Found problems: 2023
2005 Bundeswettbewerb Mathematik, 3
Let $ABC$ be a triangle with sides $a$, $b$, $c$ and (corresponding) angles $A$, $B$, $C$.
Prove that if $3A + 2B = 180^{\circ}$, then $a^2+bc=c^2$.
[b]Additional problem:[/b]
Prove that the converse also holds, i. e. prove the following:
Let $ABC$ be an arbitrary triangle. Then, $3A + 2B = 180^{\circ}$ if and only if $a^2+bc=c^2$.
[b]Similar problem:[/b]
Let $ABC$ be an arbitrary triangle. Then, $3A + 2B = 360^{\circ}$ if and only if $a^2-bc=c^2$.
2012 Kosovo Team Selection Test, 3
If $a,b,c$ are the sides of a triangle and $m_a , m_b, m_c$ are the medians prove that
\[4(m_a^2+m_b^2+m_c^2)=3(a^2+b^2+c^2)\]
2007 Iran Team Selection Test, 3
$O$ is a point inside triangle $ABC$ such that $OA=OB+OC$. Suppose $B',C'$ be midpoints of arcs $\overarc{AOC}$ and $AOB$. Prove that circumcircles $COC'$ and $BOB'$ are tangent to each other.
2009 Indonesia TST, 2
Two cirlces $ C_1$ and $ C_2$, with center $ O_1$ and $ O_2$ respectively, intersect at $ A$ and $ B$. Let $ O_1$ lies on $ C_2$. A line $ l$ passes through $ O_1$ but does not pass through $ O_2$. Let $ P$ and $ Q$ be the projection of $ A$ and $ B$ respectively on the line $ l$ and let $ M$ be the midpoint of $ \overline{AB}$. Prove that $ MPQ$ is an isoceles triangle.
2011 Balkan MO Shortlist, C2
Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.
2014 Moldova Team Selection Test, 3
Let $\triangle ABC$ be an acute triangle and $AD$ the bisector of the angle $\angle BAC$ with $D\in(BC)$. Let $E$ and $F$ denote feet of perpendiculars from $D$ to $AB$ and $AC$ respectively. If $BF\cap CE=K$ and $\odot AKE\cap BF=L$ prove that $DL\perp BF$.
2004 Romania Team Selection Test, 2
Let $\{R_i\}_{1\leq i\leq n}$ be a family of disjoint closed rectangular surfaces with total area 4 such that their projections of the $Ox$ axis is an interval. Prove that there exist a triangle with vertices in $\displaystyle \bigcup_{i=1}^n R_i$ which has an area of at least 1.
[Thanks Grobber for the correction]
2009 India IMO Training Camp, 4
Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly.
Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.
2010 ELMO Shortlist, 5
Determine all (not necessarily finite) sets $S$ of points in the plane such that given any four distinct points in $S$, there is a circle passing through all four or a line passing through some three.
[i]Carl Lian.[/i]
2005 Greece National Olympiad, 4
Let $OX_1 , OX_2$ be rays in the interior of a convex angle $XOY$ such that $\angle XOX_1=\angle YOY_1< \frac{1}{3}\angle XOY$. Points $K$ on $OX_1$ and $L$ on $OY_1$ are fixed so that $OK=OL$, and points $A$, $B$ are vary on rays $(OX , (OY$ respectively such that the area of the pentagon $OAKLB$ remains constant. Prove that the circumcircle of the triangle $OAB$ passes from a fixed point, other than $O$.
1996 Turkey Team Selection Test, 1
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ with $S_{ABC} = S_{ADC}$ intersect at $E$. The lines through $E$ parallel to $AD$, $DC$, $CB$, $BA$
meet $AB$, $BC$, $CD$, $DA$ at $K$, $L$, $M$, $N$, respectively. Compute the ratio $\frac{S_{KLMN}}{S_{ABC}}$
2013 ELMO Shortlist, 12
Let $ABC$ be a nondegenerate acute triangle with circumcircle $\omega$ and let its incircle $\gamma$ touch $AB, AC, BC$ at $X, Y, Z$ respectively. Let $XY$ hit arcs $AB, AC$ of $\omega$ at $M, N$ respectively, and let $P \neq X, Q \neq Y$ be the points on $\gamma$ such that $MP=MX, NQ=NY$. If $I$ is the center of $\gamma$, prove that $P, I, Q$ are collinear if and only if $\angle BAC=90^\circ$.
