Moscow (MMO) (hidden)

geometry problems from Moscow Mathematical Olympiads
with aops links in the names


translated by D. Leites (ed.), without figures 
written by G. Galperin, A. Tolpygo, P. Grozman, A. Shapovalov, V. Prasolov, A. Fomenko

geometry from old Moscow Mathematical Olympiads are collected here

1996 - 2018
under construction


Consider an equilateral triangle $\triangle ABC$. The points $K$ and $L$ divide the leg $BC$ into three equal parts, the point $M$ divides the leg $AC$ in the ratio $1:2$, counting from the vertex $A$. Prove that $\angle AKM+\angle ALM=30^{\circ}$. 

by V. Proizvolov

A circle is circumscribed about $\triangle ABC$; through points $A$ and $B$ tangents are drawn, they meet at $M$. Point $N$ lies on the leg $BC$, and $MN \parallel AC$. Prove that $AN=NC$. 

by I. F. Sharygin

Let $A$ and $B$ be points on a circle. They divide the circle into two parts. Find the locus of the midpoints of all chords whose endpoints lie on different arcs $AB$. 

by I. F. Sharygin.

1996 Moscow  MO grade X P
1996 Moscow  MO grade X P
1996 Moscow  MO grade XI P
1996 Moscow  MO grade XI P

1997 Moscow MO grade VIII P3
Inside acute $\angle{XOY},$ points $M$ and $N$ are taken so that $\angle{XON}=\angle{YOM}$. Point $Q$ is taken on segment $OX$ such that $\angle{NQO}=\angle{MQX}.$ Point $P$ is taken such that $\angle{NPO}=\angle{MPY}.$ Prove the lengths of the broken lines $MPN$ and $MQN$ are equal.

1997 Moscow MO grade VIII P5
Inside acute $\angle{XOY},$ points $M$ and $N$ are taken so that $\angle{XON}=\angle{YOM}$. Point $Q$ is taken on segment $OX$ such that $\angle{NQO}=\angle{MQX}.$ Point $P$ is taken such that $\angle{NPO}=\angle{MPY}.$ Prove the lengths of the broken lines $MPN$ and $MQN$ are equal.

In a triangle one side is $3$ times shorter than the sum of the other two. Prove that the angle opposite said side is the smallest of the triangle’s angles.

Convex octagon $AC_1BA_1CB_1$ satisfies: $AB_1=AC_1$, $BC_1=BA_1$, $CA_1=CB_1$ and $\angle{A}+\angle{B}+\angle{C}=\angle{A_1}+\angle{B_1}+\angle{C_1}$. Prove that the area of $\triangle{ABC}$ is equal to half the area of the octagon.


Is there a convex body distinct from ball whose three orthogonal projections on three pairwise perpendicular planes are discs?

Prove that among the quadrilaterals with given lengths of the diagonals and the angle between them, the parallelogram has the least perimeter.

A quadrilateral is rotated clockwise, and the sides are extended its length in the direction of the movement. It turns out the endpoints of the segments constructed form a square. Prove the initial quadrilateral must also be a square.

Generalization
Prove that if the same process is applied to any $n$-gon and the result is a regular $n$-gon, then the intial $n$-gon must also be regular

On sides $AB$, $BC$ and $CA$ of $\triangle ABC$, points $C'$, $A'$ and $B'$, respectively, are marked. Prove that the area of $\triangle A'B'C'$ is equal to $\frac{AB' \cdot BC' \cdot CA'+AC' \cdot CB' \cdot BA'}{4R}$, where $R$ is the radius of the circumscribed circle of $\triangle ABC$. 

by A. Zaslavsky

1998 Moscow MO grade VIII P
1998 Moscow MO grade VIII P
1998 Moscow  MO grade IX P
1998 Moscow  MO grade IX P
1998 Moscow  MO grade X P
1998 Moscow  MO grade X P
1998 Moscow  MO grade XI P
1998 Moscow  MO grade XI P

1999 Moscow MO grade VIII P
1999 Moscow MO grade VIII P
1999 Moscow  MO grade IX P
1999 Moscow  MO grade IX P
1999 Moscow  MO grade X P
1999 Moscow  MO grade X P
1999 Moscow  MO grade XI P
1999 Moscow  MO grade XI P

