Oral Sharygin 2019 12p [UC]

geometry problems
with aops links in the names from Oral Sharygin Geometry Olympiad

under construction


2019 Oral Sharygin Geometry Olympiad grades 8-9 p1
In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$

2019 Oral Sharygin Geometry Olympiad grades 8-9 p2
On the side $AC$  of the triangle $ABC$ in the external side is constructed the parallelogram $ACDE$ . Let $O$ be the intersection point of its diagonals, $N$ and $K$ be  midpoints of BC and BA respectively. Prove that lines $DK, EN$ and $BO$ intersect at one point.

2019 Oral Sharygin Geometry Olympiad grades 8-9 p3
In the acute triangle $ABC, \angle ABC = 60^o , O$ is the center of the circumscribed circle and  $H$ is the orthocenter. The bisector $BL$ intersects the circumscribed circle at the point $W, X$ is the intersection point of segments $WH$ and $AC$ . Prove that points $O, L, X$ and $H$ lie on the same circle.

2019 Oral Sharygin Geometry Olympiad grades 8-9 p4
The perpendicular bisector of  the bisector $BL$ of the triangle $ABC$ intersects the bisectors of its external angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the circle circumscribed around the triangle $PBQ$ is tangent to the circle circumscribed around the triangle $ABC$.

Given the segment$ PQ$ and a circle . A chord $AB$ moves around the circle, equal to $PQ$. Let $T$ be the intersection point of the perpendicular bisectors of the segments $AP$ and $BQ$. Prove that all points of $T$ thus obtained lie on one line.

2019 Oral Sharygin Geometry Olympiad grades 8-9 p6
In the acute triangle $ABC$, the point $I_c$ is the center of excircle on the side $AB$, $A_1$ and $B_1$ are the tangency points of the other two excircles with sides $BC$ and $CA$, respectively, $C'$  is the point on the circumcircle diametrically opposite to point $C$. Prove that the lines $I_cC'$ and $A_1B_1$ are perpendicular.

2019 Oral Sharygin Geometry Olympiad grades 10-11 p1
Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK $.

2019 Oral Sharygin Geometry Olympiad grades 10-11 p2
The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily congruent ?

2019 Oral Sharygin Geometry Olympiad grades 10-11 p3
Restore the acute triangle $ABC$ given the vertex $A$, the base of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle %ABC$).

2019 Oral Sharygin Geometry Olympiad grades 10-11 p4
Given a right triangle $ABC$ ($\angle C=90^o$). The $C$-excircle touches the hypotenuse $AB$ at point $C_1, A_1$ is the touchpoint  of $B$-excircle with  line $BC, B_1$  is the touchpoint  of $A$-excircle with line $AC$. Find the angle $\angle A_1C_1B_1$.

2019 Oral Sharygin Geometry Olympiad grades 10-11 p5
On sides $AB$ and $BC$ of a non-isosceles triangle $ABC$ are selected  points $C_1$ and $A_1$ such that the quadrilateral $AC_1A_1C$ is cyclic. Lines $CC_1$ and $AA_1$ intersect  at point $P$. Line $BP$ intersects the circumscribed circle of triangle $ABC$ at the point $Q$. Prove that the lines $QC_1$ and $CM$, where $M$ is the midpoint of $A_1C_1$, intersect at the circumscribed circles of triangle $ABC$.

The sum of the cosines of the flat angles of the trihedral angle is $-1$. Find the sum of its dihedral angles.

2018 Oral Sharygin Geometry Olympiad grades 8-9 p1
2018 Oral Sharygin Geometry Olympiad grades 8-9 p2
2018 Oral Sharygin Geometry Olympiad grades 8-9 p3
2018 Oral Sharygin Geometry Olympiad grades 8-9 p4
2018 Oral Sharygin Geometry Olympiad grades 8-9 p5
2018 Oral Sharygin Geometry Olympiad grades 8-9 p6

2018 Oral Sharygin Geometry Olympiad grades 10-11 p1
2018 Oral Sharygin Geometry Olympiad grades 10-11 p2
2018 Oral Sharygin Geometry Olympiad grades 10-11 p3
2018 Oral Sharygin Geometry Olympiad grades 10-11 p4
2018 Oral Sharygin Geometry Olympiad grades 10-11 p5

Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circumcircle of $\triangle{BOC}$ meets the  lines $AB, AC$ at points $A_1, A_2$, respectively. Let $\omega_{A}$ be the circumcircle of triangle $AA_1A_2$. Define $\omega_B$ and $\omega_C$ analogously. Prove that the circles $\omega_A, \omega_B, \omega_C$ concur on $\odot(ABC)$.

