Found problems: 2023
2007 Middle European Mathematical Olympiad, 3
A tetrahedron is called a [i]MEMO-tetrahedron[/i] if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$.
(a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)\equal{}n$.
(b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)\equal{}2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).
2014 Contests, 3
Let $l$ be the tangent line at the point $P(s,\ t)$ on a circle $C:x^2+y^2=1$. Denote by $m$ the line passing through the point $(1,\ 0)$, parallel to $l$. Let the line $m$ intersects the circle $C$ at $P'$ other than the point $(1,\ 0)$.
Note : if $m$ is the line $x=1$, then $P'$ is considered as $(1,\ 0)$.
Call $T$ the operation such that the point $P'(s',\ t')$ is obtained from the point $P(s,\ t)$ on $C$.
(1) Express $s',\ t'$ as the polynomials of $s$ and $t$ respectively.
(2) Let $P_n$ be the point obtained by $n$ operations of $T$ for $P$.
For $P\left(\frac{\sqrt{3}}{2},\ \frac{1}{2}\right)$, plot the points $P_1,\ P_2$ and $P_3$.
(3) For a positive integer $n$, find the number of $P$ such that $P_n=P$.
1997 Pre-Preparation Course Examination, 2
Two circles $O, O'$ meet each other at points $A, B$. A line from $A$ intersects the circle $O$ at $C$ and the circle $O'$ at $D$ ($A$ is between $C$ and $D$). Let $M,N$ be the midpoints of the arcs $BC, BD$, respectively (not containing $A$), and let $K$ be the midpoint of the segment $CD$. Show that $\angle KMN = 90^\circ$.
2012 Korea National Olympiad, 2
Let $ w $ be the incircle of triangle $ ABC $. Segments $ BC, CA $ meet with $ w $ at points $ D, E$. A line passing through $ B $ and parallel to $ DE $ meets $ w $ at $ F $ and $ G $. ($ F $ is nearer to $ B $ than $ G $.) Line $ CG $ meets $ w $ at $ H ( \ne G ) $. A line passing through $ G $ and parallel to $ EH $ meets with line $ AC $ at $ I $. Line $ IF $ meets with circle $ w $ at $ J (\ne F ) $. Lines $ CJ $ and $ EG $ meets at $ K $. Let $ l $ be the line passing through $ K $ and parallel to $ JD $. Prove that $ l, IF, ED $ meet at one point.
1987 Federal Competition For Advanced Students, P2, 5
Let $ P$ be a point in the interior of a convex $ n$-gon $ A_1 A_2 ... A_n$ $ (n \ge 3)$. Show that among the angles $ \beta _{ij}\equal{}\angle A_i P A_j,1 \le i \le n$, there are at least $ n\minus{}1$ angles satisfying $ 90^{\circ} \le \beta_{ij} \le 180^{\circ}$.
2010 Contests, 2
Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle.
2006 Bulgaria National Olympiad, 2
The triangle $ABC$ is such that $\angle BAC=30^{\circ},\angle ABC=45^{\circ}$. Prove that if $X$ lies on the ray $AC$, $Y$ lies on the ray $BC$ and $OX=BY$, where $O$ is the circumcentre of triangle $ABC$, then $S_{XY}$ passes through a fixed point.
[i]Emil Kolev [/i]
2012 Romania National Olympiad, 1
[color=darkred]The altitude $[BH]$ dropped onto the hypotenuse of a triangle $ABC$ intersects the bisectors $[AD]$ and $[CE]$ at $Q$ and $P$ respectively. Prove that the line passing through the midpoints of the segments $[QD]$ and $[PE]$ is parallel to the line $AC$ .[/color]
1997 Baltic Way, 14
In the triangle $ABC$, $AC^2$ is the arithmetic mean of $BC^2$ and $AB^2$. Show that $\cot^2B\ge \cot A\cdot\cot C$.
JBMO Geometry Collection, 2005
Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$.
Prove that the lines $AP$ and $CS$ are parallel.
2005 Kyiv Mathematical Festival, 2
The quadrilateral $ ABCD$ is cyclic. Points $ E$ and $ F$ are chosen at the diagonals $ AC$ and $ BD$ in such a way that $ AF\bot CD$ and $ DE\bot AB.$ Prove that $ EF \parallel BC.$
2016 Croatia Team Selection Test, Problem 3
Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.
2011 Romania Team Selection Test, 1
Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $AB$. The feet of the perpendiculars from E onto the lines $\ell$ and $CD$ are $H$ and $K$, respectively. Prove that the lines $EF$ and $HK$ are perpendicular.