2000 Moscow MO grade VIII P
2000 Moscow MO grade VIII P
2000 Moscow  MO grade IX P
2000 Moscow  MO grade IX P
2000 Moscow  MO grade X P
2000 Moscow  MO grade X P
2000 Moscow  MO grade XI P
2000 Moscow  MO grade XI P

2001 Moscow MO grade VIII P
2001 Moscow MO grade VIII P
2001 Moscow  MO grade IX P
2001 Moscow  MO grade IX P
2001 Moscow  MO grade X P
2001 Moscow  MO grade X P
2001 Moscow  MO grade XI P
2001 Moscow  MO grade XI P

2002 Moscow MO grade VIII P
2002 Moscow MO grade VIII P
2002 Moscow  MO grade IX P
2002 Moscow  MO grade IX P
2002 Moscow  MO grade X P
2002 Moscow  MO grade X P
2002 Moscow  MO grade XI P
2002 Moscow  MO grade XI P

2003 Moscow MO grade VIII P
2003 Moscow MO grade VIII P
2003 Moscow  MO grade IX P
2003 Moscow  MO grade IX P
2003 Moscow  MO grade X P
2003 Moscow  MO grade X P
2003 Moscow  MO grade XI P
2003 Moscow  MO grade XI P

2004 Moscow MO grade VIII P
2004 Moscow MO grade VIII P
2004 Moscow  MO grade IX P
2004 Moscow  MO grade IX P
2004 Moscow  MO grade X P
2004 Moscow  MO grade X P
2004 Moscow  MO grade XI P
2004 Moscow  MO grade XI P

2005 Moscow MO grade VIII P
2005 Moscow MO grade VIII P

Circle O_1 passes through the center of the circle O_2. Two tangent lines drawn from a point C on the cicle O_1 to the circle O_2 cut the circle O_1 at the points A and B. Prove that AB is perpendicular to the line passes through the centers of the circles.

 Given a point P (not intersection of altitudes) in an acute triangle ABC. Prove that the circles pass through the midpoints of the triangles PAB, PAC, PBC, ABC and the circle passes through the projections of O to the sides intersect at one point.

Is there any quadrilateral whose tangents of the interior angles are equal to each other?

Drawn the squares ABB_1A_2, BCC_1B_2 and CAA_1C_2 on the outside of triangle ABC. If the squares A_1A_2A_3A_4 and B_1B_2B_3B_4 are drawn on the line segments A_1A_2 and B_1B_2 (outside of the triangles AA_1A_2 and BB_1B_2) then prove that A_3B_4 is parallel to AB.


2005 Moscow  MO grade XI P
2005 Moscow  MO grade XI P

2006 Moscow MO grade VIII P
2006 Moscow MO grade VIII P
2006 Moscow  MO grade IX P
2006 Moscow  MO grade IX P
2006 Moscow  MO grade X P
2006 Moscow  MO grade X P
2006 Moscow  MO grade XI P
2006 Moscow  MO grade XI P

2007 Moscow MO grade VIII P
2007 Moscow MO grade VIII P
2007 Moscow  MO grade IX P
2007 Moscow  MO grade IX P
2007 Moscow  MO grade X P
2007 Moscow  MO grade X P
2007 Moscow  MO grade XI P
2007 Moscow  MO grade XI P

2008 Moscow MO grade VIII P
2008 Moscow MO grade VIII P
2008 Moscow  MO grade IX P
2008 Moscow  MO grade IX P
2008 Moscow  MO grade X P
2008 Moscow  MO grade X P
2008 Moscow  MO grade XI P
2008 Moscow  MO grade XI P

2009 Moscow MO grade VIII P
2009 Moscow MO grade VIII P
2009 Moscow  MO grade IX P
2009 Moscow  MO grade IX P
2009 Moscow  MO grade X P
2009 Moscow  MO grade X P
2009 Moscow  MO grade XI P
2009 Moscow  MO grade XI P


2010 Moscow MO grade VIII P
2010 Moscow MO grade VIII P
2010 Moscow  MO grade IX P
2010 Moscow  MO grade IX P
2010 Moscow  MO grade X P
2010 Moscow  MO grade X P
2010 Moscow  MO grade XI P


2010 Moscow  MO grade XI P
2011 Moscow  MO grade VIII P5
Denote the midpoints of the non-parallel sides $AB$ and $CD$ of the trapezoid $ABCD$ by $M$ and $N$ respectively. The perpendicular from the point $M$ to the diagonal $AC$ and the perpendicular from the point $N$ to the diagonal $BD$ intersect at the point $P$. Prove that $PA = PD$.

