Found problems: 300
2000 Tournament Of Towns, 1
The diagonals of a convex quadrilateral $ABCD$ meet at $P$. The sum of the areas of triangles $PAB$ and $PCD$ is equal to the sum of areas of triangles $PAD$ and $PCB$. Prove that $P$ is the midpoint of either $AC$ or $BD$.
(Folklore)
1953 Moscow Mathematical Olympiad, 240
Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints. Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.
2015 Junior Regional Olympiad - FBH, 3
Let $D$ be a midpoint of $BC$ of triangle $ABC$. On side $AB$ is given point $E$, and on side $AC$ is given point $F$ such that $\angle EDF = 90^{\circ}$. Prove that $BE+CF>EF$
2016 Latvia Baltic Way TST, 15
Let $ABC$ be a triangle. Let its altitudes $AD$, $BE$ and $CF$ concur at $H$. Let $K, L$ and $M$ be the midpoints of $BC$, $CA$ and $AB$, respectively. Prove that, if $\angle BAC = 60^o$, then the midpoints of the segments $AH$, $DK$, $EL$, $FM$ are concyclic.
2019 Saint Petersburg Mathematical Olympiad, 3
Prove that the distance between the midpoint of side $BC$ of triangle $ABC$ and the midpoint of the arc $ABC$ of its circumscribed circle is not less than $AB / 2$
2017 Sharygin Geometry Olympiad, P19
Let cevians $AA', BB'$ and $CC'$ of triangle $ABC$ concur at point $P.$ The circumcircle of triangle $PA'B'$ meets $AC$ and $BC$ at points $M$ and $N$ respectively, and the circumcircles of triangles $PC'B'$ and $PA'C'$ meet $AC$ and $BC$ for the second time respectively at points $K$ and $L$. The line $c$ passes through the midpoints of segments $MN$ and $KL$. The lines $a$ and $b$ are defined similarly. Prove that $a$, $b$ and $c$ concur.
2016 Czech-Polish-Slovak Junior Match, 1
Let $ABC$ be a right-angled triangle with hypotenuse $AB$. Denote by $D$ the foot of the altitude from $C$. Let $Q, R$, and $P$ be the midpoints of the segments $AD, BD$, and $CD$, respectively. Prove that $\angle AP B + \angle QCR = 180^o$.
Czech Republic
2016 Sharygin Geometry Olympiad, P1
A trapezoid $ABCD$ with bases $AD$ and $BC$ is such that $AB = BD$. Let $M$ be the midpoint of $DC$. Prove that $\angle MBC$ = $\angle BCA$.
2008 Estonia Team Selection Test, 5
Points $A$ and $B$ are fixed on a circle $c_1$. Circle $c_2$, whose centre lies on $c_1$, touches line $AB$ at $B$. Another line through $A$ intersects $c_2$ at points $D$ and $E$, where $D$ lies between $A$ and $E$. Line $BD$ intersects $c_1$ again at $F$. Prove that line $EB$ is tangent to $c_1$ if and only if $D$ is the midpoint of the segment $BF$.
1983 IMO Shortlist, 23
Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.
2018 Finnish National High School Mathematics Comp, 3
The chords $AB$ and $CD$ of a circle intersect at $M$, which is the midpoint of the chord $PQ$. The points $X$ and $Y$ are the intersections of the segments $AD$ and $PQ$, respectively, and $BC$ and $PQ$, respectively. Show that $M$ is the midpoint of $XY$.
2005 Sharygin Geometry Olympiad, 9.3
Given a circle and points $A, B$ on it. Draw the set of midpoints of the segments, one of the ends of which lies on one of the arcs $AB$, and the other on the second.
2010 Dutch BxMO TST, 1
Let $ABCD$ be a trapezoid with $AB // CD$, $2|AB| = |CD|$ and $BD \perp BC$. Let $M$ be the midpoint of $CD$ and let $E$ be the intersection $BC$ and $AD$. Let $O$ be the intersection of $AM$ and $BD$. Let $N$ be the intersection of $OE$ and $AB$.
(a) Prove that $ABMD$ is a rhombus.
(b) Prove that the line $DN$ passes through the midpoint of the line segment $BE$.
2020 New Zealand MO, 2
Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$. Let $Y$ be the point on line $CD$ such that $BX = YD$ and $D$ is between $C$ and $Y$ . Prove that the midpoint of $XY$ lies on diagonal $BD$.
2014 India PRMO, 15
Let $XOY$ be a triangle with $\angle XOY = 90^o$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Suppose that $XN = 19$ and $YM =22$. What is $XY$?
