Found problems: 300
2016 Rioplatense Mathematical Olympiad, Level 3, 3
Let $A B C$ be an acute-angled triangle of circumcenter $O$ and orthocenter $H$. Let $M$ be the midpoint of $BC, N$ be the symmetric of $H$ with respect to $A, P$ be the midpoint of $NM$ and $X$ be a point on the line A H such that $MX$ is parallel to $CH$. Prove that $BX$ and $OP$ are perpendicular.
Estonia Open Junior - geometry, 2014.1.5
In a triangle $ABC$ the midpoints of $BC, CA$ and $AB$ are $D, E$ and $F$, respectively. Prove that the circumcircles of triangles $AEF, BFD$ and $CDE$ intersect all in one point.
2013 District Olympiad, 3
On the sides $(AB)$ and $(AC)$ of the triangle $ABC$ are considered the points $M$ and $N$ respectively so that $ \angle ABC =\angle ANM$. Point $D$ is symmetric of point $A$ with respect to $B$, and $P$ and $Q$ are the midpoints of the segments $[MN]$ and $[CD]$, respectively. Prove that the points $A, P$ and $Q$ are collinear if and only if $AC = AB \sqrt {2}$
2011 Peru MO (ONEM), 3
Let $ABC$ be a right triangle, right in $B$. Inner bisectors are drawn $CM$ and $AN$ that intersect in $I$. Then, the $AMIP$ and $CNIQ$ parallelograms are constructed. Let $U$ and $V$ are the midpoints of the segments $AC$ and $PQ$, respectively. Prove that $UV$ is perpendicular to $AC$.
2005 Sharygin Geometry Olympiad, 9.4
Let $P$ be the intersection point of the diagonals of the quadrangle $ABCD$, $M$ the intersection point of the lines connecting the midpoints of its opposite sides, $O$ the intersection point of the perpendicular bisectors of the diagonals, $H$ the intersection point of the lines connecting the orthocenters of the triangles $APD$ and $BCP$, $APB$ and $CPD$. Prove that $M$ is the midpoint of $OH$.
2021 Saudi Arabia Training Tests, 18
Let $ABC$ be a triangle with $AB < AC$ and incircle $(I)$ tangent to $BC$ at $D$. Take $K$ on $AD$ such that $CD = CK$. Suppose that $AD$ cuts $(I)$ at $G$ and $BG$ cuts $CK$ at $L$. Prove that K is the midpoint of $CL$.
1998 Austrian-Polish Competition, 9
Given a triangle $ABC$, points $K,L,M$ are the midpoints of the sides $BC,CA,AB$, and points $X,Y,Z$ are the midpoints of the arcs $BC,CA,AB$ of the circumcircle not containing $A,B,C$ respectively. If $R$ denotes the circumradius and $r$ the inradius of the triangle, show that $r+KX+LY+MZ=2R$.
2008 IMAC Arhimede, 4
Let $ABCD$ be a random tetrahedron. Let $E$ and $F$ be the midpoints of segments $AB$ and $CD$, respectively. If the angle $a$ is between $AD$ and $BC$, determine $cos a$ in terms of $EF, AD$ and $BC$.
1988 Tournament Of Towns, (165) 2
We are given convex quadrilateral $ABCD$. The midpoints of $BC$ and $DA$ are $M$ and $N$ respectively. The diagonal $AC$ divides $MN$ in half. Prove that the areas of triangles $ABC$ and $ACD$ are equal .
2007 France Team Selection Test, 3
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
1990 IMO Longlists, 25
The incenter of the triangle $ ABC$ is $ K.$ The midpoint of $ AB$ is $ C_1$ and that of $ AC$ is $ B_1.$ The lines $ C_1K$ and $ AC$ meet at $ B_2,$ the lines $ B_1K$ and $ AB$ at $ C_2.$ If the areas of the triangles $ AB_2C_2$ and $ ABC$ are equal, what is the measure of angle $ \angle CAB?$
2022 Argentina National Olympiad, 3
Given a square $ABCD$, let us consider an equilateral triangle $KLM$, whose vertices $K$, $L$ and $M$ belong to the sides $AB$, $BC$ and $CD$ respectively. Find the locus of the midpoints of the sides $KL$ for all possible equilateral triangles $KLM$.
Note: The set of points that satisfy a property is called a locus.
1980 All Soviet Union Mathematical Olympiad, 287
The points $M$ and $P$ are the midpoints of $[BC]$ and $[CD]$ sides of a convex quadrangle $ABCD$. It is known that $|AM| + |AP| = a$. Prove that $ABCD$ has area less than $\frac{a^2}{2}$.