2017 Oral Sharygin Geometry Olympiad grades 8-9 p1
2017 Oral Sharygin Geometry Olympiad grades 8-9 p2
2017 Oral Sharygin Geometry Olympiad grades 8-9 p3
2017 Oral Sharygin Geometry Olympiad grades 8-9 p4
2017 Oral Sharygin Geometry Olympiad grades 8-9 p5
2017 Oral Sharygin Geometry Olympiad grades 8-9 p6

2017 Oral Sharygin Geometry Olympiad grades 10-11 p1

2017 Oral Sharygin Geometry Olympiad grades 10-11 p2
Given pyramid with base $n-gon$. How many maximum number of edges can be perpendicular to base?

2017 Oral Sharygin Geometry Olympiad grades 10-11 p3
2017 Oral Sharygin Geometry Olympiad grades 10-11 p4
2017 Oral Sharygin Geometry Olympiad grades 10-11 p5

2017 Oral Sharygin Geometry Olympiad grades 10-11 p6
Given acute angled traingle $ABC$ and altitudes $AA_1$, $BB_1$, $CC_1$.  Let $M$ midpoint of $BC$. $P$ point of intersection of circles $(AB_1C_1)$ and $(ABC)$ . $T$ is point of intersection of tangents to $(ABC)$ at $B$ and $C$. $S$ point of intersection of $AT$ and $(ABC)$. Prove that $P,A_1,S$ and midpoint of $MT$ collinear.

2016 Oral Sharygin Geometry Olympiad grades 8-9 p1
2016 Oral Sharygin Geometry Olympiad grades 8-9 p2
2016 Oral Sharygin Geometry Olympiad grades 8-9 p3
2016 Oral Sharygin Geometry Olympiad grades 8-9 p4
2016 Oral Sharygin Geometry Olympiad grades 8-9 p5
2016 Oral Sharygin Geometry Olympiad grades 8-9 p6

2016 Oral Sharygin Geometry Olympiad grades 10-11 p1
2016 Oral Sharygin Geometry Olympiad grades 10-11 p2
2016 Oral Sharygin Geometry Olympiad grades 10-11 p3
2016 Oral Sharygin Geometry Olympiad grades 10-11 p4
2016 Oral Sharygin Geometry Olympiad grades 10-11 p5
2016 Oral Sharygin Geometry Olympiad grades 10-11 p6

2015 Oral Sharygin Geometry Olympiad grades 8-9 p1
2015 Oral Sharygin Geometry Olympiad grades 8-9 p2
2015 Oral Sharygin Geometry Olympiad grades 8-9 p3
2015 Oral Sharygin Geometry Olympiad grades 8-9 p4
2015 Oral Sharygin Geometry Olympiad grades 8-9 p5
2015 Oral Sharygin Geometry Olympiad grades 8-9 p6

2015 Oral Sharygin Geometry Olympiad grades 10-11 p1
2015 Oral Sharygin Geometry Olympiad grades 10-11 p2
2015 Oral Sharygin Geometry Olympiad grades 10-11 p3
2015 Oral Sharygin Geometry Olympiad grades 10-11 p4
2015 Oral Sharygin Geometry Olympiad grades 10-11 p5
2015 Oral Sharygin Geometry Olympiad grades 10-11 p6

2014 Oral Sharygin Geometry Olympiad grades 8-9 p1
2014 Oral Sharygin Geometry Olympiad grades 8-9 p2
2014 Oral Sharygin Geometry Olympiad grades 8-9 p3
2014 Oral Sharygin Geometry Olympiad grades 8-9 p4
2014 Oral Sharygin Geometry Olympiad grades 8-9 p5
2014 Oral Sharygin Geometry Olympiad grades 8-9 p6