2011 Moscow  MO grade IX P
2011 Moscow  MO grade IX P
2011 Moscow  MO grade X P
2011 Moscow  MO grade X P
2011 Moscow  MO grade XI P
2011 Moscow  MO grade XI P
2011 Moscow  MO grade XI P
2011 Moscow  MO grade XI P


2012 Moscow  MO grade VIII P
2012 Moscow  MO grade VIII P
2012 Moscow  MO grade IX P
2012 Moscow  MO grade IX P
2012 Moscow  MO grade X P
2012 Moscow  MO grade X P
2012 Moscow  MO grade XI P
2012 Moscow  MO grade XI P
2012 Moscow  MO grade XI P
2012 Moscow  MO grade XI P


2013 Moscow  MO grade VIII P
2013 Moscow  MO grade VIII P
2013 Moscow  MO grade IX P
2013 Moscow  MO grade IX P
2013 Moscow  MO grade X P

2013 Moscow  MO grade X P6
$I$ is incenter of scalene $\triangle ABC$. $\omega, O$ -circumcircle and circumcenter of $ABC$. $A_1$ is midpoint of arc $BC$ of $\omega$, not containing $A$, $A_2$ is midpoint of arc $BAC$. Perpendicular from $A_1$ to $A_2I$ intersect $BC$ at $A'$. Same way we define $B',C'$.
a) Prove, that $A',B',C'$ lies on same line $l$
b) Prove, that $l \perp OI$

2013 Moscow  MO grade XI P
2013 Moscow  MO grade XI P
2013 Moscow  MO grade XI P
2013 Moscow  MO grade XI P


2014 Moscow MO grade VIII P
2014 Moscow MO grade VIII P
2014 Moscow MO grade IX P
2014 Moscow MO grade IX P

2014 Moscow MO grade X P3
Given triangle $ABC$. Let M be the middle side $AC$, and through $P$ - the midpoint of $CM$. described circle triangle $ABP$ intersects the segment $BC$ in interior point $Q$. Prove that $\angle ABM = \angle MQP$.

2014 Moscow MO grade X P
2014 Moscow  MO grade XI P
2014 Moscow  MO grade XI P
2014 Moscow  MO grade XI P
2014 Moscow  MO grade XI P


2015 Moscow MO grade VIII P
2015 Moscow MO grade VIII P
2015 Moscow MO grade IX P

2015 Moscow MO grade IX P4
The points $O$ and $I$ are respectively the incenter and the circumcenter of the non-isosceles triangle $ABC$. Two equal circles touch sides $AB, BC$ and $AC, BC$, respectively; in addition, they touch each other at the point $K$. Given that $K$ lies on the line $OI$, find the value of $\angle BAC$.

2015 Moscow MO grade X P
2015 Moscow MO grade X P
2015 Moscow  MO grade XI P
2015 Moscow  MO grade XI P
2015 Moscow  MO grade XI P
2015 Moscow  MO grade XI P

2016 Moscow MO grade VIII P
2016 Moscow MO grade VIII P
2016 Moscow MO grade IX P
2016 Moscow MO grade IX P
2016 Moscow MO grade X P
2016 Moscow MO grade X P
2016 Moscow MO grade XI P
2016 Moscow MO grade XI P
2016 Moscow MO grade XI P
2016 Moscow MO grade XI P

2017 Moscow MO grade VIII P
2017 Moscow MO grade VIII P
2017 Moscow MO grade IX P
2017 Moscow MO grade IX P
2017 Moscow MO grade X P
2017 Moscow MO grade X P
2017 Moscow MO grade XI P
2017 Moscow MO grade XI P

2017 Moscow MO grade XI P2
$\omega$ is incircle of $\triangle ABC$ touch $AC$ in $S$. Point $Q$ lies on $\omega$ and midpoints of $AQ$ and $QC$ lies on $\omega$ . Prove that $QS$ bisects $\angle AQC$

2017 Moscow MO grade XI P10
Point $D$ lies in $\triangle ABC$ and $BD=CD$,$\angle BDC=120$. Point $E$ lies outside $ABC$ and $AE=CE,\angle AEC=60$. Points $B$ and $E$ lies on different sides of $AC$. $F$ is midpoint $BE$. Prove, that $\angle AFD=90$

2018 Moscow MO grade VIII P3
A point $K$ is inside the parallelogram $ABCD$. Point $M$ is the midpoint of $BC$, point $P$ is the midpoint of $KM$. Prove that if  $\angle APB = \angle CPD = 90^o$, then $AK = DK$.