2016 Czech-Polish-Slovak Junior Match, 5
Let $ABC$ be a triangle with $AB : AC : BC =5:5:6$. Denote by $M$ the midpoint of $BC$ and by $N$ the point on the segment $BC$ such that $BN = 5 \cdot CN$. Prove that the circumcenter of triangle $ABN$ is the midpoint of the segment connecting the incenters of triangles $ABC$ and $ABM$.
Slovakia
VMEO III 2006, 11.2
Let $ABCD$ be an isosceles trapezoid, with a large base $CD$ and a small base $AB$. Let $M$ be any point on side $AB$ and $(d)$ be the line through $M$ and perpendicular to $AB$. Two rays $Mx$ and $My$ are said to satisfy the condition $(T)$ if they are symmetric about each other through $(d)$ and intersect the two rays $AD$ and $BC$ at $E$ and $F$ respectively. Find the locus of the midpoint of the segment $EF$ when the two rays $Mx$ and $My$ change and satisfy condition $(T)$.
Ukraine Correspondence MO - geometry, 2015.11
Let $ABC$ be an non- isosceles triangle, $H_a$, $H_b$, and $H_c$ be the feet of the altitudes drawn from the vertices $A, B$, and $C$, respectively, and $M_a$, $M_b$, and $M_c$ be the midpoints of the sides $BC$, $CA$, and $AB$, respectively. The circumscribed circles of triangles $AH_bH_c$ and $AM_bM_c$ intersect for second time at point $A'$. The circumscribed circles of triangles $BH_cH_a$ and $BM_cM_a$ intersect for second time at point $B'$. The circumscribed circles of triangles $CH_aH_b$ and $CM_aM_b$ intersect for second time at point $C'$. Prove that points $A', B'$ and $C'$ lie on the same line.
2015 Czech-Polish-Slovak Junior Match, 4
Let $ABC$ ne a right triangle with $\angle ACB=90^o$. Let $E, F$ be respecitvely the midpoints of the $BC, AC$ and $CD$ be it's altitude. Next, let $P$ be the intersection of the internal angle bisector from $A$ and the line $EF$. Prove that $P$ is the center of the circle inscribed in the triangle $CDE$ .
2016 Saudi Arabia GMO TST, 1
Let $ABC$ be an acute, non-isosceles triangle which is inscribed in a circle $(O)$. A point $I$ belongs to the segment $BC$. Denote by $H$ and $K$ the projections of $I$ on $AB$ and $AC$, respectively. Suppose that the line $HK $ intersects $(O)$ at $M, N$ ($H$ is between $M, K$ and $K$ is between $H, N$). Let $X, Y$ be the centers of the circles $(ABK),(ACH)$ respectively. Prove the
following assertions:
a) If $I$ is the projection of $A$ on $BC$, then $A$ is the center of circle $(IMN)$.
b) If $XY\parallel BC$, then the orthocenter of $XOY$ is the midpoint of $IO$.
2009 Bundeswettbewerb Mathematik, 3
Let $P$ be a point inside the triangle $ABC$ and $P_a, P_b ,P_c$ be the symmetric points wrt the midpoints of the sides $BC, CA,AB$ respectively. Prove that that the lines $AP_a, BP_b$ and $CP_c$ are concurrent.
2014 Greece Junior Math Olympiad, 1
Let $ABC$ be a triangle and let $M$ be the midpoint $BC$. On the exterior of the triangle, consider the parallelogram $BCDE$ such that $BE//AM$ and $BE=AM/2$ . Prove that line $EM$ passes through the midpoint of segment $AD$.
Novosibirsk Oral Geo Oly IX, 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]
2010 Balkan MO Shortlist, G4
Let $ABC$ be a given triangle and $\ell$ be a line that meets the lines $BC, CA$ and $AB$ in $A_1,B_1$ and $C_1$ respectively. Let $A'$ be the midpoint, of the segment connecting the projections of $A_1$ onto the lines $AB$ and $AC$. Construct, analogously the points $B'$ and $C'$.
(a) Show that the points $A', B'$ and $C'$ are collinear on some line $\ell'$.
(b) Show that if $\ell$ contains the circumcenter of the triangle $ABC$, then $\ell' $ contains the center of it's Euler circle.
2000 Estonia National Olympiad, 4
Let $E$ be the midpoint of the side $AB$ of the parallelogram $ABCD$. Let $F$ be the projection of $B$ on $AC$. Prove that the triangle $ABF$ is isosceles