2014 Oral Sharygin Geometry Olympiad grades 10-11 p1
2014 Oral Sharygin Geometry Olympiad grades 10-11 p2
2014 Oral Sharygin Geometry Olympiad grades 10-11 p3
2014 Oral Sharygin Geometry Olympiad grades 10-11 p4
2014 Oral Sharygin Geometry Olympiad grades 10-11 p5
2014 Oral Sharygin Geometry Olympiad grades 10-11 p6

2013 Oral Sharygin Geometry Olympiad grades 8-9 p1
2013 Oral Sharygin Geometry Olympiad grades 8-9 p2
2013 Oral Sharygin Geometry Olympiad grades 8-9 p3
2013 Oral Sharygin Geometry Olympiad grades 8-9 p4
2013 Oral Sharygin Geometry Olympiad grades 8-9 p5
2013 Oral Sharygin Geometry Olympiad grades 8-9 p6

2013 Oral Sharygin Geometry Olympiad grades 10-11 p1
2013 Oral Sharygin Geometry Olympiad grades 10-11 p2
2013 Oral Sharygin Geometry Olympiad grades 10-11 p3
2013 Oral Sharygin Geometry Olympiad grades 10-11 p4
2013 Oral Sharygin Geometry Olympiad grades 10-11 p5
2013 Oral Sharygin Geometry Olympiad grades 10-11 p6

2012 Oral Sharygin Geometry Olympiad grades 8-9 p1
2012 Oral Sharygin Geometry Olympiad grades 8-9 p2
2012 Oral Sharygin Geometry Olympiad grades 8-9 p3
2012 Oral Sharygin Geometry Olympiad grades 8-9 p4
2012 Oral Sharygin Geometry Olympiad grades 8-9 p5
2012 Oral Sharygin Geometry Olympiad grades 8-9 p6

2012 Oral Sharygin Geometry Olympiad grades 10-11 p1
2012 Oral Sharygin Geometry Olympiad grades 10-11 p2
2012 Oral Sharygin Geometry Olympiad grades 10-11 p3
2012 Oral Sharygin Geometry Olympiad grades 10-11 p4
2012 Oral Sharygin Geometry Olympiad grades 10-11 p5
2012 Oral Sharygin Geometry Olympiad grades 10-11 p6

2011 Oral Sharygin Geometry Olympiad grades 8-9 p1
2011 Oral Sharygin Geometry Olympiad grades 8-9 p2
2011 Oral Sharygin Geometry Olympiad grades 8-9 p3
2011 Oral Sharygin Geometry Olympiad grades 8-9 p4
2011 Oral Sharygin Geometry Olympiad grades 8-9 p5
2011 Oral Sharygin Geometry Olympiad grades 8-9 p6

2011 Oral Sharygin Geometry Olympiad grades 10-11 p1
2011 Oral Sharygin Geometry Olympiad grades 10-11 p2
2011 Oral Sharygin Geometry Olympiad grades 10-11 p3
2011 Oral Sharygin Geometry Olympiad grades 10-11 p4
2011 Oral Sharygin Geometry Olympiad grades 10-11 p5
2011 Oral Sharygin Geometry Olympiad grades 10-11 p6

2010 Oral Sharygin Geometry Olympiad grades 8-9 p1
2010 Oral Sharygin Geometry Olympiad grades 8-9 p2
2010 Oral Sharygin Geometry Olympiad grades 8-9 p3
2010 Oral Sharygin Geometry Olympiad grades 8-9 p4
2010 Oral Sharygin Geometry Olympiad grades 8-9 p5
2010 Oral Sharygin Geometry Olympiad grades 8-9 p6

2010 Oral Sharygin Geometry Olympiad grades 10-11 p1
2010 Oral Sharygin Geometry Olympiad grades 10-11 p2
2010 Oral Sharygin Geometry Olympiad grades 10-11 p3
2010 Oral Sharygin Geometry Olympiad grades 10-11 p4
2010 Oral Sharygin Geometry Olympiad grades 10-11 p5
2010 Oral Sharygin Geometry Olympiad grades 10-11 p6

2009 Oral Sharygin Geometry Olympiad grades 8-9 p1
2009 Oral Sharygin Geometry Olympiad grades 8-9 p2
2009 Oral Sharygin Geometry Olympiad grades 8-9 p3
2009 Oral Sharygin Geometry Olympiad grades 8-9 p4
2009 Oral Sharygin Geometry Olympiad grades 8-9 p5
2009 Oral Sharygin Geometry Olympiad grades 8-9 p6