2018 Moscow MO grade IX P4
$ABCD$ is convex and $AB\not \parallel CD,BC \not \parallel DA$. $P$ is variable point on $AD$. Circumcircles of $\triangle ABP$ and $\triangle CDP$ intersects at $Q$. Prove, that all lines $PQ$ goes through fixed point.

2018 Moscow MO grade X P3
$O$ is circumcircle and $AH$ is the altitude of $\triangle ABC$. $P$ is the point on line $OC$ such that $AP \perp OC$. Prove, that midpoint of $AB$ lies on the line $HP$.

2018 Moscow MO grade XI P2
There is tetrahedron and square pyramid, both with all edges equal $1$. Show how to cut them into several parts and glue together from these parts a cube (without voids and cracks, all parts must be used)

2018 Moscow MO grade VIII P6, XI P5
On the sides of the convex hexagon $ABCDEF$ into the outer side were built equilateral triangles $ABC_1$, $BCD_1$, $CDE_1$, $DEF_1$, $EFA_1$ and $FAB_1$. The triangle $B_1D_1F_1$ is equilateral too. Prove that, the triangle $A_1C_1E_1$ is also equilateral.

2018 Moscow MO grade XI P10
$ABC$ is acute-angled triangle, $AA_1,CC_1$ are altitudes. $M$ is centroid. $M$ lies on circumcircle of $A_1BC_1$. Find all values of $\angle B$


1980- 1995
1980 Moscow MO grade VIII P
1980 Moscow MO grade VIII P
1980 Moscow  MO grade IX P
1980 Moscow  MO grade IX P
1980 Moscow  MO grade X P
1980 Moscow  MO grade X P
1980 Moscow  MO grade XI P
1980 Moscow  MO grade XI P

1981 Moscow MO grade VIII P
1981 Moscow MO grade VIII P
1981 Moscow  MO grade IX P
1981 Moscow  MO grade IX P
1981 Moscow  MO grade X P
1981 Moscow  MO grade X P
1981 Moscow  MO grade XI P
1981 Moscow  MO grade XI P

1981 Moscow MO grade VIII P
1981 Moscow MO grade VIII P
1981 Moscow  MO grade IX P
1981 Moscow  MO grade IX P
1981 Moscow  MO grade X P
1981 Moscow  MO grade X P
1981Moscow  MO grade XI P
1981 Moscow  MO grade XI P

1981 Moscow MO grade VIII P
1981 Moscow MO grade VIII P
1981 Moscow  MO grade IX P
1981 Moscow  MO grade IX P
1981 Moscow  MO grade X P
1981 Moscow  MO grade X P
1981 Moscow  MO grade XI P
1981 Moscow  MO grade XI P

1982 Moscow MO grade VIII P
1982 Moscow MO grade VIII P
1982 Moscow  MO grade IX P
1982 Moscow  MO grade IX P
1982 Moscow  MO grade X P
1982 Moscow  MO grade X P
1982 Moscow  MO grade XI P
1982 Moscow  MO grade XI P

1983 Moscow MO grade VIII P
1983 Moscow MO grade VIII P
1983 Moscow  MO grade IX P
1983 Moscow  MO grade IX P
1983 Moscow  MO grade X P
1983 Moscow  MO grade X P
1983 Moscow  MO grade XI P
1983 Moscow  MO grade XI P

1985 Moscow MO grade VIII P
1985 Moscow MO grade VIII P
1985 Moscow  MO grade IX P
1985 Moscow  MO grade IX P
1985 Moscow  MO grade X P
1985 Moscow  MO grade X P
1985 Moscow  MO grade XI P
1985 Moscow  MO grade XI P

1986 Moscow MO grade VIII P
1986 Moscow MO grade VIII P
1986 Moscow  MO grade IX P
1986 Moscow  MO grade IX P
1986 Moscow  MO grade X P
1986 Moscow  MO grade X P
1986 Moscow  MO grade XI P
1986 Moscow  MO grade XI P

1987 Moscow MO grade VIII P
1987 Moscow MO grade VIII P
1987 Moscow  MO grade IX P
1987 Moscow  MO grade IX P
1987 Moscow  MO grade X P
1987 Moscow  MO grade X P
1987 Moscow  MO grade XI P
1987 Moscow  MO grade XI P

