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[Solutions] Sharygin Geometry Mathematical Olympiad 2014 (Correspondence Round)

  1. A right-angled triangle $ABC$ is given. Its catheus $AB$ is the base of a regular triangle $ADB$ lying in the exterior of $ABC$, and its hypotenuse $AC$ is the base of a regular triangle $AEC$ lying in the interior of $ABC$. Lines $DE$ and $AB$ meet at point $M$. The whole configuration except points $A$ and $B$ was erased. Restore the point $M$. 
  2. A paper square with sidelength $2$ is given. From this square, can we cut out a $12$-gon having all sidelengths equal to $1$ and all angles divisible by $45^\circ$? 
  3. Let $ABC$ be an isosceles triangle with base $AB$. Line $\ell$ touches its circumcircle at point $B$. Let $CD$ be a perpendicular from $C$ to $\ell$, and $AE$, $BF$ be the altitudes of $ABC$. Prove that $D$, $E$, and $F$ are collinear. 
  4. A square is inscribed into a triangle (one side of the triangle contains two vertices and each of two remaining sides contains one vertex. Prove that the incenter of the triangle lies inside the square. 
  5. In an acute-angled triangle $ABC$, $AM$ is a median, $AL$ is a bisector and $AH$ is an altitude ($H$ lies between $L$ and $B$). It is known that $ML=LH=HB$. Find the ratios of the sidelengths of $ABC$. 
  6. Given a circle with center $O$ and a point $P$ not lying on it, let $X$ be an arbitrary point on this circle and $Y$ be a common point of the bisector of angle $POX$ and the perpendicular bisector to segment $PX$. Find the locus of points $Y$. 
  7. A parallelogram $ABCD$ is given. The perpendicular from $C$ to $CD$ meets the perpendicular from $A$ to $BD$ at point $F$, and the perpendicular from $B$ to $AB$ meets the perpendicular bisector to $AC$ at point $E$. Find the ratio in which side $BC$ divides segment $EF$. 
  8. Let $ABCD$ be a rectangle. Two perpendicular lines pass through point $B$. One of them meets segment $AD$ at point $K$, and the second one meets the extension of side $CD$ at point $L$. Let $F$ be the common point of $KL$ and $AC$. Prove that $BF\perp KL$. 
  9. Two circles $\omega_1$ and $\omega_2$ touching externally at point $L$ are inscribed into angle $BAC$. Circle $\omega_1$ touches ray $AB$ at point $E$, and circle $\omega_2$ touches ray $AC$ at point $M$. Line $EL$ meets $\omega_2$ for the second time at point $Q$. Prove that $MQ\parallel AL$. 
  10. Two disjoint circles $\omega_1$ and $\omega_2$ are inscribed into an angle. Consider all pairs of parallel lines $l_1$ and $l_2$ such that $l_1$ touches $\omega_1$ and $l_2$ touches $\omega_2$ ($\omega_1$, $\omega_2$ lie between $l_1$ and $l_2$). Prove that the medial lines of all trapezoids formed by $l_1$ and $l_2$ and the sides of the angle touch some fixed circle. 
  11. Points $K$, $L$, $M$ and $N$ lying on the sides $AB$, $BC$, $CD$ and $DA$ of a square $ABCD$ are vertices of another square. Lines $DK$ and $N M$ meet at point $E$, and lines $KC$ and $LM$ meet at point $F$ . Prove that $EF\parallel AB$. 
  12. Circles $\omega_1$ and $\omega_2$ meet at points $A$ and $B$. Let points $K_1$ and $K_2 $ of $\omega_1$ and $\omega_2$ respectively be such that $K_1A$ touches $\omega_2$, and $K_2A$ touches $\omega_1$. The circumcircle of triangle $K_1BK_2$ meets lines $AK_1$ and $AK_2$ for the second time at points $L_1$ and $L_2$ respectively. Prove that $L_1$ and $L_2$ are equidistant from line $AB$. 