2009 Oral Sharygin Geometry Olympiad grades 10-11 p1
2009 Oral Sharygin Geometry Olympiad grades 10-11 p2
2009 Oral Sharygin Geometry Olympiad grades 10-11 p3
2009 Oral Sharygin Geometry Olympiad grades 10-11 p4
2009 Oral Sharygin Geometry Olympiad grades 10-11 p5
2009 Oral Sharygin Geometry Olympiad grades 10-11 p6

2008 Oral Sharygin Geometry Olympiad grades 8-9 p1
2008 Oral Sharygin Geometry Olympiad grades 8-9 p2
2008 Oral Sharygin Geometry Olympiad grades 8-9 p3
2008 Oral Sharygin Geometry Olympiad grades 8-9 p4
2008 Oral Sharygin Geometry Olympiad grades 8-9 p5
2008 Oral Sharygin Geometry Olympiad grades 8-9 p6

2008 Oral Sharygin Geometry Olympiad grades 10-11 p1
2008 Oral Sharygin Geometry Olympiad grades 10-11 p2
2008 Oral Sharygin Geometry Olympiad grades 10-11 p3
2008 Oral Sharygin Geometry Olympiad grades 10-11 p4
2008 Oral Sharygin Geometry Olympiad grades 10-11 p5
2008 Oral Sharygin Geometry Olympiad grades 10-11 p6

2007 Oral Sharygin Geometry Olympiad grades 8-9 p1
2007 Oral Sharygin Geometry Olympiad grades 8-9 p2
2007 Oral Sharygin Geometry Olympiad grades 8-9 p3
2007 Oral Sharygin Geometry Olympiad grades 8-9 p4
2007 Oral Sharygin Geometry Olympiad grades 8-9 p5
2007 Oral Sharygin Geometry Olympiad grades 8-9 p6

2007 Oral Sharygin Geometry Olympiad grades 10-11 p1
2007 Oral Sharygin Geometry Olympiad grades 10-11 p2
2007 Oral Sharygin Geometry Olympiad grades 10-11 p3
2007 Oral Sharygin Geometry Olympiad grades 10-11 p4
2007 Oral Sharygin Geometry Olympiad grades 10-11 p5
2007 Oral Sharygin Geometry Olympiad grades 10-11 p6

2006 Oral Sharygin Geometry Olympiad grades 8-9 p1
2006 Oral Sharygin Geometry Olympiad grades 8-9 p2
2006 Oral Sharygin Geometry Olympiad grades 8-9 p3
2006 Oral Sharygin Geometry Olympiad grades 8-9 p4
2006 Oral Sharygin Geometry Olympiad grades 8-9 p5
2006 Oral Sharygin Geometry Olympiad grades 8-9 p6

2006 Oral Sharygin Geometry Olympiad grades 10-11 p1
2006 Oral Sharygin Geometry Olympiad grades 10-11 p2
2006 Oral Sharygin Geometry Olympiad grades 10-11 p3
2006 Oral Sharygin Geometry Olympiad grades 10-11 p4
2006 Oral Sharygin Geometry Olympiad grades 10-11 p5
2006 Oral Sharygin Geometry Olympiad grades 10-11 p6

2005 Oral Sharygin Geometry Olympiad grades 8-9 p1
2005 Oral Sharygin Geometry Olympiad grades 8-9 p2
2005 Oral Sharygin Geometry Olympiad grades 8-9 p3
2005 Oral Sharygin Geometry Olympiad grades 8-9 p4
2005 Oral Sharygin Geometry Olympiad grades 8-9 p5
2005 Oral Sharygin Geometry Olympiad grades 8-9 p6

2005 Oral Sharygin Geometry Olympiad grades 10-11 p1
2005 Oral Sharygin Geometry Olympiad grades 10-11 p2
2005 Oral Sharygin Geometry Olympiad grades 10-11 p3
2005 Oral Sharygin Geometry Olympiad grades 10-11 p4
2005 Oral Sharygin Geometry Olympiad grades 10-11 p5
2005 Oral Sharygin Geometry Olympiad grades 10-11 p6

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