$.
Prove that there is a sequence of points $\displaystyle \left( p_n \right)_{n \geq 1}$ such that $\displaystyle p_n \in P_n$ and $p_n$ is the projection of $p_{n+1}$ on the plane $P_n$, for all $n$.
Does the conclusion of the problem remain true if the sets $X_n$ are infinite?
[i]Claudiu Raicu[/i]
2008 Moldova MO 11-12, 7
Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\in(HG)$ so that $ \frac {HF}{FG} \equal{} 3$. Find the locus of the point $ A$ so that $ F\in BC$.
1971 IMO Longlists, 48
The diagonals of a convex quadrilateral $ABCD$ intersect at a point $O$. Find all angles of this quadrilateral if $\measuredangle OBA=30^{\circ},\measuredangle OCB=45^{\circ},\measuredangle ODC=45^{\circ}$, and $\measuredangle OAD=30^{\circ}$.
2000 Iran MO (3rd Round), 2
Circles $ C_1$ and $ C_2$ with centers at $ O_1$ and $ O_2$ respectively meet at points $ A$ and $ B$. The radii $ O_1B$ and $ O_2B$ meet $ C_1$ and $ C_2$ at $ F$ and$ E$. The line through $ B$ parallel to $ EF$ intersects $ C_1$ again at $ M$ and $ C_2$ again at $ N$. Prove that $ MN \equal{} AE \plus{} AF$.
2002 Tournament Of Towns, 2
A cube is cut by a plane such that the cross section is a pentagon. Show there is a side of the pentagon of length $\ell$ such that the inequality holds:
\[ |\ell-1|>\frac{1}{5} \]
2006 Junior Balkan Team Selection Tests - Moldova, 2
Let $ABCD$ be a rectangle and denote by $M$ and $N$ the midpoints of $AD$ and $BC$ respectively. The point $P$ is on $(CD$ such that $D\in (CP)$, and $PM$ intersects $AC$ in $Q$. Prove that $m(\angle{MNQ})=m(\angle{MNP})$.
2024 Dutch IMO TST, 1
Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $D$ be the reflection of $A$ in $B$ and let $E$ the reflection of $A$ in $C$. Let $M$ be the midpoint of segment $DE$. Show that the tangent to $\Gamma$ in $A$ is perpendicular to $HM$.
2008 Spain Mathematical Olympiad, 2
Given a circle, two fixed points $A$ and $B$ and a variable point $P$, all of them on the circle, and a line $r$, $PA$ and $PB$ intersect $r$ at $C$ and $D$, respectively. Find two fixed points on $r$, $M$ and $N$, such that $CM\cdot DN$ is constant for all $P$.
2014 ELMO Shortlist, 11
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $D, E, F$ be on $BC, AC, AB$ respectively so that the Miquel point of $DEF$ with respect to $ABC$ is $P$. Let the reflections of $D, E, F$ over the midpoints of the sides that they lie on be $R, S, T$. Let the Miquel point of $RST$ with respect to the triangle $ABC$ be $Q$. Show that $OP = OQ$.
[i]Proposed by Yang Liu[/i]
1983 IMO Longlists, 62
$A$ circle $\gamma$ is drawn and let $AB$ be a diameter. The point $C$ on $\gamma$ is the midpoint of the line segment $BD$. The line segments $AC$ and $DO$, where $O$ is the center of $\gamma$, intersect at $P$. Prove that there is a point $E$ on $AB$ such that $P$ is on the circle with diameter $AE.$
2024 All-Russian Olympiad, 6
Let $ABCD$ be a parallelogram. Let $M$ be the midpoint of the arc $AC$ containing $B$ of the circumcircle of $ABC$ . Let $E$ be a point on segment $AD$ and $F$ a point on segment $CD$ such that $ME=MD=MF$. Show that $BMEF$ is cyclic.
[i]Proposed by A. Tereshin[/i]
2009 German National Olympiad, 3
Let $ ABCD$ be a (non-degenerate) quadrangle and $ N$ the intersection of $ AC$ and $ BD$. Denote by $ a,b,c,d$ the length of the altitudes from $ N$ to $ AB,BC,CD,DA$, respectively.
Prove that $ \frac{1}{a}\plus{}\frac{1}{c} \equal{} \frac{1}{b}\plus{}\frac{1}{d}$ if $ ABCD$ has an incircle.
Extension: Prove that the converse is true, too.
[If this has already been posted, I humbly apologize. A quick search turned up nothing.]
2016 All-Russian Olympiad, 2
$\omega$ is a circle inside angle $\measuredangle BAC$ and it is tangent to sides of this angle at $B,C$.An arbitrary line $ \ell $ intersects with $AB,AC$ at $K,L$,respectively and intersect with $\omega$ at $P,Q$.Points $S,T$ are on $BC$ such that $KS \parallel AC$ and $TL \parallel AB$.Prove that $P,Q,S,T$ are concyclic.(I.Bogdanov,P.Kozhevnikov)
2008 Vietnam Team Selection Test, 2
Let $ k$ be a positive real number. Triangle ABC is acute and not isosceles, O is its circumcenter and AD,BE,CF are the internal bisectors. On the rays AD,BE,CF, respectively, let points L,M,N such that $ \frac {AL}{AD} \equal{} \frac {BM}{BE} \equal{} \frac {CN}{CF} \equal{} k$. Denote $ (O_1),(O_2),(O_3)$ be respectively the circle through L and touches OA at A, the circle through M and touches OB at B, the circle through N and touches OC at C.
1) Prove that when $ k \equal{} \frac{1}{2}$, three circles $ (O_1),(O_2),(O_3)$ have exactly two common points, the centroid G of triangle ABC lies on that common chord of these circles.
2) Find all values of k such that three circles $ (O_1),(O_2),(O_3)$ have exactly two common points
2008 China Western Mathematical Olympiad, 2
In triangle $ ABC$, $ AB\equal{}AC$, the inscribed circle $ I$ touches $ BC, CA, AB$ at points $ D,E$ and $ F$ respectively. $ P$ is a point on arc $ EF$ opposite $ D$. Line $ BP$ intersects circle $ I$ at another point $ Q$, lines $ EP$, $ EQ$ meet line $ BC$ at $ M, N$ respectively. Prove that
(1) $ P, F, B, M$ concyclic
(2)$ \frac{EM}{EN} \equal{} \frac{BD}{BP}$
(P.S. Can anyone help me with using GeoGebra, the incircle function of the plugin doesn't work with my computer.)
2015 China Team Selection Test, 1
$\triangle{ABC}$ is isosceles with $AB = AC >BC$. Let $D$ be a point in its interior such that $DA = DB+DC$. Suppose that the perpendicular bisector of $AB$ meets the external angle bisector of $\angle{ADB}$ at $P$, and let $Q$ be the intersection of the perpendicular bisector of $AC$ and the external angle bisector of $\angle{ADC}$. Prove that $B,C,P,Q$ are concyclic.