2009 Sharygin Geometry Olympiad, 4
Let $ P$ and $ Q$ be the common points of two circles. The ray with origin $ Q$ reflects from the first circle in points $ A_1$, $ A_2$,$ \ldots$ according to the rule ''the angle of incidence is equal to the angle of reflection''. Another ray with origin $ Q$ reflects from the second circle in the points $ B_1$, $ B_2$,$ \ldots$ in the same manner. Points $ A_1$, $ B_1$ and $ P$ occurred to be collinear. Prove that all lines $ A_iB_i$ pass through P.
1999 Irish Math Olympiad, 5
A convex hexagon $ ABCDEF$ satisfies $ AB\equal{}BC, CD\equal{}DE, EF\equal{}FA$ and: $ \angle ABC\plus{}\angle CDE\plus{}\angle EFA \equal{} 360^{\circ}$. Prove that the perpendiculars from $ A,C$ and $ E$ to $ FB,BD$ and $ DF$ respectively are concurrent.
1971 IMO Longlists, 29
A rhombus with its incircle is given. At each vertex of the rhombus a circle is constructed that touches the incircle and two edges of the rhombus. These circles have radii $r_1,r_2$, while the incircle has radius $r$. Given that $r_1$ and $r_2$ are natural numbers and that $r_1r_2=r$, find $r_1,r_2,$ and $r$.
Swiss NMO - geometry, 2011.5
Let $\triangle{ABC}$ be a triangle with circumcircle $\tau$. The tangentlines to $\tau$ through $A$ and $B$ intersect at $T$. The circle through $A$, $B$ and $T$ intersects $BC$ and $AC$ again at $D$ and $E$, respectively; $CT$ and $BE$ intersect at $F$.
Suppose $D$ is the midpoint of $BC$. Calculate the ratio $BF:BE$.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 5)[/i]
2000 Vietnam National Olympiad, 2
Two circles $ (O_1)$ and $ (O_2)$ with respective centers $ O_1$, $ O_2$ are given on a plane. Let $ M_1$, $ M_2$ be points on $ (O_1)$, $ (O_2)$ respectively, and let the lines $ O_1M_1$ and $ O_2M_2$ meet at $ Q$. Starting simultaneously from these positions, the points $ M_1$ and $ M_2$ move clockwise on their own circles with the same angular velocity.
(a) Determine the locus of the midpoint of $ M_1M_2$.
(b) Prove that the circumcircle of $ \triangle M_1QM_2$ passes through a fixed point.
2006 Kyiv Mathematical Festival, 4
See all the problems from 5-th Kyiv math festival
[url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.
1995 All-Russian Olympiad, 6
In an acute-angled triangle ABC, points $A_2$, $B_2$, $C_2$ are the midpoints of the altitudes $AA_1$, $BB_1$, $CC_1$, respectively. Compute the sum of angles $B_2A_1C_2$, $C_2B_1A_2$ and $A_2C_1B_2$.
[i]D. Tereshin[/i]
1998 Iran MO (3rd Round), 5
In a triangle $ABC$, the bisector of angle $BAC$ intersects $BC$ at $D$. The circle $\Gamma$ through $A$ which is tangent to $BC$ at $D$ meets $AC$ again at $M$. Line $BM$ meets $\Gamma$ again at $P$. Prove that line $AP$ is a median of $\triangle ABD.$
2011 Korea National Olympiad, 2
Let $ABC$ be a triangle and its incircle meets $BC, AC, AB$ at $D, E$ and $F$ respectively. Let point $ P $ on the incircle and inside $ \triangle AEF $. Let $ X=PB \cap DF , Y=PC \cap DE, Q=EX \cap FY $. Prove that the points $ A$ and $Q$ lies on $DP$ simultaneously or located opposite sides from $DP$.
1997 Poland - Second Round, 2
Let P be a point inside triangle ABC such that 3<ABP = 3<ACP = <ABC + <ACB. Prove that AB/(AC + PB) = AC/(AB + PC).
2012 Middle European Mathematical Olympiad, 3
In a given trapezium $ ABCD $ with $ AB$ parallel to $ CD $ and $ AB > CD $, the line $ BD $ bisects the angle $ \angle ADC $. The line through $ C $ parallel to $ AD $ meets the segments $ BD $ and $ AB $ in $ E $ and $ F $, respectively. Let $ O $ be the circumcenter of the triangle $ BEF $. Suppose that $ \angle ACO = 60^{\circ} $. Prove the equality
\[ CF = AF + FO .\]
2011 Vietnam National Olympiad, 3
Let $AB$ be a diameter of a circle $(O)$ and let $P$ be any point on the tangent drawn at $B$ to $(O).$ Define $AP\cap (O)=C\neq A,$ and let $D$ be the point diametrically opposite to $C.$ If $DP$ meets $(O)$ second time in $E,$ then,
[b](i)[/b] Prove that $AE, BC, PO$ concur at $M.$
[b](ii)[/b] If $R$ is the radius of $(O),$ find $P$ such that the area of $\triangle AMB$ is maximum, and calculate the area in terms of $R.$