1998 Moscow MO grade VIII P
1998 Moscow MO grade VIII P
1998 Moscow  MO grade IX P
1998 Moscow  MO grade IX P
1998 Moscow  MO grade X P
1998 Moscow  MO grade X P
1998 Moscow  MO grade XI P
1998 Moscow  MO grade XI P

1989 Moscow MO grade VIII P4
1989 Moscow MO grade VIII P
1989 Moscow  MO grade IX P
1989 Moscow  MO grade IX P
1989 Moscow  MO grade X P
1989 Moscow  MO grade X P
1989 Moscow  MO grade XI P
1989 Moscow  MO grade XI P

1990 Moscow MO grade VIII P
1990 Moscow MO grade VIII P
1990 Moscow  MO grade IX P
1990 Moscow  MO grade IX P
1990 Moscow  MO grade X P
1990 Moscow  MO grade X P
1990 Moscow  MO grade XI P
1990 Moscow  MO grade XI P

1991 Moscow MO grade VIII P
1991 Moscow MO grade VIII P
1991 Moscow  MO grade IX P
1991 Moscow  MO grade IX P
1991 Moscow  MO grade X P
1991 Moscow  MO grade X P
1991 Moscow  MO grade XI P
1991 Moscow  MO grade XI P

1991 Moscow MO grade VIII P
1991 Moscow MO grade VIII P
1991 Moscow  MO grade IX P
1991 Moscow  MO grade IX P
1991 Moscow  MO grade X P
1991 Moscow  MO grade X P
1991Moscow  MO grade XI P
1991 Moscow  MO grade XI P

1991 Moscow MO grade VIII P
1991 Moscow MO grade VIII P
1991 Moscow  MO grade IX P
1991 Moscow  MO grade IX P
1991 Moscow  MO grade X P
1991 Moscow  MO grade X P
1991 Moscow  MO grade XI P
1991 Moscow  MO grade XI P

1992 Moscow MO grade VIII P
1992 Moscow MO grade VIII P
1992 Moscow  MO grade IX P
1992 Moscow  MO grade IX P
1992 Moscow  MO grade X P
1992 Moscow  MO grade X P
1992 Moscow  MO grade XI P
1992 Moscow  MO grade XI P

1993 Moscow MO grade VIII P
1993 Moscow MO grade VIII P
1993 Moscow  MO grade IX P
1993 Moscow  MO grade IX P
1993 Moscow  MO grade X P
1993 Moscow  MO grade X P
1993 Moscow  MO grade XI P
1993 Moscow  MO grade XI P

1995 Moscow MO grade VIII P
1995 Moscow MO grade VIII P
1995 Moscow  MO grade IX P
1995 Moscow  MO grade IX P
1995 Moscow  MO grade X P
1995 Moscow  MO grade X P
1995 Moscow  MO grade XI P
1995 Moscow  MO grade XI P


oldies


What maximal area can have a triangle if its sides $a,b,c$ satisfy inequality 
$0\leq a\leq 1\leq b\leq 2\leq c\leq 3$  ?

Moscow 1985
There is a tetrahedron $ABCD$. If the distance betweeen $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ is $h_1$, the distance betweeen $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$ is $h_2$, the distance betweeen $\overleftrightarrow{AD}$ and $\overleftrightarrow{BC}$ is $h_3$.

Prove that the volume of the tetrahedron is no less than $\frac{1}{3}h_1h_2h_3$.

Moscow 1993
We have a triangle with angles$140 , 20 , 20$. Each time we can take a triangle and draw one of it's bisectors. Prove that we can not reach to a triangle such that it is similar to the first triangle.

Moscow 2012
Let $l$ be a tangent to the incircle of triangle $ABC$. Let $l_{a},l_{b}$ and $l_{c}$ be the respective images of $l$ under reflection across the exterior bisector of  $\hat A,\hat B$and $\hat C$. Prove that the triangle formed by these lines is congruent to $ABC$.

Moscow Unknown year
Two circles $ w_1,w_2$ in the plane centered at $ A,B$ intersect at $ C,D$. Suppose the circumcircle of $ ABC$ intersect $ w_1$ at $ E$ and $ w_2$ at $ F$, where the arc $ EF$ not containing $ C$ lies outside $ w_1,w_2$. Prove that this arc $ EF$ is bisected by line $ CD$.



source:
olympiads.mccme.ru/mmo/

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