  13. Let $AC$ be a fixed chord of a circle $\omega$ with center $O$. Point $B$ moves along the arc $AC$. A fixed point $P$ lies on $AC$. The line passing through $P$ and parallel to $AO$ meets $BA$ at point $A_1$, the line passing through $P$ and parallel to $CO$ meets $BC$ at point $C_1$. Prove that the circumcenter of triangle $A_1BC_1$ moves along a straight line. 
  14. In a given disc, construct a subset such that its area equals the half of the disc area and its intersection with its reflection over an arbitrary diameter has the area equal to the quarter of the disc area. 
  15. Let $ABC$ be a non-isosceles triangle. The altitude from $A$, the bisector from $B$ and the median from $C$ concur at point $K$.
    a) Which of the sidelengths of the triangle is medial (intermediate in length)?.
    b) Which of the lengths of segments $AK$, $BK$, $CK$ is medial (intermediate in length)? 
  16. Given a triangle $ABC$ and an arbitrary point $D$.The lines passing through $D$ and perpendicular to segments $DA$, $DB$, $DC$ meet lines $BC$, $AC$, $AB$ at points $A_1$, $B_1$, $C_1$ respectively. Prove that the midpoints of segments $AA_1$, $BB_1$, $CC_1$ are collinear. 
  17. Let $AC$ be the hypothenuse of a right-angled triangle $ABC$. The bisector $BD$ is given, and the midpoints $E$ and $F$ of the arcs $BD$ of the circumcircles of triangles $ADB$ and $CDB$ respectively are marked (the circles are erased). Construct the centers of these circles using only a ruler. 
  18. Let $I$ be the incenter of a circumscribed quadrilateral $ABCD$. The tangents to circle $AIC$ at points $A, C$ meet at point $X$. The tangents to circle $BID$ at points $B, D$ meet at point $Y$ . Prove that $X$, $I$, $Y$ are collinear. 
  19. Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$. Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles. Let $AB$ and $AC$ be the tangents to $\omega_1$. Lines $BP$ and $CP$ meet $\omega_2$ for the second time at points $E$ and $F$. Prove that the line $EF$, the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur. 
  20. A quadrilateral $KLMN$ is given. A circle with center $O$ meets its side $KL$ at points $A$ and $A_1$, side $LM$ at points $B$ and $B_1$, etc. Prove that if the circumcircles of triangles $KDA$, $LAB$, $MBC$ and $NCD$ concur at point $P$, then
    a) the circumcircles of triangles $KD_1A_1$, $LA_1B_1$, $MB_1C_1$ and $NC1D1$ also concur at some point $Q$;
    b) point $O$ lies on the perpendicular bisector to $PQ$. 
  21. Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$. 
  22. Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges? 
  23. Let $A$, $B$, $C$ and $D$ be a triharmonic quadruple of points, i.e $$AB\cdot CD = AC \cdot BD = AD \cdot BC.$$ Let $A_1$ be a point distinct from $A$ such that the quadruple $A_1$, $B$, $C$ and $D$ is triharmonic. Points $B_1$, $C_1$ and $D_1$ are defined similarly. Prove that
    a) $A$, $B$, $C_1$, $D_1$ are concyclic;
    b) the quadruple $A_1$, $B_1$, $C_1$, $D_1$ is triharmonic. 
  24. A circumscribed pyramid $ABCDS$ is given. The opposite sidelines of its base meet at points $P$ and $Q$ in such a way that $A$ and $B$ lie on segments $PD$ and $PC$ respectively. The inscribed sphere touches faces $ABS$ and $BCS$ at points $K$ and $L$. Prove that if $PK$ and $QL$ are complanar then the touching point of the sphere with the base lies on $BD$.

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MOlympiad: [Solutions] Sharygin Geometry Mathematical Olympiad 2014 (Correspondence Round)
[Solutions] Sharygin Geometry Mathematical Olympiad 2014 (Correspondence Round)
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https://www.molympiad.net/2018/08/sharygin-geometry-mathematical-olympiad-2014-correspondence.html
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