[i]Proposed by David Stoner[/i]
2015 Vietnam Team selection test, Problem 2
sorry if this has been posted before .
given a fixed circle $(O)$ and two fixed point $B,C$ on it.point A varies on circle $(O)$. let $I$ be the midpoint of $BC$ and $H$ be the orthocenter of $\triangle ABC$. ray $IH$ meet $(O)$ at $K$ ,$AH$ meet $BC$ at $D$ ,$KD$ meet $(O)$ at $M$ .a line pass $M$ and perpendicular to $BC$ meet $AI$ at $N$.
a) prove that $N$ varies on a fixed circle.
b) a circle pass $N$ and tangent to $AK$ at $A$ cut $AB,AC$ at $P,Q$. let $J$ be the midpoint of $PQ$ .prove that $AJ$ pass through a fixed point.
1999 Romania Team Selection Test, 17
A polyhedron $P$ is given in space. Find whether there exist three edges in $P$ which can be the sides of a triangle. Justify your answer!
[i]Barbu Berceanu[/i]
2007 Italy TST, 1
Let $ABC$ an acute triangle.
(a) Find the locus of points that are centers of rectangles whose vertices lie on the sides of $ABC$;
(b) Determine if exist some points that are centers of $3$ distinct rectangles whose vertices lie on the sides of $ABC$.
2004 Bulgaria Team Selection Test, 3
Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square.
2024 Abelkonkurransen Finale, 4b
The pentagons $P_1P_2P_3P_4P_5$ and$I_1I_2I_3I_4I_5$ are cyclic, where $I_i$ is the incentre of the triangle $P_{i-1}P_iP_{i+1}$ (reckoned cyclically, that is $P_0=P_5$ and $P_6=P_1$).
Show that the lines $P_1I_1, P_2I_2, P_3I_3, P_4I_4$ and $P_5I_5$ meet in a single point.
2003 Iran MO (3rd Round), 28
There are $ n$ points in $ \mathbb R^3$ such that every three form an acute angled triangle. Find maximum of $ n$.
2004 JBMO Shortlist, 5
Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$.
1991 Brazil National Olympiad, 5
$P_0 = (1,0), P_1 = (1,1), P_2 = (0,1), P_3 = (0,0)$.
$P_{n+4}$ is the midpoint of $P_nP_{n+1}$.
$Q_n$ is the quadrilateral $P_{n}P_{n+1}P_{n+2}P_{n+3}$.
$A_n$ is the interior of $Q_n$.
Find $\cap_{n \geq 0}A_n$.
2011 Olympic Revenge, 4
Let $ABCD$ to be a quadrilateral inscribed in a circle $\Gamma$. Let $r$ and $s$ to be the tangents to $\Gamma$ through $B$ and $C$, respectively, $M$ the intersection between the lines $r$ and $AD$ and $N$ the intersection between the lines $s$ and $AD$. After all, let $E$ to be the intersection between the lines $BN$ and $CM$, $F$ the intersection between the lines $AE$ and $BC$ and $L$ the midpoint of $BC$. Prove that the circuncircle of the triangle $DLF$ is tangent to $\Gamma$.
2013 Macedonia National Olympiad, 3
Acute angle triangle is given such that $ BC $ is the longest side. Let $ E $ and $ G $ be the intersection points from the altitude from $ A $ to $ BC $ with the circumscribed circle of triangle $ ABC $ and $ BC $ respectively. Let the center $ O $ of this circle is positioned on the perpendicular line from $ A $ to $ BE $. Let $ EM $ be perpendicular to $ AC $ and $ EF $ be perpendicular to $ AB $. Prove that the area of $ FBEG $ is greater than the area of $ MFE $.
1985 IMO Longlists, 36
Determine whether there exist $100$ distinct lines in the plane having exactly $1985$ distinct points of intersection
2006 Czech and Slovak Olympiad III A, 4
Given a segment $AB$ in the plane. Let $C$ be another point in the same plane,$H,I,G$ denote the orthocenter,incenter and centroid of triangle $ABC$. Find the locus of $M$ for which $A,B,H,I$ are concyclic.
2014 Contests, 4
In triangle $ABC$ let $A'$, $B'$, $C'$ respectively be the midpoints of the sides $BC$, $CA$, $AB$. Furthermore let $L$, $M$, $N$ be the projections of the orthocenter on the three sides $BC$, $CA$, $AB$, and let $k$ denote the nine-point circle. The lines $AA'$, $BB'$, $CC'$ intersect $k$ in the points $D$, $E$, $F$. The tangent lines on $k$ in $D$, $E$, $F$ intersect the lines $MN$, $LN$ and $LM$ in the points $P$, $Q$, $R$.
Prove that $P$, $Q$ and $R$ are collinear.