2024 Brazil Team Selection Test, 2
Let \( ABC \) be an acute-angled scalene triangle with circumcenter \( O \). Denote by \( M \), \( N \), and \( P \) the midpoints of sides \( BC \), \( CA \), and \( AB \), respectively. Let \( \omega \) be the circle passing through \( A \) and tangent to \( OM \) at \( O \). The circle \( \omega \) intersects \( AB \) and \( AC \) at points \( E \) and \( F \), respectively (where \( E \) and \( F \) are distinct from \( A \)). Let \( I \) be the midpoint of segment \( EF \), and let \( K \) be the intersection of lines \( EF \) and \( NP \). Prove that \( AO = 2IK \) and that triangle \( IMO \) is isosceles.
2009 Dutch IMO TST, 2
Let $ABC$ be a triangle, $P$ the midpoint of $BC$, and $Q$ a point on segment $CA$ such that $|CQ| = 2|QA|$. Let $S$ be the intersection of $BQ$ and $AP$. Prove that $|AS| = |SP|$.
2007 Serbia National Math Olympiad, 1
A point $D$ is chosen on the side $AC$ of a triangle $ABC$ with $\angle C < \angle A < 90^\circ$ in such a way that $BD=BA$. The incircle of $ABC$ is tangent to $AB$ and $AC$ at points $K$ and $L$, respectively. Let $J$ be the incenter of triangle $BCD$. Prove that the line $KL$ intersects the line segment $AJ$ at its midpoint.
2011 Romania National Olympiad, 3
In the convex quadrilateral $ABCD$ we have that $\angle BCD = \angle ADC \ge 90 ^o$. The bisectors of $\angle BAD$ and $\angle ABC$ intersect in $M$. Prove that if $M \in CD$, then $M$ is the middle of $CD$.
2019 Saudi Arabia Pre-TST + Training Tests, 2.3
Let $ABC$ be a triangle with $A',B',C'$ are midpoints of $BC,CA,AB$ respectively. The circle $(\omega_A)$ of center $A$ has a big enough radius cuts $B'C'$ at $X_1,X_2$. Define circles $(\omega_B), (\omega_C)$ with $Y_1, Y_2,Z_1,Z_2$ similarly. Suppose that these circles have the same radius, prove that $X_1,X_2, Y_1, Y_2,Z_1,Z_2$ are concyclic.
2008 Singapore Junior Math Olympiad, 3
In the quadrilateral $PQRS, A, B, C$ and $D$ are the midpoints of the sides $PQ, QR, RS$ and $SP$ respectively, and $M$ is the midpoint of $CD$. Suppose $H$ is the point on the line $AM$ such that $HC = BC$. Prove that $\angle BHM = 90^o$.
2019 Saudi Arabia JBMO TST, 2
In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$.
Prove that $\angle A = 2\angle B$ if and only if $AC = 2MD$
2020 Abels Math Contest (Norwegian MO) Final, 4a
The midpoint of the side $AB$ in the triangle $ABC$ is called $C'$. A point on the side $BC$ is called $D$, and $E$ is the point of intersection of $AD$ and $CC'$. Assume that $AE/ED = 2$. Show that $D$ is the midpoint of $BC$.
2018 Yasinsky Geometry Olympiad, 6
Let $O$ and $I$ be the centers of the circumscribed and inscribed circle the acute-angled triangle $ABC$, respectively. It is known that line $OI$ is parallel to the side $BC$ of this triangle. Line $MI$, where $M$ is the midpoint of $BC$, intersects the altitude $AH$ at the point $T$. Find the length of the segment $IT$, if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$.
(Grigory Filippovsky)
Novosibirsk Oral Geo Oly VIII, 2016.4
The two angles of the squares are adjacent, and the extension of the diagonals of one square intersect the diagonal of another square at point $O$ (see figure). Prove that $O$ is the midpoint of $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/7/8/8daaaa55c38e15c4a8ac7492c38707f05475cc.png[/img]
2013 JBMO Shortlist, 6
Let $P$ and $Q$ be the midpoints of the sides $BC$ and $CD$, respectively in a rectangle $ABCD$. Let $K$ and $M$ be the intersections of the line $PD$ with the lines $QB$ and $QA$, respectively, and let $N$ be the intersection of the lines $PA$ and $QB$. Let $X$, $Y$ and $Z$ be the midpoints of the segments $AN$, $KN$ and $AM$, respectively. Let $\ell_1$ be the line passing through $X$ and perpendicular to $MK$, $\ell_2$ be the line passing through $Y$ and perpendicular to $AM$ and $\ell_3$ the line passing through $Z$ and perpendicular to $KN$. Prove that the lines $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent.
2008 Dutch IMO TST, 5
Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ and $|AB| > |BC|$, and let $\Gamma$ be the semicircle with diameter $AB$ that lies on the same side as $C$. Let $P$ be a point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be on $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.